LMFDB - Embedded newform 135.2.e.a.91.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,2,Mod(46,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("135.46");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	135.e (of order $3$, degree $2$, not 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:	$1.07798042729$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			91.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	135.91
Dual form			135.2.e.a.46.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{7} +3.00000 q^{8} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{7} +3.00000 q^{8} -1.00000 q^{10} +(-1.00000 - 1.73205i) q^{11} +(1.00000 - 1.73205i) q^{13} +(-1.50000 + 2.59808i) q^{14} +(0.500000 + 0.866025i) q^{16} -4.00000 q^{17} -8.00000 q^{19} +(0.500000 + 0.866025i) q^{20} +(1.00000 - 1.73205i) q^{22} +(1.50000 - 2.59808i) q^{23} +(-0.500000 - 0.866025i) q^{25} +2.00000 q^{26} +3.00000 q^{28} +(-0.500000 - 0.866025i) q^{29} +(2.50000 - 4.33013i) q^{32} +(-2.00000 - 3.46410i) q^{34} -3.00000 q^{35} -4.00000 q^{37} +(-4.00000 - 6.92820i) q^{38} +(-1.50000 + 2.59808i) q^{40} +(2.50000 - 4.33013i) q^{41} +(4.00000 + 6.92820i) q^{43} -2.00000 q^{44} +3.00000 q^{46} +(3.50000 + 6.06218i) q^{47} +(-1.00000 + 1.73205i) q^{49} +(0.500000 - 0.866025i) q^{50} +(-1.00000 - 1.73205i) q^{52} +2.00000 q^{53} +2.00000 q^{55} +(4.50000 + 7.79423i) q^{56} +(0.500000 - 0.866025i) q^{58} +(-7.00000 + 12.1244i) q^{59} +(-3.50000 - 6.06218i) q^{61} +7.00000 q^{64} +(1.00000 + 1.73205i) q^{65} +(1.50000 - 2.59808i) q^{67} +(-2.00000 + 3.46410i) q^{68} +(-1.50000 - 2.59808i) q^{70} -2.00000 q^{71} +4.00000 q^{73} +(-2.00000 - 3.46410i) q^{74} +(-4.00000 + 6.92820i) q^{76} +(3.00000 - 5.19615i) q^{77} +(3.00000 + 5.19615i) q^{79} -1.00000 q^{80} +5.00000 q^{82} +(4.50000 + 7.79423i) q^{83} +(2.00000 - 3.46410i) q^{85} +(-4.00000 + 6.92820i) q^{86} +(-3.00000 - 5.19615i) q^{88} +15.0000 q^{89} +6.00000 q^{91} +(-1.50000 - 2.59808i) q^{92} +(-3.50000 + 6.06218i) q^{94} +(4.00000 - 6.92820i) q^{95} +(-1.00000 - 1.73205i) q^{97} -2.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + q^{4} - q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + q^2 + q^4 - q^5 + 3 * q^7 + 6 * q^8	$ 2 q + q^{2} + q^{4} - q^{5} + 3 q^{7} + 6 q^{8} - 2 q^{10} - 2 q^{11} + 2 q^{13} - 3 q^{14} + q^{16} - 8 q^{17} - 16 q^{19} + q^{20} + 2 q^{22} + 3 q^{23} - q^{25} + 4 q^{26} + 6 q^{28} - q^{29} + 5 q^{32} - 4 q^{34} - 6 q^{35} - 8 q^{37} - 8 q^{38} - 3 q^{40} + 5 q^{41} + 8 q^{43} - 4 q^{44} + 6 q^{46} + 7 q^{47} - 2 q^{49} + q^{50} - 2 q^{52} + 4 q^{53} + 4 q^{55} + 9 q^{56} + q^{58} - 14 q^{59} - 7 q^{61} + 14 q^{64} + 2 q^{65} + 3 q^{67} - 4 q^{68} - 3 q^{70} - 4 q^{71} + 8 q^{73} - 4 q^{74} - 8 q^{76} + 6 q^{77} + 6 q^{79} - 2 q^{80} + 10 q^{82} + 9 q^{83} + 4 q^{85} - 8 q^{86} - 6 q^{88} + 30 q^{89} + 12 q^{91} - 3 q^{92} - 7 q^{94} + 8 q^{95} - 2 q^{97} - 4 q^{98}+O(q^{100}) $ 2 * q + q^2 + q^4 - q^5 + 3 * q^7 + 6 * q^8 - 2 * q^10 - 2 * q^11 + 2 * q^13 - 3 * q^14 + q^16 - 8 * q^17 - 16 * q^19 + q^20 + 2 * q^22 + 3 * q^23 - q^25 + 4 * q^26 + 6 * q^28 - q^29 + 5 * q^32 - 4 * q^34 - 6 * q^35 - 8 * q^37 - 8 * q^38 - 3 * q^40 + 5 * q^41 + 8 * q^43 - 4 * q^44 + 6 * q^46 + 7 * q^47 - 2 * q^49 + q^50 - 2 * q^52 + 4 * q^53 + 4 * q^55 + 9 * q^56 + q^58 - 14 * q^59 - 7 * q^61 + 14 * q^64 + 2 * q^65 + 3 * q^67 - 4 * q^68 - 3 * q^70 - 4 * q^71 + 8 * q^73 - 4 * q^74 - 8 * q^76 + 6 * q^77 + 6 * q^79 - 2 * q^80 + 10 * q^82 + 9 * q^83 + 4 * q^85 - 8 * q^86 - 6 * q^88 + 30 * q^89 + 12 * q^91 - 3 * q^92 - 7 * q^94 + 8 * q^95 - 2 * q^97 - 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$

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 $

$2$	0.500000	+	0.866025i	0.353553	+	0.612372i	0.986869	−	0.161521i	$-0.0516399\pi$
$2$	0.500000	+	0.866025i	0.353553	+	0.612372i	−0.633316	+	0.773893i	$0.718307\pi$
$3$		0			0
$3$		0			0
$4$	0.500000	−	0.866025i	0.250000	−	0.433013i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	1.50000	+	2.59808i	0.566947	+	0.981981i	0.996866	+	0.0791130i	$0.0252088\pi$
$7$	1.50000	+	2.59808i	0.566947	+	0.981981i	−0.429919	+	0.902867i	$0.641458\pi$
$8$	3.00000			1.06066
$9$		0			0
$10$	−1.00000			−0.316228
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$	1.00000	−	1.73205i	0.277350	−	0.480384i	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	1.00000	−	1.73205i	0.277350	−	0.480384i	0.970725	+	0.240192i	$0.0772105\pi$
$14$	−1.50000	+	2.59808i	−0.400892	+	0.694365i
$15$		0			0
$16$	0.500000	+	0.866025i	0.125000	+	0.216506i
$17$	−4.00000			−0.970143			−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000			−0.970143			−0.485071	+	0.874475i	$0.661206\pi$
$18$		0			0
$19$	−8.00000			−1.83533			−0.917663	−	0.397360i	$-0.869927\pi$
$19$	−8.00000			−1.83533			−0.917663	+	0.397360i	$0.869927\pi$
$20$	0.500000	+	0.866025i	0.111803	+	0.193649i
$21$		0			0
$22$	1.00000	−	1.73205i	0.213201	−	0.369274i
$23$	1.50000	−	2.59808i	0.312772	−	0.541736i	−0.666190	−	0.745782i	$-0.732076\pi$
$23$	1.50000	−	2.59808i	0.312772	−	0.541736i	0.978961	+	0.204046i	$0.0654092\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$	2.00000			0.392232
$27$		0			0
$28$	3.00000			0.566947
$29$	−0.500000	−	0.866025i	−0.0928477	−	0.160817i	0.815861	−	0.578249i	$-0.196264\pi$
$29$	−0.500000	−	0.866025i	−0.0928477	−	0.160817i	−0.908708	+	0.417432i	$0.862930\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	2.50000	−	4.33013i	0.441942	−	0.765466i
$33$		0			0
$34$	−2.00000	−	3.46410i	−0.342997	−	0.594089i
$35$	−3.00000			−0.507093
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$	−4.00000	−	6.92820i	−0.648886	−	1.12390i
$39$		0			0
$40$	−1.50000	+	2.59808i	−0.237171	+	0.410792i
$41$	2.50000	−	4.33013i	0.390434	−	0.676252i	−0.602072	−	0.798441i	$-0.705658\pi$
$41$	2.50000	−	4.33013i	0.390434	−	0.676252i	0.992507	+	0.122189i	$0.0389915\pi$
$42$		0			0
$43$	4.00000	+	6.92820i	0.609994	+	1.05654i	0.991241	+	0.132068i	$0.0421616\pi$
$43$	4.00000	+	6.92820i	0.609994	+	1.05654i	−0.381246	+	0.924473i	$0.624505\pi$
$44$	−2.00000			−0.301511
$45$		0			0
$46$	3.00000			0.442326
$47$	3.50000	+	6.06218i	0.510527	+	0.884260i	0.999926	+	0.0121990i	$0.00388317\pi$
$47$	3.50000	+	6.06218i	0.510527	+	0.884260i	−0.489398	+	0.872060i	$0.662783\pi$
$48$		0			0
$49$	−1.00000	+	1.73205i	−0.142857	+	0.247436i
$50$	0.500000	−	0.866025i	0.0707107	−	0.122474i
$51$		0			0
$52$	−1.00000	−	1.73205i	−0.138675	−	0.240192i
$53$	2.00000			0.274721			0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000			0.274721			0.137361	+	0.990521i	$0.456138\pi$
$54$		0			0
$55$	2.00000			0.269680
$56$	4.50000	+	7.79423i	0.601338	+	1.04155i
$57$		0			0
$58$	0.500000	−	0.866025i	0.0656532	−	0.113715i
$59$	−7.00000	+	12.1244i	−0.911322	+	1.57846i	−0.0991242	+	0.995075i	$0.531604\pi$
$59$	−7.00000	+	12.1244i	−0.911322	+	1.57846i	−0.812198	+	0.583382i	$0.801729\pi$
$60$		0			0
$61$	−3.50000	−	6.06218i	−0.448129	−	0.776182i	0.550135	−	0.835076i	$-0.314576\pi$
$61$	−3.50000	−	6.06218i	−0.448129	−	0.776182i	−0.998264	+	0.0588933i	$0.981243\pi$
$62$		0			0
$63$		0			0
$64$	7.00000			0.875000
$65$	1.00000	+	1.73205i	0.124035	+	0.214834i
$66$		0			0
$67$	1.50000	−	2.59808i	0.183254	−	0.317406i	−0.759733	−	0.650236i	$-0.774670\pi$
$67$	1.50000	−	2.59808i	0.183254	−	0.317406i	0.942987	+	0.332830i	$0.108004\pi$
$68$	−2.00000	+	3.46410i	−0.242536	+	0.420084i
$69$		0			0
$70$	−1.50000	−	2.59808i	−0.179284	−	0.310530i
$71$	−2.00000			−0.237356			−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000			−0.237356			−0.118678	+	0.992933i	$0.537866\pi$
$72$		0			0
$73$	4.00000			0.468165			0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000			0.468165			0.234082	+	0.972217i	$0.424791\pi$
$74$	−2.00000	−	3.46410i	−0.232495	−	0.402694i
$75$		0			0
$76$	−4.00000	+	6.92820i	−0.458831	+	0.794719i
$77$	3.00000	−	5.19615i	0.341882	−	0.592157i
$78$		0			0
$79$	3.00000	+	5.19615i	0.337526	+	0.584613i	0.983967	−	0.178352i	$-0.0570765\pi$
$79$	3.00000	+	5.19615i	0.337526	+	0.584613i	−0.646440	+	0.762964i	$0.723743\pi$
$80$	−1.00000			−0.111803
$81$		0			0
$82$	5.00000			0.552158
$83$	4.50000	+	7.79423i	0.493939	+	0.855528i	0.999976	−	0.00698436i	$-0.00222321\pi$
$83$	4.50000	+	7.79423i	0.493939	+	0.855528i	−0.506036	+	0.862512i	$0.668890\pi$
$84$		0			0
$85$	2.00000	−	3.46410i	0.216930	−	0.375735i
$86$	−4.00000	+	6.92820i	−0.431331	+	0.747087i
$87$		0			0
$88$	−3.00000	−	5.19615i	−0.319801	−	0.553912i
$89$	15.0000			1.59000			0.794998	−	0.606612i	$-0.207472\pi$
$89$	15.0000			1.59000			0.794998	+	0.606612i	$0.207472\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$	−1.50000	−	2.59808i	−0.156386	−	0.270868i
$93$		0			0
$94$	−3.50000	+	6.06218i	−0.360997	+	0.625266i
$95$	4.00000	−	6.92820i	0.410391	−	0.710819i
$96$		0			0
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	0.810782	−	0.585348i	$-0.199042\pi$
$97$	−1.00000	−	1.73205i	−0.101535	−	0.175863i	−0.912317	+	0.409484i	$0.865709\pi$
$98$	−2.00000			−0.202031
$99$		0			0
$100$	−1.00000			−0.100000
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.833143	−	0.553058i	$-0.813461\pi$
$101$	−9.00000	−	15.5885i	−0.895533	−	1.55111i	−0.0623905	−	0.998052i	$-0.519872\pi$
$102$		0			0
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	−0.992990	−	0.118199i	$-0.962288\pi$
$103$	−4.00000	+	6.92820i	−0.394132	+	0.682656i	0.598858	+	0.800855i	$0.295621\pi$
$104$	3.00000	−	5.19615i	0.294174	−	0.509525i
$105$		0			0
$106$	1.00000	+	1.73205i	0.0971286	+	0.168232i
$107$	−3.00000			−0.290021			−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000			−0.290021			−0.145010	+	0.989430i	$0.546322\pi$
$108$		0			0
$109$	5.00000			0.478913			0.239457	−	0.970907i	$-0.423031\pi$
$109$	5.00000			0.478913			0.239457	+	0.970907i	$0.423031\pi$
$110$	1.00000	+	1.73205i	0.0953463	+	0.165145i
$111$		0			0
$112$	−1.50000	+	2.59808i	−0.141737	+	0.245495i
$113$	−4.00000	+	6.92820i	−0.376288	+	0.651751i	−0.990519	−	0.137376i	$-0.956133\pi$
$113$	−4.00000	+	6.92820i	−0.376288	+	0.651751i	0.614231	+	0.789127i	$0.289466\pi$
$114$		0			0
$115$	1.50000	+	2.59808i	0.139876	+	0.242272i
$116$	−1.00000			−0.0928477
$117$		0			0
$118$	−14.0000			−1.28880
$119$	−6.00000	−	10.3923i	−0.550019	−	0.952661i
$120$		0			0
$121$	3.50000	−	6.06218i	0.318182	−	0.551107i
$122$	3.50000	−	6.06218i	0.316875	−	0.548844i
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−5.00000			−0.443678			−0.221839	−	0.975083i	$-0.571206\pi$
$127$	−5.00000			−0.443678			−0.221839	+	0.975083i	$0.571206\pi$
$128$	−1.50000	−	2.59808i	−0.132583	−	0.229640i
$129$		0			0
$130$	−1.00000	+	1.73205i	−0.0877058	+	0.151911i
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	−0.966803	−	0.255524i	$-0.917752\pi$
$131$	−3.00000	+	5.19615i	−0.262111	+	0.453990i	0.704692	+	0.709514i	$0.251085\pi$
$132$		0			0
$133$	−12.0000	−	20.7846i	−1.04053	−	1.80225i
$134$	3.00000			0.259161
$135$		0			0
$136$	−12.0000			−1.02899
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	0.999893	+	0.0146279i	$0.00465636\pi$
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	−0.487278	+	0.873247i	$0.662010\pi$
$138$		0			0
$139$	8.00000	−	13.8564i	0.678551	−	1.17529i	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	8.00000	−	13.8564i	0.678551	−	1.17529i	0.975417	−	0.220366i	$-0.0707252\pi$
$140$	−1.50000	+	2.59808i	−0.126773	+	0.219578i
$141$		0			0
$142$	−1.00000	−	1.73205i	−0.0839181	−	0.145350i
$143$	−4.00000			−0.334497
$144$		0			0
$145$	1.00000			0.0830455
$146$	2.00000	+	3.46410i	0.165521	+	0.286691i
$147$		0			0
$148$	−2.00000	+	3.46410i	−0.164399	+	0.284747i
$149$	8.50000	−	14.7224i	0.696347	−	1.20611i	−0.273377	−	0.961907i	$-0.588141\pi$
$149$	8.50000	−	14.7224i	0.696347	−	1.20611i	0.969724	−	0.244202i	$-0.0785259\pi$
$150$		0			0
$151$	1.00000	+	1.73205i	0.0813788	+	0.140952i	0.903842	−	0.427865i	$-0.140734\pi$
$151$	1.00000	+	1.73205i	0.0813788	+	0.140952i	−0.822464	+	0.568818i	$0.807401\pi$
$152$	−24.0000			−1.94666
$153$		0			0
$154$	6.00000			0.483494
$155$		0			0
$156$		0			0
$157$	−7.00000	+	12.1244i	−0.558661	+	0.967629i	0.438948	+	0.898513i	$0.355351\pi$
$157$	−7.00000	+	12.1244i	−0.558661	+	0.967629i	−0.997609	+	0.0691164i	$0.977982\pi$
$158$	−3.00000	+	5.19615i	−0.238667	+	0.413384i
$159$		0			0
$160$	2.50000	+	4.33013i	0.197642	+	0.342327i
$161$	9.00000			0.709299
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$	−2.50000	−	4.33013i	−0.195217	−	0.338126i
$165$		0			0
$166$	−4.50000	+	7.79423i	−0.349268	+	0.604949i
$167$	−4.50000	+	7.79423i	−0.348220	+	0.603136i	−0.985933	−	0.167139i	$-0.946547\pi$
$167$	−4.50000	+	7.79423i	−0.348220	+	0.603136i	0.637713	+	0.770274i	$0.279881\pi$
$168$		0			0
$169$	4.50000	+	7.79423i	0.346154	+	0.599556i
$170$	4.00000			0.306786
$171$		0			0
$172$	8.00000			0.609994
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	1.50000	−	2.59808i	0.113389	−	0.196396i
$176$	1.00000	−	1.73205i	0.0753778	−	0.130558i
$177$		0			0
$178$	7.50000	+	12.9904i	0.562149	+	0.973670i
$179$	2.00000			0.149487			0.0747435	−	0.997203i	$-0.476186\pi$
$179$	2.00000			0.149487			0.0747435	+	0.997203i	$0.476186\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$	3.00000	+	5.19615i	0.222375	+	0.385164i
$183$		0			0
$184$	4.50000	−	7.79423i	0.331744	−	0.574598i
$185$	2.00000	−	3.46410i	0.147043	−	0.254686i
$186$		0			0
$187$	4.00000	+	6.92820i	0.292509	+	0.506640i
$188$	7.00000			0.510527
$189$		0			0
$190$	8.00000			0.580381
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	0.973674	−	0.227946i	$-0.0732010\pi$
$191$	4.00000	+	6.92820i	0.289430	+	0.501307i	−0.684244	+	0.729253i	$0.739868\pi$
$192$		0			0
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	−0.628037	−	0.778183i	$-0.716141\pi$
$193$	5.00000	−	8.66025i	0.359908	−	0.623379i	0.987945	+	0.154805i	$0.0494748\pi$
$194$	1.00000	−	1.73205i	0.0717958	−	0.124354i
$195$		0			0
$196$	1.00000	+	1.73205i	0.0714286	+	0.123718i
$197$	12.0000			0.854965			0.427482	−	0.904024i	$-0.359401\pi$
$197$	12.0000			0.854965			0.427482	+	0.904024i	$0.359401\pi$
$198$		0			0
$199$	4.00000			0.283552			0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000			0.283552			0.141776	+	0.989899i	$0.454719\pi$
$200$	−1.50000	−	2.59808i	−0.106066	−	0.183712i
$201$		0			0
$202$	9.00000	−	15.5885i	0.633238	−	1.09680i
$203$	1.50000	−	2.59808i	0.105279	−	0.182349i
$204$		0			0
$205$	2.50000	+	4.33013i	0.174608	+	0.302429i
$206$	−8.00000			−0.557386
$207$		0			0
$208$	2.00000			0.138675
$209$	8.00000	+	13.8564i	0.553372	+	0.958468i
$210$		0			0
$211$	11.0000	−	19.0526i	0.757271	−	1.31163i	−0.186966	−	0.982366i	$-0.559865\pi$
$211$	11.0000	−	19.0526i	0.757271	−	1.31163i	0.944237	−	0.329266i	$-0.106801\pi$
$212$	1.00000	−	1.73205i	0.0686803	−	0.118958i
$213$		0			0
$214$	−1.50000	−	2.59808i	−0.102538	−	0.177601i
$215$	−8.00000			−0.545595
$216$		0			0
$217$		0			0
$218$	2.50000	+	4.33013i	0.169321	+	0.293273i
$219$		0			0
$220$	1.00000	−	1.73205i	0.0674200	−	0.116775i
$221$	−4.00000	+	6.92820i	−0.269069	+	0.466041i
$222$		0			0
$223$	9.50000	+	16.4545i	0.636167	+	1.10187i	0.986267	+	0.165161i	$0.0528144\pi$
$223$	9.50000	+	16.4545i	0.636167	+	1.10187i	−0.350100	+	0.936713i	$0.613852\pi$
$224$	15.0000			1.00223
$225$		0			0
$226$	−8.00000			−0.532152
$227$	−2.00000	−	3.46410i	−0.132745	−	0.229920i	0.791989	−	0.610535i	$-0.209046\pi$
$227$	−2.00000	−	3.46410i	−0.132745	−	0.229920i	−0.924734	+	0.380615i	$0.875712\pi$
$228$		0			0
$229$	−7.50000	+	12.9904i	−0.495614	+	0.858429i	−0.999987	−	0.00505719i	$-0.998390\pi$
$229$	−7.50000	+	12.9904i	−0.495614	+	0.858429i	0.504373	+	0.863486i	$0.331724\pi$
$230$	−1.50000	+	2.59808i	−0.0989071	+	0.171312i
$231$		0			0
$232$	−1.50000	−	2.59808i	−0.0984798	−	0.170572i
$233$	−24.0000			−1.57229			−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000			−1.57229			−0.786146	+	0.618041i	$0.787927\pi$
$234$		0			0
$235$	−7.00000			−0.456630
$236$	7.00000	+	12.1244i	0.455661	+	0.789228i
$237$		0			0
$238$	6.00000	−	10.3923i	0.388922	−	0.673633i
$239$	−4.00000	+	6.92820i	−0.258738	+	0.448148i	−0.965904	−	0.258900i	$-0.916640\pi$
$239$	−4.00000	+	6.92820i	−0.258738	+	0.448148i	0.707166	+	0.707048i	$0.249973\pi$
$240$		0			0
$241$	5.50000	+	9.52628i	0.354286	+	0.613642i	0.986996	−	0.160748i	$-0.0513906\pi$
$241$	5.50000	+	9.52628i	0.354286	+	0.613642i	−0.632709	+	0.774389i	$0.718057\pi$
$242$	7.00000			0.449977
$243$		0			0
$244$	−7.00000			−0.448129
$245$	−1.00000	−	1.73205i	−0.0638877	−	0.110657i
$246$		0			0
$247$	−8.00000	+	13.8564i	−0.509028	+	0.881662i
$248$		0			0
$249$		0			0
$250$	0.500000	+	0.866025i	0.0316228	+	0.0547723i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−6.00000			−0.377217
$254$	−2.50000	−	4.33013i	−0.156864	−	0.271696i
$255$		0			0
$256$	8.50000	−	14.7224i	0.531250	−	0.920152i
$257$	−3.00000	+	5.19615i	−0.187135	+	0.324127i	−0.944294	−	0.329104i	$-0.893253\pi$
$257$	−3.00000	+	5.19615i	−0.187135	+	0.324127i	0.757159	+	0.653231i	$0.226587\pi$
$258$		0			0
$259$	−6.00000	−	10.3923i	−0.372822	−	0.645746i
$260$	2.00000			0.124035
$261$		0			0
$262$	−6.00000			−0.370681
$263$	−8.00000	−	13.8564i	−0.493301	−	0.854423i	0.506669	−	0.862141i	$-0.330877\pi$
$263$	−8.00000	−	13.8564i	−0.493301	−	0.854423i	−0.999970	+	0.00771799i	$0.997543\pi$
$264$		0			0
$265$	−1.00000	+	1.73205i	−0.0614295	+	0.106399i
$266$	12.0000	−	20.7846i	0.735767	−	1.27439i
$267$		0			0
$268$	−1.50000	−	2.59808i	−0.0916271	−	0.158703i
$269$	−25.0000			−1.52428			−0.762138	−	0.647414i	$-0.775850\pi$
$269$	−25.0000			−1.52428			−0.762138	+	0.647414i	$0.775850\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$	−2.00000	−	3.46410i	−0.121268	−	0.210042i
$273$		0			0
$274$	−6.00000	+	10.3923i	−0.362473	+	0.627822i
$275$	−1.00000	+	1.73205i	−0.0603023	+	0.104447i
$276$		0			0
$277$	−6.00000	−	10.3923i	−0.360505	−	0.624413i	0.627539	−	0.778585i	$-0.284062\pi$
$277$	−6.00000	−	10.3923i	−0.360505	−	0.624413i	−0.988044	+	0.154172i	$0.950729\pi$
$278$	16.0000			0.959616
$279$		0			0
$280$	−9.00000			−0.537853
$281$	−7.50000	−	12.9904i	−0.447412	−	0.774941i	0.550804	−	0.834634i	$-0.314321\pi$
$281$	−7.50000	−	12.9904i	−0.447412	−	0.774941i	−0.998217	+	0.0596933i	$0.980988\pi$
$282$		0			0
$283$	−10.5000	+	18.1865i	−0.624160	+	1.08108i	0.364542	+	0.931187i	$0.381225\pi$
$283$	−10.5000	+	18.1865i	−0.624160	+	1.08108i	−0.988703	+	0.149890i	$0.952108\pi$
$284$	−1.00000	+	1.73205i	−0.0593391	+	0.102778i
$285$		0			0
$286$	−2.00000	−	3.46410i	−0.118262	−	0.204837i
$287$	15.0000			0.885422
$288$		0			0
$289$	−1.00000			−0.0588235
$290$	0.500000	+	0.866025i	0.0293610	+	0.0508548i
$291$		0			0
$292$	2.00000	−	3.46410i	0.117041	−	0.202721i
$293$	6.00000	−	10.3923i	0.350524	−	0.607125i	−0.635818	−	0.771839i	$-0.719337\pi$
$293$	6.00000	−	10.3923i	0.350524	−	0.607125i	0.986341	+	0.164714i	$0.0526703\pi$
$294$		0			0
$295$	−7.00000	−	12.1244i	−0.407556	−	0.705907i
$296$	−12.0000			−0.697486
$297$		0			0
$298$	17.0000			0.984784
$299$	−3.00000	−	5.19615i	−0.173494	−	0.300501i
$300$		0			0
$301$	−12.0000	+	20.7846i	−0.691669	+	1.19800i
$302$	−1.00000	+	1.73205i	−0.0575435	+	0.0996683i
$303$		0			0
$304$	−4.00000	−	6.92820i	−0.229416	−	0.397360i
$305$	7.00000			0.400819
$306$		0			0
$307$	7.00000			0.399511			0.199756	−	0.979846i	$-0.435985\pi$
$307$	7.00000			0.399511			0.199756	+	0.979846i	$0.435985\pi$
$308$	−3.00000	−	5.19615i	−0.170941	−	0.296078i
$309$		0			0
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	0.597522	−	0.801852i	$-0.296152\pi$
$313$	−7.00000	−	12.1244i	−0.395663	−	0.685309i	−0.993186	+	0.116543i	$0.962819\pi$
$314$	−14.0000			−0.790066
$315$		0			0
$316$	6.00000			0.337526
$317$	−17.0000	−	29.4449i	−0.954815	−	1.65379i	−0.734791	−	0.678294i	$-0.762720\pi$
$317$	−17.0000	−	29.4449i	−0.954815	−	1.65379i	−0.220024	−	0.975494i	$-0.570614\pi$
$318$		0			0
$319$	−1.00000	+	1.73205i	−0.0559893	+	0.0969762i
$320$	−3.50000	+	6.06218i	−0.195656	+	0.338886i
$321$		0			0
$322$	4.50000	+	7.79423i	0.250775	+	0.434355i
$323$	32.0000			1.78053
$324$		0			0
$325$	−2.00000			−0.110940
$326$	−2.00000	−	3.46410i	−0.110770	−	0.191859i
$327$		0			0
$328$	7.50000	−	12.9904i	0.414118	−	0.717274i
$329$	−10.5000	+	18.1865i	−0.578884	+	1.00266i
$330$		0			0
$331$	3.00000	+	5.19615i	0.164895	+	0.285606i	0.936618	−	0.350352i	$-0.113938\pi$
$331$	3.00000	+	5.19615i	0.164895	+	0.285606i	−0.771723	+	0.635959i	$0.780605\pi$
$332$	9.00000			0.493939
$333$		0			0
$334$	−9.00000			−0.492458
$335$	1.50000	+	2.59808i	0.0819538	+	0.141948i
$336$		0			0
$337$	4.00000	−	6.92820i	0.217894	−	0.377403i	−0.736270	−	0.676688i	$-0.763415\pi$
$337$	4.00000	−	6.92820i	0.217894	−	0.377403i	0.954164	+	0.299285i	$0.0967480\pi$
$338$	−4.50000	+	7.79423i	−0.244768	+	0.423950i
$339$		0			0
$340$	−2.00000	−	3.46410i	−0.108465	−	0.187867i
$341$		0			0
$342$		0			0
$343$	15.0000			0.809924
$344$	12.0000	+	20.7846i	0.646997	+	1.12063i
$345$		0			0
$346$		0			0
$347$	2.00000	−	3.46410i	0.107366	−	0.185963i	−0.807337	−	0.590091i	$-0.799092\pi$
$347$	2.00000	−	3.46410i	0.107366	−	0.185963i	0.914702	+	0.404128i	$0.132425\pi$
$348$		0			0
$349$	2.50000	+	4.33013i	0.133822	+	0.231786i	0.925147	−	0.379610i	$-0.123942\pi$
$349$	2.50000	+	4.33013i	0.133822	+	0.231786i	−0.791325	+	0.611396i	$0.790608\pi$
$350$	3.00000			0.160357
$351$		0			0
$352$	−10.0000			−0.533002
$353$	12.0000	+	20.7846i	0.638696	+	1.10625i	0.985719	+	0.168397i	$0.0538590\pi$
$353$	12.0000	+	20.7846i	0.638696	+	1.10625i	−0.347024	+	0.937856i	$0.612808\pi$
$354$		0			0
$355$	1.00000	−	1.73205i	0.0530745	−	0.0919277i
$356$	7.50000	−	12.9904i	0.397499	−	0.688489i
$357$		0			0
$358$	1.00000	+	1.73205i	0.0528516	+	0.0915417i
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	45.0000			2.36842
$362$	−3.50000	−	6.06218i	−0.183956	−	0.318621i
$363$		0			0
$364$	3.00000	−	5.19615i	0.157243	−	0.272352i
$365$	−2.00000	+	3.46410i	−0.104685	+	0.181319i
$366$		0			0
$367$	−12.0000	−	20.7846i	−0.626395	−	1.08495i	−0.988269	−	0.152721i	$-0.951196\pi$
$367$	−12.0000	−	20.7846i	−0.626395	−	1.08495i	0.361874	−	0.932227i	$-0.382137\pi$
$368$	3.00000			0.156386
$369$		0			0
$370$	4.00000			0.207950
$371$	3.00000	+	5.19615i	0.155752	+	0.269771i
$372$		0			0
$373$	−5.00000	+	8.66025i	−0.258890	+	0.448411i	−0.965945	−	0.258748i	$-0.916690\pi$
$373$	−5.00000	+	8.66025i	−0.258890	+	0.448411i	0.707055	+	0.707159i	$0.250023\pi$
$374$	−4.00000	+	6.92820i	−0.206835	+	0.358249i
$375$		0			0
$376$	10.5000	+	18.1865i	0.541496	+	0.937899i
$377$	−2.00000			−0.103005
$378$		0			0
$379$	−26.0000			−1.33553			−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000			−1.33553			−0.667765	+	0.744372i	$0.732749\pi$
$380$	−4.00000	−	6.92820i	−0.205196	−	0.355409i
$381$		0			0
$382$	−4.00000	+	6.92820i	−0.204658	+	0.354478i
$383$	18.0000	−	31.1769i	0.919757	−	1.59307i	0.119974	−	0.992777i	$-0.461719\pi$
$383$	18.0000	−	31.1769i	0.919757	−	1.59307i	0.799783	−	0.600289i	$-0.204948\pi$
$384$		0			0
$385$	3.00000	+	5.19615i	0.152894	+	0.264820i
$386$	10.0000			0.508987
$387$		0			0
$388$	−2.00000			−0.101535
$389$	16.5000	+	28.5788i	0.836583	+	1.44900i	0.892735	+	0.450582i	$0.148784\pi$
$389$	16.5000	+	28.5788i	0.836583	+	1.44900i	−0.0561516	+	0.998422i	$0.517883\pi$
$390$		0			0
$391$	−6.00000	+	10.3923i	−0.303433	+	0.525561i
$392$	−3.00000	+	5.19615i	−0.151523	+	0.262445i
$393$		0			0
$394$	6.00000	+	10.3923i	0.302276	+	0.523557i
$395$	−6.00000			−0.301893
$396$		0			0
$397$	34.0000			1.70641			0.853206	−	0.521575i	$-0.174655\pi$
$397$	34.0000			1.70641			0.853206	+	0.521575i	$0.174655\pi$
$398$	2.00000	+	3.46410i	0.100251	+	0.173640i
$399$		0			0
$400$	0.500000	−	0.866025i	0.0250000	−	0.0433013i
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	−0.998350	−	0.0574304i	$-0.981709\pi$
$401$	−9.00000	+	15.5885i	−0.449439	+	0.778450i	0.548911	+	0.835881i	$0.315043\pi$
$402$		0			0
$403$		0			0
$404$	−18.0000			−0.895533
$405$		0			0
$406$	3.00000			0.148888
$407$	4.00000	+	6.92820i	0.198273	+	0.343418i
$408$		0			0
$409$	−7.00000	+	12.1244i	−0.346128	+	0.599511i	−0.985558	−	0.169338i	$-0.945837\pi$
$409$	−7.00000	+	12.1244i	−0.346128	+	0.599511i	0.639430	+	0.768849i	$0.279170\pi$
$410$	−2.50000	+	4.33013i	−0.123466	+	0.213850i
$411$		0			0
$412$	4.00000	+	6.92820i	0.197066	+	0.341328i
$413$	−42.0000			−2.06668
$414$		0			0
$415$	−9.00000			−0.441793
$416$	−5.00000	−	8.66025i	−0.245145	−	0.424604i
$417$		0			0
$418$	−8.00000	+	13.8564i	−0.391293	+	0.677739i
$419$	13.0000	−	22.5167i	0.635092	−	1.10001i	−0.351404	−	0.936224i	$-0.614296\pi$
$419$	13.0000	−	22.5167i	0.635092	−	1.10001i	0.986496	−	0.163787i	$-0.0523710\pi$
$420$		0			0
$421$	−17.0000	−	29.4449i	−0.828529	−	1.43505i	−0.899192	−	0.437555i	$-0.855845\pi$
$421$	−17.0000	−	29.4449i	−0.828529	−	1.43505i	0.0706626	−	0.997500i	$-0.477489\pi$
$422$	22.0000			1.07094
$423$		0			0
$424$	6.00000			0.291386
$425$	2.00000	+	3.46410i	0.0970143	+	0.168034i
$426$		0			0
$427$	10.5000	−	18.1865i	0.508131	−	0.880108i
$428$	−1.50000	+	2.59808i	−0.0725052	+	0.125583i
$429$		0			0
$430$	−4.00000	−	6.92820i	−0.192897	−	0.334108i
$431$	30.0000			1.44505			0.722525	−	0.691345i	$-0.242982\pi$
$431$	30.0000			1.44505			0.722525	+	0.691345i	$0.242982\pi$
$432$		0			0
$433$	28.0000			1.34559			0.672797	−	0.739827i	$-0.265093\pi$
$433$	28.0000			1.34559			0.672797	+	0.739827i	$0.265093\pi$
$434$		0			0
$435$		0			0
$436$	2.50000	−	4.33013i	0.119728	−	0.207375i
$437$	−12.0000	+	20.7846i	−0.574038	+	0.994263i
$438$		0			0
$439$	−14.0000	−	24.2487i	−0.668184	−	1.15733i	−0.978412	−	0.206666i	$-0.933739\pi$
$439$	−14.0000	−	24.2487i	−0.668184	−	1.15733i	0.310228	−	0.950662i	$-0.399595\pi$
$440$	6.00000			0.286039
$441$		0			0
$442$	−8.00000			−0.380521
$443$	7.50000	+	12.9904i	0.356336	+	0.617192i	0.987346	−	0.158583i	$-0.0506926\pi$
$443$	7.50000	+	12.9904i	0.356336	+	0.617192i	−0.631010	+	0.775775i	$0.717359\pi$
$444$		0			0
$445$	−7.50000	+	12.9904i	−0.355534	+	0.615803i
$446$	−9.50000	+	16.4545i	−0.449838	+	0.779142i
$447$		0			0
$448$	10.5000	+	18.1865i	0.496078	+	0.859233i
$449$	26.0000			1.22702			0.613508	−	0.789689i	$-0.289758\pi$
$449$	26.0000			1.22702			0.613508	+	0.789689i	$0.289758\pi$
$450$		0			0
$451$	−10.0000			−0.470882
$452$	4.00000	+	6.92820i	0.188144	+	0.325875i
$453$		0			0
$454$	2.00000	−	3.46410i	0.0938647	−	0.162578i
$455$	−3.00000	+	5.19615i	−0.140642	+	0.243599i
$456$		0			0
$457$	10.0000	+	17.3205i	0.467780	+	0.810219i	0.999322	−	0.0368128i	$-0.0117205\pi$
$457$	10.0000	+	17.3205i	0.467780	+	0.810219i	−0.531542	+	0.847032i	$0.678387\pi$
$458$	−15.0000			−0.700904
$459$		0			0
$460$	3.00000			0.139876
$461$	4.50000	+	7.79423i	0.209586	+	0.363013i	0.951584	−	0.307388i	$-0.0994551\pi$
$461$	4.50000	+	7.79423i	0.209586	+	0.363013i	−0.741998	+	0.670402i	$0.766122\pi$
$462$		0			0
$463$	18.0000	−	31.1769i	0.836531	−	1.44891i	−0.0562469	−	0.998417i	$-0.517913\pi$
$463$	18.0000	−	31.1769i	0.836531	−	1.44891i	0.892778	−	0.450497i	$-0.148753\pi$
$464$	0.500000	−	0.866025i	0.0232119	−	0.0402042i
$465$		0			0
$466$	−12.0000	−	20.7846i	−0.555889	−	0.962828i
$467$	−20.0000			−0.925490			−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000			−0.925490			−0.462745	+	0.886492i	$0.653135\pi$
$468$		0			0
$469$	9.00000			0.415581
$470$	−3.50000	−	6.06218i	−0.161443	−	0.279627i
$471$		0			0
$472$	−21.0000	+	36.3731i	−0.966603	+	1.67421i
$473$	8.00000	−	13.8564i	0.367840	−	0.637118i
$474$		0			0
$475$	4.00000	+	6.92820i	0.183533	+	0.317888i
$476$	−12.0000			−0.550019
$477$		0			0
$478$	−8.00000			−0.365911
$479$	−9.00000	−	15.5885i	−0.411220	−	0.712255i	0.583803	−	0.811895i	$-0.301564\pi$
$479$	−9.00000	−	15.5885i	−0.411220	−	0.712255i	−0.995023	+	0.0996406i	$0.968231\pi$
$480$		0			0
$481$	−4.00000	+	6.92820i	−0.182384	+	0.315899i
$482$	−5.50000	+	9.52628i	−0.250518	+	0.433910i
$483$		0			0
$484$	−3.50000	−	6.06218i	−0.159091	−	0.275554i
$485$	2.00000			0.0908153
$486$		0			0
$487$	−16.0000			−0.725029			−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000			−0.725029			−0.362515	+	0.931978i	$0.618082\pi$
$488$	−10.5000	−	18.1865i	−0.475313	−	0.823266i
$489$		0			0
$490$	1.00000	−	1.73205i	0.0451754	−	0.0782461i
$491$	10.0000	−	17.3205i	0.451294	−	0.781664i	−0.547173	−	0.837020i	$-0.684296\pi$
$491$	10.0000	−	17.3205i	0.451294	−	0.781664i	0.998467	+	0.0553560i	$0.0176294\pi$
$492$		0			0
$493$	2.00000	+	3.46410i	0.0900755	+	0.156015i
$494$	−16.0000			−0.719874
$495$		0			0
$496$		0			0
$497$	−3.00000	−	5.19615i	−0.134568	−	0.233079i
$498$		0			0
$499$	16.0000	−	27.7128i	0.716258	−	1.24060i	−0.246214	−	0.969216i	$-0.579187\pi$
$499$	16.0000	−	27.7128i	0.716258	−	1.24060i	0.962472	−	0.271380i	$-0.0874801\pi$
$500$	0.500000	−	0.866025i	0.0223607	−	0.0387298i
$501$		0			0
$502$		0			0
$503$	7.00000			0.312115			0.156057	−	0.987748i	$-0.450122\pi$
$503$	7.00000			0.312115			0.156057	+	0.987748i	$0.450122\pi$
$504$		0			0
$505$	18.0000			0.800989
$506$	−3.00000	−	5.19615i	−0.133366	−	0.230997i
$507$		0			0
$508$	−2.50000	+	4.33013i	−0.110920	+	0.192118i
$509$	21.5000	−	37.2391i	0.952971	−	1.65059i	0.214026	−	0.976828i	$-0.431342\pi$
$509$	21.5000	−	37.2391i	0.952971	−	1.65059i	0.738945	−	0.673766i	$-0.235324\pi$
$510$		0			0
$511$	6.00000	+	10.3923i	0.265424	+	0.459728i
$512$	11.0000			0.486136
$513$		0			0
$514$	−6.00000			−0.264649
$515$	−4.00000	−	6.92820i	−0.176261	−	0.305293i
$516$		0			0
$517$	7.00000	−	12.1244i	0.307860	−	0.533229i
$518$	6.00000	−	10.3923i	0.263625	−	0.456612i
$519$		0			0
$520$	3.00000	+	5.19615i	0.131559	+	0.227866i
$521$	−11.0000			−0.481919			−0.240959	−	0.970535i	$-0.577462\pi$
$521$	−11.0000			−0.481919			−0.240959	+	0.970535i	$0.577462\pi$
$522$		0			0
$523$	−29.0000			−1.26808			−0.634041	−	0.773300i	$-0.718605\pi$
$523$	−29.0000			−1.26808			−0.634041	+	0.773300i	$0.718605\pi$
$524$	3.00000	+	5.19615i	0.131056	+	0.226995i
$525$		0			0
$526$	8.00000	−	13.8564i	0.348817	−	0.604168i
$527$		0			0
$528$		0			0
$529$	7.00000	+	12.1244i	0.304348	+	0.527146i
$530$	−2.00000			−0.0868744
$531$		0			0
$532$	−24.0000			−1.04053
$533$	−5.00000	−	8.66025i	−0.216574	−	0.375117i
$534$		0			0
$535$	1.50000	−	2.59808i	0.0648507	−	0.112325i
$536$	4.50000	−	7.79423i	0.194370	−	0.336659i
$537$		0			0
$538$	−12.5000	−	21.6506i	−0.538913	−	0.933425i
$539$	4.00000			0.172292
$540$		0			0
$541$	−39.0000			−1.67674			−0.838370	−	0.545101i	$-0.816491\pi$
$541$	−39.0000			−1.67674			−0.838370	+	0.545101i	$0.816491\pi$
$542$	−4.00000	−	6.92820i	−0.171815	−	0.297592i
$543$		0			0
$544$	−10.0000	+	17.3205i	−0.428746	+	0.742611i
$545$	−2.50000	+	4.33013i	−0.107088	+	0.185482i
$546$		0			0
$547$	14.5000	+	25.1147i	0.619975	+	1.07383i	0.989490	+	0.144604i	$0.0461907\pi$
$547$	14.5000	+	25.1147i	0.619975	+	1.07383i	−0.369514	+	0.929225i	$0.620476\pi$
$548$	12.0000			0.512615
$549$		0			0
$550$	−2.00000			−0.0852803
$551$	4.00000	+	6.92820i	0.170406	+	0.295151i
$552$		0			0
$553$	−9.00000	+	15.5885i	−0.382719	+	0.662889i
$554$	6.00000	−	10.3923i	0.254916	−	0.441527i
$555$		0			0
$556$	−8.00000	−	13.8564i	−0.339276	−	0.587643i
$557$	30.0000			1.27114			0.635570	−	0.772043i	$-0.280765\pi$
$557$	30.0000			1.27114			0.635570	+	0.772043i	$0.280765\pi$
$558$		0			0
$559$	16.0000			0.676728
$560$	−1.50000	−	2.59808i	−0.0633866	−	0.109789i
$561$		0			0
$562$	7.50000	−	12.9904i	0.316368	−	0.547966i
$563$	−10.5000	+	18.1865i	−0.442522	+	0.766471i	−0.997876	−	0.0651433i	$-0.979250\pi$
$563$	−10.5000	+	18.1865i	−0.442522	+	0.766471i	0.555354	+	0.831614i	$0.312583\pi$
$564$		0			0
$565$	−4.00000	−	6.92820i	−0.168281	−	0.291472i
$566$	−21.0000			−0.882696
$567$		0			0
$568$	−6.00000			−0.251754
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	0.796266	−	0.604947i	$-0.206806\pi$
$569$	−3.00000	−	5.19615i	−0.125767	−	0.217834i	−0.922032	+	0.387113i	$0.873472\pi$
$570$		0			0
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	0.308443	+	0.951243i	$0.400192\pi$
$571$	−16.0000	+	27.7128i	−0.669579	+	1.15975i	−0.978022	+	0.208502i	$0.933141\pi$
$572$	−2.00000	+	3.46410i	−0.0836242	+	0.144841i
$573$		0			0
$574$	7.50000	+	12.9904i	0.313044	+	0.542208i
$575$	−3.00000			−0.125109
$576$		0			0
$577$	−10.0000			−0.416305			−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000			−0.416305			−0.208153	+	0.978096i	$0.566745\pi$
$578$	−0.500000	−	0.866025i	−0.0207973	−	0.0360219i
$579$		0			0
$580$	0.500000	−	0.866025i	0.0207614	−	0.0359597i
$581$	−13.5000	+	23.3827i	−0.560074	+	0.970077i
$582$		0			0
$583$	−2.00000	−	3.46410i	−0.0828315	−	0.143468i
$584$	12.0000			0.496564
$585$		0			0
$586$	12.0000			0.495715
$587$	−16.5000	−	28.5788i	−0.681028	−	1.17957i	−0.974668	−	0.223659i	$-0.928200\pi$
$587$	−16.5000	−	28.5788i	−0.681028	−	1.17957i	0.293640	−	0.955916i	$-0.405133\pi$
$588$		0			0
$589$		0			0
$590$	7.00000	−	12.1244i	0.288185	−	0.499152i
$591$		0			0
$592$	−2.00000	−	3.46410i	−0.0821995	−	0.142374i
$593$	−20.0000			−0.821302			−0.410651	−	0.911793i	$-0.634698\pi$
$593$	−20.0000			−0.821302			−0.410651	+	0.911793i	$0.634698\pi$
$594$		0			0
$595$	12.0000			0.491952
$596$	−8.50000	−	14.7224i	−0.348174	−	0.603054i
$597$		0			0
$598$	3.00000	−	5.19615i	0.122679	−	0.212486i
$599$	−5.00000	+	8.66025i	−0.204294	+	0.353848i	−0.949908	−	0.312531i	$-0.898823\pi$
$599$	−5.00000	+	8.66025i	−0.204294	+	0.353848i	0.745613	+	0.666379i	$0.232157\pi$
$600$		0			0
$601$	−1.00000	−	1.73205i	−0.0407909	−	0.0706518i	0.844909	−	0.534910i	$-0.179654\pi$
$601$	−1.00000	−	1.73205i	−0.0407909	−	0.0706518i	−0.885700	+	0.464258i	$0.846321\pi$
$602$	−24.0000			−0.978167
$603$		0			0
$604$	2.00000			0.0813788
$605$	3.50000	+	6.06218i	0.142295	+	0.246463i
$606$		0			0
$607$	20.5000	−	35.5070i	0.832069	−	1.44119i	−0.0643251	−	0.997929i	$-0.520489\pi$
$607$	20.5000	−	35.5070i	0.832069	−	1.44119i	0.896394	−	0.443257i	$-0.146177\pi$
$608$	−20.0000	+	34.6410i	−0.811107	+	1.40488i
$609$		0			0
$610$	3.50000	+	6.06218i	0.141711	+	0.245450i
$611$	14.0000			0.566379
$612$		0			0
$613$	−44.0000			−1.77714			−0.888572	−	0.458738i	$-0.848302\pi$
$613$	−44.0000			−1.77714			−0.888572	+	0.458738i	$0.848302\pi$
$614$	3.50000	+	6.06218i	0.141249	+	0.244650i
$615$		0			0
$616$	9.00000	−	15.5885i	0.362620	−	0.628077i
$617$	−18.0000	+	31.1769i	−0.724653	+	1.25514i	0.234464	+	0.972125i	$0.424666\pi$
$617$	−18.0000	+	31.1769i	−0.724653	+	1.25514i	−0.959117	+	0.283011i	$0.908667\pi$
$618$		0			0
$619$	2.00000	+	3.46410i	0.0803868	+	0.139234i	0.903416	−	0.428765i	$-0.141051\pi$
$619$	2.00000	+	3.46410i	0.0803868	+	0.139234i	−0.823029	+	0.567999i	$0.807718\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	22.5000	+	38.9711i	0.901443	+	1.56135i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$	7.00000	−	12.1244i	0.279776	−	0.484587i
$627$		0			0
$628$	7.00000	+	12.1244i	0.279330	+	0.483814i
$629$	16.0000			0.637962
$630$		0			0
$631$	16.0000			0.636950			0.318475	−	0.947931i	$-0.396829\pi$
$631$	16.0000			0.636950			0.318475	+	0.947931i	$0.396829\pi$
$632$	9.00000	+	15.5885i	0.358001	+	0.620076i
$633$		0			0
$634$	17.0000	−	29.4449i	0.675156	−	1.16940i
$635$	2.50000	−	4.33013i	0.0992095	−	0.171836i
$636$		0			0
$637$	2.00000	+	3.46410i	0.0792429	+	0.137253i
$638$	−2.00000			−0.0791808
$639$		0			0
$640$	3.00000			0.118585
$641$	16.5000	+	28.5788i	0.651711	+	1.12880i	0.982708	+	0.185164i	$0.0592817\pi$
$641$	16.5000	+	28.5788i	0.651711	+	1.12880i	−0.330997	+	0.943632i	$0.607385\pi$
$642$		0			0
$643$	4.50000	−	7.79423i	0.177463	−	0.307374i	−0.763548	−	0.645751i	$-0.776544\pi$
$643$	4.50000	−	7.79423i	0.177463	−	0.307374i	0.941011	+	0.338377i	$0.109878\pi$
$644$	4.50000	−	7.79423i	0.177325	−	0.307136i
$645$		0			0
$646$	16.0000	+	27.7128i	0.629512	+	1.09035i
$647$	17.0000			0.668339			0.334169	−	0.942513i	$-0.391544\pi$
$647$	17.0000			0.668339			0.334169	+	0.942513i	$0.391544\pi$
$648$		0			0
$649$	28.0000			1.09910
$650$	−1.00000	−	1.73205i	−0.0392232	−	0.0679366i
$651$		0			0
$652$	−2.00000	+	3.46410i	−0.0783260	+	0.135665i
$653$	−2.00000	+	3.46410i	−0.0782660	+	0.135561i	−0.902502	−	0.430686i	$-0.858272\pi$
$653$	−2.00000	+	3.46410i	−0.0782660	+	0.135561i	0.824236	+	0.566247i	$0.191605\pi$
$654$		0			0
$655$	−3.00000	−	5.19615i	−0.117220	−	0.203030i
$656$	5.00000			0.195217
$657$		0			0
$658$	−21.0000			−0.818665
$659$	−4.00000	−	6.92820i	−0.155818	−	0.269884i	0.777539	−	0.628835i	$-0.216468\pi$
$659$	−4.00000	−	6.92820i	−0.155818	−	0.269884i	−0.933357	+	0.358951i	$0.883135\pi$
$660$		0			0
$661$	7.00000	−	12.1244i	0.272268	−	0.471583i	−0.697174	−	0.716902i	$-0.745559\pi$
$661$	7.00000	−	12.1244i	0.272268	−	0.471583i	0.969442	+	0.245319i	$0.0788928\pi$
$662$	−3.00000	+	5.19615i	−0.116598	+	0.201954i
$663$		0			0
$664$	13.5000	+	23.3827i	0.523902	+	0.907424i
$665$	24.0000			0.930680
$666$		0			0
$667$	−3.00000			−0.116160
$668$	4.50000	+	7.79423i	0.174110	+	0.301568i
$669$		0			0
$670$	−1.50000	+	2.59808i	−0.0579501	+	0.100372i
$671$	−7.00000	+	12.1244i	−0.270232	+	0.468056i
$672$		0			0
$673$	−3.00000	−	5.19615i	−0.115642	−	0.200297i	0.802395	−	0.596794i	$-0.203559\pi$
$673$	−3.00000	−	5.19615i	−0.115642	−	0.200297i	−0.918036	+	0.396497i	$0.870226\pi$
$674$	8.00000			0.308148
$675$		0			0
$676$	9.00000			0.346154
$677$	−21.0000	−	36.3731i	−0.807096	−	1.39793i	−0.914867	−	0.403755i	$-0.867705\pi$
$677$	−21.0000	−	36.3731i	−0.807096	−	1.39793i	0.107772	−	0.994176i	$-0.465628\pi$
$678$		0			0
$679$	3.00000	−	5.19615i	0.115129	−	0.199410i
$680$	6.00000	−	10.3923i	0.230089	−	0.398527i
$681$		0			0
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$	−12.0000			−0.458496
$686$	7.50000	+	12.9904i	0.286351	+	0.495975i
$687$		0			0
$688$	−4.00000	+	6.92820i	−0.152499	+	0.264135i
$689$	2.00000	−	3.46410i	0.0761939	−	0.131972i
$690$		0			0
$691$	7.00000	+	12.1244i	0.266293	+	0.461232i	0.967901	−	0.251330i	$-0.0808679\pi$
$691$	7.00000	+	12.1244i	0.266293	+	0.461232i	−0.701609	+	0.712562i	$0.747535\pi$
$692$		0			0
$693$		0			0
$694$	4.00000			0.151838
$695$	8.00000	+	13.8564i	0.303457	+	0.525603i
$696$		0			0
$697$	−10.0000	+	17.3205i	−0.378777	+	0.656061i
$698$	−2.50000	+	4.33013i	−0.0946264	+	0.163898i
$699$		0			0
$700$	−1.50000	−	2.59808i	−0.0566947	−	0.0981981i
$701$	−23.0000			−0.868698			−0.434349	−	0.900745i	$-0.643022\pi$
$701$	−23.0000			−0.868698			−0.434349	+	0.900745i	$0.643022\pi$
$702$		0			0
$703$	32.0000			1.20690
$704$	−7.00000	−	12.1244i	−0.263822	−	0.456954i
$705$		0			0
$706$	−12.0000	+	20.7846i	−0.451626	+	0.782239i
$707$	27.0000	−	46.7654i	1.01544	−	1.75879i
$708$		0			0
$709$	20.5000	+	35.5070i	0.769894	+	1.33349i	0.937620	+	0.347661i	$0.113024\pi$
$709$	20.5000	+	35.5070i	0.769894	+	1.33349i	−0.167727	+	0.985834i	$0.553643\pi$
$710$	2.00000			0.0750587
$711$		0			0
$712$	45.0000			1.68645
$713$		0			0
$714$		0			0
$715$	2.00000	−	3.46410i	0.0747958	−	0.129550i
$716$	1.00000	−	1.73205i	0.0373718	−	0.0647298i
$717$		0			0
$718$	−12.0000	−	20.7846i	−0.447836	−	0.775675i
$719$	−6.00000			−0.223762			−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000			−0.223762			−0.111881	+	0.993722i	$0.535688\pi$
$720$		0			0
$721$	−24.0000			−0.893807
$722$	22.5000	+	38.9711i	0.837363	+	1.45036i
$723$		0			0
$724$	−3.50000	+	6.06218i	−0.130076	+	0.225299i
$725$	−0.500000	+	0.866025i	−0.0185695	+	0.0321634i
$726$		0			0
$727$	−11.5000	−	19.9186i	−0.426511	−	0.738739i	0.570049	−	0.821611i	$-0.306924\pi$
$727$	−11.5000	−	19.9186i	−0.426511	−	0.738739i	−0.996560	+	0.0828714i	$0.973591\pi$
$728$	18.0000			0.667124
$729$		0			0
$730$	−4.00000			−0.148047
$731$	−16.0000	−	27.7128i	−0.591781	−	1.02500i
$732$		0			0
$733$	17.0000	−	29.4449i	0.627909	−	1.08757i	−0.360061	−	0.932929i	$-0.617244\pi$
$733$	17.0000	−	29.4449i	0.627909	−	1.08757i	0.987971	−	0.154642i	$-0.0494225\pi$
$734$	12.0000	−	20.7846i	0.442928	−	0.767174i
$735$		0			0
$736$	−7.50000	−	12.9904i	−0.276454	−	0.478832i
$737$	−6.00000			−0.221013
$738$		0			0
$739$	−2.00000			−0.0735712			−0.0367856	−	0.999323i	$-0.511712\pi$
$739$	−2.00000			−0.0735712			−0.0367856	+	0.999323i	$0.511712\pi$
$740$	−2.00000	−	3.46410i	−0.0735215	−	0.127343i
$741$		0			0
$742$	−3.00000	+	5.19615i	−0.110133	+	0.190757i
$743$	−14.5000	+	25.1147i	−0.531953	+	0.921370i	0.467351	+	0.884072i	$0.345209\pi$
$743$	−14.5000	+	25.1147i	−0.531953	+	0.921370i	−0.999304	+	0.0372984i	$0.988125\pi$
$744$		0			0
$745$	8.50000	+	14.7224i	0.311416	+	0.539388i
$746$	−10.0000			−0.366126
$747$		0			0
$748$	8.00000			0.292509
$749$	−4.50000	−	7.79423i	−0.164426	−	0.284795i
$750$		0			0
$751$	−5.00000	+	8.66025i	−0.182453	+	0.316017i	−0.942715	−	0.333599i	$-0.891737\pi$
$751$	−5.00000	+	8.66025i	−0.182453	+	0.316017i	0.760263	+	0.649616i	$0.225070\pi$
$752$	−3.50000	+	6.06218i	−0.127632	+	0.221065i
$753$		0			0
$754$	−1.00000	−	1.73205i	−0.0364179	−	0.0630776i
$755$	−2.00000			−0.0727875
$756$		0			0
$757$	−26.0000			−0.944986			−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000			−0.944986			−0.472493	+	0.881334i	$0.656646\pi$
$758$	−13.0000	−	22.5167i	−0.472181	−	0.817842i
$759$		0			0
$760$	12.0000	−	20.7846i	0.435286	−	0.753937i
$761$	7.50000	−	12.9904i	0.271875	−	0.470901i	−0.697467	−	0.716617i	$-0.745690\pi$
$761$	7.50000	−	12.9904i	0.271875	−	0.470901i	0.969342	+	0.245716i	$0.0790230\pi$
$762$		0			0
$763$	7.50000	+	12.9904i	0.271518	+	0.470283i
$764$	8.00000			0.289430
$765$		0			0
$766$	36.0000			1.30073
$767$	14.0000	+	24.2487i	0.505511	+	0.875570i
$768$		0			0
$769$	−2.50000	+	4.33013i	−0.0901523	+	0.156148i	−0.907575	−	0.419890i	$-0.862069\pi$
$769$	−2.50000	+	4.33013i	−0.0901523	+	0.156148i	0.817423	+	0.576038i	$0.195402\pi$
$770$	−3.00000	+	5.19615i	−0.108112	+	0.187256i
$771$		0			0
$772$	−5.00000	−	8.66025i	−0.179954	−	0.311689i
$773$	24.0000			0.863220			0.431610	−	0.902060i	$-0.357946\pi$
$773$	24.0000			0.863220			0.431610	+	0.902060i	$0.357946\pi$
$774$		0			0
$775$		0			0
$776$	−3.00000	−	5.19615i	−0.107694	−	0.186531i
$777$		0			0
$778$	−16.5000	+	28.5788i	−0.591554	+	1.02460i
$779$	−20.0000	+	34.6410i	−0.716574	+	1.24114i
$780$		0			0
$781$	2.00000	+	3.46410i	0.0715656	+	0.123955i
$782$	−12.0000			−0.429119
$783$		0			0
$784$	−2.00000			−0.0714286
$785$	−7.00000	−	12.1244i	−0.249841	−	0.432737i
$786$		0			0
$787$	−14.0000	+	24.2487i	−0.499046	+	0.864373i	−0.999999	−	0.00110111i	$-0.999650\pi$
$787$	−14.0000	+	24.2487i	−0.499046	+	0.864373i	0.500953	+	0.865474i	$0.332983\pi$
$788$	6.00000	−	10.3923i	0.213741	−	0.370211i
$789$		0			0
$790$	−3.00000	−	5.19615i	−0.106735	−	0.184871i
$791$	−24.0000			−0.853342
$792$		0			0
$793$	−14.0000			−0.497155
$794$	17.0000	+	29.4449i	0.603307	+	1.04496i
$795$		0			0
$796$	2.00000	−	3.46410i	0.0708881	−	0.122782i
$797$	13.0000	−	22.5167i	0.460484	−	0.797581i	−0.538501	−	0.842625i	$-0.681009\pi$
$797$	13.0000	−	22.5167i	0.460484	−	0.797581i	0.998985	+	0.0450436i	$0.0143427\pi$
$798$		0			0
$799$	−14.0000	−	24.2487i	−0.495284	−	0.857858i
$800$	−5.00000			−0.176777
$801$		0			0
$802$	−18.0000			−0.635602
$803$	−4.00000	−	6.92820i	−0.141157	−	0.244491i
$804$		0			0
$805$	−4.50000	+	7.79423i	−0.158604	+	0.274710i
$806$		0			0
$807$		0			0
$808$	−27.0000	−	46.7654i	−0.949857	−	1.64520i
$809$	−26.0000			−0.914111			−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000			−0.914111			−0.457056	+	0.889438i	$0.651096\pi$
$810$		0			0
$811$	−42.0000			−1.47482			−0.737410	−	0.675446i	$-0.763951\pi$
$811$	−42.0000			−1.47482			−0.737410	+	0.675446i	$0.763951\pi$
$812$	−1.50000	−	2.59808i	−0.0526397	−	0.0911746i
$813$		0			0
$814$	−4.00000	+	6.92820i	−0.140200	+	0.242833i
$815$	2.00000	−	3.46410i	0.0700569	−	0.121342i
$816$		0			0
$817$	−32.0000	−	55.4256i	−1.11954	−	1.93910i
$818$	−14.0000			−0.489499
$819$		0			0
$820$	5.00000			0.174608
$821$	7.50000	+	12.9904i	0.261752	+	0.453367i	0.966708	−	0.255884i	$-0.0823665\pi$
$821$	7.50000	+	12.9904i	0.261752	+	0.453367i	−0.704956	+	0.709251i	$0.749033\pi$
$822$		0			0
$823$	−26.5000	+	45.8993i	−0.923732	+	1.59995i	−0.130144	+	0.991495i	$0.541544\pi$
$823$	−26.5000	+	45.8993i	−0.923732	+	1.59995i	−0.793588	+	0.608456i	$0.791789\pi$
$824$	−12.0000	+	20.7846i	−0.418040	+	0.724066i
$825$		0			0
$826$	−21.0000	−	36.3731i	−0.730683	−	1.26558i
$827$	−37.0000			−1.28662			−0.643308	−	0.765607i	$-0.722439\pi$
$827$	−37.0000			−1.28662			−0.643308	+	0.765607i	$0.722439\pi$
$828$		0			0
$829$	−3.00000			−0.104194			−0.0520972	−	0.998642i	$-0.516591\pi$
$829$	−3.00000			−0.104194			−0.0520972	+	0.998642i	$0.516591\pi$
$830$	−4.50000	−	7.79423i	−0.156197	−	0.270542i
$831$		0			0
$832$	7.00000	−	12.1244i	0.242681	−	0.420336i
$833$	4.00000	−	6.92820i	0.138592	−	0.240048i
$834$		0			0
$835$	−4.50000	−	7.79423i	−0.155729	−	0.269730i
$836$	16.0000			0.553372
$837$		0			0
$838$	26.0000			0.898155
$839$	−20.0000	−	34.6410i	−0.690477	−	1.19594i	−0.971682	−	0.236293i	$-0.924067\pi$
$839$	−20.0000	−	34.6410i	−0.690477	−	1.19594i	0.281205	−	0.959648i	$-0.409266\pi$
$840$		0			0
$841$	14.0000	−	24.2487i	0.482759	−	0.836162i
$842$	17.0000	−	29.4449i	0.585859	−	1.01474i
$843$		0			0
$844$	−11.0000	−	19.0526i	−0.378636	−	0.655816i
$845$	−9.00000			−0.309609
$846$		0			0
$847$	21.0000			0.721569
$848$	1.00000	+	1.73205i	0.0343401	+	0.0594789i
$849$		0			0
$850$	−2.00000	+	3.46410i	−0.0685994	+	0.118818i
$851$	−6.00000	+	10.3923i	−0.205677	+	0.356244i
$852$		0			0
$853$	27.0000	+	46.7654i	0.924462	+	1.60122i	0.792424	+	0.609971i	$0.208819\pi$
$853$	27.0000	+	46.7654i	0.924462	+	1.60122i	0.132039	+	0.991245i	$0.457848\pi$
$854$	21.0000			0.718605
$855$		0			0
$856$	−9.00000			−0.307614
$857$	5.00000	+	8.66025i	0.170797	+	0.295829i	0.938699	−	0.344739i	$-0.112033\pi$
$857$	5.00000	+	8.66025i	0.170797	+	0.295829i	−0.767902	+	0.640567i	$0.778699\pi$
$858$		0			0
$859$	−11.0000	+	19.0526i	−0.375315	+	0.650065i	−0.990374	−	0.138416i	$-0.955799\pi$
$859$	−11.0000	+	19.0526i	−0.375315	+	0.650065i	0.615059	+	0.788481i	$0.289132\pi$
$860$	−4.00000	+	6.92820i	−0.136399	+	0.236250i
$861$		0			0
$862$	15.0000	+	25.9808i	0.510902	+	0.884908i
$863$	−17.0000			−0.578687			−0.289343	−	0.957225i	$-0.593437\pi$
$863$	−17.0000			−0.578687			−0.289343	+	0.957225i	$0.593437\pi$
$864$		0			0
$865$		0			0
$866$	14.0000	+	24.2487i	0.475739	+	0.824005i
$867$		0			0
$868$		0			0
$869$	6.00000	−	10.3923i	0.203536	−	0.352535i
$870$		0			0
$871$	−3.00000	−	5.19615i	−0.101651	−	0.176065i
$872$	15.0000			0.507964
$873$		0			0
$874$	−24.0000			−0.811812
$875$	1.50000	+	2.59808i	0.0507093	+	0.0878310i
$876$		0			0
$877$	9.00000	−	15.5885i	0.303908	−	0.526385i	−0.673109	−	0.739543i	$-0.735042\pi$
$877$	9.00000	−	15.5885i	0.303908	−	0.526385i	0.977018	+	0.213158i	$0.0683750\pi$
$878$	14.0000	−	24.2487i	0.472477	−	0.818354i
$879$		0			0
$880$	1.00000	+	1.73205i	0.0337100	+	0.0583874i
$881$	35.0000			1.17918			0.589590	−	0.807703i	$-0.299289\pi$
$881$	35.0000			1.17918			0.589590	+	0.807703i	$0.299289\pi$
$882$		0			0
$883$	23.0000			0.774012			0.387006	−	0.922077i	$-0.373509\pi$
$883$	23.0000			0.774012			0.387006	+	0.922077i	$0.373509\pi$
$884$	4.00000	+	6.92820i	0.134535	+	0.233021i
$885$		0			0
$886$	−7.50000	+	12.9904i	−0.251967	+	0.436420i
$887$	18.0000	−	31.1769i	0.604381	−	1.04682i	−0.387768	−	0.921757i	$-0.626754\pi$
$887$	18.0000	−	31.1769i	0.604381	−	1.04682i	0.992149	−	0.125061i	$-0.0399128\pi$
$888$		0			0
$889$	−7.50000	−	12.9904i	−0.251542	−	0.435683i
$890$	−15.0000			−0.502801
$891$		0			0
$892$	19.0000			0.636167
$893$	−28.0000	−	48.4974i	−0.936984	−	1.62290i
$894$		0			0
$895$	−1.00000	+	1.73205i	−0.0334263	+	0.0578961i
$896$	4.50000	−	7.79423i	0.150334	−	0.260387i
$897$		0			0
$898$	13.0000	+	22.5167i	0.433816	+	0.751391i
$899$		0			0
$900$		0			0
$901$	−8.00000			−0.266519
$902$	−5.00000	−	8.66025i	−0.166482	−	0.288355i
$903$		0			0
$904$	−12.0000	+	20.7846i	−0.399114	+	0.691286i
$905$	3.50000	−	6.06218i	0.116344	−	0.201514i
$906$		0			0
$907$	25.5000	+	44.1673i	0.846714	+	1.46655i	0.884125	+	0.467251i	$0.154756\pi$
$907$	25.5000	+	44.1673i	0.846714	+	1.46655i	−0.0374111	+	0.999300i	$0.511911\pi$
$908$	−4.00000			−0.132745
$909$		0			0
$910$	−6.00000			−0.198898
$911$	−25.0000	−	43.3013i	−0.828287	−	1.43464i	−0.899381	−	0.437165i	$-0.855982\pi$
$911$	−25.0000	−	43.3013i	−0.828287	−	1.43464i	0.0710941	−	0.997470i	$-0.477351\pi$
$912$		0			0
$913$	9.00000	−	15.5885i	0.297857	−	0.515903i
$914$	−10.0000	+	17.3205i	−0.330771	+	0.572911i
$915$		0			0
$916$	7.50000	+	12.9904i	0.247807	+	0.429214i
$917$	−18.0000			−0.594412
$918$		0			0
$919$	10.0000			0.329870			0.164935	−	0.986304i	$-0.447259\pi$
$919$	10.0000			0.329870			0.164935	+	0.986304i	$0.447259\pi$
$920$	4.50000	+	7.79423i	0.148361	+	0.256968i
$921$		0			0
$922$	−4.50000	+	7.79423i	−0.148200	+	0.256689i
$923$	−2.00000	+	3.46410i	−0.0658308	+	0.114022i
$924$		0			0
$925$	2.00000	+	3.46410i	0.0657596	+	0.113899i
$926$	36.0000			1.18303
$927$		0			0
$928$	−5.00000			−0.164133
$929$	7.00000	+	12.1244i	0.229663	+	0.397787i	0.957708	−	0.287742i	$-0.0929044\pi$
$929$	7.00000	+	12.1244i	0.229663	+	0.397787i	−0.728046	+	0.685529i	$0.759571\pi$
$930$		0			0
$931$	8.00000	−	13.8564i	0.262189	−	0.454125i
$932$	−12.0000	+	20.7846i	−0.393073	+	0.680823i
$933$		0			0
$934$	−10.0000	−	17.3205i	−0.327210	−	0.566744i
$935$	−8.00000			−0.261628
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$	4.50000	+	7.79423i	0.146930	+	0.254491i
$939$		0			0
$940$	−3.50000	+	6.06218i	−0.114157	+	0.197726i
$941$	−3.50000	+	6.06218i	−0.114097	+	0.197621i	−0.917418	−	0.397924i	$-0.869731\pi$
$941$	−3.50000	+	6.06218i	−0.114097	+	0.197621i	0.803322	+	0.595545i	$0.203064\pi$
$942$		0			0
$943$	−7.50000	−	12.9904i	−0.244234	−	0.423025i
$944$	−14.0000			−0.455661
$945$		0			0
$946$	16.0000			0.520205
$947$	28.5000	+	49.3634i	0.926126	+	1.60410i	0.789741	+	0.613441i	$0.210215\pi$
$947$	28.5000	+	49.3634i	0.926126	+	1.60410i	0.136385	+	0.990656i	$0.456452\pi$
$948$		0			0
$949$	4.00000	−	6.92820i	0.129845	−	0.224899i
$950$	−4.00000	+	6.92820i	−0.129777	+	0.224781i
$951$		0			0
$952$	−18.0000	−	31.1769i	−0.583383	−	1.01045i
$953$	26.0000			0.842223			0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000			0.842223			0.421111	+	0.907009i	$0.361640\pi$
$954$		0			0
$955$	−8.00000			−0.258874
$956$	4.00000	+	6.92820i	0.129369	+	0.224074i
$957$		0			0
$958$	9.00000	−	15.5885i	0.290777	−	0.503640i
$959$	−18.0000	+	31.1769i	−0.581250	+	1.00676i
$960$		0			0
$961$	15.5000	+	26.8468i	0.500000	+	0.866025i
$962$	−8.00000			−0.257930
$963$		0			0
$964$	11.0000			0.354286
$965$	5.00000	+	8.66025i	0.160956	+	0.278783i
$966$		0			0
$967$	−20.5000	+	35.5070i	−0.659236	+	1.14183i	0.321578	+	0.946883i	$0.395787\pi$
$967$	−20.5000	+	35.5070i	−0.659236	+	1.14183i	−0.980814	+	0.194946i	$0.937547\pi$
$968$	10.5000	−	18.1865i	0.337483	−	0.584537i
$969$		0			0
$970$	1.00000	+	1.73205i	0.0321081	+	0.0556128i
$971$	36.0000			1.15529			0.577647	−	0.816286i	$-0.303971\pi$
$971$	36.0000			1.15529			0.577647	+	0.816286i	$0.303971\pi$
$972$		0			0
$973$	48.0000			1.53881
$974$	−8.00000	−	13.8564i	−0.256337	−	0.443988i
$975$		0			0
$976$	3.50000	−	6.06218i	0.112032	−	0.194046i
$977$	−19.0000	+	32.9090i	−0.607864	+	1.05285i	0.383728	+	0.923446i	$0.374640\pi$
$977$	−19.0000	+	32.9090i	−0.607864	+	1.05285i	−0.991592	+	0.129405i	$0.958693\pi$
$978$		0			0
$979$	−15.0000	−	25.9808i	−0.479402	−	0.830349i
$980$	−2.00000			−0.0638877
$981$		0			0
$982$	20.0000			0.638226
$983$	1.50000	+	2.59808i	0.0478426	+	0.0828658i	0.888955	−	0.457995i	$-0.151432\pi$
$983$	1.50000	+	2.59808i	0.0478426	+	0.0828658i	−0.841112	+	0.540860i	$0.818099\pi$
$984$		0			0
$985$	−6.00000	+	10.3923i	−0.191176	+	0.331126i
$986$	−2.00000	+	3.46410i	−0.0636930	+	0.110319i
$987$		0			0
$988$	8.00000	+	13.8564i	0.254514	+	0.440831i
$989$	24.0000			0.763156
$990$		0			0
$991$	26.0000			0.825917			0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000			0.825917			0.412959	+	0.910750i	$0.364495\pi$
$992$		0			0
$993$		0			0
$994$	3.00000	−	5.19615i	0.0951542	−	0.164812i
$995$	−2.00000	+	3.46410i	−0.0634043	+	0.109819i
$996$		0			0
$997$	9.00000	+	15.5885i	0.285033	+	0.493691i	0.972617	−	0.232413i	$-0.0746622\pi$
$997$	9.00000	+	15.5885i	0.285033	+	0.493691i	−0.687584	+	0.726105i	$0.741329\pi$
$998$	32.0000			1.01294
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.2.e.a.91.1		2
3.2	odd	2		45.2.e.a.31.1	yes	2
4.3	odd	2		2160.2.q.a.1441.1		2
5.2	odd	4		675.2.k.a.199.1		4
5.3	odd	4		675.2.k.a.199.2		4
5.4	even	2		675.2.e.a.226.1		2
9.2	odd	6		45.2.e.a.16.1	✓	2
9.4	even	3		405.2.a.b.1.1		1
9.5	odd	6		405.2.a.e.1.1		1
9.7	even	3	inner	135.2.e.a.46.1		2
12.11	even	2		720.2.q.d.481.1		2
15.2	even	4		225.2.k.a.49.2		4
15.8	even	4		225.2.k.a.49.1		4
15.14	odd	2		225.2.e.a.76.1		2
36.7	odd	6		2160.2.q.a.721.1		2
36.11	even	6		720.2.q.d.241.1		2
36.23	even	6		6480.2.a.k.1.1		1
36.31	odd	6		6480.2.a.x.1.1		1
45.2	even	12		225.2.k.a.124.1		4
45.4	even	6		2025.2.a.e.1.1		1
45.7	odd	12		675.2.k.a.424.2		4
45.13	odd	12		2025.2.b.d.649.2		2
45.14	odd	6		2025.2.a.b.1.1		1
45.22	odd	12		2025.2.b.d.649.1		2
45.23	even	12		2025.2.b.c.649.1		2
45.29	odd	6		225.2.e.a.151.1		2
45.32	even	12		2025.2.b.c.649.2		2
45.34	even	6		675.2.e.a.451.1		2
45.38	even	12		225.2.k.a.124.2		4
45.43	odd	12		675.2.k.a.424.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.a.16.1	✓	2	9.2	odd	6
45.2.e.a.31.1	yes	2	3.2	odd	2
135.2.e.a.46.1		2	9.7	even	3	inner
135.2.e.a.91.1		2	1.1	even	1	trivial
225.2.e.a.76.1		2	15.14	odd	2
225.2.e.a.151.1		2	45.29	odd	6
225.2.k.a.49.1		4	15.8	even	4
225.2.k.a.49.2		4	15.2	even	4
225.2.k.a.124.1		4	45.2	even	12
225.2.k.a.124.2		4	45.38	even	12
405.2.a.b.1.1		1	9.4	even	3
405.2.a.e.1.1		1	9.5	odd	6
675.2.e.a.226.1		2	5.4	even	2
675.2.e.a.451.1		2	45.34	even	6
675.2.k.a.199.1		4	5.2	odd	4
675.2.k.a.199.2		4	5.3	odd	4
675.2.k.a.424.1		4	45.43	odd	12
675.2.k.a.424.2		4	45.7	odd	12
720.2.q.d.241.1		2	36.11	even	6
720.2.q.d.481.1		2	12.11	even	2
2025.2.a.b.1.1		1	45.14	odd	6
2025.2.a.e.1.1		1	45.4	even	6
2025.2.b.c.649.1		2	45.23	even	12
2025.2.b.c.649.2		2	45.32	even	12
2025.2.b.d.649.1		2	45.22	odd	12
2025.2.b.d.649.2		2	45.13	odd	12
2160.2.q.a.721.1		2	36.7	odd	6
2160.2.q.a.1441.1		2	4.3	odd	2
6480.2.a.k.1.1		1	36.23	even	6
6480.2.a.x.1.1		1	36.31	odd	6

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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	135.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{7} +3.00000 q^{8} -1.00000 q^{10} +(-1.00000 - 1.73205i) q^{11} +(1.00000 - 1.73205i) q^{13} +(-1.50000 + 2.59808i) q^{14} +(0.500000 + 0.866025i) q^{16} -4.00000 q^{17} -8.00000 q^{19} +(0.500000 + 0.866025i) q^{20} +(1.00000 - 1.73205i) q^{22} +(1.50000 - 2.59808i) q^{23} +(-0.500000 - 0.866025i) q^{25} +2.00000 q^{26} +3.00000 q^{28} +(-0.500000 - 0.866025i) q^{29} +(2.50000 - 4.33013i) q^{32} +(-2.00000 - 3.46410i) q^{34} -3.00000 q^{35} -4.00000 q^{37} +(-4.00000 - 6.92820i) q^{38} +(-1.50000 + 2.59808i) q^{40} +(2.50000 - 4.33013i) q^{41} +(4.00000 + 6.92820i) q^{43} -2.00000 q^{44} +3.00000 q^{46} +(3.50000 + 6.06218i) q^{47} +(-1.00000 + 1.73205i) q^{49} +(0.500000 - 0.866025i) q^{50} +(-1.00000 - 1.73205i) q^{52} +2.00000 q^{53} +2.00000 q^{55} +(4.50000 + 7.79423i) q^{56} +(0.500000 - 0.866025i) q^{58} +(-7.00000 + 12.1244i) q^{59} +(-3.50000 - 6.06218i) q^{61} +7.00000 q^{64} +(1.00000 + 1.73205i) q^{65} +(1.50000 - 2.59808i) q^{67} +(-2.00000 + 3.46410i) q^{68} +(-1.50000 - 2.59808i) q^{70} -2.00000 q^{71} +4.00000 q^{73} +(-2.00000 - 3.46410i) q^{74} +(-4.00000 + 6.92820i) q^{76} +(3.00000 - 5.19615i) q^{77} +(3.00000 + 5.19615i) q^{79} -1.00000 q^{80} +5.00000 q^{82} +(4.50000 + 7.79423i) q^{83} +(2.00000 - 3.46410i) q^{85} +(-4.00000 + 6.92820i) q^{86} +(-3.00000 - 5.19615i) q^{88} +15.0000 q^{89} +6.00000 q^{91} +(-1.50000 - 2.59808i) q^{92} +(-3.50000 + 6.06218i) q^{94} +(4.00000 - 6.92820i) q^{95} +(-1.00000 - 1.73205i) q^{97} -2.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{4} - q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + q^2 + q^4 - q^5 + 3 * q^7 + 6 * q^8	\( 2 q + q^{2} + q^{4} - q^{5} + 3 q^{7} + 6 q^{8} - 2 q^{10} - 2 q^{11} + 2 q^{13} - 3 q^{14} + q^{16} - 8 q^{17} - 16 q^{19} + q^{20} + 2 q^{22} + 3 q^{23} - q^{25} + 4 q^{26} + 6 q^{28} - q^{29} + 5 q^{32} - 4 q^{34} - 6 q^{35} - 8 q^{37} - 8 q^{38} - 3 q^{40} + 5 q^{41} + 8 q^{43} - 4 q^{44} + 6 q^{46} + 7 q^{47} - 2 q^{49} + q^{50} - 2 q^{52} + 4 q^{53} + 4 q^{55} + 9 q^{56} + q^{58} - 14 q^{59} - 7 q^{61} + 14 q^{64} + 2 q^{65} + 3 q^{67} - 4 q^{68} - 3 q^{70} - 4 q^{71} + 8 q^{73} - 4 q^{74} - 8 q^{76} + 6 q^{77} + 6 q^{79} - 2 q^{80} + 10 q^{82} + 9 q^{83} + 4 q^{85} - 8 q^{86} - 6 q^{88} + 30 q^{89} + 12 q^{91} - 3 q^{92} - 7 q^{94} + 8 q^{95} - 2 q^{97} - 4 q^{98}+O(q^{100}) \) 2 * q + q^2 + q^4 - q^5 + 3 * q^7 + 6 * q^8 - 2 * q^10 - 2 * q^11 + 2 * q^13 - 3 * q^14 + q^16 - 8 * q^17 - 16 * q^19 + q^20 + 2 * q^22 + 3 * q^23 - q^25 + 4 * q^26 + 6 * q^28 - q^29 + 5 * q^32 - 4 * q^34 - 6 * q^35 - 8 * q^37 - 8 * q^38 - 3 * q^40 + 5 * q^41 + 8 * q^43 - 4 * q^44 + 6 * q^46 + 7 * q^47 - 2 * q^49 + q^50 - 2 * q^52 + 4 * q^53 + 4 * q^55 + 9 * q^56 + q^58 - 14 * q^59 - 7 * q^61 + 14 * q^64 + 2 * q^65 + 3 * q^67 - 4 * q^68 - 3 * q^70 - 4 * q^71 + 8 * q^73 - 4 * q^74 - 8 * q^76 + 6 * q^77 + 6 * q^79 - 2 * q^80 + 10 * q^82 + 9 * q^83 + 4 * q^85 - 8 * q^86 - 6 * q^88 + 30 * q^89 + 12 * q^91 - 3 * q^92 - 7 * q^94 + 8 * q^95 - 2 * q^97 - 4 * q^98

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