LMFDB - Embedded newform 270.2.i.a.19.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [270,2,Mod(19,270)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(270, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 3]))

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

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

chi := DirichletCharacter("270.19");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	270.i (of order $6$, 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:	$2.15596085457$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			19.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	270.19
Dual form			270.2.i.a.199.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-2.23205 + 0.133975i) q^{5} +(-0.866025 - 0.500000i) q^{7} -1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +(-1.00000 + 1.73205i) q^{11} +(-5.19615 + 3.00000i) q^{13} +(0.500000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -2.00000i q^{17} -6.00000 q^{19} +(-1.23205 - 1.86603i) q^{20} +(1.73205 - 1.00000i) q^{22} +(-0.866025 + 0.500000i) q^{23} +(4.96410 - 0.598076i) q^{25} +6.00000 q^{26} -1.00000i q^{28} +(-4.50000 + 7.79423i) q^{29} +(1.00000 + 1.73205i) q^{31} +(0.866025 - 0.500000i) q^{32} +(-1.00000 + 1.73205i) q^{34} +(2.00000 + 1.00000i) q^{35} -2.00000i q^{37} +(5.19615 + 3.00000i) q^{38} +(0.133975 + 2.23205i) q^{40} +(-5.50000 - 9.52628i) q^{41} +(3.46410 + 2.00000i) q^{43} -2.00000 q^{44} +1.00000 q^{46} +(6.06218 + 3.50000i) q^{47} +(-3.00000 - 5.19615i) q^{49} +(-4.59808 - 1.96410i) q^{50} +(-5.19615 - 3.00000i) q^{52} +(2.00000 - 4.00000i) q^{55} +(-0.500000 + 0.866025i) q^{56} +(7.79423 - 4.50000i) q^{58} +(2.00000 + 3.46410i) q^{59} +(3.50000 - 6.06218i) q^{61} -2.00000i q^{62} -1.00000 q^{64} +(11.1962 - 7.39230i) q^{65} +(9.52628 - 5.50000i) q^{67} +(1.73205 - 1.00000i) q^{68} +(-1.23205 - 1.86603i) q^{70} +6.00000 q^{71} -4.00000i q^{73} +(-1.00000 + 1.73205i) q^{74} +(-3.00000 - 5.19615i) q^{76} +(1.73205 - 1.00000i) q^{77} +(-6.00000 + 10.3923i) q^{79} +(1.00000 - 2.00000i) q^{80} +11.0000i q^{82} +(-9.52628 - 5.50000i) q^{83} +(0.267949 + 4.46410i) q^{85} +(-2.00000 - 3.46410i) q^{86} +(1.73205 + 1.00000i) q^{88} +1.00000 q^{89} +6.00000 q^{91} +(-0.866025 - 0.500000i) q^{92} +(-3.50000 - 6.06218i) q^{94} +(13.3923 - 0.803848i) q^{95} +(-6.92820 - 4.00000i) q^{97} +6.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{5} + 8 q^{10} - 4 q^{11} + 2 q^{14} - 2 q^{16} - 24 q^{19} + 2 q^{20} + 6 q^{25} + 24 q^{26} - 18 q^{29} + 4 q^{31} - 4 q^{34} + 8 q^{35} + 4 q^{40} - 22 q^{41} - 8 q^{44} + 4 q^{46}+ \cdots + 12 q^{95}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^5 + 8 * q^10 - 4 * q^11 + 2 * q^14 - 2 * q^16 - 24 * q^19 + 2 * q^20 + 6 * q^25 + 24 * q^26 - 18 * q^29 + 4 * q^31 - 4 * q^34 + 8 * q^35 + 4 * q^40 - 22 * q^41 - 8 * q^44 + 4 * q^46 - 12 * q^49 - 8 * q^50 + 8 * q^55 - 2 * q^56 + 8 * q^59 + 14 * q^61 - 4 * q^64 + 24 * q^65 + 2 * q^70 + 24 * q^71 - 4 * q^74 - 12 * q^76 - 24 * q^79 + 4 * q^80 + 8 * q^85 - 8 * q^86 + 4 * q^89 + 24 * q^91 - 14 * q^94 + 12 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/270\mathbb{Z}\right)^\times$.

$n$	$191$	$217$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$

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 $

$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i
$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$	−2.23205	+	0.133975i	−0.998203	+	0.0599153i
$5$	−2.23205	+	0.133975i	−0.998203	+	0.0599153i
$6$		0			0
$7$	−0.866025	−	0.500000i	−0.327327	−	0.188982i	0.327327	−	0.944911i	$-0.393852\pi$
$7$	−0.866025	−	0.500000i	−0.327327	−	0.188982i	−0.654654	+	0.755929i	$0.727186\pi$
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	2.00000	+	1.00000i	0.632456	+	0.316228i
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	−5.19615	+	3.00000i	−1.44115	+	0.832050i	−0.997927	−	0.0643593i	$-0.979500\pi$
$13$	−5.19615	+	3.00000i	−1.44115	+	0.832050i	−0.443227	+	0.896410i	$0.646166\pi$
$14$	0.500000	+	0.866025i	0.133631	+	0.231455i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$		−	2.00000i		−	0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$		−	2.00000i		−	0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$		0			0
$19$	−6.00000			−1.37649			−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000			−1.37649			−0.688247	+	0.725476i	$0.741620\pi$
$20$	−1.23205	−	1.86603i	−0.275495	−	0.417256i
$21$		0			0
$22$	1.73205	−	1.00000i	0.369274	−	0.213201i
$23$	−0.866025	+	0.500000i	−0.180579	+	0.104257i	−0.587565	−	0.809177i	$-0.699913\pi$
$23$	−0.866025	+	0.500000i	−0.180579	+	0.104257i	0.406986	+	0.913434i	$0.366580\pi$
$24$		0			0
$25$	4.96410	−	0.598076i	0.992820	−	0.119615i
$26$	6.00000			1.17670
$27$		0			0
$28$		−	1.00000i		−	0.188982i
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	0.0578882	+	0.998323i	$0.481563\pi$
$29$	−4.50000	+	7.79423i	−0.835629	+	1.44735i	−0.893517	+	0.449029i	$0.851770\pi$
$30$		0			0
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	0.941745	−	0.336327i	$-0.109185\pi$
$31$	1.00000	+	1.73205i	0.179605	+	0.311086i	−0.762140	+	0.647412i	$0.775851\pi$
$32$	0.866025	−	0.500000i	0.153093	−	0.0883883i
$33$		0			0
$34$	−1.00000	+	1.73205i	−0.171499	+	0.297044i
$35$	2.00000	+	1.00000i	0.338062	+	0.169031i
$36$		0			0
$37$		−	2.00000i		−	0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$		−	2.00000i		−	0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	5.19615	+	3.00000i	0.842927	+	0.486664i
$39$		0			0
$40$	0.133975	+	2.23205i	0.0211832	+	0.352918i
$41$	−5.50000	−	9.52628i	−0.858956	−	1.48775i	−0.872926	−	0.487852i	$-0.837780\pi$
$41$	−5.50000	−	9.52628i	−0.858956	−	1.48775i	0.0139704	−	0.999902i	$-0.495553\pi$
$42$		0			0
$43$	3.46410	+	2.00000i	0.528271	+	0.304997i	0.740312	−	0.672264i	$-0.234678\pi$
$43$	3.46410	+	2.00000i	0.528271	+	0.304997i	−0.212041	+	0.977261i	$0.568011\pi$
$44$	−2.00000			−0.301511
$45$		0			0
$46$	1.00000			0.147442
$47$	6.06218	+	3.50000i	0.884260	+	0.510527i	0.872060	−	0.489398i	$-0.162783\pi$
$47$	6.06218	+	3.50000i	0.884260	+	0.510527i	0.0121990	+	0.999926i	$0.496117\pi$
$48$		0			0
$49$	−3.00000	−	5.19615i	−0.428571	−	0.742307i
$50$	−4.59808	−	1.96410i	−0.650266	−	0.277766i
$51$		0			0
$52$	−5.19615	−	3.00000i	−0.720577	−	0.416025i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$	2.00000	−	4.00000i	0.269680	−	0.539360i
$56$	−0.500000	+	0.866025i	−0.0668153	+	0.115728i
$57$		0			0
$58$	7.79423	−	4.50000i	1.02343	−	0.590879i
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	0.966342	−	0.257260i	$-0.0828195\pi$
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	−0.705965	+	0.708247i	$0.749486\pi$
$60$		0			0
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	−0.550135	−	0.835076i	$-0.685424\pi$
$61$	3.50000	−	6.06218i	0.448129	−	0.776182i	0.998264	+	0.0588933i	$0.0187572\pi$
$62$		−	2.00000i		−	0.254000i
$63$		0			0
$64$	−1.00000			−0.125000
$65$	11.1962	−	7.39230i	1.38871	−	0.916903i
$66$		0			0
$67$	9.52628	−	5.50000i	1.16382	−	0.671932i	0.211604	−	0.977356i	$-0.432131\pi$
$67$	9.52628	−	5.50000i	1.16382	−	0.671932i	0.952217	+	0.305424i	$0.0987981\pi$
$68$	1.73205	−	1.00000i	0.210042	−	0.121268i
$69$		0			0
$70$	−1.23205	−	1.86603i	−0.147258	−	0.223033i
$71$	6.00000			0.712069			0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000			0.712069			0.356034	+	0.934473i	$0.384129\pi$
$72$		0			0
$73$		−	4.00000i		−	0.468165i	−0.972217	−	0.234082i	$-0.924791\pi$
$73$		−	4.00000i		−	0.468165i	0.972217	−	0.234082i	$-0.0752085\pi$
$74$	−1.00000	+	1.73205i	−0.116248	+	0.201347i
$75$		0			0
$76$	−3.00000	−	5.19615i	−0.344124	−	0.596040i
$77$	1.73205	−	1.00000i	0.197386	−	0.113961i
$78$		0			0
$79$	−6.00000	+	10.3923i	−0.675053	+	1.16923i	0.301401	+	0.953498i	$0.402546\pi$
$79$	−6.00000	+	10.3923i	−0.675053	+	1.16923i	−0.976453	+	0.215728i	$0.930788\pi$
$80$	1.00000	−	2.00000i	0.111803	−	0.223607i
$81$		0			0
$82$			11.0000i			1.21475i
$83$	−9.52628	−	5.50000i	−1.04565	−	0.603703i	−0.124218	−	0.992255i	$-0.539642\pi$
$83$	−9.52628	−	5.50000i	−1.04565	−	0.603703i	−0.921427	+	0.388552i	$0.872976\pi$
$84$		0			0
$85$	0.267949	+	4.46410i	0.0290632	+	0.484200i
$86$	−2.00000	−	3.46410i	−0.215666	−	0.373544i
$87$		0			0
$88$	1.73205	+	1.00000i	0.184637	+	0.106600i
$89$	1.00000			0.106000			0.0529999	−	0.998595i	$-0.483122\pi$
$89$	1.00000			0.106000			0.0529999	+	0.998595i	$0.483122\pi$
$90$		0			0
$91$	6.00000			0.628971
$92$	−0.866025	−	0.500000i	−0.0902894	−	0.0521286i
$93$		0			0
$94$	−3.50000	−	6.06218i	−0.360997	−	0.625266i
$95$	13.3923	−	0.803848i	1.37402	−	0.0824730i
$96$		0			0
$97$	−6.92820	−	4.00000i	−0.703452	−	0.406138i	0.105180	−	0.994453i	$-0.466458\pi$
$97$	−6.92820	−	4.00000i	−0.703452	−	0.406138i	−0.808632	+	0.588315i	$0.799792\pi$
$98$			6.00000i			0.606092i
$99$		0			0
$100$	3.00000	+	4.00000i	0.300000	+	0.400000i
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	−0.811976	−	0.583691i	$-0.801608\pi$
$101$	1.00000	−	1.73205i	0.0995037	−	0.172345i	0.911479	+	0.411346i	$0.134941\pi$
$102$		0			0
$103$	−6.92820	+	4.00000i	−0.682656	+	0.394132i	−0.800855	−	0.598858i	$-0.795621\pi$
$103$	−6.92820	+	4.00000i	−0.682656	+	0.394132i	0.118199	+	0.992990i	$0.462288\pi$
$104$	3.00000	+	5.19615i	0.294174	+	0.509525i
$105$		0			0
$106$		0			0
$107$			3.00000i			0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$			3.00000i			0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$		0			0
$109$	−7.00000			−0.670478			−0.335239	−	0.942133i	$-0.608817\pi$
$109$	−7.00000			−0.670478			−0.335239	+	0.942133i	$0.608817\pi$
$110$	−3.73205	+	2.46410i	−0.355837	+	0.234943i
$111$		0			0
$112$	0.866025	−	0.500000i	0.0818317	−	0.0472456i
$113$	−10.3923	+	6.00000i	−0.977626	+	0.564433i	−0.901553	−	0.432670i	$-0.857572\pi$
$113$	−10.3923	+	6.00000i	−0.977626	+	0.564433i	−0.0760733	+	0.997102i	$0.524238\pi$
$114$		0			0
$115$	1.86603	−	1.23205i	0.174008	−	0.114889i
$116$	−9.00000			−0.835629
$117$		0			0
$118$		−	4.00000i		−	0.368230i
$119$	−1.00000	+	1.73205i	−0.0916698	+	0.158777i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$	−6.06218	+	3.50000i	−0.548844	+	0.316875i
$123$		0			0
$124$	−1.00000	+	1.73205i	−0.0898027	+	0.155543i
$125$	−11.0000	+	2.00000i	−0.983870	+	0.178885i
$126$		0			0
$127$			19.0000i			1.68598i	0.537931	+	0.842989i	$0.319206\pi$
$127$			19.0000i			1.68598i	−0.537931	+	0.842989i	$0.680794\pi$
$128$	0.866025	+	0.500000i	0.0765466	+	0.0441942i
$129$		0			0
$130$	−13.3923	+	0.803848i	−1.17458	+	0.0705021i
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	0.999602	+	0.0281993i	$0.00897729\pi$
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	−0.475380	+	0.879781i	$0.657689\pi$
$132$		0			0
$133$	5.19615	+	3.00000i	0.450564	+	0.260133i
$134$	−11.0000			−0.950255
$135$		0			0
$136$	−2.00000			−0.171499
$137$	−10.3923	−	6.00000i	−0.887875	−	0.512615i	−0.0146279	−	0.999893i	$-0.504656\pi$
$137$	−10.3923	−	6.00000i	−0.887875	−	0.512615i	−0.873247	+	0.487278i	$0.837990\pi$
$138$		0			0
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	0.975417	+	0.220366i	$0.0707252\pi$
$139$	8.00000	+	13.8564i	0.678551	+	1.17529i	−0.296866	+	0.954919i	$0.595942\pi$
$140$	0.133975	+	2.23205i	0.0113229	+	0.188643i
$141$		0			0
$142$	−5.19615	−	3.00000i	−0.436051	−	0.251754i
$143$		−	12.0000i		−	1.00349i
$144$		0			0
$145$	9.00000	−	18.0000i	0.747409	−	1.49482i
$146$	−2.00000	+	3.46410i	−0.165521	+	0.286691i
$147$		0			0
$148$	1.73205	−	1.00000i	0.142374	−	0.0821995i
$149$	0.500000	+	0.866025i	0.0409616	+	0.0709476i	0.885779	−	0.464107i	$-0.153625\pi$
$149$	0.500000	+	0.866025i	0.0409616	+	0.0709476i	−0.844818	+	0.535054i	$0.820291\pi$
$150$		0			0
$151$	−5.00000	+	8.66025i	−0.406894	+	0.704761i	−0.994540	−	0.104357i	$-0.966722\pi$
$151$	−5.00000	+	8.66025i	−0.406894	+	0.704761i	0.587646	+	0.809118i	$0.300055\pi$
$152$			6.00000i			0.486664i
$153$		0			0
$154$	−2.00000			−0.161165
$155$	−2.46410	−	3.73205i	−0.197921	−	0.299766i
$156$		0			0
$157$	−3.46410	+	2.00000i	−0.276465	+	0.159617i	−0.631822	−	0.775113i	$-0.717693\pi$
$157$	−3.46410	+	2.00000i	−0.276465	+	0.159617i	0.355357	+	0.934731i	$0.384359\pi$
$158$	10.3923	−	6.00000i	0.826767	−	0.477334i
$159$		0			0
$160$	−1.86603	+	1.23205i	−0.147522	+	0.0974022i
$161$	1.00000			0.0788110
$162$		0			0
$163$			4.00000i			0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$			4.00000i			0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	5.50000	−	9.52628i	0.429478	−	0.743877i
$165$		0			0
$166$	5.50000	+	9.52628i	0.426883	+	0.739383i
$167$	−2.59808	+	1.50000i	−0.201045	+	0.116073i	−0.597143	−	0.802135i	$-0.703697\pi$
$167$	−2.59808	+	1.50000i	−0.201045	+	0.116073i	0.396098	+	0.918208i	$0.370364\pi$
$168$		0			0
$169$	11.5000	−	19.9186i	0.884615	−	1.53220i
$170$	2.00000	−	4.00000i	0.153393	−	0.306786i
$171$		0			0
$172$			4.00000i			0.304997i
$173$	−3.46410	−	2.00000i	−0.263371	−	0.152057i	0.362500	−	0.931984i	$-0.381923\pi$
$173$	−3.46410	−	2.00000i	−0.263371	−	0.152057i	−0.625871	+	0.779926i	$0.715256\pi$
$174$		0			0
$175$	−4.59808	−	1.96410i	−0.347582	−	0.148472i
$176$	−1.00000	−	1.73205i	−0.0753778	−	0.130558i
$177$		0			0
$178$	−0.866025	−	0.500000i	−0.0649113	−	0.0374766i
$179$	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
$181$	−13.0000			−0.966282			−0.483141	−	0.875542i	$-0.660504\pi$
$181$	−13.0000			−0.966282			−0.483141	+	0.875542i	$0.660504\pi$
$182$	−5.19615	−	3.00000i	−0.385164	−	0.222375i
$183$		0			0
$184$	0.500000	+	0.866025i	0.0368605	+	0.0638442i
$185$	0.267949	+	4.46410i	0.0197000	+	0.328207i
$186$		0			0
$187$	3.46410	+	2.00000i	0.253320	+	0.146254i
$188$			7.00000i			0.510527i
$189$		0			0
$190$	−12.0000	−	6.00000i	−0.870572	−	0.435286i
$191$	3.00000	−	5.19615i	0.217072	−	0.375980i	−0.736839	−	0.676068i	$-0.763683\pi$
$191$	3.00000	−	5.19615i	0.217072	−	0.375980i	0.953912	+	0.300088i	$0.0970159\pi$
$192$		0			0
$193$	−8.66025	+	5.00000i	−0.623379	+	0.359908i	−0.778183	−	0.628037i	$-0.783859\pi$
$193$	−8.66025	+	5.00000i	−0.623379	+	0.359908i	0.154805	+	0.987945i	$0.450525\pi$
$194$	4.00000	+	6.92820i	0.287183	+	0.497416i
$195$		0			0
$196$	3.00000	−	5.19615i	0.214286	−	0.371154i
$197$			8.00000i			0.569976i	0.958531	+	0.284988i	$0.0919897\pi$
$197$			8.00000i			0.569976i	−0.958531	+	0.284988i	$0.908010\pi$
$198$		0			0
$199$	18.0000			1.27599			0.637993	−	0.770042i	$-0.279765\pi$
$199$	18.0000			1.27599			0.637993	+	0.770042i	$0.279765\pi$
$200$	−0.598076	−	4.96410i	−0.0422904	−	0.351015i
$201$		0			0
$202$	−1.73205	+	1.00000i	−0.121867	+	0.0703598i
$203$	7.79423	−	4.50000i	0.547048	−	0.315838i
$204$		0			0
$205$	13.5526	+	20.5263i	0.946552	+	1.43362i
$206$	8.00000			0.557386
$207$		0			0
$208$		−	6.00000i		−	0.416025i
$209$	6.00000	−	10.3923i	0.415029	−	0.718851i
$210$		0			0
$211$	−9.00000	−	15.5885i	−0.619586	−	1.07315i	−0.989561	−	0.144112i	$-0.953967\pi$
$211$	−9.00000	−	15.5885i	−0.619586	−	1.07315i	0.369976	−	0.929041i	$-0.379366\pi$
$212$		0			0
$213$		0			0
$214$	1.50000	−	2.59808i	0.102538	−	0.177601i
$215$	−8.00000	−	4.00000i	−0.545595	−	0.272798i
$216$		0			0
$217$		−	2.00000i		−	0.135769i
$218$	6.06218	+	3.50000i	0.410582	+	0.237050i
$219$		0			0
$220$	4.46410	−	0.267949i	0.300970	−	0.0180651i
$221$	6.00000	+	10.3923i	0.403604	+	0.699062i
$222$		0			0
$223$	−19.9186	−	11.5000i	−1.33385	−	0.770097i	−0.347960	−	0.937509i	$-0.613126\pi$
$223$	−19.9186	−	11.5000i	−1.33385	−	0.770097i	−0.985887	+	0.167412i	$0.946459\pi$
$224$	−1.00000			−0.0668153
$225$		0			0
$226$	12.0000			0.798228
$227$	6.92820	+	4.00000i	0.459841	+	0.265489i	0.711977	−	0.702202i	$-0.247800\pi$
$227$	6.92820	+	4.00000i	0.459841	+	0.265489i	−0.252136	+	0.967692i	$0.581133\pi$
$228$		0			0
$229$	−3.50000	−	6.06218i	−0.231287	−	0.400600i	0.726900	−	0.686743i	$-0.240960\pi$
$229$	−3.50000	−	6.06218i	−0.231287	−	0.400600i	−0.958187	+	0.286143i	$0.907627\pi$
$230$	−2.23205	+	0.133975i	−0.147177	+	0.00883402i
$231$		0			0
$232$	7.79423	+	4.50000i	0.511716	+	0.295439i
$233$			10.0000i			0.655122i	0.944830	+	0.327561i	$0.106227\pi$
$233$			10.0000i			0.655122i	−0.944830	+	0.327561i	$0.893773\pi$
$234$		0			0
$235$	−14.0000	−	7.00000i	−0.913259	−	0.456630i
$236$	−2.00000	+	3.46410i	−0.130189	+	0.225494i
$237$		0			0
$238$	1.73205	−	1.00000i	0.112272	−	0.0648204i
$239$	−14.0000	−	24.2487i	−0.905585	−	1.56852i	−0.820130	−	0.572177i	$-0.806099\pi$
$239$	−14.0000	−	24.2487i	−0.905585	−	1.56852i	−0.0854543	−	0.996342i	$-0.527234\pi$
$240$		0			0
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	−0.881680	−	0.471848i	$-0.843587\pi$
$241$	−0.500000	+	0.866025i	−0.0322078	+	0.0557856i	0.849472	+	0.527633i	$0.176921\pi$
$242$		−	7.00000i		−	0.449977i
$243$		0			0
$244$	7.00000			0.448129
$245$	7.39230	+	11.1962i	0.472277	+	0.715296i
$246$		0			0
$247$	31.1769	−	18.0000i	1.98374	−	1.14531i
$248$	1.73205	−	1.00000i	0.109985	−	0.0635001i
$249$		0			0
$250$	10.5263	+	3.76795i	0.665740	+	0.238306i
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$		−	2.00000i		−	0.125739i
$254$	9.50000	−	16.4545i	0.596083	−	1.03245i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	10.3923	−	6.00000i	0.648254	−	0.374270i	−0.139533	−	0.990217i	$-0.544560\pi$
$257$	10.3923	−	6.00000i	0.648254	−	0.374270i	0.787787	+	0.615948i	$0.211227\pi$
$258$		0			0
$259$	−1.00000	+	1.73205i	−0.0621370	+	0.107624i
$260$	12.0000	+	6.00000i	0.744208	+	0.372104i
$261$		0			0
$262$		−	12.0000i		−	0.741362i
$263$	13.8564	+	8.00000i	0.854423	+	0.493301i	0.862141	−	0.506669i	$-0.169123\pi$
$263$	13.8564	+	8.00000i	0.854423	+	0.493301i	−0.00771799	+	0.999970i	$0.502457\pi$
$264$		0			0
$265$		0			0
$266$	−3.00000	−	5.19615i	−0.183942	−	0.318597i
$267$		0			0
$268$	9.52628	+	5.50000i	0.581910	+	0.335966i
$269$	3.00000			0.182913			0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000			0.182913			0.0914566	+	0.995809i	$0.470848\pi$
$270$		0			0
$271$	−14.0000			−0.850439			−0.425220	−	0.905090i	$-0.639803\pi$
$271$	−14.0000			−0.850439			−0.425220	+	0.905090i	$0.639803\pi$
$272$	1.73205	+	1.00000i	0.105021	+	0.0606339i
$273$		0			0
$274$	6.00000	+	10.3923i	0.362473	+	0.627822i
$275$	−3.92820	+	9.19615i	−0.236880	+	0.554549i
$276$		0			0
$277$	−19.0526	−	11.0000i	−1.14476	−	0.660926i	−0.197153	−	0.980373i	$-0.563170\pi$
$277$	−19.0526	−	11.0000i	−1.14476	−	0.660926i	−0.947604	+	0.319447i	$0.896503\pi$
$278$		−	16.0000i		−	0.959616i
$279$		0			0
$280$	1.00000	−	2.00000i	0.0597614	−	0.119523i
$281$	−1.50000	+	2.59808i	−0.0894825	+	0.154988i	−0.907293	−	0.420500i	$-0.861855\pi$
$281$	−1.50000	+	2.59808i	−0.0894825	+	0.154988i	0.817810	+	0.575488i	$0.195188\pi$
$282$		0			0
$283$	0.866025	−	0.500000i	0.0514799	−	0.0297219i	−0.474039	−	0.880504i	$-0.657204\pi$
$283$	0.866025	−	0.500000i	0.0514799	−	0.0297219i	0.525519	+	0.850782i	$0.323871\pi$
$284$	3.00000	+	5.19615i	0.178017	+	0.308335i
$285$		0			0
$286$	−6.00000	+	10.3923i	−0.354787	+	0.614510i
$287$			11.0000i			0.649309i
$288$		0			0
$289$	13.0000			0.764706
$290$	−16.7942	+	11.0885i	−0.986191	+	0.651137i
$291$		0			0
$292$	3.46410	−	2.00000i	0.202721	−	0.117041i
$293$	15.5885	−	9.00000i	0.910687	−	0.525786i	0.0300351	−	0.999549i	$-0.490438\pi$
$293$	15.5885	−	9.00000i	0.910687	−	0.525786i	0.880652	+	0.473763i	$0.157105\pi$
$294$		0			0
$295$	−4.92820	−	7.46410i	−0.286931	−	0.434577i
$296$	−2.00000			−0.116248
$297$		0			0
$298$		−	1.00000i		−	0.0579284i
$299$	3.00000	−	5.19615i	0.173494	−	0.300501i
$300$		0			0
$301$	−2.00000	−	3.46410i	−0.115278	−	0.199667i
$302$	8.66025	−	5.00000i	0.498342	−	0.287718i
$303$		0			0
$304$	3.00000	−	5.19615i	0.172062	−	0.298020i
$305$	−7.00000	+	14.0000i	−0.400819	+	0.801638i
$306$		0			0
$307$			9.00000i			0.513657i	0.966457	+	0.256829i	$0.0826776\pi$
$307$			9.00000i			0.513657i	−0.966457	+	0.256829i	$0.917322\pi$
$308$	1.73205	+	1.00000i	0.0986928	+	0.0569803i
$309$		0			0
$310$	0.267949	+	4.46410i	0.0152185	+	0.253544i
$311$	3.00000	+	5.19615i	0.170114	+	0.294647i	0.938460	−	0.345389i	$-0.112253\pi$
$311$	3.00000	+	5.19615i	0.170114	+	0.294647i	−0.768345	+	0.640036i	$0.778920\pi$
$312$		0			0
$313$	19.0526	+	11.0000i	1.07691	+	0.621757i	0.930062	−	0.367402i	$-0.119753\pi$
$313$	19.0526	+	11.0000i	1.07691	+	0.621757i	0.146852	+	0.989158i	$0.453086\pi$
$314$	4.00000			0.225733
$315$		0			0
$316$	−12.0000			−0.675053
$317$	1.73205	+	1.00000i	0.0972817	+	0.0561656i	0.547852	−	0.836576i	$-0.315446\pi$
$317$	1.73205	+	1.00000i	0.0972817	+	0.0561656i	−0.450570	+	0.892741i	$0.648779\pi$
$318$		0			0
$319$	−9.00000	−	15.5885i	−0.503903	−	0.872786i
$320$	2.23205	−	0.133975i	0.124775	−	0.00748941i
$321$		0			0
$322$	−0.866025	−	0.500000i	−0.0482617	−	0.0278639i
$323$			12.0000i			0.667698i
$324$		0			0
$325$	−24.0000	+	18.0000i	−1.33128	+	0.998460i
$326$	2.00000	−	3.46410i	0.110770	−	0.191859i
$327$		0			0
$328$	−9.52628	+	5.50000i	−0.526001	+	0.303687i
$329$	−3.50000	−	6.06218i	−0.192961	−	0.334219i
$330$		0			0
$331$	4.00000	−	6.92820i	0.219860	−	0.380808i	−0.734905	−	0.678170i	$-0.762773\pi$
$331$	4.00000	−	6.92820i	0.219860	−	0.380808i	0.954765	+	0.297361i	$0.0961066\pi$
$332$		−	11.0000i		−	0.603703i
$333$		0			0
$334$	3.00000			0.164153
$335$	−20.5263	+	13.5526i	−1.12147	+	0.740455i
$336$		0			0
$337$	−6.92820	+	4.00000i	−0.377403	+	0.217894i	−0.676688	−	0.736270i	$-0.736585\pi$
$337$	−6.92820	+	4.00000i	−0.377403	+	0.217894i	0.299285	+	0.954164i	$0.403252\pi$
$338$	−19.9186	+	11.5000i	−1.08343	+	0.625518i
$339$		0			0
$340$	−3.73205	+	2.46410i	−0.202399	+	0.133635i
$341$	−4.00000			−0.216612
$342$		0			0
$343$			13.0000i			0.701934i
$344$	2.00000	−	3.46410i	0.107833	−	0.186772i
$345$		0			0
$346$	2.00000	+	3.46410i	0.107521	+	0.186231i
$347$	−10.3923	+	6.00000i	−0.557888	+	0.322097i	−0.752297	−	0.658824i	$-0.771054\pi$
$347$	−10.3923	+	6.00000i	−0.557888	+	0.322097i	0.194409	+	0.980921i	$0.437721\pi$
$348$		0			0
$349$	−5.50000	+	9.52628i	−0.294408	+	0.509930i	−0.974847	−	0.222875i	$-0.928456\pi$
$349$	−5.50000	+	9.52628i	−0.294408	+	0.509930i	0.680439	+	0.732805i	$0.261789\pi$
$350$	3.00000	+	4.00000i	0.160357	+	0.213809i
$351$		0			0
$352$			2.00000i			0.106600i
$353$	−13.8564	−	8.00000i	−0.737502	−	0.425797i	0.0836583	−	0.996495i	$-0.473340\pi$
$353$	−13.8564	−	8.00000i	−0.737502	−	0.425797i	−0.821160	+	0.570697i	$0.806673\pi$
$354$		0			0
$355$	−13.3923	+	0.803848i	−0.710790	+	0.0426638i
$356$	0.500000	+	0.866025i	0.0264999	+	0.0458993i
$357$		0			0
$358$	−1.73205	−	1.00000i	−0.0915417	−	0.0528516i
$359$	30.0000			1.58334			0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000			1.58334			0.791670	+	0.610949i	$0.209212\pi$
$360$		0			0
$361$	17.0000			0.894737
$362$	11.2583	+	6.50000i	0.591725	+	0.341632i
$363$		0			0
$364$	3.00000	+	5.19615i	0.157243	+	0.272352i
$365$	0.535898	+	8.92820i	0.0280502	+	0.467324i
$366$		0			0
$367$	−13.8564	−	8.00000i	−0.723299	−	0.417597i	0.0926670	−	0.995697i	$-0.470461\pi$
$367$	−13.8564	−	8.00000i	−0.723299	−	0.417597i	−0.815966	+	0.578101i	$0.803794\pi$
$368$		−	1.00000i		−	0.0521286i
$369$		0			0
$370$	2.00000	−	4.00000i	0.103975	−	0.207950i
$371$		0			0
$372$		0			0
$373$	10.3923	−	6.00000i	0.538093	−	0.310668i	−0.206213	−	0.978507i	$-0.566114\pi$
$373$	10.3923	−	6.00000i	0.538093	−	0.310668i	0.744306	+	0.667839i	$0.232781\pi$
$374$	−2.00000	−	3.46410i	−0.103418	−	0.179124i
$375$		0			0
$376$	3.50000	−	6.06218i	0.180499	−	0.312633i
$377$		−	54.0000i		−	2.78114i
$378$		0			0
$379$	16.0000			0.821865			0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000			0.821865			0.410932	+	0.911666i	$0.365203\pi$
$380$	7.39230	+	11.1962i	0.379217	+	0.574351i
$381$		0			0
$382$	−5.19615	+	3.00000i	−0.265858	+	0.153493i
$383$	−27.7128	+	16.0000i	−1.41606	+	0.817562i	−0.995950	−	0.0899119i	$-0.971341\pi$
$383$	−27.7128	+	16.0000i	−1.41606	+	0.817562i	−0.420109	+	0.907474i	$0.638008\pi$
$384$		0			0
$385$	−3.73205	+	2.46410i	−0.190203	+	0.125582i
$386$	10.0000			0.508987
$387$		0			0
$388$		−	8.00000i		−	0.406138i
$389$	−9.50000	+	16.4545i	−0.481669	+	0.834275i	−0.999779	−	0.0210389i	$-0.993303\pi$
$389$	−9.50000	+	16.4545i	−0.481669	+	0.834275i	0.518110	+	0.855314i	$0.326636\pi$
$390$		0			0
$391$	1.00000	+	1.73205i	0.0505722	+	0.0875936i
$392$	−5.19615	+	3.00000i	−0.262445	+	0.151523i
$393$		0			0
$394$	4.00000	−	6.92820i	0.201517	−	0.349038i
$395$	12.0000	−	24.0000i	0.603786	−	1.20757i
$396$		0			0
$397$			4.00000i			0.200754i	0.994949	+	0.100377i	$0.0320049\pi$
$397$			4.00000i			0.200754i	−0.994949	+	0.100377i	$0.967995\pi$
$398$	−15.5885	−	9.00000i	−0.781379	−	0.451129i
$399$		0			0
$400$	−1.96410	+	4.59808i	−0.0982051	+	0.229904i
$401$	−5.00000	−	8.66025i	−0.249688	−	0.432472i	0.713751	−	0.700399i	$-0.246995\pi$
$401$	−5.00000	−	8.66025i	−0.249688	−	0.432472i	−0.963439	+	0.267927i	$0.913661\pi$
$402$		0			0
$403$	−10.3923	−	6.00000i	−0.517678	−	0.298881i
$404$	2.00000			0.0995037
$405$		0			0
$406$	−9.00000			−0.446663
$407$	3.46410	+	2.00000i	0.171709	+	0.0991363i
$408$		0			0
$409$	19.0000	+	32.9090i	0.939490	+	1.62724i	0.766426	+	0.642333i	$0.222033\pi$
$409$	19.0000	+	32.9090i	0.939490	+	1.62724i	0.173064	+	0.984911i	$0.444633\pi$
$410$	−1.47372	−	24.5526i	−0.0727819	−	1.21256i
$411$		0			0
$412$	−6.92820	−	4.00000i	−0.341328	−	0.197066i
$413$		−	4.00000i		−	0.196827i
$414$		0			0
$415$	22.0000	+	11.0000i	1.07994	+	0.539969i
$416$	−3.00000	+	5.19615i	−0.147087	+	0.254762i
$417$		0			0
$418$	−10.3923	+	6.00000i	−0.508304	+	0.293470i
$419$	−17.0000	−	29.4449i	−0.830504	−	1.43848i	−0.897639	−	0.440732i	$-0.854719\pi$
$419$	−17.0000	−	29.4449i	−0.830504	−	1.43848i	0.0671345	−	0.997744i	$-0.478614\pi$
$420$		0			0
$421$	−11.0000	+	19.0526i	−0.536107	+	0.928565i	0.463002	+	0.886357i	$0.346772\pi$
$421$	−11.0000	+	19.0526i	−0.536107	+	0.928565i	−0.999109	+	0.0422075i	$0.986561\pi$
$422$			18.0000i			0.876226i
$423$		0			0
$424$		0			0
$425$	−1.19615	−	9.92820i	−0.0580219	−	0.481589i
$426$		0			0
$427$	−6.06218	+	3.50000i	−0.293369	+	0.169377i
$428$	−2.59808	+	1.50000i	−0.125583	+	0.0725052i
$429$		0			0
$430$	4.92820	+	7.46410i	0.237659	+	0.359951i
$431$	−16.0000			−0.770693			−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000			−0.770693			−0.385346	+	0.922772i	$0.625918\pi$
$432$		0			0
$433$			2.00000i			0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$			2.00000i			0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	−1.00000	+	1.73205i	−0.0480015	+	0.0831411i
$435$		0			0
$436$	−3.50000	−	6.06218i	−0.167620	−	0.290326i
$437$	5.19615	−	3.00000i	0.248566	−	0.143509i
$438$		0			0
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	−0.423556	−	0.905870i	$-0.639218\pi$
$439$	12.0000	−	20.7846i	0.572729	−	0.991995i	0.996284	−	0.0861252i	$-0.0274485\pi$
$440$	−4.00000	−	2.00000i	−0.190693	−	0.0953463i
$441$		0			0
$442$		−	12.0000i		−	0.570782i
$443$	−7.79423	−	4.50000i	−0.370315	−	0.213801i	0.303281	−	0.952901i	$-0.401918\pi$
$443$	−7.79423	−	4.50000i	−0.370315	−	0.213801i	−0.673596	+	0.739100i	$0.735251\pi$
$444$		0			0
$445$	−2.23205	+	0.133975i	−0.105809	+	0.00635100i
$446$	11.5000	+	19.9186i	0.544541	+	0.943172i
$447$		0			0
$448$	0.866025	+	0.500000i	0.0409159	+	0.0236228i
$449$	18.0000			0.849473			0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000			0.849473			0.424736	+	0.905317i	$0.360367\pi$
$450$		0			0
$451$	22.0000			1.03594
$452$	−10.3923	−	6.00000i	−0.488813	−	0.282216i
$453$		0			0
$454$	−4.00000	−	6.92820i	−0.187729	−	0.325157i
$455$	−13.3923	+	0.803848i	−0.627841	+	0.0376850i
$456$		0			0
$457$	−8.66025	−	5.00000i	−0.405110	−	0.233890i	0.283577	−	0.958950i	$-0.408479\pi$
$457$	−8.66025	−	5.00000i	−0.405110	−	0.233890i	−0.688686	+	0.725059i	$0.741812\pi$
$458$			7.00000i			0.327089i
$459$		0			0
$460$	2.00000	+	1.00000i	0.0932505	+	0.0466252i
$461$	−10.5000	+	18.1865i	−0.489034	+	0.847031i	−0.999920	−	0.0126168i	$-0.995984\pi$
$461$	−10.5000	+	18.1865i	−0.489034	+	0.847031i	0.510887	+	0.859648i	$0.329317\pi$
$462$		0			0
$463$	31.1769	−	18.0000i	1.44891	−	0.836531i	0.450497	−	0.892778i	$-0.351247\pi$
$463$	31.1769	−	18.0000i	1.44891	−	0.836531i	0.998417	+	0.0562469i	$0.0179134\pi$
$464$	−4.50000	−	7.79423i	−0.208907	−	0.361838i
$465$		0			0
$466$	5.00000	−	8.66025i	0.231621	−	0.401179i
$467$			36.0000i			1.66588i	0.553362	+	0.832941i	$0.313345\pi$
$467$			36.0000i			1.66588i	−0.553362	+	0.832941i	$0.686655\pi$
$468$		0			0
$469$	−11.0000			−0.507933
$470$	8.62436	+	13.0622i	0.397812	+	0.602513i
$471$		0			0
$472$	3.46410	−	2.00000i	0.159448	−	0.0920575i
$473$	−6.92820	+	4.00000i	−0.318559	+	0.183920i
$474$		0			0
$475$	−29.7846	+	3.58846i	−1.36661	+	0.164650i
$476$	−2.00000			−0.0916698
$477$		0			0
$478$			28.0000i			1.28069i
$479$	−14.0000	+	24.2487i	−0.639676	+	1.10795i	0.345827	+	0.938298i	$0.387598\pi$
$479$	−14.0000	+	24.2487i	−0.639676	+	1.10795i	−0.985504	+	0.169654i	$0.945735\pi$
$480$		0			0
$481$	6.00000	+	10.3923i	0.273576	+	0.473848i
$482$	0.866025	−	0.500000i	0.0394464	−	0.0227744i
$483$		0			0
$484$	−3.50000	+	6.06218i	−0.159091	+	0.275554i
$485$	16.0000	+	8.00000i	0.726523	+	0.363261i
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	−6.06218	−	3.50000i	−0.274422	−	0.158438i
$489$		0			0
$490$	−0.803848	−	13.3923i	−0.0363141	−	0.605003i
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$	15.5885	+	9.00000i	0.702069	+	0.405340i
$494$	−36.0000			−1.61972
$495$		0			0
$496$	−2.00000			−0.0898027
$497$	−5.19615	−	3.00000i	−0.233079	−	0.134568i
$498$		0			0
$499$	−12.0000	−	20.7846i	−0.537194	−	0.930447i	−0.999054	−	0.0434940i	$-0.986151\pi$
$499$	−12.0000	−	20.7846i	−0.537194	−	0.930447i	0.461860	−	0.886953i	$-0.347182\pi$
$500$	−7.23205	−	8.52628i	−0.323427	−	0.381307i
$501$		0			0
$502$	15.5885	+	9.00000i	0.695747	+	0.401690i
$503$		−	27.0000i		−	1.20387i	−0.798545	−	0.601935i	$-0.794397\pi$
$503$		−	27.0000i		−	1.20387i	0.798545	−	0.601935i	$-0.205603\pi$
$504$		0			0
$505$	−2.00000	+	4.00000i	−0.0889988	+	0.177998i
$506$	−1.00000	+	1.73205i	−0.0444554	+	0.0769991i
$507$		0			0
$508$	−16.4545	+	9.50000i	−0.730050	+	0.421494i
$509$	7.50000	+	12.9904i	0.332432	+	0.575789i	0.982988	−	0.183669i	$-0.0587976\pi$
$509$	7.50000	+	12.9904i	0.332432	+	0.575789i	−0.650556	+	0.759458i	$0.725464\pi$
$510$		0			0
$511$	−2.00000	+	3.46410i	−0.0884748	+	0.153243i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−12.0000			−0.529297
$515$	14.9282	−	9.85641i	0.657815	−	0.434325i
$516$		0			0
$517$	−12.1244	+	7.00000i	−0.533229	+	0.307860i
$518$	1.73205	−	1.00000i	0.0761019	−	0.0439375i
$519$		0			0
$520$	−7.39230	−	11.1962i	−0.324174	−	0.490984i
$521$	37.0000			1.62100			0.810500	−	0.585739i	$-0.199196\pi$
$521$	37.0000			1.62100			0.810500	+	0.585739i	$0.199196\pi$
$522$		0			0
$523$		−	29.0000i		−	1.26808i	−0.773300	−	0.634041i	$-0.781395\pi$
$523$		−	29.0000i		−	1.26808i	0.773300	−	0.634041i	$-0.218605\pi$
$524$	−6.00000	+	10.3923i	−0.262111	+	0.453990i
$525$		0			0
$526$	−8.00000	−	13.8564i	−0.348817	−	0.604168i
$527$	3.46410	−	2.00000i	0.150899	−	0.0871214i
$528$		0			0
$529$	−11.0000	+	19.0526i	−0.478261	+	0.828372i
$530$		0			0
$531$		0			0
$532$			6.00000i			0.260133i
$533$	57.1577	+	33.0000i	2.47577	+	1.42939i
$534$		0			0
$535$	−0.401924	−	6.69615i	−0.0173767	−	0.289500i
$536$	−5.50000	−	9.52628i	−0.237564	−	0.411473i
$537$		0			0
$538$	−2.59808	−	1.50000i	−0.112011	−	0.0646696i
$539$	12.0000			0.516877
$540$		0			0
$541$	−17.0000			−0.730887			−0.365444	−	0.930834i	$-0.619083\pi$
$541$	−17.0000			−0.730887			−0.365444	+	0.930834i	$0.619083\pi$
$542$	12.1244	+	7.00000i	0.520786	+	0.300676i
$543$		0			0
$544$	−1.00000	−	1.73205i	−0.0428746	−	0.0742611i
$545$	15.6244	−	0.937822i	0.669274	−	0.0401719i
$546$		0			0
$547$	30.3109	+	17.5000i	1.29600	+	0.748246i	0.979711	−	0.200417i	$-0.0642296\pi$
$547$	30.3109	+	17.5000i	1.29600	+	0.748246i	0.316289	+	0.948663i	$0.397563\pi$
$548$		−	12.0000i		−	0.512615i
$549$		0			0
$550$	8.00000	−	6.00000i	0.341121	−	0.255841i
$551$	27.0000	−	46.7654i	1.15024	−	1.99227i
$552$		0			0
$553$	10.3923	−	6.00000i	0.441926	−	0.255146i
$554$	11.0000	+	19.0526i	0.467345	+	0.809466i
$555$		0			0
$556$	−8.00000	+	13.8564i	−0.339276	+	0.587643i
$557$		−	24.0000i		−	1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$		−	24.0000i		−	1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$		0			0
$559$	−24.0000			−1.01509
$560$	−1.86603	+	1.23205i	−0.0788540	+	0.0520636i
$561$		0			0
$562$	2.59808	−	1.50000i	0.109593	−	0.0632737i
$563$	32.0429	−	18.5000i	1.35045	−	0.779682i	0.362137	−	0.932125i	$-0.382047\pi$
$563$	32.0429	−	18.5000i	1.35045	−	0.779682i	0.988312	+	0.152443i	$0.0487140\pi$
$564$		0			0
$565$	22.3923	−	14.7846i	0.942051	−	0.621993i
$566$	−1.00000			−0.0420331
$567$		0			0
$568$		−	6.00000i		−	0.251754i
$569$	1.00000	−	1.73205i	0.0419222	−	0.0726113i	−0.844303	−	0.535866i	$-0.819985\pi$
$569$	1.00000	−	1.73205i	0.0419222	−	0.0726113i	0.886225	+	0.463255i	$0.153319\pi$
$570$		0			0
$571$	10.0000	+	17.3205i	0.418487	+	0.724841i	0.995788	−	0.0916910i	$-0.0292272\pi$
$571$	10.0000	+	17.3205i	0.418487	+	0.724841i	−0.577301	+	0.816532i	$0.695894\pi$
$572$	10.3923	−	6.00000i	0.434524	−	0.250873i
$573$		0			0
$574$	5.50000	−	9.52628i	0.229566	−	0.397619i
$575$	−4.00000	+	3.00000i	−0.166812	+	0.125109i
$576$		0			0
$577$		−	32.0000i		−	1.33218i	−0.745873	−	0.666089i	$-0.767967\pi$
$577$		−	32.0000i		−	1.33218i	0.745873	−	0.666089i	$-0.232033\pi$
$578$	−11.2583	−	6.50000i	−0.468285	−	0.270364i
$579$		0			0
$580$	20.0885	−	1.20577i	0.834128	−	0.0500669i
$581$	5.50000	+	9.52628i	0.228178	+	0.395217i
$582$		0			0
$583$		0			0
$584$	−4.00000			−0.165521
$585$		0			0
$586$	−18.0000			−0.743573
$587$	−2.59808	−	1.50000i	−0.107234	−	0.0619116i	0.445424	−	0.895320i	$-0.353053\pi$
$587$	−2.59808	−	1.50000i	−0.107234	−	0.0619116i	−0.552658	+	0.833408i	$0.686386\pi$
$588$		0			0
$589$	−6.00000	−	10.3923i	−0.247226	−	0.428207i
$590$	0.535898	+	8.92820i	0.0220626	+	0.367568i
$591$		0			0
$592$	1.73205	+	1.00000i	0.0711868	+	0.0410997i
$593$			30.0000i			1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$			30.0000i			1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$		0			0
$595$	2.00000	−	4.00000i	0.0819920	−	0.163984i
$596$	−0.500000	+	0.866025i	−0.0204808	+	0.0354738i
$597$		0			0
$598$	−5.19615	+	3.00000i	−0.212486	+	0.122679i
$599$	6.00000	+	10.3923i	0.245153	+	0.424618i	0.962175	−	0.272433i	$-0.0878284\pi$
$599$	6.00000	+	10.3923i	0.245153	+	0.424618i	−0.717021	+	0.697051i	$0.754495\pi$
$600$		0			0
$601$	11.0000	−	19.0526i	0.448699	−	0.777170i	−0.549602	−	0.835426i	$-0.685221\pi$
$601$	11.0000	−	19.0526i	0.448699	−	0.777170i	0.998302	+	0.0582563i	$0.0185541\pi$
$602$			4.00000i			0.163028i
$603$		0			0
$604$	−10.0000			−0.406894
$605$	−8.62436	−	13.0622i	−0.350630	−	0.531053i
$606$		0			0
$607$	−0.866025	+	0.500000i	−0.0351509	+	0.0202944i	−0.517472	−	0.855700i	$-0.673127\pi$
$607$	−0.866025	+	0.500000i	−0.0351509	+	0.0202944i	0.482322	+	0.875994i	$0.339794\pi$
$608$	−5.19615	+	3.00000i	−0.210732	+	0.121666i
$609$		0			0
$610$	13.0622	−	8.62436i	0.528872	−	0.349190i
$611$	−42.0000			−1.69914
$612$		0			0
$613$		−	34.0000i		−	1.37325i	−0.727013	−	0.686624i	$-0.759092\pi$
$613$		−	34.0000i		−	1.37325i	0.727013	−	0.686624i	$-0.240908\pi$
$614$	4.50000	−	7.79423i	0.181605	−	0.314549i
$615$		0			0
$616$	−1.00000	−	1.73205i	−0.0402911	−	0.0697863i
$617$	−27.7128	+	16.0000i	−1.11568	+	0.644136i	−0.940294	−	0.340365i	$-0.889449\pi$
$617$	−27.7128	+	16.0000i	−1.11568	+	0.644136i	−0.175382	+	0.984500i	$0.556116\pi$
$618$		0			0
$619$	−5.00000	+	8.66025i	−0.200967	+	0.348085i	−0.948840	−	0.315757i	$-0.897742\pi$
$619$	−5.00000	+	8.66025i	−0.200967	+	0.348085i	0.747873	+	0.663842i	$0.231075\pi$
$620$	2.00000	−	4.00000i	0.0803219	−	0.160644i
$621$		0			0
$622$		−	6.00000i		−	0.240578i
$623$	−0.866025	−	0.500000i	−0.0346966	−	0.0200321i
$624$		0			0
$625$	24.2846	−	5.93782i	0.971384	−	0.237513i
$626$	−11.0000	−	19.0526i	−0.439648	−	0.761493i
$627$		0			0
$628$	−3.46410	−	2.00000i	−0.138233	−	0.0798087i
$629$	−4.00000			−0.159490
$630$		0			0
$631$	8.00000			0.318475			0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000			0.318475			0.159237	+	0.987240i	$0.449096\pi$
$632$	10.3923	+	6.00000i	0.413384	+	0.238667i
$633$		0			0
$634$	−1.00000	−	1.73205i	−0.0397151	−	0.0687885i
$635$	−2.54552	−	42.4090i	−0.101016	−	1.68295i
$636$		0			0
$637$	31.1769	+	18.0000i	1.23527	+	0.713186i
$638$			18.0000i			0.712627i
$639$		0			0
$640$	−2.00000	−	1.00000i	−0.0790569	−	0.0395285i
$641$	6.50000	−	11.2583i	0.256735	−	0.444677i	−0.708631	−	0.705580i	$-0.750687\pi$
$641$	6.50000	−	11.2583i	0.256735	−	0.444677i	0.965365	+	0.260902i	$0.0840201\pi$
$642$		0			0
$643$	−28.5788	+	16.5000i	−1.12704	+	0.650696i	−0.943189	−	0.332258i	$-0.892190\pi$
$643$	−28.5788	+	16.5000i	−1.12704	+	0.650696i	−0.183851	+	0.982954i	$0.558856\pi$
$644$	0.500000	+	0.866025i	0.0197028	+	0.0341262i
$645$		0			0
$646$	6.00000	−	10.3923i	0.236067	−	0.408880i
$647$			33.0000i			1.29736i	0.761060	+	0.648682i	$0.224679\pi$
$647$			33.0000i			1.29736i	−0.761060	+	0.648682i	$0.775321\pi$
$648$		0			0
$649$	−8.00000			−0.314027
$650$	29.7846	−	3.58846i	1.16825	−	0.140751i
$651$		0			0
$652$	−3.46410	+	2.00000i	−0.135665	+	0.0783260i
$653$	22.5167	−	13.0000i	0.881145	−	0.508729i	0.0101092	−	0.999949i	$-0.496782\pi$
$653$	22.5167	−	13.0000i	0.881145	−	0.508729i	0.871036	+	0.491220i	$0.163449\pi$
$654$		0			0
$655$	−14.7846	−	22.3923i	−0.577683	−	0.874940i
$656$	11.0000			0.429478
$657$		0			0
$658$			7.00000i			0.272888i
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	−0.992388	−	0.123148i	$-0.960701\pi$
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	0.602844	+	0.797859i	$0.294034\pi$
$660$		0			0
$661$	5.00000	+	8.66025i	0.194477	+	0.336845i	0.946729	−	0.322031i	$-0.104366\pi$
$661$	5.00000	+	8.66025i	0.194477	+	0.336845i	−0.752252	+	0.658876i	$0.771032\pi$
$662$	−6.92820	+	4.00000i	−0.269272	+	0.155464i
$663$		0			0
$664$	−5.50000	+	9.52628i	−0.213441	+	0.369691i
$665$	−12.0000	−	6.00000i	−0.465340	−	0.232670i
$666$		0			0
$667$		−	9.00000i		−	0.348481i
$668$	−2.59808	−	1.50000i	−0.100523	−	0.0580367i
$669$		0			0
$670$	24.5526	−	1.47372i	0.948548	−	0.0569348i
$671$	7.00000	+	12.1244i	0.270232	+	0.468056i
$672$		0			0
$673$	−5.19615	−	3.00000i	−0.200297	−	0.115642i	0.396497	−	0.918036i	$-0.370226\pi$
$673$	−5.19615	−	3.00000i	−0.200297	−	0.115642i	−0.596794	+	0.802395i	$0.703559\pi$
$674$	8.00000			0.308148
$675$		0			0
$676$	23.0000			0.884615
$677$	−19.0526	−	11.0000i	−0.732249	−	0.422764i	0.0869952	−	0.996209i	$-0.472274\pi$
$677$	−19.0526	−	11.0000i	−0.732249	−	0.422764i	−0.819244	+	0.573444i	$0.805607\pi$
$678$		0			0
$679$	4.00000	+	6.92820i	0.153506	+	0.265880i
$680$	4.46410	−	0.267949i	0.171190	−	0.0102754i
$681$		0			0
$682$	3.46410	+	2.00000i	0.132647	+	0.0765840i
$683$		−	4.00000i		−	0.153056i	−0.997067	−	0.0765279i	$-0.975617\pi$
$683$		−	4.00000i		−	0.153056i	0.997067	−	0.0765279i	$-0.0243834\pi$
$684$		0			0
$685$	24.0000	+	12.0000i	0.916993	+	0.458496i
$686$	6.50000	−	11.2583i	0.248171	−	0.429845i
$687$		0			0
$688$	−3.46410	+	2.00000i	−0.132068	+	0.0762493i
$689$		0			0
$690$		0			0
$691$	22.0000	−	38.1051i	0.836919	−	1.44959i	−0.0555386	−	0.998457i	$-0.517688\pi$
$691$	22.0000	−	38.1051i	0.836919	−	1.44959i	0.892458	−	0.451130i	$-0.148979\pi$
$692$		−	4.00000i		−	0.152057i
$693$		0			0
$694$	12.0000			0.455514
$695$	−19.7128	−	29.8564i	−0.747750	−	1.13252i
$696$		0			0
$697$	−19.0526	+	11.0000i	−0.721667	+	0.416655i
$698$	9.52628	−	5.50000i	0.360575	−	0.208178i
$699$		0			0
$700$	−0.598076	−	4.96410i	−0.0226052	−	0.187625i
$701$	−13.0000			−0.491003			−0.245502	−	0.969396i	$-0.578953\pi$
$701$	−13.0000			−0.491003			−0.245502	+	0.969396i	$0.578953\pi$
$702$		0			0
$703$			12.0000i			0.452589i
$704$	1.00000	−	1.73205i	0.0376889	−	0.0652791i
$705$		0			0
$706$	8.00000	+	13.8564i	0.301084	+	0.521493i
$707$	−1.73205	+	1.00000i	−0.0651405	+	0.0376089i
$708$		0			0
$709$	−13.5000	+	23.3827i	−0.507003	+	0.878155i	0.492964	+	0.870050i	$0.335913\pi$
$709$	−13.5000	+	23.3827i	−0.507003	+	0.878155i	−0.999967	+	0.00810550i	$0.997420\pi$
$710$	12.0000	+	6.00000i	0.450352	+	0.225176i
$711$		0			0
$712$		−	1.00000i		−	0.0374766i
$713$	−1.73205	−	1.00000i	−0.0648658	−	0.0374503i
$714$		0			0
$715$	1.60770	+	26.7846i	0.0601244	+	1.00169i
$716$	1.00000	+	1.73205i	0.0373718	+	0.0647298i
$717$		0			0
$718$	−25.9808	−	15.0000i	−0.969593	−	0.559795i
$719$	−44.0000			−1.64092			−0.820462	−	0.571702i	$-0.806283\pi$
$719$	−44.0000			−1.64092			−0.820462	+	0.571702i	$0.806283\pi$
$720$		0			0
$721$	8.00000			0.297936
$722$	−14.7224	−	8.50000i	−0.547912	−	0.316337i
$723$		0			0
$724$	−6.50000	−	11.2583i	−0.241571	−	0.418413i
$725$	−17.6769	+	41.3827i	−0.656504	+	1.53691i
$726$		0			0
$727$	18.1865	+	10.5000i	0.674501	+	0.389423i	0.797780	−	0.602949i	$-0.206008\pi$
$727$	18.1865	+	10.5000i	0.674501	+	0.389423i	−0.123279	+	0.992372i	$0.539341\pi$
$728$		−	6.00000i		−	0.222375i
$729$		0			0
$730$	4.00000	−	8.00000i	0.148047	−	0.296093i
$731$	4.00000	−	6.92820i	0.147945	−	0.256249i
$732$		0			0
$733$	3.46410	−	2.00000i	0.127950	−	0.0738717i	−0.434659	−	0.900595i	$-0.643131\pi$
$733$	3.46410	−	2.00000i	0.127950	−	0.0738717i	0.562609	+	0.826723i	$0.309798\pi$
$734$	8.00000	+	13.8564i	0.295285	+	0.511449i
$735$		0			0
$736$	−0.500000	+	0.866025i	−0.0184302	+	0.0319221i
$737$			22.0000i			0.810380i
$738$		0			0
$739$	−40.0000			−1.47142			−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000			−1.47142			−0.735712	+	0.677295i	$0.763152\pi$
$740$	−3.73205	+	2.46410i	−0.137193	+	0.0905822i
$741$		0			0
$742$		0			0
$743$	12.9904	−	7.50000i	0.476571	−	0.275148i	−0.242415	−	0.970173i	$-0.577940\pi$
$743$	12.9904	−	7.50000i	0.476571	−	0.275148i	0.718986	+	0.695024i	$0.244606\pi$
$744$		0			0
$745$	−1.23205	−	1.86603i	−0.0451388	−	0.0683659i
$746$	−12.0000			−0.439351
$747$		0			0
$748$			4.00000i			0.146254i
$749$	1.50000	−	2.59808i	0.0548088	−	0.0949316i
$750$		0			0
$751$	−13.0000	−	22.5167i	−0.474377	−	0.821645i	0.525193	−	0.850983i	$-0.323993\pi$
$751$	−13.0000	−	22.5167i	−0.474377	−	0.821645i	−0.999570	+	0.0293387i	$0.990660\pi$
$752$	−6.06218	+	3.50000i	−0.221065	+	0.127632i
$753$		0			0
$754$	−27.0000	+	46.7654i	−0.983282	+	1.70309i
$755$	10.0000	−	20.0000i	0.363937	−	0.727875i
$756$		0			0
$757$			10.0000i			0.363456i	0.983349	+	0.181728i	$0.0581691\pi$
$757$			10.0000i			0.363456i	−0.983349	+	0.181728i	$0.941831\pi$
$758$	−13.8564	−	8.00000i	−0.503287	−	0.290573i
$759$		0			0
$760$	−0.803848	−	13.3923i	−0.0291586	−	0.485790i
$761$	−4.50000	−	7.79423i	−0.163125	−	0.282541i	0.772863	−	0.634573i	$-0.218824\pi$
$761$	−4.50000	−	7.79423i	−0.163125	−	0.282541i	−0.935988	+	0.352032i	$0.885491\pi$
$762$		0			0
$763$	6.06218	+	3.50000i	0.219466	+	0.126709i
$764$	6.00000			0.217072
$765$		0			0
$766$	32.0000			1.15621
$767$	−20.7846	−	12.0000i	−0.750489	−	0.433295i
$768$		0			0
$769$	−7.50000	−	12.9904i	−0.270457	−	0.468445i	0.698522	−	0.715589i	$-0.253841\pi$
$769$	−7.50000	−	12.9904i	−0.270457	−	0.468445i	−0.968979	+	0.247143i	$0.920508\pi$
$770$	4.46410	−	0.267949i	0.160875	−	0.00965622i
$771$		0			0
$772$	−8.66025	−	5.00000i	−0.311689	−	0.179954i
$773$		−	12.0000i		−	0.431610i	−0.976436	−	0.215805i	$-0.930762\pi$
$773$		−	12.0000i		−	0.431610i	0.976436	−	0.215805i	$-0.0692376\pi$
$774$		0			0
$775$	6.00000	+	8.00000i	0.215526	+	0.287368i
$776$	−4.00000	+	6.92820i	−0.143592	+	0.248708i
$777$		0			0
$778$	16.4545	−	9.50000i	0.589922	−	0.340592i
$779$	33.0000	+	57.1577i	1.18235	+	2.04789i
$780$		0			0
$781$	−6.00000	+	10.3923i	−0.214697	+	0.371866i
$782$		−	2.00000i		−	0.0715199i
$783$		0			0
$784$	6.00000			0.214286
$785$	7.46410	−	4.92820i	0.266405	−	0.175895i
$786$		0			0
$787$	−38.1051	+	22.0000i	−1.35830	+	0.784215i	−0.989395	−	0.145251i	$-0.953601\pi$
$787$	−38.1051	+	22.0000i	−1.35830	+	0.784215i	−0.368906	+	0.929467i	$0.620268\pi$
$788$	−6.92820	+	4.00000i	−0.246807	+	0.142494i
$789$		0			0
$790$	−22.3923	+	14.7846i	−0.796682	+	0.526013i
$791$	12.0000			0.426671
$792$		0			0
$793$			42.0000i			1.49146i
$794$	2.00000	−	3.46410i	0.0709773	−	0.122936i
$795$		0			0
$796$	9.00000	+	15.5885i	0.318997	+	0.552518i
$797$	−1.73205	+	1.00000i	−0.0613524	+	0.0354218i	−0.530362	−	0.847771i	$-0.677944\pi$
$797$	−1.73205	+	1.00000i	−0.0613524	+	0.0354218i	0.469010	+	0.883193i	$0.344611\pi$
$798$		0			0
$799$	7.00000	−	12.1244i	0.247642	−	0.428929i
$800$	4.00000	−	3.00000i	0.141421	−	0.106066i
$801$		0			0
$802$			10.0000i			0.353112i
$803$	6.92820	+	4.00000i	0.244491	+	0.141157i
$804$		0			0
$805$	−2.23205	+	0.133975i	−0.0786695	+	0.00472198i
$806$	6.00000	+	10.3923i	0.211341	+	0.366053i
$807$		0			0
$808$	−1.73205	−	1.00000i	−0.0609333	−	0.0351799i
$809$	−30.0000			−1.05474			−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000			−1.05474			−0.527372	+	0.849635i	$0.676823\pi$
$810$		0			0
$811$	−20.0000			−0.702295			−0.351147	−	0.936320i	$-0.614208\pi$
$811$	−20.0000			−0.702295			−0.351147	+	0.936320i	$0.614208\pi$
$812$	7.79423	+	4.50000i	0.273524	+	0.157919i
$813$		0			0
$814$	−2.00000	−	3.46410i	−0.0701000	−	0.121417i
$815$	−0.535898	−	8.92820i	−0.0187717	−	0.312741i
$816$		0			0
$817$	−20.7846	−	12.0000i	−0.727161	−	0.419827i
$818$		−	38.0000i		−	1.32864i
$819$		0			0
$820$	−11.0000	+	22.0000i	−0.384137	+	0.768273i
$821$	−15.5000	+	26.8468i	−0.540954	+	0.936959i	0.457896	+	0.889006i	$0.348603\pi$
$821$	−15.5000	+	26.8468i	−0.540954	+	0.936959i	−0.998850	+	0.0479535i	$0.984730\pi$
$822$		0			0
$823$	12.9904	−	7.50000i	0.452816	−	0.261434i	−0.256203	−	0.966623i	$-0.582471\pi$
$823$	12.9904	−	7.50000i	0.452816	−	0.261434i	0.709019	+	0.705190i	$0.249138\pi$
$824$	4.00000	+	6.92820i	0.139347	+	0.241355i
$825$		0			0
$826$	−2.00000	+	3.46410i	−0.0695889	+	0.120532i
$827$		−	35.0000i		−	1.21707i	−0.793527	−	0.608535i	$-0.791758\pi$
$827$		−	35.0000i		−	1.21707i	0.793527	−	0.608535i	$-0.208242\pi$
$828$		0			0
$829$	−15.0000			−0.520972			−0.260486	−	0.965478i	$-0.583883\pi$
$829$	−15.0000			−0.520972			−0.260486	+	0.965478i	$0.583883\pi$
$830$	−13.5526	−	20.5263i	−0.470416	−	0.712478i
$831$		0			0
$832$	5.19615	−	3.00000i	0.180144	−	0.104006i
$833$	−10.3923	+	6.00000i	−0.360072	+	0.207888i
$834$		0			0
$835$	5.59808	−	3.69615i	0.193729	−	0.127911i
$836$	12.0000			0.415029
$837$		0			0
$838$			34.0000i			1.17451i
$839$	−19.0000	+	32.9090i	−0.655953	+	1.13614i	0.325701	+	0.945473i	$0.394400\pi$
$839$	−19.0000	+	32.9090i	−0.655953	+	1.13614i	−0.981654	+	0.190671i	$0.938934\pi$
$840$		0			0
$841$	−26.0000	−	45.0333i	−0.896552	−	1.55287i
$842$	19.0526	−	11.0000i	0.656595	−	0.379085i
$843$		0			0
$844$	9.00000	−	15.5885i	0.309793	−	0.536577i
$845$	−23.0000	+	46.0000i	−0.791224	+	1.58245i
$846$		0			0
$847$		−	7.00000i		−	0.240523i
$848$		0			0
$849$		0			0
$850$	−3.92820	+	9.19615i	−0.134736	+	0.315425i
$851$	1.00000	+	1.73205i	0.0342796	+	0.0593739i
$852$		0			0
$853$	−39.8372	−	23.0000i	−1.36400	−	0.787505i	−0.373845	−	0.927491i	$-0.621961\pi$
$853$	−39.8372	−	23.0000i	−1.36400	−	0.787505i	−0.990153	+	0.139986i	$0.955294\pi$
$854$	7.00000			0.239535
$855$		0			0
$856$	3.00000			0.102538
$857$	15.5885	+	9.00000i	0.532492	+	0.307434i	0.742030	−	0.670366i	$-0.233863\pi$
$857$	15.5885	+	9.00000i	0.532492	+	0.307434i	−0.209539	+	0.977800i	$0.567196\pi$
$858$		0			0
$859$	7.00000	+	12.1244i	0.238837	+	0.413678i	0.960381	−	0.278691i	$-0.0899005\pi$
$859$	7.00000	+	12.1244i	0.238837	+	0.413678i	−0.721544	+	0.692369i	$0.756567\pi$
$860$	−0.535898	−	8.92820i	−0.0182740	−	0.304449i
$861$		0			0
$862$	13.8564	+	8.00000i	0.471951	+	0.272481i
$863$			45.0000i			1.53182i	0.642949	+	0.765909i	$0.277711\pi$
$863$			45.0000i			1.53182i	−0.642949	+	0.765909i	$0.722289\pi$
$864$		0			0
$865$	8.00000	+	4.00000i	0.272008	+	0.136004i
$866$	1.00000	−	1.73205i	0.0339814	−	0.0588575i
$867$		0			0
$868$	1.73205	−	1.00000i	0.0587896	−	0.0339422i
$869$	−12.0000	−	20.7846i	−0.407072	−	0.705070i
$870$		0			0
$871$	−33.0000	+	57.1577i	−1.11816	+	1.93671i
$872$			7.00000i			0.237050i
$873$		0			0
$874$	−6.00000			−0.202953
$875$	10.5263	+	3.76795i	0.355853	+	0.127380i
$876$		0			0
$877$	−39.8372	+	23.0000i	−1.34521	+	0.776655i	−0.987566	−	0.157205i	$-0.949752\pi$
$877$	−39.8372	+	23.0000i	−1.34521	+	0.776655i	−0.357640	+	0.933860i	$0.616418\pi$
$878$	−20.7846	+	12.0000i	−0.701447	+	0.404980i
$879$		0			0
$880$	2.46410	+	3.73205i	0.0830648	+	0.125807i
$881$	15.0000			0.505363			0.252681	−	0.967550i	$-0.418688\pi$
$881$	15.0000			0.505363			0.252681	+	0.967550i	$0.418688\pi$
$882$		0			0
$883$			3.00000i			0.100958i	0.998725	+	0.0504790i	$0.0160748\pi$
$883$			3.00000i			0.100958i	−0.998725	+	0.0504790i	$0.983925\pi$
$884$	−6.00000	+	10.3923i	−0.201802	+	0.349531i
$885$		0			0
$886$	4.50000	+	7.79423i	0.151180	+	0.261852i
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	9.50000	−	16.4545i	0.318620	−	0.551866i
$890$	2.00000	+	1.00000i	0.0670402	+	0.0335201i
$891$		0			0
$892$		−	23.0000i		−	0.770097i
$893$	−36.3731	−	21.0000i	−1.21718	−	0.702738i
$894$		0			0
$895$	−4.46410	+	0.267949i	−0.149218	+	0.00895655i
$896$	−0.500000	−	0.866025i	−0.0167038	−	0.0289319i
$897$		0			0
$898$	−15.5885	−	9.00000i	−0.520194	−	0.300334i
$899$	−18.0000			−0.600334
$900$		0			0
$901$		0			0
$902$	−19.0526	−	11.0000i	−0.634381	−	0.366260i
$903$		0			0
$904$	6.00000	+	10.3923i	0.199557	+	0.345643i
$905$	29.0167	−	1.74167i	0.964546	−	0.0578951i
$906$		0			0
$907$	28.5788	+	16.5000i	0.948945	+	0.547874i	0.892753	−	0.450546i	$-0.148771\pi$
$907$	28.5788	+	16.5000i	0.948945	+	0.547874i	0.0561918	+	0.998420i	$0.482104\pi$
$908$			8.00000i			0.265489i
$909$		0			0
$910$	12.0000	+	6.00000i	0.397796	+	0.198898i
$911$	6.00000	−	10.3923i	0.198789	−	0.344312i	−0.749347	−	0.662177i	$-0.769633\pi$
$911$	6.00000	−	10.3923i	0.198789	−	0.344312i	0.948136	+	0.317865i	$0.102966\pi$
$912$		0			0
$913$	19.0526	−	11.0000i	0.630548	−	0.364047i
$914$	5.00000	+	8.66025i	0.165385	+	0.286456i
$915$		0			0
$916$	3.50000	−	6.06218i	0.115643	−	0.200300i
$917$		−	12.0000i		−	0.396275i
$918$		0			0
$919$	36.0000			1.18753			0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000			1.18753			0.593765	+	0.804638i	$0.297641\pi$
$920$	−1.23205	−	1.86603i	−0.0406195	−	0.0615210i
$921$		0			0
$922$	18.1865	−	10.5000i	0.598942	−	0.345799i
$923$	−31.1769	+	18.0000i	−1.02620	+	0.592477i
$924$		0			0
$925$	−1.19615	−	9.92820i	−0.0393292	−	0.326437i
$926$	−36.0000			−1.18303
$927$		0			0
$928$			9.00000i			0.295439i
$929$	9.00000	−	15.5885i	0.295280	−	0.511441i	−0.679770	−	0.733426i	$-0.737920\pi$
$929$	9.00000	−	15.5885i	0.295280	−	0.511441i	0.975050	+	0.221985i	$0.0712536\pi$
$930$		0			0
$931$	18.0000	+	31.1769i	0.589926	+	1.02178i
$932$	−8.66025	+	5.00000i	−0.283676	+	0.163780i
$933$		0			0
$934$	18.0000	−	31.1769i	0.588978	−	1.02014i
$935$	−8.00000	−	4.00000i	−0.261628	−	0.130814i
$936$		0			0
$937$		−	52.0000i		−	1.69877i	−0.527777	−	0.849383i	$-0.676974\pi$
$937$		−	52.0000i		−	1.69877i	0.527777	−	0.849383i	$-0.323026\pi$
$938$	9.52628	+	5.50000i	0.311044	+	0.179581i
$939$		0			0
$940$	−0.937822	−	15.6244i	−0.0305884	−	0.509610i
$941$	−20.5000	−	35.5070i	−0.668281	−	1.15750i	−0.978385	−	0.206794i	$-0.933697\pi$
$941$	−20.5000	−	35.5070i	−0.668281	−	1.15750i	0.310104	−	0.950703i	$-0.399636\pi$
$942$		0			0
$943$	9.52628	+	5.50000i	0.310218	+	0.179105i
$944$	−4.00000			−0.130189
$945$		0			0
$946$	8.00000			0.260102
$947$	44.1673	+	25.5000i	1.43524	+	0.828639i	0.997514	−	0.0704677i	$-0.0224492\pi$
$947$	44.1673	+	25.5000i	1.43524	+	0.828639i	0.437730	+	0.899106i	$0.355783\pi$
$948$		0			0
$949$	12.0000	+	20.7846i	0.389536	+	0.674697i
$950$	27.5885	+	11.7846i	0.895088	+	0.382343i
$951$		0			0
$952$	1.73205	+	1.00000i	0.0561361	+	0.0324102i
$953$		−	2.00000i		−	0.0647864i	−0.999475	−	0.0323932i	$-0.989687\pi$
$953$		−	2.00000i		−	0.0647864i	0.999475	−	0.0323932i	$-0.0103129\pi$
$954$		0			0
$955$	−6.00000	+	12.0000i	−0.194155	+	0.388311i
$956$	14.0000	−	24.2487i	0.452792	−	0.784259i
$957$		0			0
$958$	24.2487	−	14.0000i	0.783440	−	0.452319i
$959$	6.00000	+	10.3923i	0.193750	+	0.335585i
$960$		0			0
$961$	13.5000	−	23.3827i	0.435484	−	0.754280i
$962$		−	12.0000i		−	0.386896i
$963$		0			0
$964$	−1.00000			−0.0322078
$965$	18.6603	−	12.3205i	0.600695	−	0.396611i
$966$		0			0
$967$	14.7224	−	8.50000i	0.473441	−	0.273342i	−0.244238	−	0.969715i	$-0.578538\pi$
$967$	14.7224	−	8.50000i	0.473441	−	0.273342i	0.717679	+	0.696374i	$0.245204\pi$
$968$	6.06218	−	3.50000i	0.194846	−	0.112494i
$969$		0			0
$970$	−9.85641	−	14.9282i	−0.316470	−	0.479316i
$971$	42.0000			1.34784			0.673922	−	0.738802i	$-0.264608\pi$
$971$	42.0000			1.34784			0.673922	+	0.738802i	$0.264608\pi$
$972$		0			0
$973$		−	16.0000i		−	0.512936i
$974$	6.00000	−	10.3923i	0.192252	−	0.332991i
$975$		0			0
$976$	3.50000	+	6.06218i	0.112032	+	0.194046i
$977$	46.7654	−	27.0000i	1.49616	−	0.863807i	0.496167	−	0.868227i	$-0.334741\pi$
$977$	46.7654	−	27.0000i	1.49616	−	0.863807i	0.999990	+	0.00442082i	$0.00140720\pi$
$978$		0			0
$979$	−1.00000	+	1.73205i	−0.0319601	+	0.0553566i
$980$	−6.00000	+	12.0000i	−0.191663	+	0.383326i
$981$		0			0
$982$		0			0
$983$	−16.4545	−	9.50000i	−0.524816	−	0.303003i	0.214087	−	0.976815i	$-0.431323\pi$
$983$	−16.4545	−	9.50000i	−0.524816	−	0.303003i	−0.738903	+	0.673812i	$0.764656\pi$
$984$		0			0
$985$	−1.07180	−	17.8564i	−0.0341503	−	0.568952i
$986$	−9.00000	−	15.5885i	−0.286618	−	0.496438i
$987$		0			0
$988$	31.1769	+	18.0000i	0.991870	+	0.572656i
$989$	−4.00000			−0.127193
$990$		0			0
$991$	4.00000			0.127064			0.0635321	−	0.997980i	$-0.479763\pi$
$991$	4.00000			0.127064			0.0635321	+	0.997980i	$0.479763\pi$
$992$	1.73205	+	1.00000i	0.0549927	+	0.0317500i
$993$		0			0
$994$	3.00000	+	5.19615i	0.0951542	+	0.164812i
$995$	−40.1769	+	2.41154i	−1.27369	+	0.0764510i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$			24.0000i			0.759707i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.2.i.a.19.1		4
3.2	odd	2		90.2.i.a.79.2	yes	4
4.3	odd	2		2160.2.by.b.289.1		4
5.2	odd	4		1350.2.e.i.451.1		2
5.3	odd	4		1350.2.e.a.451.1		2
5.4	even	2	inner	270.2.i.a.19.2		4
9.2	odd	6		810.2.c.b.649.1		2
9.4	even	3	inner	270.2.i.a.199.2		4
9.5	odd	6		90.2.i.a.49.1	✓	4
9.7	even	3		810.2.c.c.649.2		2
12.11	even	2		720.2.by.a.529.2		4
15.2	even	4		450.2.e.b.151.1		2
15.8	even	4		450.2.e.g.151.1		2
15.14	odd	2		90.2.i.a.79.1	yes	4
20.19	odd	2		2160.2.by.b.289.2		4
36.23	even	6		720.2.by.a.49.1		4
36.31	odd	6		2160.2.by.b.1009.2		4
45.2	even	12		4050.2.a.x.1.1		1
45.4	even	6	inner	270.2.i.a.199.1		4
45.7	odd	12		4050.2.a.g.1.1		1
45.13	odd	12		1350.2.e.a.901.1		2
45.14	odd	6		90.2.i.a.49.2	yes	4
45.22	odd	12		1350.2.e.i.901.1		2
45.23	even	12		450.2.e.g.301.1		2
45.29	odd	6		810.2.c.b.649.2		2
45.32	even	12		450.2.e.b.301.1		2
45.34	even	6		810.2.c.c.649.1		2
45.38	even	12		4050.2.a.j.1.1		1
45.43	odd	12		4050.2.a.be.1.1		1
60.59	even	2		720.2.by.a.529.1		4
180.59	even	6		720.2.by.a.49.2		4
180.139	odd	6		2160.2.by.b.1009.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.i.a.49.1	✓	4	9.5	odd	6
90.2.i.a.49.2	yes	4	45.14	odd	6
90.2.i.a.79.1	yes	4	15.14	odd	2
90.2.i.a.79.2	yes	4	3.2	odd	2
270.2.i.a.19.1		4	1.1	even	1	trivial
270.2.i.a.19.2		4	5.4	even	2	inner
270.2.i.a.199.1		4	45.4	even	6	inner
270.2.i.a.199.2		4	9.4	even	3	inner
450.2.e.b.151.1		2	15.2	even	4
450.2.e.b.301.1		2	45.32	even	12
450.2.e.g.151.1		2	15.8	even	4
450.2.e.g.301.1		2	45.23	even	12
720.2.by.a.49.1		4	36.23	even	6
720.2.by.a.49.2		4	180.59	even	6
720.2.by.a.529.1		4	60.59	even	2
720.2.by.a.529.2		4	12.11	even	2
810.2.c.b.649.1		2	9.2	odd	6
810.2.c.b.649.2		2	45.29	odd	6
810.2.c.c.649.1		2	45.34	even	6
810.2.c.c.649.2		2	9.7	even	3
1350.2.e.a.451.1		2	5.3	odd	4
1350.2.e.a.901.1		2	45.13	odd	12
1350.2.e.i.451.1		2	5.2	odd	4
1350.2.e.i.901.1		2	45.22	odd	12
2160.2.by.b.289.1		4	4.3	odd	2
2160.2.by.b.289.2		4	20.19	odd	2
2160.2.by.b.1009.1		4	180.139	odd	6
2160.2.by.b.1009.2		4	36.31	odd	6
4050.2.a.g.1.1		1	45.7	odd	12
4050.2.a.j.1.1		1	45.38	even	12
4050.2.a.x.1.1		1	45.2	even	12
4050.2.a.be.1.1		1	45.43	odd	12

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:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	270.i (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-2.23205 + 0.133975i) q^{5} +(-0.866025 - 0.500000i) q^{7} -1.00000i q^{8} +(2.00000 + 1.00000i) q^{10} +(-1.00000 + 1.73205i) q^{11} +(-5.19615 + 3.00000i) q^{13} +(0.500000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -2.00000i q^{17} -6.00000 q^{19} +(-1.23205 - 1.86603i) q^{20} +(1.73205 - 1.00000i) q^{22} +(-0.866025 + 0.500000i) q^{23} +(4.96410 - 0.598076i) q^{25} +6.00000 q^{26} -1.00000i q^{28} +(-4.50000 + 7.79423i) q^{29} +(1.00000 + 1.73205i) q^{31} +(0.866025 - 0.500000i) q^{32} +(-1.00000 + 1.73205i) q^{34} +(2.00000 + 1.00000i) q^{35} -2.00000i q^{37} +(5.19615 + 3.00000i) q^{38} +(0.133975 + 2.23205i) q^{40} +(-5.50000 - 9.52628i) q^{41} +(3.46410 + 2.00000i) q^{43} -2.00000 q^{44} +1.00000 q^{46} +(6.06218 + 3.50000i) q^{47} +(-3.00000 - 5.19615i) q^{49} +(-4.59808 - 1.96410i) q^{50} +(-5.19615 - 3.00000i) q^{52} +(2.00000 - 4.00000i) q^{55} +(-0.500000 + 0.866025i) q^{56} +(7.79423 - 4.50000i) q^{58} +(2.00000 + 3.46410i) q^{59} +(3.50000 - 6.06218i) q^{61} -2.00000i q^{62} -1.00000 q^{64} +(11.1962 - 7.39230i) q^{65} +(9.52628 - 5.50000i) q^{67} +(1.73205 - 1.00000i) q^{68} +(-1.23205 - 1.86603i) q^{70} +6.00000 q^{71} -4.00000i q^{73} +(-1.00000 + 1.73205i) q^{74} +(-3.00000 - 5.19615i) q^{76} +(1.73205 - 1.00000i) q^{77} +(-6.00000 + 10.3923i) q^{79} +(1.00000 - 2.00000i) q^{80} +11.0000i q^{82} +(-9.52628 - 5.50000i) q^{83} +(0.267949 + 4.46410i) q^{85} +(-2.00000 - 3.46410i) q^{86} +(1.73205 + 1.00000i) q^{88} +1.00000 q^{89} +6.00000 q^{91} +(-0.866025 - 0.500000i) q^{92} +(-3.50000 - 6.06218i) q^{94} +(13.3923 - 0.803848i) q^{95} +(-6.92820 - 4.00000i) q^{97} +6.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{5} + 8 q^{10} - 4 q^{11} + 2 q^{14} - 2 q^{16} - 24 q^{19} + 2 q^{20} + 6 q^{25} + 24 q^{26} - 18 q^{29} + 4 q^{31} - 4 q^{34} + 8 q^{35} + 4 q^{40} - 22 q^{41} - 8 q^{44} + 4 q^{46}+ \cdots + 12 q^{95}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^5 + 8 * q^10 - 4 * q^11 + 2 * q^14 - 2 * q^16 - 24 * q^19 + 2 * q^20 + 6 * q^25 + 24 * q^26 - 18 * q^29 + 4 * q^31 - 4 * q^34 + 8 * q^35 + 4 * q^40 - 22 * q^41 - 8 * q^44 + 4 * q^46 - 12 * q^49 - 8 * q^50 + 8 * q^55 - 2 * q^56 + 8 * q^59 + 14 * q^61 - 4 * q^64 + 24 * q^65 + 2 * q^70 + 24 * q^71 - 4 * q^74 - 12 * q^76 - 24 * q^79 + 4 * q^80 + 8 * q^85 - 8 * q^86 + 4 * q^89 + 24 * q^91 - 14 * q^94 + 12 * q^95

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