LMFDB - Embedded newform 85.2.j.a.4.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [85,2,Mod(4,85)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(85, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("85.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 85 = 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	85.j (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.678728417181$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			4.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	85.4
Dual form			85.2.j.a.64.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +(-1.00000 - 1.00000i) q^{3} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +(1.00000 + 1.00000i) q^{6} +(-3.00000 + 3.00000i) q^{7} +3.00000 q^{8} -1.00000i q^{9} +(2.00000 - 1.00000i) q^{10} +(-3.00000 - 3.00000i) q^{11} +(1.00000 + 1.00000i) q^{12} +(3.00000 - 3.00000i) q^{14} +(3.00000 + 1.00000i) q^{15} -1.00000 q^{16} +(-1.00000 - 4.00000i) q^{17} +1.00000i q^{18} +6.00000i q^{19} +(2.00000 - 1.00000i) q^{20} +6.00000 q^{21} +(3.00000 + 3.00000i) q^{22} +(1.00000 - 1.00000i) q^{23} +(-3.00000 - 3.00000i) q^{24} +(3.00000 - 4.00000i) q^{25} +(-4.00000 + 4.00000i) q^{27} +(3.00000 - 3.00000i) q^{28} +(-3.00000 + 3.00000i) q^{29} +(-3.00000 - 1.00000i) q^{30} +(-1.00000 + 1.00000i) q^{31} -5.00000 q^{32} +6.00000i q^{33} +(1.00000 + 4.00000i) q^{34} +(3.00000 - 9.00000i) q^{35} +1.00000i q^{36} +(3.00000 + 3.00000i) q^{37} -6.00000i q^{38} +(-6.00000 + 3.00000i) q^{40} +(-3.00000 - 3.00000i) q^{41} -6.00000 q^{42} -12.0000 q^{43} +(3.00000 + 3.00000i) q^{44} +(1.00000 + 2.00000i) q^{45} +(-1.00000 + 1.00000i) q^{46} -2.00000i q^{47} +(1.00000 + 1.00000i) q^{48} -11.0000i q^{49} +(-3.00000 + 4.00000i) q^{50} +(-3.00000 + 5.00000i) q^{51} +2.00000 q^{53} +(4.00000 - 4.00000i) q^{54} +(9.00000 + 3.00000i) q^{55} +(-9.00000 + 9.00000i) q^{56} +(6.00000 - 6.00000i) q^{57} +(3.00000 - 3.00000i) q^{58} +6.00000i q^{59} +(-3.00000 - 1.00000i) q^{60} +(1.00000 + 1.00000i) q^{61} +(1.00000 - 1.00000i) q^{62} +(3.00000 + 3.00000i) q^{63} +7.00000 q^{64} -6.00000i q^{66} -6.00000i q^{67} +(1.00000 + 4.00000i) q^{68} -2.00000 q^{69} +(-3.00000 + 9.00000i) q^{70} +(3.00000 - 3.00000i) q^{71} -3.00000i q^{72} +(3.00000 + 3.00000i) q^{73} +(-3.00000 - 3.00000i) q^{74} +(-7.00000 + 1.00000i) q^{75} -6.00000i q^{76} +18.0000 q^{77} +(-7.00000 - 7.00000i) q^{79} +(2.00000 - 1.00000i) q^{80} +5.00000 q^{81} +(3.00000 + 3.00000i) q^{82} -4.00000 q^{83} -6.00000 q^{84} +(6.00000 + 7.00000i) q^{85} +12.0000 q^{86} +6.00000 q^{87} +(-9.00000 - 9.00000i) q^{88} +6.00000 q^{89} +(-1.00000 - 2.00000i) q^{90} +(-1.00000 + 1.00000i) q^{92} +2.00000 q^{93} +2.00000i q^{94} +(-6.00000 - 12.0000i) q^{95} +(5.00000 + 5.00000i) q^{96} +(3.00000 + 3.00000i) q^{97} +11.0000i q^{98} +(-3.00000 + 3.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{7} + 6 q^{8} + 4 q^{10} - 6 q^{11} + 2 q^{12} + 6 q^{14} + 6 q^{15} - 2 q^{16} - 2 q^{17} + 4 q^{20} + 12 q^{21} + 6 q^{22} + 2 q^{23} - 6 q^{24}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 - 4 * q^5 + 2 * q^6 - 6 * q^7 + 6 * q^8 + 4 * q^10 - 6 * q^11 + 2 * q^12 + 6 * q^14 + 6 * q^15 - 2 * q^16 - 2 * q^17 + 4 * q^20 + 12 * q^21 + 6 * q^22 + 2 * q^23 - 6 * q^24 + 6 * q^25 - 8 * q^27 + 6 * q^28 - 6 * q^29 - 6 * q^30 - 2 * q^31 - 10 * q^32 + 2 * q^34 + 6 * q^35 + 6 * q^37 - 12 * q^40 - 6 * q^41 - 12 * q^42 - 24 * q^43 + 6 * q^44 + 2 * q^45 - 2 * q^46 + 2 * q^48 - 6 * q^50 - 6 * q^51 + 4 * q^53 + 8 * q^54 + 18 * q^55 - 18 * q^56 + 12 * q^57 + 6 * q^58 - 6 * q^60 + 2 * q^61 + 2 * q^62 + 6 * q^63 + 14 * q^64 + 2 * q^68 - 4 * q^69 - 6 * q^70 + 6 * q^71 + 6 * q^73 - 6 * q^74 - 14 * q^75 + 36 * q^77 - 14 * q^79 + 4 * q^80 + 10 * q^81 + 6 * q^82 - 8 * q^83 - 12 * q^84 + 12 * q^85 + 24 * q^86 + 12 * q^87 - 18 * q^88 + 12 * q^89 - 2 * q^90 - 2 * q^92 + 4 * q^93 - 12 * q^95 + 10 * q^96 + 6 * q^97 - 6 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/85\mathbb{Z}\right)^\times$.

$n$	$52$	$71$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.00000			−0.707107			−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000			−0.707107			−0.353553	+	0.935414i	$0.615027\pi$
$3$	−1.00000	−	1.00000i	−0.577350	−	0.577350i	0.356822	−	0.934172i	$-0.383860\pi$
$3$	−1.00000	−	1.00000i	−0.577350	−	0.577350i	−0.934172	+	0.356822i	$0.883860\pi$
$4$	−1.00000			−0.500000
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$5$	−2.00000	+	1.00000i	−0.894427	+	0.447214i
$6$	1.00000	+	1.00000i	0.408248	+	0.408248i
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.144370	+	0.989524i	$0.546115\pi$
$7$	−3.00000	+	3.00000i	−1.13389	+	1.13389i	−0.989524	+	0.144370i	$0.953885\pi$
$8$	3.00000			1.06066
$9$		−	1.00000i		−	0.333333i
$10$	2.00000	−	1.00000i	0.632456	−	0.316228i
$11$	−3.00000	−	3.00000i	−0.904534	−	0.904534i	0.0912903	−	0.995824i	$-0.470901\pi$
$11$	−3.00000	−	3.00000i	−0.904534	−	0.904534i	−0.995824	+	0.0912903i	$0.970901\pi$
$12$	1.00000	+	1.00000i	0.288675	+	0.288675i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	3.00000	−	3.00000i	0.801784	−	0.801784i
$15$	3.00000	+	1.00000i	0.774597	+	0.258199i
$16$	−1.00000			−0.250000
$17$	−1.00000	−	4.00000i	−0.242536	−	0.970143i
$17$	−1.00000	−	4.00000i	−0.242536	−	0.970143i
$18$			1.00000i			0.235702i
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$	2.00000	−	1.00000i	0.447214	−	0.223607i
$21$	6.00000			1.30931
$22$	3.00000	+	3.00000i	0.639602	+	0.639602i
$23$	1.00000	−	1.00000i	0.208514	−	0.208514i	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	1.00000	−	1.00000i	0.208514	−	0.208514i	0.803636	+	0.595121i	$0.202896\pi$
$24$	−3.00000	−	3.00000i	−0.612372	−	0.612372i
$25$	3.00000	−	4.00000i	0.600000	−	0.800000i
$26$		0			0
$27$	−4.00000	+	4.00000i	−0.769800	+	0.769800i
$28$	3.00000	−	3.00000i	0.566947	−	0.566947i
$29$	−3.00000	+	3.00000i	−0.557086	+	0.557086i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	−3.00000	+	3.00000i	−0.557086	+	0.557086i	0.371391	+	0.928477i	$0.378881\pi$
$30$	−3.00000	−	1.00000i	−0.547723	−	0.182574i
$31$	−1.00000	+	1.00000i	−0.179605	+	0.179605i	−0.791184	−	0.611578i	$-0.790535\pi$
$31$	−1.00000	+	1.00000i	−0.179605	+	0.179605i	0.611578	+	0.791184i	$0.290535\pi$
$32$	−5.00000			−0.883883
$33$			6.00000i			1.04447i
$34$	1.00000	+	4.00000i	0.171499	+	0.685994i
$35$	3.00000	−	9.00000i	0.507093	−	1.52128i
$36$			1.00000i			0.166667i
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	0.909312	−	0.416115i	$-0.136609\pi$
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	−0.416115	+	0.909312i	$0.636609\pi$
$38$		−	6.00000i		−	0.973329i
$39$		0			0
$40$	−6.00000	+	3.00000i	−0.948683	+	0.474342i
$41$	−3.00000	−	3.00000i	−0.468521	−	0.468521i	0.432914	−	0.901435i	$-0.357485\pi$
$41$	−3.00000	−	3.00000i	−0.468521	−	0.468521i	−0.901435	+	0.432914i	$0.857485\pi$
$42$	−6.00000			−0.925820
$43$	−12.0000			−1.82998			−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000			−1.82998			−0.914991	+	0.403473i	$0.867803\pi$
$44$	3.00000	+	3.00000i	0.452267	+	0.452267i
$45$	1.00000	+	2.00000i	0.149071	+	0.298142i
$46$	−1.00000	+	1.00000i	−0.147442	+	0.147442i
$47$		−	2.00000i		−	0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$		−	2.00000i		−	0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	1.00000	+	1.00000i	0.144338	+	0.144338i
$49$		−	11.0000i		−	1.57143i
$50$	−3.00000	+	4.00000i	−0.424264	+	0.565685i
$51$	−3.00000	+	5.00000i	−0.420084	+	0.700140i
$52$		0			0
$53$	2.00000			0.274721			0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000			0.274721			0.137361	+	0.990521i	$0.456138\pi$
$54$	4.00000	−	4.00000i	0.544331	−	0.544331i
$55$	9.00000	+	3.00000i	1.21356	+	0.404520i
$56$	−9.00000	+	9.00000i	−1.20268	+	1.20268i
$57$	6.00000	−	6.00000i	0.794719	−	0.794719i
$58$	3.00000	−	3.00000i	0.393919	−	0.393919i
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$	−3.00000	−	1.00000i	−0.387298	−	0.129099i
$61$	1.00000	+	1.00000i	0.128037	+	0.128037i	0.768221	−	0.640184i	$-0.221142\pi$
$61$	1.00000	+	1.00000i	0.128037	+	0.128037i	−0.640184	+	0.768221i	$0.721142\pi$
$62$	1.00000	−	1.00000i	0.127000	−	0.127000i
$63$	3.00000	+	3.00000i	0.377964	+	0.377964i
$64$	7.00000			0.875000
$65$		0			0
$66$		−	6.00000i		−	0.738549i
$67$		−	6.00000i		−	0.733017i	−0.930415	−	0.366508i	$-0.880553\pi$
$67$		−	6.00000i		−	0.733017i	0.930415	−	0.366508i	$-0.119447\pi$
$68$	1.00000	+	4.00000i	0.121268	+	0.485071i
$69$	−2.00000			−0.240772
$70$	−3.00000	+	9.00000i	−0.358569	+	1.07571i
$71$	3.00000	−	3.00000i	0.356034	−	0.356034i	−0.506314	−	0.862349i	$-0.668992\pi$
$71$	3.00000	−	3.00000i	0.356034	−	0.356034i	0.862349	+	0.506314i	$0.168992\pi$
$72$		−	3.00000i		−	0.353553i
$73$	3.00000	+	3.00000i	0.351123	+	0.351123i	0.860527	−	0.509404i	$-0.170134\pi$
$73$	3.00000	+	3.00000i	0.351123	+	0.351123i	−0.509404	+	0.860527i	$0.670134\pi$
$74$	−3.00000	−	3.00000i	−0.348743	−	0.348743i
$75$	−7.00000	+	1.00000i	−0.808290	+	0.115470i
$76$		−	6.00000i		−	0.688247i
$77$	18.0000			2.05129
$78$		0			0
$79$	−7.00000	−	7.00000i	−0.787562	−	0.787562i	0.193532	−	0.981094i	$-0.438006\pi$
$79$	−7.00000	−	7.00000i	−0.787562	−	0.787562i	−0.981094	+	0.193532i	$0.938006\pi$
$80$	2.00000	−	1.00000i	0.223607	−	0.111803i
$81$	5.00000			0.555556
$82$	3.00000	+	3.00000i	0.331295	+	0.331295i
$83$	−4.00000			−0.439057			−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000			−0.439057			−0.219529	+	0.975606i	$0.570452\pi$
$84$	−6.00000			−0.654654
$85$	6.00000	+	7.00000i	0.650791	+	0.759257i
$86$	12.0000			1.29399
$87$	6.00000			0.643268
$88$	−9.00000	−	9.00000i	−0.959403	−	0.959403i
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−	2.00000i	−0.105409	−	0.210819i
$91$		0			0
$92$	−1.00000	+	1.00000i	−0.104257	+	0.104257i
$93$	2.00000			0.207390
$94$			2.00000i			0.206284i
$95$	−6.00000	−	12.0000i	−0.615587	−	1.23117i
$96$	5.00000	+	5.00000i	0.510310	+	0.510310i
$97$	3.00000	+	3.00000i	0.304604	+	0.304604i	0.842812	−	0.538208i	$-0.180899\pi$
$97$	3.00000	+	3.00000i	0.304604	+	0.304604i	−0.538208	+	0.842812i	$0.680899\pi$
$98$			11.0000i			1.11117i
$99$	−3.00000	+	3.00000i	−0.301511	+	0.301511i
$100$	−3.00000	+	4.00000i	−0.300000	+	0.400000i
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$	3.00000	−	5.00000i	0.297044	−	0.495074i
$103$			6.00000i			0.591198i	0.955312	+	0.295599i	$0.0955191\pi$
$103$			6.00000i			0.591198i	−0.955312	+	0.295599i	$0.904481\pi$
$104$		0			0
$105$	−12.0000	+	6.00000i	−1.17108	+	0.585540i
$106$	−2.00000			−0.194257
$107$	−9.00000	−	9.00000i	−0.870063	−	0.870063i	0.122416	−	0.992479i	$-0.460936\pi$
$107$	−9.00000	−	9.00000i	−0.870063	−	0.870063i	−0.992479	+	0.122416i	$0.960936\pi$
$108$	4.00000	−	4.00000i	0.384900	−	0.384900i
$109$	−7.00000	−	7.00000i	−0.670478	−	0.670478i	0.287348	−	0.957826i	$-0.407226\pi$
$109$	−7.00000	−	7.00000i	−0.670478	−	0.670478i	−0.957826	+	0.287348i	$0.907226\pi$
$110$	−9.00000	−	3.00000i	−0.858116	−	0.286039i
$111$		−	6.00000i		−	0.569495i
$112$	3.00000	−	3.00000i	0.283473	−	0.283473i
$113$	−9.00000	+	9.00000i	−0.846649	+	0.846649i	−0.989713	−	0.143065i	$-0.954304\pi$
$113$	−9.00000	+	9.00000i	−0.846649	+	0.846649i	0.143065	+	0.989713i	$0.454304\pi$
$114$	−6.00000	+	6.00000i	−0.561951	+	0.561951i
$115$	−1.00000	+	3.00000i	−0.0932505	+	0.279751i
$116$	3.00000	−	3.00000i	0.278543	−	0.278543i
$117$		0			0
$118$		−	6.00000i		−	0.552345i
$119$	15.0000	+	9.00000i	1.37505	+	0.825029i
$120$	9.00000	+	3.00000i	0.821584	+	0.273861i
$121$			7.00000i			0.636364i
$122$	−1.00000	−	1.00000i	−0.0905357	−	0.0905357i
$123$			6.00000i			0.541002i
$124$	1.00000	−	1.00000i	0.0898027	−	0.0898027i
$125$	−2.00000	+	11.0000i	−0.178885	+	0.983870i
$126$	−3.00000	−	3.00000i	−0.267261	−	0.267261i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	3.00000			0.265165
$129$	12.0000	+	12.0000i	1.05654	+	1.05654i
$130$		0			0
$131$	3.00000	−	3.00000i	0.262111	−	0.262111i	−0.563800	−	0.825911i	$-0.690661\pi$
$131$	3.00000	−	3.00000i	0.262111	−	0.262111i	0.825911	+	0.563800i	$0.190661\pi$
$132$		−	6.00000i		−	0.522233i
$133$	−18.0000	−	18.0000i	−1.56080	−	1.56080i
$134$			6.00000i			0.518321i
$135$	4.00000	−	12.0000i	0.344265	−	1.03280i
$136$	−3.00000	−	12.0000i	−0.257248	−	1.02899i
$137$		−	4.00000i		−	0.341743i	−0.985293	−	0.170872i	$-0.945342\pi$
$137$		−	4.00000i		−	0.341743i	0.985293	−	0.170872i	$-0.0546583\pi$
$138$	2.00000			0.170251
$139$	7.00000	−	7.00000i	0.593732	−	0.593732i	−0.344905	−	0.938638i	$-0.612089\pi$
$139$	7.00000	−	7.00000i	0.593732	−	0.593732i	0.938638	+	0.344905i	$0.112089\pi$
$140$	−3.00000	+	9.00000i	−0.253546	+	0.760639i
$141$	−2.00000	+	2.00000i	−0.168430	+	0.168430i
$142$	−3.00000	+	3.00000i	−0.251754	+	0.251754i
$143$		0			0
$144$			1.00000i			0.0833333i
$145$	3.00000	−	9.00000i	0.249136	−	0.747409i
$146$	−3.00000	−	3.00000i	−0.248282	−	0.248282i
$147$	−11.0000	+	11.0000i	−0.907265	+	0.907265i
$148$	−3.00000	−	3.00000i	−0.246598	−	0.246598i
$149$	−18.0000			−1.47462			−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000			−1.47462			−0.737309	+	0.675556i	$0.763904\pi$
$150$	7.00000	−	1.00000i	0.571548	−	0.0816497i
$151$			10.0000i			0.813788i	0.913475	+	0.406894i	$0.133388\pi$
$151$			10.0000i			0.813788i	−0.913475	+	0.406894i	$0.866612\pi$
$152$			18.0000i			1.45999i
$153$	−4.00000	+	1.00000i	−0.323381	+	0.0808452i
$154$	−18.0000			−1.45048
$155$	1.00000	−	3.00000i	0.0803219	−	0.240966i
$156$		0			0
$157$		−	12.0000i		−	0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$		−	12.0000i		−	0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$	7.00000	+	7.00000i	0.556890	+	0.556890i
$159$	−2.00000	−	2.00000i	−0.158610	−	0.158610i
$160$	10.0000	−	5.00000i	0.790569	−	0.395285i
$161$			6.00000i			0.472866i
$162$	−5.00000			−0.392837
$163$	9.00000	−	9.00000i	0.704934	−	0.704934i	−0.260531	−	0.965465i	$-0.583898\pi$
$163$	9.00000	−	9.00000i	0.704934	−	0.704934i	0.965465	+	0.260531i	$0.0838976\pi$
$164$	3.00000	+	3.00000i	0.234261	+	0.234261i
$165$	−6.00000	−	12.0000i	−0.467099	−	0.934199i
$166$	4.00000			0.310460
$167$	3.00000	+	3.00000i	0.232147	+	0.232147i	0.813588	−	0.581441i	$-0.197511\pi$
$167$	3.00000	+	3.00000i	0.232147	+	0.232147i	−0.581441	+	0.813588i	$0.697511\pi$
$168$	18.0000			1.38873
$169$	13.0000			1.00000
$170$	−6.00000	−	7.00000i	−0.460179	−	0.536875i
$171$	6.00000			0.458831
$172$	12.0000			0.914991
$173$	15.0000	+	15.0000i	1.14043	+	1.14043i	0.988372	+	0.152057i	$0.0485898\pi$
$173$	15.0000	+	15.0000i	1.14043	+	1.14043i	0.152057	+	0.988372i	$0.451410\pi$
$174$	−6.00000			−0.454859
$175$	3.00000	+	21.0000i	0.226779	+	1.58745i
$176$	3.00000	+	3.00000i	0.226134	+	0.226134i
$177$	6.00000	−	6.00000i	0.450988	−	0.450988i
$178$	−6.00000			−0.449719
$179$			6.00000i			0.448461i	0.974536	+	0.224231i	$0.0719869\pi$
$179$			6.00000i			0.448461i	−0.974536	+	0.224231i	$0.928013\pi$
$180$	−1.00000	−	2.00000i	−0.0745356	−	0.149071i
$181$	−11.0000	−	11.0000i	−0.817624	−	0.817624i	0.168140	−	0.985763i	$-0.446224\pi$
$181$	−11.0000	−	11.0000i	−0.817624	−	0.817624i	−0.985763	+	0.168140i	$0.946224\pi$
$182$		0			0
$183$		−	2.00000i		−	0.147844i
$184$	3.00000	−	3.00000i	0.221163	−	0.221163i
$185$	−9.00000	−	3.00000i	−0.661693	−	0.220564i
$186$	−2.00000			−0.146647
$187$	−9.00000	+	15.0000i	−0.658145	+	1.09691i
$188$			2.00000i			0.145865i
$189$		−	24.0000i		−	1.74574i
$190$	6.00000	+	12.0000i	0.435286	+	0.870572i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−7.00000	−	7.00000i	−0.505181	−	0.505181i
$193$	3.00000	−	3.00000i	0.215945	−	0.215945i	−0.590842	−	0.806787i	$-0.701204\pi$
$193$	3.00000	−	3.00000i	0.215945	−	0.215945i	0.806787	+	0.590842i	$0.201204\pi$
$194$	−3.00000	−	3.00000i	−0.215387	−	0.215387i
$195$		0			0
$196$			11.0000i			0.785714i
$197$	−13.0000	+	13.0000i	−0.926212	+	0.926212i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	−13.0000	+	13.0000i	−0.926212	+	0.926212i	0.0712470	+	0.997459i	$0.477302\pi$
$198$	3.00000	−	3.00000i	0.213201	−	0.213201i
$199$	−1.00000	+	1.00000i	−0.0708881	+	0.0708881i	−0.741662	−	0.670774i	$-0.765962\pi$
$199$	−1.00000	+	1.00000i	−0.0708881	+	0.0708881i	0.670774	+	0.741662i	$0.265962\pi$
$200$	9.00000	−	12.0000i	0.636396	−	0.848528i
$201$	−6.00000	+	6.00000i	−0.423207	+	0.423207i
$202$	6.00000			0.422159
$203$		−	18.0000i		−	1.26335i
$204$	3.00000	−	5.00000i	0.210042	−	0.350070i
$205$	9.00000	+	3.00000i	0.628587	+	0.209529i
$206$		−	6.00000i		−	0.418040i
$207$	−1.00000	−	1.00000i	−0.0695048	−	0.0695048i
$208$		0			0
$209$	18.0000	−	18.0000i	1.24509	−	1.24509i
$210$	12.0000	−	6.00000i	0.828079	−	0.414039i
$211$	17.0000	+	17.0000i	1.17033	+	1.17033i	0.982131	+	0.188197i	$0.0602643\pi$
$211$	17.0000	+	17.0000i	1.17033	+	1.17033i	0.188197	+	0.982131i	$0.439736\pi$
$212$	−2.00000			−0.137361
$213$	−6.00000			−0.411113
$214$	9.00000	+	9.00000i	0.615227	+	0.615227i
$215$	24.0000	−	12.0000i	1.63679	−	0.818393i
$216$	−12.0000	+	12.0000i	−0.816497	+	0.816497i
$217$		−	6.00000i		−	0.407307i
$218$	7.00000	+	7.00000i	0.474100	+	0.474100i
$219$		−	6.00000i		−	0.405442i
$220$	−9.00000	−	3.00000i	−0.606780	−	0.202260i
$221$		0			0
$222$			6.00000i			0.402694i
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	15.0000	−	15.0000i	1.00223	−	1.00223i
$225$	−4.00000	−	3.00000i	−0.266667	−	0.200000i
$226$	9.00000	−	9.00000i	0.598671	−	0.598671i
$227$	−15.0000	+	15.0000i	−0.995585	+	0.995585i	−0.999990	−	0.00440533i	$-0.998598\pi$
$227$	−15.0000	+	15.0000i	−0.995585	+	0.995585i	0.00440533	+	0.999990i	$0.498598\pi$
$228$	−6.00000	+	6.00000i	−0.397360	+	0.397360i
$229$			24.0000i			1.58596i	0.609245	+	0.792982i	$0.291473\pi$
$229$			24.0000i			1.58596i	−0.609245	+	0.792982i	$0.708527\pi$
$230$	1.00000	−	3.00000i	0.0659380	−	0.197814i
$231$	−18.0000	−	18.0000i	−1.18431	−	1.18431i
$232$	−9.00000	+	9.00000i	−0.590879	+	0.590879i
$233$	−5.00000	−	5.00000i	−0.327561	−	0.327561i	0.524097	−	0.851658i	$-0.324403\pi$
$233$	−5.00000	−	5.00000i	−0.327561	−	0.327561i	−0.851658	+	0.524097i	$0.824403\pi$
$234$		0			0
$235$	2.00000	+	4.00000i	0.130466	+	0.260931i
$236$		−	6.00000i		−	0.390567i
$237$			14.0000i			0.909398i
$238$	−15.0000	−	9.00000i	−0.972306	−	0.583383i
$239$	−24.0000			−1.55243			−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000			−1.55243			−0.776215	+	0.630468i	$0.782863\pi$
$240$	−3.00000	−	1.00000i	−0.193649	−	0.0645497i
$241$	−19.0000	+	19.0000i	−1.22390	+	1.22390i	−0.257663	+	0.966235i	$0.582952\pi$
$241$	−19.0000	+	19.0000i	−1.22390	+	1.22390i	−0.966235	+	0.257663i	$0.917048\pi$
$242$		−	7.00000i		−	0.449977i
$243$	7.00000	+	7.00000i	0.449050	+	0.449050i
$244$	−1.00000	−	1.00000i	−0.0640184	−	0.0640184i
$245$	11.0000	+	22.0000i	0.702764	+	1.40553i
$246$		−	6.00000i		−	0.382546i
$247$		0			0
$248$	−3.00000	+	3.00000i	−0.190500	+	0.190500i
$249$	4.00000	+	4.00000i	0.253490	+	0.253490i
$250$	2.00000	−	11.0000i	0.126491	−	0.695701i
$251$	−12.0000			−0.757433			−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000			−0.757433			−0.378717	+	0.925513i	$0.623635\pi$
$252$	−3.00000	−	3.00000i	−0.188982	−	0.188982i
$253$	−6.00000			−0.377217
$254$		0			0
$255$	1.00000	−	13.0000i	0.0626224	−	0.814092i
$256$	−17.0000			−1.06250
$257$	−14.0000			−0.873296			−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000			−0.873296			−0.436648	+	0.899632i	$0.643834\pi$
$258$	−12.0000	−	12.0000i	−0.747087	−	0.747087i
$259$	−18.0000			−1.11847
$260$		0			0
$261$	3.00000	+	3.00000i	0.185695	+	0.185695i
$262$	−3.00000	+	3.00000i	−0.185341	+	0.185341i
$263$	16.0000			0.986602			0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000			0.986602			0.493301	+	0.869859i	$0.335790\pi$
$264$			18.0000i			1.10782i
$265$	−4.00000	+	2.00000i	−0.245718	+	0.122859i
$266$	18.0000	+	18.0000i	1.10365	+	1.10365i
$267$	−6.00000	−	6.00000i	−0.367194	−	0.367194i
$268$			6.00000i			0.366508i
$269$	−3.00000	+	3.00000i	−0.182913	+	0.182913i	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−3.00000	+	3.00000i	−0.182913	+	0.182913i	0.609711	+	0.792624i	$0.291286\pi$
$270$	−4.00000	+	12.0000i	−0.243432	+	0.730297i
$271$	24.0000			1.45790			0.728948	−	0.684569i	$-0.240010\pi$
$271$	24.0000			1.45790			0.728948	+	0.684569i	$0.240010\pi$
$272$	1.00000	+	4.00000i	0.0606339	+	0.242536i
$273$		0			0
$274$			4.00000i			0.241649i
$275$	−21.0000	+	3.00000i	−1.26635	+	0.180907i
$276$	2.00000			0.120386
$277$	−21.0000	−	21.0000i	−1.26177	−	1.26177i	−0.950236	−	0.311532i	$-0.899158\pi$
$277$	−21.0000	−	21.0000i	−1.26177	−	1.26177i	−0.311532	−	0.950236i	$-0.600842\pi$
$278$	−7.00000	+	7.00000i	−0.419832	+	0.419832i
$279$	1.00000	+	1.00000i	0.0598684	+	0.0598684i
$280$	9.00000	−	27.0000i	0.537853	−	1.61356i
$281$		−	24.0000i		−	1.43172i	−0.698244	−	0.715860i	$-0.746035\pi$
$281$		−	24.0000i		−	1.43172i	0.698244	−	0.715860i	$-0.253965\pi$
$282$	2.00000	−	2.00000i	0.119098	−	0.119098i
$283$	9.00000	−	9.00000i	0.534994	−	0.534994i	−0.387060	−	0.922055i	$-0.626509\pi$
$283$	9.00000	−	9.00000i	0.534994	−	0.534994i	0.922055	+	0.387060i	$0.126509\pi$
$284$	−3.00000	+	3.00000i	−0.178017	+	0.178017i
$285$	−6.00000	+	18.0000i	−0.355409	+	1.06623i
$286$		0			0
$287$	18.0000			1.06251
$288$			5.00000i			0.294628i
$289$	−15.0000	+	8.00000i	−0.882353	+	0.470588i
$290$	−3.00000	+	9.00000i	−0.176166	+	0.528498i
$291$		−	6.00000i		−	0.351726i
$292$	−3.00000	−	3.00000i	−0.175562	−	0.175562i
$293$			4.00000i			0.233682i	0.993151	+	0.116841i	$0.0372769\pi$
$293$			4.00000i			0.233682i	−0.993151	+	0.116841i	$0.962723\pi$
$294$	11.0000	−	11.0000i	0.641533	−	0.641533i
$295$	−6.00000	−	12.0000i	−0.349334	−	0.698667i
$296$	9.00000	+	9.00000i	0.523114	+	0.523114i
$297$	24.0000			1.39262
$298$	18.0000			1.04271
$299$		0			0
$300$	7.00000	−	1.00000i	0.404145	−	0.0577350i
$301$	36.0000	−	36.0000i	2.07501	−	2.07501i
$302$		−	10.0000i		−	0.575435i
$303$	6.00000	+	6.00000i	0.344691	+	0.344691i
$304$		−	6.00000i		−	0.344124i
$305$	−3.00000	−	1.00000i	−0.171780	−	0.0572598i
$306$	4.00000	−	1.00000i	0.228665	−	0.0571662i
$307$			18.0000i			1.02731i	0.857996	+	0.513657i	$0.171710\pi$
$307$			18.0000i			1.02731i	−0.857996	+	0.513657i	$0.828290\pi$
$308$	−18.0000			−1.02565
$309$	6.00000	−	6.00000i	0.341328	−	0.341328i
$310$	−1.00000	+	3.00000i	−0.0567962	+	0.170389i
$311$	3.00000	−	3.00000i	0.170114	−	0.170114i	−0.616915	−	0.787030i	$-0.711618\pi$
$311$	3.00000	−	3.00000i	0.170114	−	0.170114i	0.787030	+	0.616915i	$0.211618\pi$
$312$		0			0
$313$	3.00000	−	3.00000i	0.169570	−	0.169570i	−0.617220	−	0.786790i	$-0.711741\pi$
$313$	3.00000	−	3.00000i	0.169570	−	0.169570i	0.786790	+	0.617220i	$0.211741\pi$
$314$			12.0000i			0.677199i
$315$	−9.00000	−	3.00000i	−0.507093	−	0.169031i
$316$	7.00000	+	7.00000i	0.393781	+	0.393781i
$317$	15.0000	−	15.0000i	0.842484	−	0.842484i	−0.146697	−	0.989181i	$-0.546864\pi$
$317$	15.0000	−	15.0000i	0.842484	−	0.842484i	0.989181	+	0.146697i	$0.0468644\pi$
$318$	2.00000	+	2.00000i	0.112154	+	0.112154i
$319$	18.0000			1.00781
$320$	−14.0000	+	7.00000i	−0.782624	+	0.391312i
$321$			18.0000i			1.00466i
$322$		−	6.00000i		−	0.334367i
$323$	24.0000	−	6.00000i	1.33540	−	0.333849i
$324$	−5.00000			−0.277778
$325$		0			0
$326$	−9.00000	+	9.00000i	−0.498464	+	0.498464i
$327$			14.0000i			0.774202i
$328$	−9.00000	−	9.00000i	−0.496942	−	0.496942i
$329$	6.00000	+	6.00000i	0.330791	+	0.330791i
$330$	6.00000	+	12.0000i	0.330289	+	0.660578i
$331$		−	18.0000i		−	0.989369i	−0.869072	−	0.494685i	$-0.835284\pi$
$331$		−	18.0000i		−	0.989369i	0.869072	−	0.494685i	$-0.164716\pi$
$332$	4.00000			0.219529
$333$	3.00000	−	3.00000i	0.164399	−	0.164399i
$334$	−3.00000	−	3.00000i	−0.164153	−	0.164153i
$335$	6.00000	+	12.0000i	0.327815	+	0.655630i
$336$	−6.00000			−0.327327
$337$	15.0000	+	15.0000i	0.817102	+	0.817102i	0.985687	−	0.168585i	$-0.0539198\pi$
$337$	15.0000	+	15.0000i	0.817102	+	0.817102i	−0.168585	+	0.985687i	$0.553920\pi$
$338$	−13.0000			−0.707107
$339$	18.0000			0.977626
$340$	−6.00000	−	7.00000i	−0.325396	−	0.379628i
$341$	6.00000			0.324918
$342$	−6.00000			−0.324443
$343$	12.0000	+	12.0000i	0.647939	+	0.647939i
$344$	−36.0000			−1.94099
$345$	4.00000	−	2.00000i	0.215353	−	0.107676i
$346$	−15.0000	−	15.0000i	−0.806405	−	0.806405i
$347$	9.00000	−	9.00000i	0.483145	−	0.483145i	−0.422989	−	0.906135i	$-0.639019\pi$
$347$	9.00000	−	9.00000i	0.483145	−	0.483145i	0.906135	+	0.422989i	$0.139019\pi$
$348$	−6.00000			−0.321634
$349$			28.0000i			1.49881i	0.662114	+	0.749403i	$0.269659\pi$
$349$			28.0000i			1.49881i	−0.662114	+	0.749403i	$0.730341\pi$
$350$	−3.00000	−	21.0000i	−0.160357	−	1.12250i
$351$		0			0
$352$	15.0000	+	15.0000i	0.799503	+	0.799503i
$353$		−	16.0000i		−	0.851594i	−0.904819	−	0.425797i	$-0.859994\pi$
$353$		−	16.0000i		−	0.851594i	0.904819	−	0.425797i	$-0.140006\pi$
$354$	−6.00000	+	6.00000i	−0.318896	+	0.318896i
$355$	−3.00000	+	9.00000i	−0.159223	+	0.477670i
$356$	−6.00000			−0.317999
$357$	−6.00000	−	24.0000i	−0.317554	−	1.27021i
$358$		−	6.00000i		−	0.317110i
$359$		−	30.0000i		−	1.58334i	−0.610949	−	0.791670i	$-0.709212\pi$
$359$		−	30.0000i		−	1.58334i	0.610949	−	0.791670i	$-0.290788\pi$
$360$	3.00000	+	6.00000i	0.158114	+	0.316228i
$361$	−17.0000			−0.894737
$362$	11.0000	+	11.0000i	0.578147	+	0.578147i
$363$	7.00000	−	7.00000i	0.367405	−	0.367405i
$364$		0			0
$365$	−9.00000	−	3.00000i	−0.471082	−	0.157027i
$366$			2.00000i			0.104542i
$367$	−3.00000	+	3.00000i	−0.156599	+	0.156599i	−0.781058	−	0.624459i	$-0.785320\pi$
$367$	−3.00000	+	3.00000i	−0.156599	+	0.156599i	0.624459	+	0.781058i	$0.285320\pi$
$368$	−1.00000	+	1.00000i	−0.0521286	+	0.0521286i
$369$	−3.00000	+	3.00000i	−0.156174	+	0.156174i
$370$	9.00000	+	3.00000i	0.467888	+	0.155963i
$371$	−6.00000	+	6.00000i	−0.311504	+	0.311504i
$372$	−2.00000			−0.103695
$373$			24.0000i			1.24267i	0.783544	+	0.621336i	$0.213410\pi$
$373$			24.0000i			1.24267i	−0.783544	+	0.621336i	$0.786590\pi$
$374$	9.00000	−	15.0000i	0.465379	−	0.775632i
$375$	13.0000	−	9.00000i	0.671317	−	0.464758i
$376$		−	6.00000i		−	0.309426i
$377$		0			0
$378$			24.0000i			1.23443i
$379$	11.0000	−	11.0000i	0.565032	−	0.565032i	−0.365701	−	0.930733i	$-0.619171\pi$
$379$	11.0000	−	11.0000i	0.565032	−	0.565032i	0.930733	+	0.365701i	$0.119171\pi$
$380$	6.00000	+	12.0000i	0.307794	+	0.615587i
$381$		0			0
$382$		0			0
$383$	−16.0000			−0.817562			−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000			−0.817562			−0.408781	+	0.912633i	$0.634046\pi$
$384$	−3.00000	−	3.00000i	−0.153093	−	0.153093i
$385$	−36.0000	+	18.0000i	−1.83473	+	0.917365i
$386$	−3.00000	+	3.00000i	−0.152696	+	0.152696i
$387$			12.0000i			0.609994i
$388$	−3.00000	−	3.00000i	−0.152302	−	0.152302i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$	−5.00000	−	3.00000i	−0.252861	−	0.151717i
$392$		−	33.0000i		−	1.66675i
$393$	−6.00000			−0.302660
$394$	13.0000	−	13.0000i	0.654931	−	0.654931i
$395$	21.0000	+	7.00000i	1.05662	+	0.352208i
$396$	3.00000	−	3.00000i	0.150756	−	0.150756i
$397$	27.0000	−	27.0000i	1.35509	−	1.35509i	0.475229	−	0.879862i	$-0.342365\pi$
$397$	27.0000	−	27.0000i	1.35509	−	1.35509i	0.879862	−	0.475229i	$-0.157635\pi$
$398$	1.00000	−	1.00000i	0.0501255	−	0.0501255i
$399$			36.0000i			1.80225i
$400$	−3.00000	+	4.00000i	−0.150000	+	0.200000i
$401$	9.00000	+	9.00000i	0.449439	+	0.449439i	0.895168	−	0.445729i	$-0.147056\pi$
$401$	9.00000	+	9.00000i	0.449439	+	0.449439i	−0.445729	+	0.895168i	$0.647056\pi$
$402$	6.00000	−	6.00000i	0.299253	−	0.299253i
$403$		0			0
$404$	6.00000			0.298511
$405$	−10.0000	+	5.00000i	−0.496904	+	0.248452i
$406$			18.0000i			0.893325i
$407$		−	18.0000i		−	0.892227i
$408$	−9.00000	+	15.0000i	−0.445566	+	0.742611i
$409$	−30.0000			−1.48340			−0.741702	−	0.670729i	$-0.765981\pi$
$409$	−30.0000			−1.48340			−0.741702	+	0.670729i	$0.765981\pi$
$410$	−9.00000	−	3.00000i	−0.444478	−	0.148159i
$411$	−4.00000	+	4.00000i	−0.197305	+	0.197305i
$412$		−	6.00000i		−	0.295599i
$413$	−18.0000	−	18.0000i	−0.885722	−	0.885722i
$414$	1.00000	+	1.00000i	0.0491473	+	0.0491473i
$415$	8.00000	−	4.00000i	0.392705	−	0.196352i
$416$		0			0
$417$	−14.0000			−0.685583
$418$	−18.0000	+	18.0000i	−0.880409	+	0.880409i
$419$	−15.0000	−	15.0000i	−0.732798	−	0.732798i	0.238375	−	0.971173i	$-0.423385\pi$
$419$	−15.0000	−	15.0000i	−0.732798	−	0.732798i	−0.971173	+	0.238375i	$0.923385\pi$
$420$	12.0000	−	6.00000i	0.585540	−	0.292770i
$421$	−6.00000			−0.292422			−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000			−0.292422			−0.146211	+	0.989253i	$0.546708\pi$
$422$	−17.0000	−	17.0000i	−0.827547	−	0.827547i
$423$	−2.00000			−0.0972433
$424$	6.00000			0.291386
$425$	−19.0000	−	8.00000i	−0.921635	−	0.388057i
$426$	6.00000			0.290701
$427$	−6.00000			−0.290360
$428$	9.00000	+	9.00000i	0.435031	+	0.435031i
$429$		0			0
$430$	−24.0000	+	12.0000i	−1.15738	+	0.578691i
$431$	9.00000	+	9.00000i	0.433515	+	0.433515i	0.889822	−	0.456307i	$-0.150828\pi$
$431$	9.00000	+	9.00000i	0.433515	+	0.433515i	−0.456307	+	0.889822i	$0.650828\pi$
$432$	4.00000	−	4.00000i	0.192450	−	0.192450i
$433$	−30.0000			−1.44171			−0.720854	−	0.693087i	$-0.756250\pi$
$433$	−30.0000			−1.44171			−0.720854	+	0.693087i	$0.756250\pi$
$434$			6.00000i			0.288009i
$435$	−12.0000	+	6.00000i	−0.575356	+	0.287678i
$436$	7.00000	+	7.00000i	0.335239	+	0.335239i
$437$	6.00000	+	6.00000i	0.287019	+	0.287019i
$438$			6.00000i			0.286691i
$439$	19.0000	−	19.0000i	0.906821	−	0.906821i	−0.0891938	−	0.996014i	$-0.528429\pi$
$439$	19.0000	−	19.0000i	0.906821	−	0.906821i	0.996014	+	0.0891938i	$0.0284290\pi$
$440$	27.0000	+	9.00000i	1.28717	+	0.429058i
$441$	−11.0000			−0.523810
$442$		0			0
$443$		−	22.0000i		−	1.04525i	−0.852562	−	0.522626i	$-0.824953\pi$
$443$		−	22.0000i		−	1.04525i	0.852562	−	0.522626i	$-0.175047\pi$
$444$			6.00000i			0.284747i
$445$	−12.0000	+	6.00000i	−0.568855	+	0.284427i
$446$		0			0
$447$	18.0000	+	18.0000i	0.851371	+	0.851371i
$448$	−21.0000	+	21.0000i	−0.992157	+	0.992157i
$449$	−15.0000	−	15.0000i	−0.707894	−	0.707894i	0.258198	−	0.966092i	$-0.416871\pi$
$449$	−15.0000	−	15.0000i	−0.707894	−	0.707894i	−0.966092	+	0.258198i	$0.916871\pi$
$450$	4.00000	+	3.00000i	0.188562	+	0.141421i
$451$			18.0000i			0.847587i
$452$	9.00000	−	9.00000i	0.423324	−	0.423324i
$453$	10.0000	−	10.0000i	0.469841	−	0.469841i
$454$	15.0000	−	15.0000i	0.703985	−	0.703985i
$455$		0			0
$456$	18.0000	−	18.0000i	0.842927	−	0.842927i
$457$	−6.00000			−0.280668			−0.140334	−	0.990104i	$-0.544818\pi$
$457$	−6.00000			−0.280668			−0.140334	+	0.990104i	$0.544818\pi$
$458$		−	24.0000i		−	1.12145i
$459$	20.0000	+	12.0000i	0.933520	+	0.560112i
$460$	1.00000	−	3.00000i	0.0466252	−	0.139876i
$461$			12.0000i			0.558896i	0.960161	+	0.279448i	$0.0901514\pi$
$461$			12.0000i			0.558896i	−0.960161	+	0.279448i	$0.909849\pi$
$462$	18.0000	+	18.0000i	0.837436	+	0.837436i
$463$		−	18.0000i		−	0.836531i	−0.908325	−	0.418265i	$-0.862638\pi$
$463$		−	18.0000i		−	0.836531i	0.908325	−	0.418265i	$-0.137362\pi$
$464$	3.00000	−	3.00000i	0.139272	−	0.139272i
$465$	−4.00000	+	2.00000i	−0.185496	+	0.0927478i
$466$	5.00000	+	5.00000i	0.231621	+	0.231621i
$467$	−28.0000			−1.29569			−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000			−1.29569			−0.647843	+	0.761774i	$0.724329\pi$
$468$		0			0
$469$	18.0000	+	18.0000i	0.831163	+	0.831163i
$470$	−2.00000	−	4.00000i	−0.0922531	−	0.184506i
$471$	−12.0000	+	12.0000i	−0.552931	+	0.552931i
$472$			18.0000i			0.828517i
$473$	36.0000	+	36.0000i	1.65528	+	1.65528i
$474$		−	14.0000i		−	0.643041i
$475$	24.0000	+	18.0000i	1.10120	+	0.825897i
$476$	−15.0000	−	9.00000i	−0.687524	−	0.412514i
$477$		−	2.00000i		−	0.0915737i
$478$	24.0000			1.09773
$479$	−9.00000	+	9.00000i	−0.411220	+	0.411220i	−0.882164	−	0.470943i	$-0.843914\pi$
$479$	−9.00000	+	9.00000i	−0.411220	+	0.411220i	0.470943	+	0.882164i	$0.343914\pi$
$480$	−15.0000	−	5.00000i	−0.684653	−	0.228218i
$481$		0			0
$482$	19.0000	−	19.0000i	0.865426	−	0.865426i
$483$	6.00000	−	6.00000i	0.273009	−	0.273009i
$484$		−	7.00000i		−	0.318182i
$485$	−9.00000	−	3.00000i	−0.408669	−	0.136223i
$486$	−7.00000	−	7.00000i	−0.317526	−	0.317526i
$487$	−3.00000	+	3.00000i	−0.135943	+	0.135943i	−0.771804	−	0.635861i	$-0.780645\pi$
$487$	−3.00000	+	3.00000i	−0.135943	+	0.135943i	0.635861	+	0.771804i	$0.280645\pi$
$488$	3.00000	+	3.00000i	0.135804	+	0.135804i
$489$	−18.0000			−0.813988
$490$	−11.0000	−	22.0000i	−0.496929	−	0.993859i
$491$			6.00000i			0.270776i	0.990793	+	0.135388i	$0.0432281\pi$
$491$			6.00000i			0.270776i	−0.990793	+	0.135388i	$0.956772\pi$
$492$		−	6.00000i		−	0.270501i
$493$	15.0000	+	9.00000i	0.675566	+	0.405340i
$494$		0			0
$495$	3.00000	−	9.00000i	0.134840	−	0.404520i
$496$	1.00000	−	1.00000i	0.0449013	−	0.0449013i
$497$			18.0000i			0.807410i
$498$	−4.00000	−	4.00000i	−0.179244	−	0.179244i
$499$	−23.0000	−	23.0000i	−1.02962	−	1.02962i	−0.999548	−	0.0300737i	$-0.990426\pi$
$499$	−23.0000	−	23.0000i	−1.02962	−	1.02962i	−0.0300737	−	0.999548i	$-0.509574\pi$
$500$	2.00000	−	11.0000i	0.0894427	−	0.491935i
$501$		−	6.00000i		−	0.268060i
$502$	12.0000			0.535586
$503$	−3.00000	+	3.00000i	−0.133763	+	0.133763i	−0.770818	−	0.637055i	$-0.780152\pi$
$503$	−3.00000	+	3.00000i	−0.133763	+	0.133763i	0.637055	+	0.770818i	$0.280152\pi$
$504$	9.00000	+	9.00000i	0.400892	+	0.400892i
$505$	12.0000	−	6.00000i	0.533993	−	0.266996i
$506$	6.00000			0.266733
$507$	−13.0000	−	13.0000i	−0.577350	−	0.577350i
$508$		0			0
$509$	6.00000			0.265945			0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000			0.265945			0.132973	+	0.991120i	$0.457548\pi$
$510$	−1.00000	+	13.0000i	−0.0442807	+	0.575650i
$511$	−18.0000			−0.796273
$512$	11.0000			0.486136
$513$	−24.0000	−	24.0000i	−1.05963	−	1.05963i
$514$	14.0000			0.617514
$515$	−6.00000	−	12.0000i	−0.264392	−	0.528783i
$516$	−12.0000	−	12.0000i	−0.528271	−	0.528271i
$517$	−6.00000	+	6.00000i	−0.263880	+	0.263880i
$518$	18.0000			0.790875
$519$		−	30.0000i		−	1.31685i
$520$		0			0
$521$	9.00000	+	9.00000i	0.394297	+	0.394297i	0.876216	−	0.481919i	$-0.160060\pi$
$521$	9.00000	+	9.00000i	0.394297	+	0.394297i	−0.481919	+	0.876216i	$0.660060\pi$
$522$	−3.00000	−	3.00000i	−0.131306	−	0.131306i
$523$		−	6.00000i		−	0.262362i	−0.991358	−	0.131181i	$-0.958123\pi$
$523$		−	6.00000i		−	0.262362i	0.991358	−	0.131181i	$-0.0418769\pi$
$524$	−3.00000	+	3.00000i	−0.131056	+	0.131056i
$525$	18.0000	−	24.0000i	0.785584	−	1.04745i
$526$	−16.0000			−0.697633
$527$	5.00000	+	3.00000i	0.217803	+	0.130682i
$528$		−	6.00000i		−	0.261116i
$529$			21.0000i			0.913043i
$530$	4.00000	−	2.00000i	0.173749	−	0.0868744i
$531$	6.00000			0.260378
$532$	18.0000	+	18.0000i	0.780399	+	0.780399i
$533$		0			0
$534$	6.00000	+	6.00000i	0.259645	+	0.259645i
$535$	27.0000	+	9.00000i	1.16731	+	0.389104i
$536$		−	18.0000i		−	0.777482i
$537$	6.00000	−	6.00000i	0.258919	−	0.258919i
$538$	3.00000	−	3.00000i	0.129339	−	0.129339i
$539$	−33.0000	+	33.0000i	−1.42141	+	1.42141i
$540$	−4.00000	+	12.0000i	−0.172133	+	0.516398i
$541$	1.00000	−	1.00000i	0.0429934	−	0.0429934i	−0.685283	−	0.728277i	$-0.740322\pi$
$541$	1.00000	−	1.00000i	0.0429934	−	0.0429934i	0.728277	+	0.685283i	$0.240322\pi$
$542$	−24.0000			−1.03089
$543$			22.0000i			0.944110i
$544$	5.00000	+	20.0000i	0.214373	+	0.857493i
$545$	21.0000	+	7.00000i	0.899541	+	0.299847i
$546$		0			0
$547$	−9.00000	−	9.00000i	−0.384812	−	0.384812i	0.488020	−	0.872832i	$-0.337719\pi$
$547$	−9.00000	−	9.00000i	−0.384812	−	0.384812i	−0.872832	+	0.488020i	$0.837719\pi$
$548$			4.00000i			0.170872i
$549$	1.00000	−	1.00000i	0.0426790	−	0.0426790i
$550$	21.0000	−	3.00000i	0.895443	−	0.127920i
$551$	−18.0000	−	18.0000i	−0.766826	−	0.766826i
$552$	−6.00000			−0.255377
$553$	42.0000			1.78602
$554$	21.0000	+	21.0000i	0.892205	+	0.892205i
$555$	6.00000	+	12.0000i	0.254686	+	0.509372i
$556$	−7.00000	+	7.00000i	−0.296866	+	0.296866i
$557$			32.0000i			1.35588i	0.735116	+	0.677942i	$0.237128\pi$
$557$			32.0000i			1.35588i	−0.735116	+	0.677942i	$0.762872\pi$
$558$	−1.00000	−	1.00000i	−0.0423334	−	0.0423334i
$559$		0			0
$560$	−3.00000	+	9.00000i	−0.126773	+	0.380319i
$561$	24.0000	−	6.00000i	1.01328	−	0.253320i
$562$			24.0000i			1.01238i
$563$	−28.0000			−1.18006			−0.590030	−	0.807382i	$-0.700884\pi$
$563$	−28.0000			−1.18006			−0.590030	+	0.807382i	$0.700884\pi$
$564$	2.00000	−	2.00000i	0.0842152	−	0.0842152i
$565$	9.00000	−	27.0000i	0.378633	−	1.13590i
$566$	−9.00000	+	9.00000i	−0.378298	+	0.378298i
$567$	−15.0000	+	15.0000i	−0.629941	+	0.629941i
$568$	9.00000	−	9.00000i	0.377632	−	0.377632i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$	6.00000	−	18.0000i	0.251312	−	0.753937i
$571$	−23.0000	−	23.0000i	−0.962520	−	0.962520i	0.0368025	−	0.999323i	$-0.488283\pi$
$571$	−23.0000	−	23.0000i	−0.962520	−	0.962520i	−0.999323	+	0.0368025i	$0.988283\pi$
$572$		0			0
$573$		0			0
$574$	−18.0000			−0.751305
$575$	−1.00000	−	7.00000i	−0.0417029	−	0.291920i
$576$		−	7.00000i		−	0.291667i
$577$		−	36.0000i		−	1.49870i	−0.662174	−	0.749350i	$-0.730366\pi$
$577$		−	36.0000i		−	1.49870i	0.662174	−	0.749350i	$-0.269634\pi$
$578$	15.0000	−	8.00000i	0.623918	−	0.332756i
$579$	−6.00000			−0.249351
$580$	−3.00000	+	9.00000i	−0.124568	+	0.373705i
$581$	12.0000	−	12.0000i	0.497844	−	0.497844i
$582$			6.00000i			0.248708i
$583$	−6.00000	−	6.00000i	−0.248495	−	0.248495i
$584$	9.00000	+	9.00000i	0.372423	+	0.372423i
$585$		0			0
$586$		−	4.00000i		−	0.165238i
$587$	4.00000			0.165098			0.0825488	−	0.996587i	$-0.473694\pi$
$587$	4.00000			0.165098			0.0825488	+	0.996587i	$0.473694\pi$
$588$	11.0000	−	11.0000i	0.453632	−	0.453632i
$589$	−6.00000	−	6.00000i	−0.247226	−	0.247226i
$590$	6.00000	+	12.0000i	0.247016	+	0.494032i
$591$	26.0000			1.06950
$592$	−3.00000	−	3.00000i	−0.123299	−	0.123299i
$593$	−2.00000			−0.0821302			−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000			−0.0821302			−0.0410651	+	0.999156i	$0.513075\pi$
$594$	−24.0000			−0.984732
$595$	−39.0000	−	3.00000i	−1.59884	−	0.122988i
$596$	18.0000			0.737309
$597$	2.00000			0.0818546
$598$		0			0
$599$	24.0000			0.980613			0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000			0.980613			0.490307	+	0.871550i	$0.336885\pi$
$600$	−21.0000	+	3.00000i	−0.857321	+	0.122474i
$601$	29.0000	+	29.0000i	1.18293	+	1.18293i	0.978980	+	0.203954i	$0.0653794\pi$
$601$	29.0000	+	29.0000i	1.18293	+	1.18293i	0.203954	+	0.978980i	$0.434621\pi$
$602$	−36.0000	+	36.0000i	−1.46725	+	1.46725i
$603$	−6.00000			−0.244339
$604$		−	10.0000i		−	0.406894i
$605$	−7.00000	−	14.0000i	−0.284590	−	0.569181i
$606$	−6.00000	−	6.00000i	−0.243733	−	0.243733i
$607$	−33.0000	−	33.0000i	−1.33943	−	1.33943i	−0.896612	−	0.442816i	$-0.853979\pi$
$607$	−33.0000	−	33.0000i	−1.33943	−	1.33943i	−0.442816	−	0.896612i	$-0.646021\pi$
$608$		−	30.0000i		−	1.21666i
$609$	−18.0000	+	18.0000i	−0.729397	+	0.729397i
$610$	3.00000	+	1.00000i	0.121466	+	0.0404888i
$611$		0			0
$612$	4.00000	−	1.00000i	0.161690	−	0.0404226i
$613$			36.0000i			1.45403i	0.686624	+	0.727013i	$0.259092\pi$
$613$			36.0000i			1.45403i	−0.686624	+	0.727013i	$0.740908\pi$
$614$		−	18.0000i		−	0.726421i
$615$	−6.00000	−	12.0000i	−0.241943	−	0.483887i
$616$	54.0000			2.17572
$617$	15.0000	+	15.0000i	0.603877	+	0.603877i	0.941339	−	0.337462i	$-0.109568\pi$
$617$	15.0000	+	15.0000i	0.603877	+	0.603877i	−0.337462	+	0.941339i	$0.609568\pi$
$618$	−6.00000	+	6.00000i	−0.241355	+	0.241355i
$619$	13.0000	+	13.0000i	0.522514	+	0.522514i	0.918330	−	0.395816i	$-0.129538\pi$
$619$	13.0000	+	13.0000i	0.522514	+	0.522514i	−0.395816	+	0.918330i	$0.629538\pi$
$620$	−1.00000	+	3.00000i	−0.0401610	+	0.120483i
$621$			8.00000i			0.321029i
$622$	−3.00000	+	3.00000i	−0.120289	+	0.120289i
$623$	−18.0000	+	18.0000i	−0.721155	+	0.721155i
$624$		0			0
$625$	−7.00000	−	24.0000i	−0.280000	−	0.960000i
$626$	−3.00000	+	3.00000i	−0.119904	+	0.119904i
$627$	−36.0000			−1.43770
$628$			12.0000i			0.478852i
$629$	9.00000	−	15.0000i	0.358854	−	0.598089i
$630$	9.00000	+	3.00000i	0.358569	+	0.119523i
$631$			34.0000i			1.35352i	0.736204	+	0.676759i	$0.236616\pi$
$631$			34.0000i			1.35352i	−0.736204	+	0.676759i	$0.763384\pi$
$632$	−21.0000	−	21.0000i	−0.835335	−	0.835335i
$633$		−	34.0000i		−	1.35138i
$634$	−15.0000	+	15.0000i	−0.595726	+	0.595726i
$635$		0			0
$636$	2.00000	+	2.00000i	0.0793052	+	0.0793052i
$637$		0			0
$638$	−18.0000			−0.712627
$639$	−3.00000	−	3.00000i	−0.118678	−	0.118678i
$640$	−6.00000	+	3.00000i	−0.237171	+	0.118585i
$641$	9.00000	−	9.00000i	0.355479	−	0.355479i	−0.506665	−	0.862143i	$-0.669122\pi$
$641$	9.00000	−	9.00000i	0.355479	−	0.355479i	0.862143	+	0.506665i	$0.169122\pi$
$642$		−	18.0000i		−	0.710403i
$643$	−9.00000	−	9.00000i	−0.354925	−	0.354925i	0.507013	−	0.861938i	$-0.330750\pi$
$643$	−9.00000	−	9.00000i	−0.354925	−	0.354925i	−0.861938	+	0.507013i	$0.830750\pi$
$644$		−	6.00000i		−	0.236433i
$645$	−36.0000	−	12.0000i	−1.41750	−	0.472500i
$646$	−24.0000	+	6.00000i	−0.944267	+	0.236067i
$647$			46.0000i			1.80845i	0.427060	+	0.904223i	$0.359549\pi$
$647$			46.0000i			1.80845i	−0.427060	+	0.904223i	$0.640451\pi$
$648$	15.0000			0.589256
$649$	18.0000	−	18.0000i	0.706562	−	0.706562i
$650$		0			0
$651$	−6.00000	+	6.00000i	−0.235159	+	0.235159i
$652$	−9.00000	+	9.00000i	−0.352467	+	0.352467i
$653$	−21.0000	+	21.0000i	−0.821794	+	0.821794i	−0.986365	−	0.164572i	$-0.947376\pi$
$653$	−21.0000	+	21.0000i	−0.821794	+	0.821794i	0.164572	+	0.986365i	$0.447376\pi$
$654$		−	14.0000i		−	0.547443i
$655$	−3.00000	+	9.00000i	−0.117220	+	0.351659i
$656$	3.00000	+	3.00000i	0.117130	+	0.117130i
$657$	3.00000	−	3.00000i	0.117041	−	0.117041i
$658$	−6.00000	−	6.00000i	−0.233904	−	0.233904i
$659$	36.0000			1.40236			0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000			1.40236			0.701180	+	0.712984i	$0.252657\pi$
$660$	6.00000	+	12.0000i	0.233550	+	0.467099i
$661$		−	36.0000i		−	1.40024i	−0.714026	−	0.700119i	$-0.753130\pi$
$661$		−	36.0000i		−	1.40024i	0.714026	−	0.700119i	$-0.246870\pi$
$662$			18.0000i			0.699590i
$663$		0			0
$664$	−12.0000			−0.465690
$665$	54.0000	+	18.0000i	2.09403	+	0.698010i
$666$	−3.00000	+	3.00000i	−0.116248	+	0.116248i
$667$			6.00000i			0.232321i
$668$	−3.00000	−	3.00000i	−0.116073	−	0.116073i
$669$		0			0
$670$	−6.00000	−	12.0000i	−0.231800	−	0.463600i
$671$		−	6.00000i		−	0.231627i
$672$	−30.0000			−1.15728
$673$	15.0000	−	15.0000i	0.578208	−	0.578208i	−0.356202	−	0.934409i	$-0.615928\pi$
$673$	15.0000	−	15.0000i	0.578208	−	0.578208i	0.934409	+	0.356202i	$0.115928\pi$
$674$	−15.0000	−	15.0000i	−0.577778	−	0.577778i
$675$	4.00000	+	28.0000i	0.153960	+	1.07772i
$676$	−13.0000			−0.500000
$677$	11.0000	+	11.0000i	0.422764	+	0.422764i	0.886154	−	0.463390i	$-0.153367\pi$
$677$	11.0000	+	11.0000i	0.422764	+	0.422764i	−0.463390	+	0.886154i	$0.653367\pi$
$678$	−18.0000			−0.691286
$679$	−18.0000			−0.690777
$680$	18.0000	+	21.0000i	0.690268	+	0.805313i
$681$	30.0000			1.14960
$682$	−6.00000			−0.229752
$683$	15.0000	+	15.0000i	0.573959	+	0.573959i	0.933232	−	0.359273i	$-0.116975\pi$
$683$	15.0000	+	15.0000i	0.573959	+	0.573959i	−0.359273	+	0.933232i	$0.616975\pi$
$684$	−6.00000			−0.229416
$685$	4.00000	+	8.00000i	0.152832	+	0.305664i
$686$	−12.0000	−	12.0000i	−0.458162	−	0.458162i
$687$	24.0000	−	24.0000i	0.915657	−	0.915657i
$688$	12.0000			0.457496
$689$		0			0
$690$	−4.00000	+	2.00000i	−0.152277	+	0.0761387i
$691$	−19.0000	−	19.0000i	−0.722794	−	0.722794i	0.246379	−	0.969173i	$-0.420759\pi$
$691$	−19.0000	−	19.0000i	−0.722794	−	0.722794i	−0.969173	+	0.246379i	$0.920759\pi$
$692$	−15.0000	−	15.0000i	−0.570214	−	0.570214i
$693$		−	18.0000i		−	0.683763i
$694$	−9.00000	+	9.00000i	−0.341635	+	0.341635i
$695$	−7.00000	+	21.0000i	−0.265525	+	0.796575i
$696$	18.0000			0.682288
$697$	−9.00000	+	15.0000i	−0.340899	+	0.568166i
$698$		−	28.0000i		−	1.05982i
$699$			10.0000i			0.378235i
$700$	−3.00000	−	21.0000i	−0.113389	−	0.793725i
$701$	−30.0000			−1.13308			−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000			−1.13308			−0.566542	+	0.824033i	$0.691719\pi$
$702$		0			0
$703$	−18.0000	+	18.0000i	−0.678883	+	0.678883i
$704$	−21.0000	−	21.0000i	−0.791467	−	0.791467i
$705$	2.00000	−	6.00000i	0.0753244	−	0.225973i
$706$			16.0000i			0.602168i
$707$	18.0000	−	18.0000i	0.676960	−	0.676960i
$708$	−6.00000	+	6.00000i	−0.225494	+	0.225494i
$709$	5.00000	−	5.00000i	0.187779	−	0.187779i	−0.606956	−	0.794735i	$-0.707610\pi$
$709$	5.00000	−	5.00000i	0.187779	−	0.187779i	0.794735	+	0.606956i	$0.207610\pi$
$710$	3.00000	−	9.00000i	0.112588	−	0.337764i
$711$	−7.00000	+	7.00000i	−0.262521	+	0.262521i
$712$	18.0000			0.674579
$713$			2.00000i			0.0749006i
$714$	6.00000	+	24.0000i	0.224544	+	0.898177i
$715$		0			0
$716$		−	6.00000i		−	0.224231i
$717$	24.0000	+	24.0000i	0.896296	+	0.896296i
$718$			30.0000i			1.11959i
$719$	−21.0000	+	21.0000i	−0.783168	+	0.783168i	−0.980364	−	0.197196i	$-0.936816\pi$
$719$	−21.0000	+	21.0000i	−0.783168	+	0.783168i	0.197196	+	0.980364i	$0.436816\pi$
$720$	−1.00000	−	2.00000i	−0.0372678	−	0.0745356i
$721$	−18.0000	−	18.0000i	−0.670355	−	0.670355i
$722$	17.0000			0.632674
$723$	38.0000			1.41324
$724$	11.0000	+	11.0000i	0.408812	+	0.408812i
$725$	3.00000	+	21.0000i	0.111417	+	0.779920i
$726$	−7.00000	+	7.00000i	−0.259794	+	0.259794i
$727$		−	18.0000i		−	0.667583i	−0.942647	−	0.333792i	$-0.891672\pi$
$727$		−	18.0000i		−	0.667583i	0.942647	−	0.333792i	$-0.108328\pi$
$728$		0			0
$729$		−	29.0000i		−	1.07407i
$730$	9.00000	+	3.00000i	0.333105	+	0.111035i
$731$	12.0000	+	48.0000i	0.443836	+	1.77534i
$732$			2.00000i			0.0739221i
$733$	42.0000			1.55131			0.775653	−	0.631160i	$-0.217421\pi$
$733$	42.0000			1.55131			0.775653	+	0.631160i	$0.217421\pi$
$734$	3.00000	−	3.00000i	0.110732	−	0.110732i
$735$	11.0000	−	33.0000i	0.405741	−	1.21722i
$736$	−5.00000	+	5.00000i	−0.184302	+	0.184302i
$737$	−18.0000	+	18.0000i	−0.663039	+	0.663039i
$738$	3.00000	−	3.00000i	0.110432	−	0.110432i
$739$		−	34.0000i		−	1.25071i	−0.780340	−	0.625355i	$-0.784954\pi$
$739$		−	34.0000i		−	1.25071i	0.780340	−	0.625355i	$-0.215046\pi$
$740$	9.00000	+	3.00000i	0.330847	+	0.110282i
$741$		0			0
$742$	6.00000	−	6.00000i	0.220267	−	0.220267i
$743$	−9.00000	−	9.00000i	−0.330178	−	0.330178i	0.522476	−	0.852654i	$-0.325008\pi$
$743$	−9.00000	−	9.00000i	−0.330178	−	0.330178i	−0.852654	+	0.522476i	$0.825008\pi$
$744$	6.00000			0.219971
$745$	36.0000	−	18.0000i	1.31894	−	0.659469i
$746$		−	24.0000i		−	0.878702i
$747$			4.00000i			0.146352i
$748$	9.00000	−	15.0000i	0.329073	−	0.548454i
$749$	54.0000			1.97312
$750$	−13.0000	+	9.00000i	−0.474693	+	0.328634i
$751$	19.0000	−	19.0000i	0.693320	−	0.693320i	−0.269641	−	0.962961i	$-0.586905\pi$
$751$	19.0000	−	19.0000i	0.693320	−	0.693320i	0.962961	+	0.269641i	$0.0869050\pi$
$752$			2.00000i			0.0729325i
$753$	12.0000	+	12.0000i	0.437304	+	0.437304i
$754$		0			0
$755$	−10.0000	−	20.0000i	−0.363937	−	0.727875i
$756$			24.0000i			0.872872i
$757$	−42.0000			−1.52652			−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000			−1.52652			−0.763258	+	0.646094i	$0.776401\pi$
$758$	−11.0000	+	11.0000i	−0.399538	+	0.399538i
$759$	6.00000	+	6.00000i	0.217786	+	0.217786i
$760$	−18.0000	−	36.0000i	−0.652929	−	1.30586i
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$	42.0000			1.52050
$764$		0			0
$765$	7.00000	−	6.00000i	0.253086	−	0.216930i
$766$	16.0000			0.578103
$767$		0			0
$768$	17.0000	+	17.0000i	0.613435	+	0.613435i
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$	36.0000	−	18.0000i	1.29735	−	0.648675i
$771$	14.0000	+	14.0000i	0.504198	+	0.504198i
$772$	−3.00000	+	3.00000i	−0.107972	+	0.107972i
$773$	−26.0000			−0.935155			−0.467578	−	0.883952i	$-0.654873\pi$
$773$	−26.0000			−0.935155			−0.467578	+	0.883952i	$0.654873\pi$
$774$		−	12.0000i		−	0.431331i
$775$	1.00000	+	7.00000i	0.0359211	+	0.251447i
$776$	9.00000	+	9.00000i	0.323081	+	0.323081i
$777$	18.0000	+	18.0000i	0.645746	+	0.645746i
$778$		0			0
$779$	18.0000	−	18.0000i	0.644917	−	0.644917i
$780$		0			0
$781$	−18.0000			−0.644091
$782$	5.00000	+	3.00000i	0.178800	+	0.107280i
$783$		−	24.0000i		−	0.857690i
$784$			11.0000i			0.392857i
$785$	12.0000	+	24.0000i	0.428298	+	0.856597i
$786$	6.00000			0.214013
$787$	27.0000	+	27.0000i	0.962446	+	0.962446i	0.999320	−	0.0368739i	$-0.0117400\pi$
$787$	27.0000	+	27.0000i	0.962446	+	0.962446i	−0.0368739	+	0.999320i	$0.511740\pi$
$788$	13.0000	−	13.0000i	0.463106	−	0.463106i
$789$	−16.0000	−	16.0000i	−0.569615	−	0.569615i
$790$	−21.0000	−	7.00000i	−0.747146	−	0.249049i
$791$		−	54.0000i		−	1.92002i
$792$	−9.00000	+	9.00000i	−0.319801	+	0.319801i
$793$		0			0
$794$	−27.0000	+	27.0000i	−0.958194	+	0.958194i
$795$	6.00000	+	2.00000i	0.212798	+	0.0709327i
$796$	1.00000	−	1.00000i	0.0354441	−	0.0354441i
$797$	34.0000			1.20434			0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000			1.20434			0.602171	+	0.798367i	$0.294303\pi$
$798$		−	36.0000i		−	1.27439i
$799$	−8.00000	+	2.00000i	−0.283020	+	0.0707549i
$800$	−15.0000	+	20.0000i	−0.530330	+	0.707107i
$801$		−	6.00000i		−	0.212000i
$802$	−9.00000	−	9.00000i	−0.317801	−	0.317801i
$803$		−	18.0000i		−	0.635206i
$804$	6.00000	−	6.00000i	0.211604	−	0.211604i
$805$	−6.00000	−	12.0000i	−0.211472	−	0.422944i
$806$		0			0
$807$	6.00000			0.211210
$808$	−18.0000			−0.633238
$809$	−3.00000	−	3.00000i	−0.105474	−	0.105474i	0.652400	−	0.757875i	$-0.273762\pi$
$809$	−3.00000	−	3.00000i	−0.105474	−	0.105474i	−0.757875	+	0.652400i	$0.773762\pi$
$810$	10.0000	−	5.00000i	0.351364	−	0.175682i
$811$	−1.00000	+	1.00000i	−0.0351147	+	0.0351147i	−0.724446	−	0.689331i	$-0.757904\pi$
$811$	−1.00000	+	1.00000i	−0.0351147	+	0.0351147i	0.689331	+	0.724446i	$0.257904\pi$
$812$			18.0000i			0.631676i
$813$	−24.0000	−	24.0000i	−0.841717	−	0.841717i
$814$			18.0000i			0.630900i
$815$	−9.00000	+	27.0000i	−0.315256	+	0.945769i
$816$	3.00000	−	5.00000i	0.105021	−	0.175035i
$817$		−	72.0000i		−	2.51896i
$818$	30.0000			1.04893
$819$		0			0
$820$	−9.00000	−	3.00000i	−0.314294	−	0.104765i
$821$	−15.0000	+	15.0000i	−0.523504	+	0.523504i	−0.918628	−	0.395124i	$-0.870702\pi$
$821$	−15.0000	+	15.0000i	−0.523504	+	0.523504i	0.395124	+	0.918628i	$0.370702\pi$
$822$	4.00000	−	4.00000i	0.139516	−	0.139516i
$823$	−39.0000	+	39.0000i	−1.35945	+	1.35945i	−0.484866	+	0.874588i	$0.661132\pi$
$823$	−39.0000	+	39.0000i	−1.35945	+	1.35945i	−0.874588	+	0.484866i	$0.838868\pi$
$824$			18.0000i			0.627060i
$825$	24.0000	+	18.0000i	0.835573	+	0.626680i
$826$	18.0000	+	18.0000i	0.626300	+	0.626300i
$827$	13.0000	−	13.0000i	0.452054	−	0.452054i	−0.443982	−	0.896036i	$-0.646434\pi$
$827$	13.0000	−	13.0000i	0.452054	−	0.452054i	0.896036	+	0.443982i	$0.146434\pi$
$828$	1.00000	+	1.00000i	0.0347524	+	0.0347524i
$829$	−2.00000			−0.0694629			−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000			−0.0694629			−0.0347314	+	0.999397i	$0.511058\pi$
$830$	−8.00000	+	4.00000i	−0.277684	+	0.138842i
$831$			42.0000i			1.45696i
$832$		0			0
$833$	−44.0000	+	11.0000i	−1.52451	+	0.381127i
$834$	14.0000			0.484780
$835$	−9.00000	−	3.00000i	−0.311458	−	0.103819i
$836$	−18.0000	+	18.0000i	−0.622543	+	0.622543i
$837$		−	8.00000i		−	0.276520i
$838$	15.0000	+	15.0000i	0.518166	+	0.518166i
$839$	−3.00000	−	3.00000i	−0.103572	−	0.103572i	0.653422	−	0.756994i	$-0.273333\pi$
$839$	−3.00000	−	3.00000i	−0.103572	−	0.103572i	−0.756994	+	0.653422i	$0.773333\pi$
$840$	−36.0000	+	18.0000i	−1.24212	+	0.621059i
$841$			11.0000i			0.379310i
$842$	6.00000			0.206774
$843$	−24.0000	+	24.0000i	−0.826604	+	0.826604i
$844$	−17.0000	−	17.0000i	−0.585164	−	0.585164i
$845$	−26.0000	+	13.0000i	−0.894427	+	0.447214i
$846$	2.00000			0.0687614
$847$	−21.0000	−	21.0000i	−0.721569	−	0.721569i
$848$	−2.00000			−0.0686803
$849$	−18.0000			−0.617758
$850$	19.0000	+	8.00000i	0.651695	+	0.274398i
$851$	6.00000			0.205677
$852$	6.00000			0.205557
$853$	3.00000	+	3.00000i	0.102718	+	0.102718i	0.756598	−	0.653880i	$-0.226860\pi$
$853$	3.00000	+	3.00000i	0.102718	+	0.102718i	−0.653880	+	0.756598i	$0.726860\pi$
$854$	6.00000			0.205316
$855$	−12.0000	+	6.00000i	−0.410391	+	0.205196i
$856$	−27.0000	−	27.0000i	−0.922841	−	0.922841i
$857$	11.0000	−	11.0000i	0.375753	−	0.375753i	−0.493814	−	0.869567i	$-0.664398\pi$
$857$	11.0000	−	11.0000i	0.375753	−	0.375753i	0.869567	+	0.493814i	$0.164398\pi$
$858$		0			0
$859$			30.0000i			1.02359i	0.859109	+	0.511793i	$0.171019\pi$
$859$			30.0000i			1.02359i	−0.859109	+	0.511793i	$0.828981\pi$
$860$	−24.0000	+	12.0000i	−0.818393	+	0.409197i
$861$	−18.0000	−	18.0000i	−0.613438	−	0.613438i
$862$	−9.00000	−	9.00000i	−0.306541	−	0.306541i
$863$			38.0000i			1.29354i	0.762687	+	0.646768i	$0.223880\pi$
$863$			38.0000i			1.29354i	−0.762687	+	0.646768i	$0.776120\pi$
$864$	20.0000	−	20.0000i	0.680414	−	0.680414i
$865$	−45.0000	−	15.0000i	−1.53005	−	0.510015i
$866$	30.0000			1.01944
$867$	23.0000	+	7.00000i	0.781121	+	0.237732i
$868$			6.00000i			0.203653i
$869$			42.0000i			1.42475i
$870$	12.0000	−	6.00000i	0.406838	−	0.203419i
$871$		0			0
$872$	−21.0000	−	21.0000i	−0.711150	−	0.711150i
$873$	3.00000	−	3.00000i	0.101535	−	0.101535i
$874$	−6.00000	−	6.00000i	−0.202953	−	0.202953i
$875$	−27.0000	−	39.0000i	−0.912767	−	1.31844i
$876$			6.00000i			0.202721i
$877$	27.0000	−	27.0000i	0.911725	−	0.911725i	−0.0846827	−	0.996408i	$-0.526988\pi$
$877$	27.0000	−	27.0000i	0.911725	−	0.911725i	0.996408	+	0.0846827i	$0.0269877\pi$
$878$	−19.0000	+	19.0000i	−0.641219	+	0.641219i
$879$	4.00000	−	4.00000i	0.134917	−	0.134917i
$880$	−9.00000	−	3.00000i	−0.303390	−	0.101130i
$881$	−3.00000	+	3.00000i	−0.101073	+	0.101073i	−0.755835	−	0.654762i	$-0.772769\pi$
$881$	−3.00000	+	3.00000i	−0.101073	+	0.101073i	0.654762	+	0.755835i	$0.272769\pi$
$882$	11.0000			0.370389
$883$		−	6.00000i		−	0.201916i	−0.994891	−	0.100958i	$-0.967809\pi$
$883$		−	6.00000i		−	0.201916i	0.994891	−	0.100958i	$-0.0321908\pi$
$884$		0			0
$885$	−6.00000	+	18.0000i	−0.201688	+	0.605063i
$886$			22.0000i			0.739104i
$887$	−17.0000	−	17.0000i	−0.570804	−	0.570804i	0.361549	−	0.932353i	$-0.382248\pi$
$887$	−17.0000	−	17.0000i	−0.570804	−	0.570804i	−0.932353	+	0.361549i	$0.882248\pi$
$888$		−	18.0000i		−	0.604040i
$889$		0			0
$890$	12.0000	−	6.00000i	0.402241	−	0.201120i
$891$	−15.0000	−	15.0000i	−0.502519	−	0.502519i
$892$		0			0
$893$	12.0000			0.401565
$894$	−18.0000	−	18.0000i	−0.602010	−	0.602010i
$895$	−6.00000	−	12.0000i	−0.200558	−	0.401116i
$896$	−9.00000	+	9.00000i	−0.300669	+	0.300669i
$897$		0			0
$898$	15.0000	+	15.0000i	0.500556	+	0.500556i
$899$		−	6.00000i		−	0.200111i
$900$	4.00000	+	3.00000i	0.133333	+	0.100000i
$901$	−2.00000	−	8.00000i	−0.0666297	−	0.266519i
$902$		−	18.0000i		−	0.599334i
$903$	−72.0000			−2.39601
$904$	−27.0000	+	27.0000i	−0.898007	+	0.898007i
$905$	33.0000	+	11.0000i	1.09696	+	0.365652i
$906$	−10.0000	+	10.0000i	−0.332228	+	0.332228i
$907$	−15.0000	+	15.0000i	−0.498067	+	0.498067i	−0.910836	−	0.412769i	$-0.864562\pi$
$907$	−15.0000	+	15.0000i	−0.498067	+	0.498067i	0.412769	+	0.910836i	$0.364562\pi$
$908$	15.0000	−	15.0000i	0.497792	−	0.497792i
$909$			6.00000i			0.199007i
$910$		0			0
$911$	21.0000	+	21.0000i	0.695761	+	0.695761i	0.963493	−	0.267732i	$-0.0862743\pi$
$911$	21.0000	+	21.0000i	0.695761	+	0.695761i	−0.267732	+	0.963493i	$0.586274\pi$
$912$	−6.00000	+	6.00000i	−0.198680	+	0.198680i
$913$	12.0000	+	12.0000i	0.397142	+	0.397142i
$914$	6.00000			0.198462
$915$	2.00000	+	4.00000i	0.0661180	+	0.132236i
$916$		−	24.0000i		−	0.792982i
$917$			18.0000i			0.594412i
$918$	−20.0000	−	12.0000i	−0.660098	−	0.396059i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$	−3.00000	+	9.00000i	−0.0989071	+	0.296721i
$921$	18.0000	−	18.0000i	0.593120	−	0.593120i
$922$		−	12.0000i		−	0.395199i
$923$		0			0
$924$	18.0000	+	18.0000i	0.592157	+	0.592157i
$925$	21.0000	−	3.00000i	0.690476	−	0.0986394i
$926$			18.0000i			0.591517i
$927$	6.00000			0.197066
$928$	15.0000	−	15.0000i	0.492399	−	0.492399i
$929$	21.0000	+	21.0000i	0.688988	+	0.688988i	0.962008	−	0.273021i	$-0.0880227\pi$
$929$	21.0000	+	21.0000i	0.688988	+	0.688988i	−0.273021	+	0.962008i	$0.588023\pi$
$930$	4.00000	−	2.00000i	0.131165	−	0.0655826i
$931$	66.0000			2.16306
$932$	5.00000	+	5.00000i	0.163780	+	0.163780i
$933$	−6.00000			−0.196431
$934$	28.0000			0.916188
$935$	3.00000	−	39.0000i	0.0981105	−	1.27544i
$936$		0			0
$937$	−18.0000			−0.588034			−0.294017	−	0.955800i	$-0.594992\pi$
$937$	−18.0000			−0.588034			−0.294017	+	0.955800i	$0.594992\pi$
$938$	−18.0000	−	18.0000i	−0.587721	−	0.587721i
$939$	−6.00000			−0.195803
$940$	−2.00000	−	4.00000i	−0.0652328	−	0.130466i
$941$	−15.0000	−	15.0000i	−0.488986	−	0.488986i	0.419000	−	0.907986i	$-0.362381\pi$
$941$	−15.0000	−	15.0000i	−0.488986	−	0.488986i	−0.907986	+	0.419000i	$0.862381\pi$
$942$	12.0000	−	12.0000i	0.390981	−	0.390981i
$943$	−6.00000			−0.195387
$944$		−	6.00000i		−	0.195283i
$945$	24.0000	+	48.0000i	0.780720	+	1.56144i
$946$	−36.0000	−	36.0000i	−1.17046	−	1.17046i
$947$	11.0000	+	11.0000i	0.357452	+	0.357452i	0.862873	−	0.505421i	$-0.168663\pi$
$947$	11.0000	+	11.0000i	0.357452	+	0.357452i	−0.505421	+	0.862873i	$0.668663\pi$
$948$		−	14.0000i		−	0.454699i
$949$		0			0
$950$	−24.0000	−	18.0000i	−0.778663	−	0.583997i
$951$	−30.0000			−0.972817
$952$	45.0000	+	27.0000i	1.45846	+	0.875075i
$953$		−	4.00000i		−	0.129573i	−0.997899	−	0.0647864i	$-0.979363\pi$
$953$		−	4.00000i		−	0.129573i	0.997899	−	0.0647864i	$-0.0206366\pi$
$954$			2.00000i			0.0647524i
$955$		0			0
$956$	24.0000			0.776215
$957$	−18.0000	−	18.0000i	−0.581857	−	0.581857i
$958$	9.00000	−	9.00000i	0.290777	−	0.290777i
$959$	12.0000	+	12.0000i	0.387500	+	0.387500i
$960$	21.0000	+	7.00000i	0.677772	+	0.225924i
$961$			29.0000i			0.935484i
$962$		0			0
$963$	−9.00000	+	9.00000i	−0.290021	+	0.290021i
$964$	19.0000	−	19.0000i	0.611949	−	0.611949i
$965$	−3.00000	+	9.00000i	−0.0965734	+	0.289720i
$966$	−6.00000	+	6.00000i	−0.193047	+	0.193047i
$967$	24.0000			0.771788			0.385894	−	0.922543i	$-0.373893\pi$
$967$	24.0000			0.771788			0.385894	+	0.922543i	$0.373893\pi$
$968$			21.0000i			0.674966i
$969$	−30.0000	−	18.0000i	−0.963739	−	0.578243i
$970$	9.00000	+	3.00000i	0.288973	+	0.0963242i
$971$		−	42.0000i		−	1.34784i	−0.738802	−	0.673922i	$-0.764608\pi$
$971$		−	42.0000i		−	1.34784i	0.738802	−	0.673922i	$-0.235392\pi$
$972$	−7.00000	−	7.00000i	−0.224525	−	0.224525i
$973$			42.0000i			1.34646i
$974$	3.00000	−	3.00000i	0.0961262	−	0.0961262i
$975$		0			0
$976$	−1.00000	−	1.00000i	−0.0320092	−	0.0320092i
$977$	46.0000			1.47167			0.735835	−	0.677161i	$-0.236790\pi$
$977$	46.0000			1.47167			0.735835	+	0.677161i	$0.236790\pi$
$978$	18.0000			0.575577
$979$	−18.0000	−	18.0000i	−0.575282	−	0.575282i
$980$	−11.0000	−	22.0000i	−0.351382	−	0.702764i
$981$	−7.00000	+	7.00000i	−0.223493	+	0.223493i
$982$		−	6.00000i		−	0.191468i
$983$	−21.0000	−	21.0000i	−0.669796	−	0.669796i	0.287873	−	0.957669i	$-0.407052\pi$
$983$	−21.0000	−	21.0000i	−0.669796	−	0.669796i	−0.957669	+	0.287873i	$0.907052\pi$
$984$			18.0000i			0.573819i
$985$	13.0000	−	39.0000i	0.414214	−	1.24264i
$986$	−15.0000	−	9.00000i	−0.477697	−	0.286618i
$987$		−	12.0000i		−	0.381964i
$988$		0			0
$989$	−12.0000	+	12.0000i	−0.381578	+	0.381578i
$990$	−3.00000	+	9.00000i	−0.0953463	+	0.286039i
$991$	−13.0000	+	13.0000i	−0.412959	+	0.412959i	−0.882768	−	0.469809i	$-0.844323\pi$
$991$	−13.0000	+	13.0000i	−0.412959	+	0.412959i	0.469809	+	0.882768i	$0.344323\pi$
$992$	5.00000	−	5.00000i	0.158750	−	0.158750i
$993$	−18.0000	+	18.0000i	−0.571213	+	0.571213i
$994$		−	18.0000i		−	0.570925i
$995$	1.00000	−	3.00000i	0.0317021	−	0.0951064i
$996$	−4.00000	−	4.00000i	−0.126745	−	0.126745i
$997$	3.00000	−	3.00000i	0.0950110	−	0.0950110i	−0.658004	−	0.753015i	$-0.728599\pi$
$997$	3.00000	−	3.00000i	0.0950110	−	0.0950110i	0.753015	+	0.658004i	$0.228599\pi$
$998$	23.0000	+	23.0000i	0.728052	+	0.728052i
$999$	−24.0000			−0.759326

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	85.2.j.a.4.1	✓	2
3.2	odd	2		765.2.t.b.514.1		2
5.2	odd	4		425.2.e.a.276.1		2
5.3	odd	4		425.2.e.b.276.1		2
5.4	even	2		85.2.j.b.4.1	yes	2
15.14	odd	2		765.2.t.a.514.1		2
17.8	even	8		1445.2.b.a.579.4		4
17.9	even	8		1445.2.b.a.579.3		4
17.13	even	4		85.2.j.b.64.1	yes	2
51.47	odd	4		765.2.t.a.64.1		2
85.8	odd	8		7225.2.a.p.1.1		2
85.9	even	8		1445.2.b.a.579.2		4
85.13	odd	4		425.2.e.b.251.1		2
85.42	odd	8		7225.2.a.i.1.2		2
85.43	odd	8		7225.2.a.p.1.2		2
85.47	odd	4		425.2.e.a.251.1		2
85.59	even	8		1445.2.b.a.579.1		4
85.64	even	4	inner	85.2.j.a.64.1	yes	2
85.77	odd	8		7225.2.a.i.1.1		2
255.149	odd	4		765.2.t.b.64.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.j.a.4.1	✓	2	1.1	even	1	trivial
85.2.j.a.64.1	yes	2	85.64	even	4	inner
85.2.j.b.4.1	yes	2	5.4	even	2
85.2.j.b.64.1	yes	2	17.13	even	4
425.2.e.a.251.1		2	85.47	odd	4
425.2.e.a.276.1		2	5.2	odd	4
425.2.e.b.251.1		2	85.13	odd	4
425.2.e.b.276.1		2	5.3	odd	4
765.2.t.a.64.1		2	51.47	odd	4
765.2.t.a.514.1		2	15.14	odd	2
765.2.t.b.64.1		2	255.149	odd	4
765.2.t.b.514.1		2	3.2	odd	2
1445.2.b.a.579.1		4	85.59	even	8
1445.2.b.a.579.2		4	85.9	even	8
1445.2.b.a.579.3		4	17.9	even	8
1445.2.b.a.579.4		4	17.8	even	8
7225.2.a.i.1.1		2	85.77	odd	8
7225.2.a.i.1.2		2	85.42	odd	8
7225.2.a.p.1.1		2	85.8	odd	8
7225.2.a.p.1.2		2	85.43	odd	8

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +(-1.00000 - 1.00000i) q^{3} -1.00000 q^{4} +(-2.00000 + 1.00000i) q^{5} +(1.00000 + 1.00000i) q^{6} +(-3.00000 + 3.00000i) q^{7} +3.00000 q^{8} -1.00000i q^{9} +(2.00000 - 1.00000i) q^{10} +(-3.00000 - 3.00000i) q^{11} +(1.00000 + 1.00000i) q^{12} +(3.00000 - 3.00000i) q^{14} +(3.00000 + 1.00000i) q^{15} -1.00000 q^{16} +(-1.00000 - 4.00000i) q^{17} +1.00000i q^{18} +6.00000i q^{19} +(2.00000 - 1.00000i) q^{20} +6.00000 q^{21} +(3.00000 + 3.00000i) q^{22} +(1.00000 - 1.00000i) q^{23} +(-3.00000 - 3.00000i) q^{24} +(3.00000 - 4.00000i) q^{25} +(-4.00000 + 4.00000i) q^{27} +(3.00000 - 3.00000i) q^{28} +(-3.00000 + 3.00000i) q^{29} +(-3.00000 - 1.00000i) q^{30} +(-1.00000 + 1.00000i) q^{31} -5.00000 q^{32} +6.00000i q^{33} +(1.00000 + 4.00000i) q^{34} +(3.00000 - 9.00000i) q^{35} +1.00000i q^{36} +(3.00000 + 3.00000i) q^{37} -6.00000i q^{38} +(-6.00000 + 3.00000i) q^{40} +(-3.00000 - 3.00000i) q^{41} -6.00000 q^{42} -12.0000 q^{43} +(3.00000 + 3.00000i) q^{44} +(1.00000 + 2.00000i) q^{45} +(-1.00000 + 1.00000i) q^{46} -2.00000i q^{47} +(1.00000 + 1.00000i) q^{48} -11.0000i q^{49} +(-3.00000 + 4.00000i) q^{50} +(-3.00000 + 5.00000i) q^{51} +2.00000 q^{53} +(4.00000 - 4.00000i) q^{54} +(9.00000 + 3.00000i) q^{55} +(-9.00000 + 9.00000i) q^{56} +(6.00000 - 6.00000i) q^{57} +(3.00000 - 3.00000i) q^{58} +6.00000i q^{59} +(-3.00000 - 1.00000i) q^{60} +(1.00000 + 1.00000i) q^{61} +(1.00000 - 1.00000i) q^{62} +(3.00000 + 3.00000i) q^{63} +7.00000 q^{64} -6.00000i q^{66} -6.00000i q^{67} +(1.00000 + 4.00000i) q^{68} -2.00000 q^{69} +(-3.00000 + 9.00000i) q^{70} +(3.00000 - 3.00000i) q^{71} -3.00000i q^{72} +(3.00000 + 3.00000i) q^{73} +(-3.00000 - 3.00000i) q^{74} +(-7.00000 + 1.00000i) q^{75} -6.00000i q^{76} +18.0000 q^{77} +(-7.00000 - 7.00000i) q^{79} +(2.00000 - 1.00000i) q^{80} +5.00000 q^{81} +(3.00000 + 3.00000i) q^{82} -4.00000 q^{83} -6.00000 q^{84} +(6.00000 + 7.00000i) q^{85} +12.0000 q^{86} +6.00000 q^{87} +(-9.00000 - 9.00000i) q^{88} +6.00000 q^{89} +(-1.00000 - 2.00000i) q^{90} +(-1.00000 + 1.00000i) q^{92} +2.00000 q^{93} +2.00000i q^{94} +(-6.00000 - 12.0000i) q^{95} +(5.00000 + 5.00000i) q^{96} +(3.00000 + 3.00000i) q^{97} +11.0000i q^{98} +(-3.00000 + 3.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 4 q^{5} + 2 q^{6} - 6 q^{7} + 6 q^{8} + 4 q^{10} - 6 q^{11} + 2 q^{12} + 6 q^{14} + 6 q^{15} - 2 q^{16} - 2 q^{17} + 4 q^{20} + 12 q^{21} + 6 q^{22} + 2 q^{23} - 6 q^{24}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 - 2 * q^4 - 4 * q^5 + 2 * q^6 - 6 * q^7 + 6 * q^8 + 4 * q^10 - 6 * q^11 + 2 * q^12 + 6 * q^14 + 6 * q^15 - 2 * q^16 - 2 * q^17 + 4 * q^20 + 12 * q^21 + 6 * q^22 + 2 * q^23 - 6 * q^24 + 6 * q^25 - 8 * q^27 + 6 * q^28 - 6 * q^29 - 6 * q^30 - 2 * q^31 - 10 * q^32 + 2 * q^34 + 6 * q^35 + 6 * q^37 - 12 * q^40 - 6 * q^41 - 12 * q^42 - 24 * q^43 + 6 * q^44 + 2 * q^45 - 2 * q^46 + 2 * q^48 - 6 * q^50 - 6 * q^51 + 4 * q^53 + 8 * q^54 + 18 * q^55 - 18 * q^56 + 12 * q^57 + 6 * q^58 - 6 * q^60 + 2 * q^61 + 2 * q^62 + 6 * q^63 + 14 * q^64 + 2 * q^68 - 4 * q^69 - 6 * q^70 + 6 * q^71 + 6 * q^73 - 6 * q^74 - 14 * q^75 + 36 * q^77 - 14 * q^79 + 4 * q^80 + 10 * q^81 + 6 * q^82 - 8 * q^83 - 12 * q^84 + 12 * q^85 + 24 * q^86 + 12 * q^87 - 18 * q^88 + 12 * q^89 - 2 * q^90 - 2 * q^92 + 4 * q^93 - 12 * q^95 + 10 * q^96 + 6 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more