LMFDB - Embedded newform 2808.2.cw.b.2521.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2808,2,Mod(1585,2808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2808, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2808.1585"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2808 = 2^{3} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2808.cw (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$22.4219928876$
Analytic rank:	$0$
Dimension:	$80$
Relative dimension:	$40$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2521.19
Character	$\chi$	$=$	2808.2521
Dual form			2808.2.cw.b.1585.19

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.107804 - 0.0622405i) q^{5} +(1.23873 - 0.715181i) q^{7} +(4.42238 - 2.55326i) q^{11} +(-1.32079 - 3.35493i) q^{13} +5.55695 q^{17} +6.33272i q^{19} +(0.310585 - 0.537949i) q^{23} +(-2.49225 - 4.31671i) q^{25} +(2.82232 + 4.88840i) q^{29} +(0.0848405 + 0.0489827i) q^{31} -0.178053 q^{35} -2.19755i q^{37} +(-3.86373 - 2.23073i) q^{41} +(-3.36274 - 5.82443i) q^{43} +(1.69881 - 0.980810i) q^{47} +(-2.47703 + 4.29034i) q^{49} +8.47447 q^{53} -0.635665 q^{55} +(-5.54284 - 3.20016i) q^{59} +(1.96107 + 3.39667i) q^{61} +(-0.0664265 + 0.443880i) q^{65} +(-10.5326 - 6.08102i) q^{67} -11.5162i q^{71} -7.57155i q^{73} +(3.65209 - 6.32560i) q^{77} +(6.67133 + 11.5551i) q^{79} +(5.92857 - 3.42286i) q^{83} +(-0.599059 - 0.345867i) q^{85} -4.80570i q^{89} +(-4.03548 - 3.21125i) q^{91} +(0.394152 - 0.682691i) q^{95} +(-1.15415 + 0.666350i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 80 q + 6 q^{13} + 4 q^{17} - 10 q^{23} + 44 q^{25} + 52 q^{35} - 26 q^{43} + 48 q^{49} - 60 q^{53} - 16 q^{55} - 10 q^{61} + 26 q^{65} - 32 q^{77} + 6 q^{79} - 4 q^{91} - 8 q^{95}+O(q^{100}) $ 80 * q + 6 * q^13 + 4 * q^17 - 10 * q^23 + 44 * q^25 + 52 * q^35 - 26 * q^43 + 48 * q^49 - 60 * q^53 - 16 * q^55 - 10 * q^61 + 26 * q^65 - 32 * q^77 + 6 * q^79 - 4 * q^91 - 8 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2808\mathbb{Z}\right)^\times$.

$n$	$703$	$1081$	$1405$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.107804	−	0.0622405i	−0.0482113	−	0.0278348i	0.475701	−	0.879607i	$-0.342195\pi$
$5$	−0.107804	−	0.0622405i	−0.0482113	−	0.0278348i	−0.523912	+	0.851772i	$0.675528\pi$
$6$		0			0
$7$	1.23873	−	0.715181i	0.468196	−	0.270313i	−0.247288	−	0.968942i	$-0.579539\pi$
$7$	1.23873	−	0.715181i	0.468196	−	0.270313i	0.715484	+	0.698629i	$0.246206\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.42238	−	2.55326i	1.33340	−	0.769837i	0.347578	−	0.937651i	$-0.387004\pi$
$11$	4.42238	−	2.55326i	1.33340	−	0.769837i	0.985819	+	0.167814i	$0.0536708\pi$
$12$		0			0
$13$	−1.32079	−	3.35493i	−0.366320	−	0.930489i
$13$	−1.32079	−	3.35493i	−0.366320	−	0.930489i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.55695			1.34776			0.673879	−	0.738842i	$-0.264627\pi$
$17$	5.55695			1.34776			0.673879	+	0.738842i	$0.264627\pi$
$18$		0			0
$19$			6.33272i			1.45283i	0.687258	+	0.726413i	$0.258814\pi$
$19$			6.33272i			1.45283i	−0.687258	+	0.726413i	$0.741186\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.310585	−	0.537949i	0.0647615	−	0.112170i	−0.831827	−	0.555035i	$-0.812705\pi$
$23$	0.310585	−	0.537949i	0.0647615	−	0.112170i	0.896588	+	0.442865i	$0.146038\pi$
$24$		0			0
$25$	−2.49225	−	4.31671i	−0.498450	−	0.863342i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.82232	+	4.88840i	0.524092	+	0.907754i	0.999607	+	0.0280461i	$0.00892853\pi$
$29$	2.82232	+	4.88840i	0.524092	+	0.907754i	−0.475515	+	0.879708i	$0.657738\pi$
$30$		0			0
$31$	0.0848405	+	0.0489827i	0.0152378	+	0.00879755i	0.507600	−	0.861593i	$-0.330533\pi$
$31$	0.0848405	+	0.0489827i	0.0152378	+	0.00879755i	−0.492362	+	0.870391i	$0.663866\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.178053			−0.0300964
$36$		0			0
$37$		−	2.19755i		−	0.361276i	−0.983550	−	0.180638i	$-0.942184\pi$
$37$		−	2.19755i		−	0.361276i	0.983550	−	0.180638i	$-0.0578162\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−3.86373	−	2.23073i	−0.603413	−	0.348381i	0.166970	−	0.985962i	$-0.446602\pi$
$41$	−3.86373	−	2.23073i	−0.603413	−	0.348381i	−0.770383	+	0.637581i	$0.779935\pi$
$42$		0			0
$43$	−3.36274	−	5.82443i	−0.512813	−	0.888218i	−0.999890	−	0.0148587i	$-0.995270\pi$
$43$	−3.36274	−	5.82443i	−0.512813	−	0.888218i	0.487077	−	0.873359i	$-0.338063\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.69881	−	0.980810i	0.247797	−	0.143066i	−0.370958	−	0.928650i	$-0.620971\pi$
$47$	1.69881	−	0.980810i	0.247797	−	0.143066i	0.618755	+	0.785584i	$0.287637\pi$
$48$		0			0
$49$	−2.47703	+	4.29034i	−0.353862	+	0.612906i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	8.47447			1.16406			0.582029	−	0.813168i	$-0.302259\pi$
$53$	8.47447			1.16406			0.582029	+	0.813168i	$0.302259\pi$
$54$		0			0
$55$	−0.635665			−0.0857130
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−5.54284	−	3.20016i	−0.721616	−	0.416625i	0.0937314	−	0.995598i	$-0.470121\pi$
$59$	−5.54284	−	3.20016i	−0.721616	−	0.416625i	−0.815347	+	0.578973i	$0.803454\pi$
$60$		0			0
$61$	1.96107	+	3.39667i	0.251089	+	0.434899i	0.963826	−	0.266533i	$-0.0858780\pi$
$61$	1.96107	+	3.39667i	0.251089	+	0.434899i	−0.712737	+	0.701431i	$0.752545\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.0664265	+	0.443880i	−0.00823920	+	0.0550565i
$66$		0			0
$67$	−10.5326	−	6.08102i	−1.28677	−	0.742915i	−0.308690	−	0.951163i	$-0.599891\pi$
$67$	−10.5326	−	6.08102i	−1.28677	−	0.742915i	−0.978076	+	0.208248i	$0.933224\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.5162i		−	1.36672i	−0.730083	−	0.683359i	$-0.760519\pi$
$71$		−	11.5162i		−	1.36672i	0.730083	−	0.683359i	$-0.239481\pi$
$72$		0			0
$73$		−	7.57155i		−	0.886183i	−0.896476	−	0.443092i	$-0.853882\pi$
$73$		−	7.57155i		−	0.886183i	0.896476	−	0.443092i	$-0.146118\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.65209	−	6.32560i	0.416194	−	0.720869i
$78$		0			0
$79$	6.67133	+	11.5551i	0.750584	+	1.30005i	0.947540	+	0.319637i	$0.103561\pi$
$79$	6.67133	+	11.5551i	0.750584	+	1.30005i	−0.196956	+	0.980412i	$0.563106\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	5.92857	−	3.42286i	0.650745	−	0.375708i	−0.137997	−	0.990433i	$-0.544066\pi$
$83$	5.92857	−	3.42286i	0.650745	−	0.375708i	0.788741	+	0.614725i	$0.210733\pi$
$84$		0			0
$85$	−0.599059	−	0.345867i	−0.0649771	−	0.0375145i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	4.80570i		−	0.509403i	−0.967020	−	0.254701i	$-0.918023\pi$
$89$		−	4.80570i		−	0.509403i	0.967020	−	0.254701i	$-0.0819772\pi$
$90$		0			0
$91$	−4.03548	−	3.21125i	−0.423033	−	0.336630i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.394152	−	0.682691i	0.0404391	−	0.0700426i
$96$		0			0
$97$	−1.15415	+	0.666350i	−0.117186	+	0.0676576i	−0.557447	−	0.830212i	$-0.688219\pi$
$97$	−1.15415	+	0.666350i	−0.117186	+	0.0676576i	0.440261	+	0.897870i	$0.354886\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.10949	+	8.84989i	0.508413	+	0.880597i	0.999953	+	0.00974180i	$0.00310096\pi$
$101$	5.10949	+	8.84989i	0.508413	+	0.880597i	−0.491540	+	0.870855i	$0.663566\pi$
$102$		0			0
$103$	9.07403	−	15.7167i	0.894091	−	1.54861i	0.0591648	−	0.998248i	$-0.481156\pi$
$103$	9.07403	−	15.7167i	0.894091	−	1.54861i	0.834926	−	0.550362i	$-0.185510\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	17.2761			1.67014			0.835072	−	0.550140i	$-0.185426\pi$
$107$	17.2761			1.67014			0.835072	+	0.550140i	$0.185426\pi$
$108$		0			0
$109$			10.1231i			0.969618i	0.874620	+	0.484809i	$0.161111\pi$
$109$			10.1231i			0.969618i	−0.874620	+	0.484809i	$0.838889\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−6.85914	+	11.8804i	−0.645253	+	1.11761i	0.338990	+	0.940790i	$0.389915\pi$
$113$	−6.85914	+	11.8804i	−0.645253	+	1.11761i	−0.984243	+	0.176821i	$0.943418\pi$
$114$		0			0
$115$	−0.0669644	+	0.0386619i	−0.00624447	+	0.00360524i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.88356	−	3.97423i	0.631015	−	0.364317i
$120$		0			0
$121$	7.53828	−	13.0567i	0.685298	−	1.18697i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1.24288i			0.111167i
$126$		0			0
$127$	13.3132			1.18136			0.590680	−	0.806906i	$-0.298860\pi$
$127$	13.3132			1.18136			0.590680	+	0.806906i	$0.298860\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.984731	−	1.70560i	0.0860363	−	0.149019i	−0.819796	−	0.572656i	$-0.805913\pi$
$131$	0.984731	−	1.70560i	0.0860363	−	0.149019i	0.905832	+	0.423636i	$0.139247\pi$
$132$		0			0
$133$	4.52905	+	7.84454i	0.392718	+	0.680208i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−14.0419	+	8.10710i	−1.19968	+	0.692636i	−0.960484	−	0.278335i	$-0.910218\pi$
$137$	−14.0419	+	8.10710i	−1.19968	+	0.692636i	−0.239197	+	0.970971i	$0.576884\pi$
$138$		0			0
$139$	8.10810	−	14.0436i	0.687720	−	1.19117i	−0.284853	−	0.958571i	$-0.591945\pi$
$139$	8.10810	−	14.0436i	0.687720	−	1.19117i	0.972574	−	0.232595i	$-0.0747218\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−14.4070	−	11.4644i	−1.20477	−	0.958704i
$144$		0			0
$145$		−	0.702651i		−	0.0583520i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	15.5377	+	8.97068i	1.27290	+	0.734907i	0.975532	−	0.219857i	$-0.0705592\pi$
$149$	15.5377	+	8.97068i	1.27290	+	0.734907i	0.297364	+	0.954764i	$0.403893\pi$
$150$		0			0
$151$	5.26643	−	3.04058i	0.428576	−	0.247439i	−0.270164	−	0.962814i	$-0.587078\pi$
$151$	5.26643	−	3.04058i	0.428576	−	0.247439i	0.698740	+	0.715376i	$0.253744\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.00609741	−	0.0105610i	−0.000489756	−	0.000848282i
$156$		0			0
$157$	4.96233	−	8.59500i	0.396037	−	0.685956i	−0.597196	−	0.802095i	$-0.703719\pi$
$157$	4.96233	−	8.59500i	0.396037	−	0.685956i	0.993233	+	0.116140i	$0.0370520\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	0.888499i		−	0.0700235i
$162$		0			0
$163$		−	12.4422i		−	0.974545i	−0.873250	−	0.487273i	$-0.837992\pi$
$163$		−	12.4422i		−	0.974545i	0.873250	−	0.487273i	$-0.162008\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	18.3062	+	10.5691i	1.41657	+	0.817859i	0.995996	−	0.0893970i	$-0.0284940\pi$
$167$	18.3062	+	10.5691i	1.41657	+	0.817859i	0.420578	+	0.907256i	$0.361827\pi$
$168$		0			0
$169$	−9.51105	+	8.86228i	−0.731619	+	0.681714i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.05528	−	7.02395i	−0.308317	−	0.534021i	0.669677	−	0.742652i	$-0.266432\pi$
$173$	−4.05528	−	7.02395i	−0.308317	−	0.534021i	−0.977994	+	0.208631i	$0.933099\pi$
$174$		0			0
$175$	−6.17446	−	3.56482i	−0.466745	−	0.269475i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.534744			−0.0399687			−0.0199843	−	0.999800i	$-0.506362\pi$
$179$	−0.534744			−0.0399687			−0.0199843	+	0.999800i	$0.506362\pi$
$180$		0			0
$181$	3.02076			0.224531			0.112266	−	0.993678i	$-0.464189\pi$
$181$	3.02076			0.224531			0.112266	+	0.993678i	$0.464189\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.136777	+	0.236904i	−0.0100560	+	0.0174176i
$186$		0			0
$187$	24.5749	−	14.1883i	1.79710	−	1.03755i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.93227	−	3.34679i	−0.139814	−	0.242165i	0.787612	−	0.616171i	$-0.211317\pi$
$191$	−1.93227	−	3.34679i	−0.139814	−	0.242165i	−0.927426	+	0.374006i	$0.877984\pi$
$192$		0			0
$193$	−17.1010	−	9.87328i	−1.23096	−	0.710694i	−0.263729	−	0.964597i	$-0.584952\pi$
$193$	−17.1010	−	9.87328i	−1.23096	−	0.710694i	−0.967230	+	0.253903i	$0.918286\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			8.73846i			0.622589i	0.950313	+	0.311295i	$0.100763\pi$
$197$			8.73846i			0.622589i	−0.950313	+	0.311295i	$0.899237\pi$
$198$		0			0
$199$	−13.5343			−0.959423			−0.479712	−	0.877426i	$-0.659259\pi$
$199$	−13.5343			−0.959423			−0.479712	+	0.877426i	$0.659259\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	6.99219	+	4.03694i	0.490756	+	0.283338i
$204$		0			0
$205$	0.277683	+	0.480961i	0.0193942	+	0.0335918i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	16.1691	+	28.0057i	1.11844	+	1.93719i
$210$		0			0
$211$	−3.73277	+	6.46534i	−0.256974	+	0.445092i	−0.965430	−	0.260663i	$-0.916059\pi$
$211$	−3.73277	+	6.46534i	−0.256974	+	0.445092i	0.708456	+	0.705755i	$0.249392\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			0.837194i			0.0570961i
$216$		0			0
$217$	0.140126			0.00951237
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.33954	−	18.6431i	−0.493711	−	1.25407i
$222$		0			0
$223$	−10.8722	+	6.27709i	−0.728059	+	0.420345i	−0.817712	−	0.575628i	$-0.804758\pi$
$223$	−10.8722	+	6.27709i	−0.728059	+	0.420345i	0.0896528	+	0.995973i	$0.471424\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.2633	−	6.50285i	0.747569	−	0.431609i	−0.0772456	−	0.997012i	$-0.524613\pi$
$227$	11.2633	−	6.50285i	0.747569	−	0.431609i	0.824815	+	0.565403i	$0.191279\pi$
$228$		0			0
$229$	−25.5552	−	14.7543i	−1.68874	−	0.974992i	−0.955488	−	0.295031i	$-0.904670\pi$
$229$	−25.5552	−	14.7543i	−1.68874	−	0.974992i	−0.733248	−	0.679961i	$-0.761997\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−10.1835			−0.667145			−0.333573	−	0.942724i	$-0.608254\pi$
$233$	−10.1835			−0.667145			−0.333573	+	0.942724i	$0.608254\pi$
$234$		0			0
$235$	−0.244184			−0.0159288
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−24.6974	−	14.2591i	−1.59755	−	0.922343i	−0.991958	−	0.126564i	$-0.959605\pi$
$239$	−24.6974	−	14.2591i	−1.59755	−	0.922343i	−0.605587	−	0.795779i	$-0.707061\pi$
$240$		0			0
$241$	23.4478	−	13.5376i	1.51040	−	0.872031i	0.510476	−	0.859892i	$-0.329469\pi$
$241$	23.4478	−	13.5376i	1.51040	−	0.872031i	0.999926	−	0.0121394i	$-0.00386420\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.534066	−	0.308343i	0.0341202	−	0.0196993i
$246$		0			0
$247$	21.2458	−	8.36418i	1.35184	−	0.532200i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.75511			−0.300140			−0.150070	−	0.988675i	$-0.547950\pi$
$251$	−4.75511			−0.300140			−0.150070	+	0.988675i	$0.547950\pi$
$252$		0			0
$253$		−	3.17202i		−	0.199423i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	7.31065	−	12.6624i	0.456026	−	0.789860i	−0.542720	−	0.839913i	$-0.682606\pi$
$257$	7.31065	−	12.6624i	0.456026	−	0.789860i	0.998747	+	0.0500530i	$0.0159390\pi$
$258$		0			0
$259$	−1.57165	−	2.72218i	−0.0976575	−	0.169148i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.59569	−	11.4241i	−0.406708	−	0.704438i	0.587811	−	0.808998i	$-0.299990\pi$
$263$	−6.59569	−	11.4241i	−0.406708	−	0.704438i	−0.994519	+	0.104560i	$0.966657\pi$
$264$		0			0
$265$	−0.913579	−	0.527455i	−0.0561207	−	0.0324013i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.7284			0.654124			0.327062	−	0.945003i	$-0.393941\pi$
$269$	10.7284			0.654124			0.327062	+	0.945003i	$0.393941\pi$
$270$		0			0
$271$		−	6.83605i		−	0.415261i	−0.978207	−	0.207630i	$-0.933425\pi$
$271$		−	6.83605i		−	0.415261i	0.978207	−	0.207630i	$-0.0665751\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−22.0434	−	12.7267i	−1.32926	−	0.767451i
$276$		0			0
$277$	16.1542	+	27.9798i	0.970609	+	1.68114i	0.693724	+	0.720241i	$0.255969\pi$
$277$	16.1542	+	27.9798i	0.970609	+	1.68114i	0.276885	+	0.960903i	$0.410698\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.2814	+	9.40008i	−0.971267	+	0.560761i	−0.899622	−	0.436669i	$-0.856158\pi$
$281$	−16.2814	+	9.40008i	−0.971267	+	0.560761i	−0.0716450	+	0.997430i	$0.522825\pi$
$282$		0			0
$283$	0.598221	−	1.03615i	0.0355606	−	0.0615927i	−0.847697	−	0.530480i	$-0.822012\pi$
$283$	0.598221	−	1.03615i	0.0355606	−	0.0615927i	0.883258	+	0.468887i	$0.155345\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−6.38149			−0.376688
$288$		0			0
$289$	13.8797			0.816451
$290$		0			0
$291$		0			0
$292$		0			0
$293$	23.9635	+	13.8353i	1.39996	+	0.808269i	0.994388	−	0.105793i	$-0.0337382\pi$
$293$	23.9635	+	13.8353i	1.39996	+	0.808269i	0.405574	+	0.914062i	$0.367071\pi$
$294$		0			0
$295$	0.398359	+	0.689977i	0.0231933	+	0.0401720i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−2.21500	−	0.331474i	−0.128097	−	0.0191696i
$300$		0			0
$301$	−8.33105	−	4.80994i	−0.480194	−	0.277240i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	0.488231i		−	0.0279560i
$306$		0			0
$307$			9.49426i			0.541866i	0.962598	+	0.270933i	$0.0873322\pi$
$307$			9.49426i			0.541866i	−0.962598	+	0.270933i	$0.912668\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−10.7524	+	18.6237i	−0.609713	+	1.05605i	0.381574	+	0.924338i	$0.375382\pi$
$311$	−10.7524	+	18.6237i	−0.609713	+	1.05605i	−0.991288	+	0.131716i	$0.957951\pi$
$312$		0			0
$313$	3.05449	+	5.29053i	0.172650	+	0.299038i	0.939345	−	0.342973i	$-0.111434\pi$
$313$	3.05449	+	5.29053i	0.172650	+	0.299038i	−0.766696	+	0.642011i	$0.778100\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−5.09446	+	2.94129i	−0.286133	+	0.165199i	−0.636197	−	0.771527i	$-0.719493\pi$
$317$	−5.09446	+	2.94129i	−0.286133	+	0.165199i	0.350064	+	0.936726i	$0.386160\pi$
$318$		0			0
$319$	24.9627	+	14.4122i	1.39765	+	0.806931i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			35.1906i			1.95806i
$324$		0			0
$325$	−11.1905	+	14.0628i	−0.620737	+	0.780062i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.40291	−	2.42992i	0.0773452	−	0.133966i
$330$		0			0
$331$	7.17112	−	4.14025i	0.394160	−	0.227568i	−0.289801	−	0.957087i	$-0.593589\pi$
$331$	7.17112	−	4.14025i	0.394160	−	0.227568i	0.683961	+	0.729518i	$0.260256\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	0.756971	+	1.31111i	0.0413578	+	0.0716337i
$336$		0			0
$337$	−9.29585	+	16.1009i	−0.506377	+	0.877071i	0.493595	+	0.869692i	$0.335682\pi$
$337$	−9.29585	+	16.1009i	−0.506377	+	0.877071i	−0.999973	+	0.00737973i	$0.997651\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.500262			0.0270907
$342$		0			0
$343$			17.0986i			0.923240i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−3.79875	+	6.57964i	−0.203928	+	0.353213i	−0.949791	−	0.312886i	$-0.898704\pi$
$347$	−3.79875	+	6.57964i	−0.203928	+	0.353213i	0.745863	+	0.666100i	$0.232037\pi$
$348$		0			0
$349$	−13.5732	+	7.83649i	−0.726556	+	0.419478i	−0.817161	−	0.576409i	$-0.804453\pi$
$349$	−13.5732	+	7.83649i	−0.726556	+	0.419478i	0.0906046	+	0.995887i	$0.471120\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	19.5570	−	11.2912i	1.04091	−	0.600972i	0.120822	−	0.992674i	$-0.461447\pi$
$353$	19.5570	−	11.2912i	1.04091	−	0.600972i	0.920092	+	0.391702i	$0.128114\pi$
$354$		0			0
$355$	−0.716772	+	1.24148i	−0.0380423	+	0.0658912i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	1.73683i		−	0.0916662i	−0.998949	−	0.0458331i	$-0.985406\pi$
$359$		−	1.73683i		−	0.0916662i	0.998949	−	0.0458331i	$-0.0145942\pi$
$360$		0			0
$361$	−21.1034			−1.11071
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.471257	+	0.816241i	−0.0246667	+	0.0427240i
$366$		0			0
$367$	−2.67182	−	4.62772i	−0.139468	−	0.241565i	0.787828	−	0.615896i	$-0.211206\pi$
$367$	−2.67182	−	4.62772i	−0.139468	−	0.241565i	−0.927295	+	0.374331i	$0.877872\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	10.4976	−	6.06078i	0.545007	−	0.314660i
$372$		0			0
$373$	3.74634	−	6.48885i	0.193978	−	0.335980i	−0.752587	−	0.658493i	$-0.771194\pi$
$373$	3.74634	−	6.48885i	0.193978	−	0.335980i	0.946565	+	0.322513i	$0.104528\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.6725	−	15.9252i	0.652669	−	0.820190i
$378$		0			0
$379$		−	4.96945i		−	0.255263i	−0.991822	−	0.127632i	$-0.959262\pi$
$379$		−	4.96945i		−	0.255263i	0.991822	−	0.127632i	$-0.0407375\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.3855	−	7.72812i	−0.683967	−	0.394889i	0.117381	−	0.993087i	$-0.462550\pi$
$383$	−13.3855	−	7.72812i	−0.683967	−	0.394889i	−0.801348	+	0.598198i	$0.795883\pi$
$384$		0			0
$385$	−0.787417	+	0.454615i	−0.0401305	+	0.0231693i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.0196973	+	0.0341167i	0.000998691	+	0.00172978i	0.866524	−	0.499135i	$-0.166349\pi$
$389$	0.0196973	+	0.0341167i	0.000998691	+	0.00172978i	−0.865526	+	0.500865i	$0.833015\pi$
$390$		0			0
$391$	1.72591	−	2.98936i	0.0872828	−	0.151178i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	1.66091i		−	0.0835694i
$396$		0			0
$397$			28.3519i			1.42294i	0.702716	+	0.711470i	$0.251970\pi$
$397$			28.3519i			1.42294i	−0.702716	+	0.711470i	$0.748030\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	21.5623	+	12.4490i	1.07677	+	0.621673i	0.930023	−	0.367501i	$-0.119787\pi$
$401$	21.5623	+	12.4490i	1.07677	+	0.621673i	0.146746	+	0.989174i	$0.453120\pi$
$402$		0			0
$403$	0.0522771	−	0.349329i	0.00260411	−	0.0174013i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.61093	−	9.71841i	−0.278123	−	0.481724i
$408$		0			0
$409$	13.6728	+	7.89398i	0.676075	+	0.390332i	0.798375	−	0.602161i	$-0.205693\pi$
$409$	13.6728	+	7.89398i	0.676075	+	0.390332i	−0.122299	+	0.992493i	$0.539027\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−9.15477			−0.450477
$414$		0			0
$415$	−0.852161			−0.0418310
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.12759	+	7.14919i	−0.201646	+	0.349261i	−0.949059	−	0.315099i	$-0.897962\pi$
$419$	−4.12759	+	7.14919i	−0.201646	+	0.349261i	0.747413	+	0.664360i	$0.231296\pi$
$420$		0			0
$421$	−12.5328	+	7.23583i	−0.610812	+	0.352653i	−0.773283	−	0.634061i	$-0.781387\pi$
$421$	−12.5328	+	7.23583i	−0.610812	+	0.352653i	0.162471	+	0.986713i	$0.448054\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−13.8493	−	23.9877i	−0.671790	−	1.16358i
$426$		0			0
$427$	4.85847	+	2.80504i	0.235118	+	0.135745i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	17.9574i		−	0.864978i	−0.901639	−	0.432489i	$-0.857635\pi$
$431$		−	17.9574i		−	0.864978i	0.901639	−	0.432489i	$-0.142365\pi$
$432$		0			0
$433$	−24.5493			−1.17976			−0.589882	−	0.807489i	$-0.700826\pi$
$433$	−24.5493			−1.17976			−0.589882	+	0.807489i	$0.700826\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.40669	+	1.96685i	0.162964	+	0.0940872i
$438$		0			0
$439$	−8.39074	−	14.5332i	−0.400468	−	0.693631i	0.593314	−	0.804971i	$-0.297819\pi$
$439$	−8.39074	−	14.5332i	−0.400468	−	0.693631i	−0.993782	+	0.111340i	$0.964486\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.10192	−	8.83678i	−0.242399	−	0.419848i	0.718998	−	0.695012i	$-0.244601\pi$
$443$	−5.10192	−	8.83678i	−0.242399	−	0.419848i	−0.961397	+	0.275164i	$0.911268\pi$
$444$		0			0
$445$	−0.299109	+	0.518072i	−0.0141791	+	0.0245590i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			2.90395i			0.137046i	0.997650	+	0.0685230i	$0.0218287\pi$
$449$			2.90395i			0.137046i	−0.997650	+	0.0685230i	$0.978171\pi$
$450$		0			0
$451$	−22.7825			−1.07279
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.235170	+	0.597354i	0.0110249	+	0.0280044i
$456$		0			0
$457$	−7.21692	+	4.16669i	−0.337593	+	0.194910i	−0.659207	−	0.751961i	$-0.729108\pi$
$457$	−7.21692	+	4.16669i	−0.337593	+	0.194910i	0.321614	+	0.946871i	$0.395775\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.8461	+	14.9223i	−1.20377	+	0.695000i	−0.961392	−	0.275181i	$-0.911262\pi$
$461$	−25.8461	+	14.9223i	−1.20377	+	0.695000i	−0.242382	+	0.970181i	$0.577929\pi$
$462$		0			0
$463$	−29.0805	−	16.7896i	−1.35149	−	0.780280i	−0.363028	−	0.931778i	$-0.618257\pi$
$463$	−29.0805	−	16.7896i	−1.35149	−	0.780280i	−0.988458	+	0.151498i	$0.951590\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	7.04842			0.326162			0.163081	−	0.986613i	$-0.447857\pi$
$467$	7.04842			0.326162			0.163081	+	0.986613i	$0.447857\pi$
$468$		0			0
$469$	−17.3961			−0.803279
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−29.7426	−	17.1719i	−1.36757	−	0.789565i
$474$		0			0
$475$	27.3365	−	15.7827i	1.25429	−	0.724162i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	28.3125	−	16.3462i	1.29363	−	0.746879i	0.314336	−	0.949312i	$-0.398218\pi$
$479$	28.3125	−	16.3462i	1.29363	−	0.746879i	0.979296	+	0.202433i	$0.0648847\pi$
$480$		0			0
$481$	−7.37263	+	2.90250i	−0.336163	+	0.132343i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.165896			0.00753294
$486$		0			0
$487$			4.64327i			0.210407i	0.994451	+	0.105203i	$0.0335493\pi$
$487$			4.64327i			0.210407i	−0.994451	+	0.105203i	$0.966451\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.11404	−	1.92957i	0.0502758	−	0.0870803i	−0.839792	−	0.542908i	$-0.817323\pi$
$491$	1.11404	−	1.92957i	0.0502758	−	0.0870803i	0.890068	+	0.455828i	$0.150657\pi$
$492$		0			0
$493$	15.6835	+	27.1646i	0.706349	+	1.22343i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.23615	−	14.2654i	−0.369442	−	0.639892i
$498$		0			0
$499$	6.18239	+	3.56940i	0.276762	+	0.159788i	0.631957	−	0.775004i	$-0.282252\pi$
$499$	6.18239	+	3.56940i	0.276762	+	0.159788i	−0.355195	+	0.934792i	$0.615585\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	40.5832			1.80952			0.904759	−	0.425924i	$-0.140051\pi$
$503$	40.5832			1.80952			0.904759	+	0.425924i	$0.140051\pi$
$504$		0			0
$505$		−	1.27207i		−	0.0566063i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	14.9806	+	8.64905i	0.664003	+	0.383362i	0.793801	−	0.608178i	$-0.208099\pi$
$509$	14.9806	+	8.64905i	0.664003	+	0.383362i	−0.129797	+	0.991541i	$0.541433\pi$
$510$		0			0
$511$	−5.41503	−	9.37911i	−0.239547	−	0.414908i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.95643	+	1.12954i	−0.0862105	+	0.0497736i
$516$		0			0
$517$	5.00853	−	8.67503i	0.220275	−	0.381527i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−40.2648			−1.76403			−0.882017	−	0.471217i	$-0.843815\pi$
$521$	−40.2648			−1.76403			−0.882017	+	0.471217i	$0.843815\pi$
$522$		0			0
$523$	−37.6724			−1.64730			−0.823649	−	0.567100i	$-0.808065\pi$
$523$	−37.6724			−1.64730			−0.823649	+	0.567100i	$0.808065\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.471454	+	0.272194i	0.0205369	+	0.0118570i
$528$		0			0
$529$	11.3071	+	19.5844i	0.491612	+	0.851497i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.38076	+	15.9088i	−0.103122	+	0.689088i
$534$		0			0
$535$	−1.86243	−	1.07527i	−0.0805198	−	0.0464881i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			25.2980i			1.08966i
$540$		0			0
$541$			20.8974i			0.898451i	0.893418	+	0.449226i	$0.148300\pi$
$541$			20.8974i			0.898451i	−0.893418	+	0.449226i	$0.851700\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.630067	−	1.09131i	0.0269891	−	0.0467465i
$546$		0			0
$547$	7.35397	+	12.7375i	0.314433	+	0.544614i	0.979317	−	0.202333i	$-0.0648523\pi$
$547$	7.35397	+	12.7375i	0.314433	+	0.544614i	−0.664884	+	0.746947i	$0.731519\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−30.9569	+	17.8730i	−1.31881	+	0.761415i
$552$		0			0
$553$	16.5280	+	9.54243i	0.702841	+	0.405785i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			29.5223i			1.25090i	0.780265	+	0.625450i	$0.215084\pi$
$557$			29.5223i			1.25090i	−0.780265	+	0.625450i	$0.784916\pi$
$558$		0			0
$559$	−15.0991	+	18.9746i	−0.638623	+	0.802539i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−15.9484	+	27.6234i	−0.672144	+	1.16419i	0.305151	+	0.952304i	$0.401293\pi$
$563$	−15.9484	+	27.6234i	−0.672144	+	1.16419i	−0.977295	+	0.211884i	$0.932040\pi$
$564$		0			0
$565$	1.47888	−	0.853832i	0.0622170	−	0.0359210i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.5077	+	21.6640i	0.524351	+	0.908202i	0.999598	+	0.0283499i	$0.00902527\pi$
$569$	12.5077	+	21.6640i	0.524351	+	0.908202i	−0.475247	+	0.879852i	$0.657641\pi$
$570$		0			0
$571$	−3.27326	+	5.66946i	−0.136982	+	0.237259i	−0.926353	−	0.376657i	$-0.877074\pi$
$571$	−3.27326	+	5.66946i	−0.136982	+	0.237259i	0.789371	+	0.613916i	$0.210407\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−3.09623			−0.129122
$576$		0			0
$577$		−	6.43675i		−	0.267965i	−0.990984	−	0.133983i	$-0.957223\pi$
$577$		−	6.43675i		−	0.267965i	0.990984	−	0.133983i	$-0.0427767\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	4.89593	−	8.48000i	0.203117	−	0.351810i
$582$		0			0
$583$	37.4773	−	21.6375i	1.55215	−	0.896135i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.5761	+	9.57022i	−0.684169	+	0.395005i	−0.801424	−	0.598097i	$-0.795924\pi$
$587$	−16.5761	+	9.57022i	−0.684169	+	0.395005i	0.117255	+	0.993102i	$0.462591\pi$
$588$		0			0
$589$	−0.310194	+	0.537271i	−0.0127813	+	0.0221379i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	16.3476i		−	0.671315i	−0.941984	−	0.335658i	$-0.891041\pi$
$593$		−	16.3476i		−	0.671315i	0.941984	−	0.335658i	$-0.108959\pi$
$594$		0			0
$595$	−0.989431			−0.0405627
$596$		0			0
$597$		0			0
$598$		0			0
$599$	13.2978	−	23.0325i	0.543334	−	0.941081i	−0.455376	−	0.890299i	$-0.650495\pi$
$599$	13.2978	−	23.0325i	0.543334	−	0.941081i	0.998710	−	0.0507823i	$-0.0161715\pi$
$600$		0			0
$601$	−17.1713	−	29.7415i	−0.700431	−	1.21318i	−0.968315	−	0.249731i	$-0.919658\pi$
$601$	−17.1713	−	29.7415i	−0.700431	−	1.21318i	0.267885	−	0.963451i	$-0.413675\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.62531	+	0.938372i	−0.0660782	+	0.0381503i
$606$		0			0
$607$	−9.09912	+	15.7601i	−0.369322	+	0.639685i	−0.989460	−	0.144808i	$-0.953743\pi$
$607$	−9.09912	+	15.7601i	−0.369322	+	0.639685i	0.620138	+	0.784493i	$0.287077\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5.53432	−	4.40395i	−0.223894	−	0.178165i
$612$		0			0
$613$			29.1208i			1.17618i	0.808797	+	0.588088i	$0.200119\pi$
$613$			29.1208i			1.17618i	−0.808797	+	0.588088i	$0.799881\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	8.73626	+	5.04388i	0.351708	+	0.203059i	0.665437	−	0.746454i	$-0.268245\pi$
$617$	8.73626	+	5.04388i	0.351708	+	0.203059i	−0.313729	+	0.949513i	$0.601578\pi$
$618$		0			0
$619$	−8.17164	+	4.71790i	−0.328446	+	0.189628i	−0.655151	−	0.755498i	$-0.727395\pi$
$619$	−8.17164	+	4.71790i	−0.328446	+	0.189628i	0.326705	+	0.945126i	$0.394062\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3.43694	−	5.95296i	−0.137698	−	0.238500i
$624$		0			0
$625$	−12.3839	+	21.4496i	−0.495356	+	0.857982i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	12.2117i		−	0.486912i
$630$		0			0
$631$			45.1898i			1.79898i	0.436943	+	0.899489i	$0.356061\pi$
$631$			45.1898i			1.79898i	−0.436943	+	0.899489i	$0.643939\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.43522	−	0.828622i	−0.0569548	−	0.0328829i
$636$		0			0
$637$	17.6654	+	2.64363i	0.699929	+	0.104744i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6.52825	+	11.3073i	0.257850	+	0.446610i	0.965666	−	0.259788i	$-0.0836525\pi$
$641$	6.52825	+	11.3073i	0.257850	+	0.446610i	−0.707816	+	0.706397i	$0.750319\pi$
$642$		0			0
$643$	4.07575	+	2.35313i	0.160732	+	0.0927985i	0.578208	−	0.815889i	$-0.303752\pi$
$643$	4.07575	+	2.35313i	0.160732	+	0.0927985i	−0.417477	+	0.908688i	$0.637085\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−42.6817			−1.67799			−0.838995	−	0.544140i	$-0.816856\pi$
$647$	−42.6817			−1.67799			−0.838995	+	0.544140i	$0.816856\pi$
$648$		0			0
$649$	−32.6833			−1.28293
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−6.78464	+	11.7513i	−0.265504	+	0.459866i	−0.967695	−	0.252122i	$-0.918872\pi$
$653$	−6.78464	+	11.7513i	−0.265504	+	0.459866i	0.702192	+	0.711988i	$0.252205\pi$
$654$		0			0
$655$	−0.212315	+	0.122580i	−0.00829584	+	0.00478961i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.13615	+	7.16402i	0.161121	+	0.279071i	0.935271	−	0.353932i	$-0.115156\pi$
$659$	4.13615	+	7.16402i	0.161121	+	0.279071i	−0.774150	+	0.633003i	$0.781822\pi$
$660$		0			0
$661$	14.8793	+	8.59059i	0.578739	+	0.334135i	0.760632	−	0.649183i	$-0.224889\pi$
$661$	14.8793	+	8.59059i	0.578739	+	0.334135i	−0.181893	+	0.983318i	$0.558222\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	1.12756i		−	0.0437249i
$666$		0			0
$667$	3.50629			0.135764
$668$		0			0
$669$		0			0
$670$		0			0
$671$	17.3452	+	10.0142i	0.669602	+	0.386595i
$672$		0			0
$673$	0.105568	+	0.182849i	0.00406933	+	0.00704829i	0.868053	−	0.496472i	$-0.165371\pi$
$673$	0.105568	+	0.182849i	0.00406933	+	0.00704829i	−0.863984	+	0.503520i	$0.832038\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−15.2201	−	26.3619i	−0.584954	−	1.01317i	−0.994881	−	0.101052i	$-0.967779\pi$
$677$	−15.2201	−	26.3619i	−0.584954	−	1.01317i	0.409927	−	0.912118i	$-0.365554\pi$
$678$		0			0
$679$	−0.953122	+	1.65086i	−0.0365775	+	0.0633540i
$680$		0			0
$681$		0			0
$682$		0			0
$683$			39.1443i			1.49782i	0.662674	+	0.748908i	$0.269421\pi$
$683$			39.1443i			1.49782i	−0.662674	+	0.748908i	$0.730579\pi$
$684$		0			0
$685$	2.01836			0.0771175
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−11.1930	−	28.4312i	−0.426418	−	1.08314i
$690$		0			0
$691$	−31.8371	+	18.3811i	−1.21114	+	0.699251i	−0.963007	−	0.269475i	$-0.913150\pi$
$691$	−31.8371	+	18.3811i	−1.21114	+	0.699251i	−0.248131	+	0.968726i	$0.579816\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.74817	+	1.00930i	−0.0663117	+	0.0382851i
$696$		0			0
$697$	−21.4705	−	12.3960i	−0.813255	−	0.469533i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	10.4630			0.395184			0.197592	−	0.980284i	$-0.436688\pi$
$701$	10.4630			0.395184			0.197592	+	0.980284i	$0.436688\pi$
$702$		0			0
$703$	13.9165			0.524871
$704$		0			0
$705$		0			0
$706$		0			0
$707$	12.6586	+	7.30842i	0.476074	+	0.274861i
$708$		0			0
$709$	24.0129	−	13.8638i	0.901822	−	0.520667i	0.0240312	−	0.999711i	$-0.492350\pi$
$709$	24.0129	−	13.8638i	0.901822	−	0.520667i	0.877791	+	0.479044i	$0.159017\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.0527004	−	0.0304266i	0.00197365	−	0.00113948i
$714$		0			0
$715$	0.839577	+	2.13261i	0.0313984	+	0.0797550i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−3.34049			−0.124579			−0.0622896	−	0.998058i	$-0.519840\pi$
$719$	−3.34049			−0.124579			−0.0622896	+	0.998058i	$0.519840\pi$
$720$		0			0
$721$		−	25.9583i		−	0.966738i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	14.0679	−	24.3663i	0.522468	−	0.904941i
$726$		0			0
$727$	21.3278	+	36.9409i	0.791005	+	1.37006i	0.925345	+	0.379126i	$0.123775\pi$
$727$	21.3278	+	36.9409i	0.791005	+	1.37006i	−0.134340	+	0.990935i	$0.542891\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−18.6866	−	32.3661i	−0.691148	−	1.19710i
$732$		0			0
$733$	35.9640	+	20.7638i	1.32836	+	0.766929i	0.985046	−	0.172291i	$-0.0551169\pi$
$733$	35.9640	+	20.7638i	1.32836	+	0.766929i	0.343315	+	0.939220i	$0.388450\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−62.1057			−2.28769
$738$		0			0
$739$		−	16.5772i		−	0.609802i	−0.952384	−	0.304901i	$-0.901377\pi$
$739$		−	16.5772i		−	0.609802i	0.952384	−	0.304901i	$-0.0986234\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−7.42474	−	4.28667i	−0.272387	−	0.157263i	0.357585	−	0.933881i	$-0.383600\pi$
$743$	−7.42474	−	4.28667i	−0.272387	−	0.157263i	−0.629972	+	0.776618i	$0.716934\pi$
$744$		0			0
$745$	−1.11668	−	1.93414i	−0.0409119	−	0.0708616i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	21.4004	−	12.3556i	0.781955	−	0.451462i
$750$		0			0
$751$	−6.05873	+	10.4940i	−0.221086	+	0.382932i	−0.955138	−	0.296161i	$-0.904294\pi$
$751$	−6.05873	+	10.4940i	−0.221086	+	0.382932i	0.734052	+	0.679093i	$0.237627\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.756988			−0.0275496
$756$		0			0
$757$	38.6116			1.40336			0.701682	−	0.712490i	$-0.252433\pi$
$757$	38.6116			1.40336			0.701682	+	0.712490i	$0.252433\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.678687	+	0.391840i	0.0246024	+	0.0142042i	0.512251	−	0.858836i	$-0.328812\pi$
$761$	0.678687	+	0.391840i	0.0246024	+	0.0142042i	−0.487648	+	0.873040i	$0.662145\pi$
$762$		0			0
$763$	7.23986	+	12.5398i	0.262101	+	0.453971i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3.41539	+	22.8225i	−0.123323	+	0.824074i
$768$		0			0
$769$	−31.7357	−	18.3226i	−1.14442	−	0.660730i	−0.196897	−	0.980424i	$-0.563087\pi$
$769$	−31.7357	−	18.3226i	−1.14442	−	0.660730i	−0.947521	+	0.319694i	$0.896420\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			2.36509i			0.0850664i	0.999095	+	0.0425332i	$0.0135428\pi$
$773$			2.36509i			0.0850664i	−0.999095	+	0.0425332i	$0.986457\pi$
$774$		0			0
$775$		−	0.488309i		−	0.0175406i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	14.1266	−	24.4679i	0.506137	−	0.876655i
$780$		0			0
$781$	−29.4038	−	50.9288i	−1.05215	−	1.82238i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.06991	+	0.617715i	−0.0381869	+	0.0220472i
$786$		0			0
$787$	12.5486	+	7.24495i	0.447310	+	0.258254i	0.706693	−	0.707520i	$-0.250186\pi$
$787$	12.5486	+	7.24495i	0.447310	+	0.258254i	−0.259384	+	0.965774i	$0.583519\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			19.6221i			0.697682i
$792$		0			0
$793$	8.80542	−	11.0655i	0.312690	−	0.392948i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.62409	−	13.2053i	0.270059	−	0.467756i	−0.698818	−	0.715300i	$-0.746290\pi$
$797$	7.62409	−	13.2053i	0.270059	−	0.467756i	0.968877	+	0.247544i	$0.0796234\pi$
$798$		0			0
$799$	9.44022	−	5.45031i	0.333971	−	0.192818i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−19.3322	−	33.4843i	−0.682217	−	1.18163i
$804$		0			0
$805$	−0.0553006	+	0.0957834i	−0.00194909	+	0.00337592i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−37.0260			−1.30176			−0.650882	−	0.759179i	$-0.725601\pi$
$809$	−37.0260			−1.30176			−0.650882	+	0.759179i	$0.725601\pi$
$810$		0			0
$811$			26.4852i			0.930020i	0.885305	+	0.465010i	$0.153949\pi$
$811$			26.4852i			0.930020i	−0.885305	+	0.465010i	$0.846051\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.774406	+	1.34131i	−0.0271263	+	0.0469841i
$816$		0			0
$817$	36.8845	−	21.2953i	1.29043	−	0.745028i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−30.2740	+	17.4787i	−1.05657	+	0.610011i	−0.924482	−	0.381227i	$-0.875502\pi$
$821$	−30.2740	+	17.4787i	−1.05657	+	0.610011i	−0.132089	+	0.991238i	$0.542168\pi$
$822$		0			0
$823$	−13.4726	+	23.3352i	−0.469625	+	0.813415i	−0.999397	−	0.0347253i	$-0.988944\pi$
$823$	−13.4726	+	23.3352i	−0.469625	+	0.813415i	0.529771	+	0.848140i	$0.322278\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	23.2041i		−	0.806886i	−0.915005	−	0.403443i	$-0.867813\pi$
$827$		−	23.2041i		−	0.806886i	0.915005	−	0.403443i	$-0.132187\pi$
$828$		0			0
$829$	−3.87791			−0.134685			−0.0673427	−	0.997730i	$-0.521452\pi$
$829$	−3.87791			−0.134685			−0.0673427	+	0.997730i	$0.521452\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−13.7647	+	23.8412i	−0.476920	+	0.826049i
$834$		0			0
$835$	−1.31565	−	2.27877i	−0.0455299	−	0.0788601i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	9.33328	−	5.38857i	0.322221	−	0.186034i	−0.330161	−	0.943925i	$-0.607103\pi$
$839$	9.33328	−	5.38857i	0.322221	−	0.186034i	0.652382	+	0.757890i	$0.273770\pi$
$840$		0			0
$841$	−1.43100	+	2.47856i	−0.0493448	+	0.0854677i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.57692	−	0.363414i	0.0542476	−	0.0125018i
$846$		0			0
$847$		−	21.5649i		−	0.740980i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.18217	−	0.682528i	−0.0405244	−	0.0233967i
$852$		0			0
$853$	−31.4188	+	18.1397i	−1.07576	+	0.621090i	−0.929750	−	0.368193i	$-0.879977\pi$
$853$	−31.4188	+	18.1397i	−1.07576	+	0.621090i	−0.146011	+	0.989283i	$0.546643\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	15.2301	+	26.3794i	0.520252	+	0.901102i	0.999723	+	0.0235447i	$0.00749522\pi$
$857$	15.2301	+	26.3794i	0.520252	+	0.901102i	−0.479471	+	0.877558i	$0.659171\pi$
$858$		0			0
$859$	21.1701	−	36.6676i	0.722313	−	1.25108i	−0.237757	−	0.971325i	$-0.576412\pi$
$859$	21.1701	−	36.6676i	0.722313	−	1.25108i	0.960070	−	0.279759i	$-0.0902544\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	0.175513i		−	0.00597452i	−0.999996	−	0.00298726i	$-0.999049\pi$
$863$		−	0.175513i		−	0.00597452i	0.999996	−	0.00298726i	$-0.000950876\pi$
$864$		0			0
$865$			1.00961i			0.0343278i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	59.0063	+	34.0673i	2.00165	+	1.15565i
$870$		0			0
$871$	−6.49001	+	43.3679i	−0.219905	+	1.46947i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.888885	+	1.53959i	0.0300498	+	0.0520478i
$876$		0			0
$877$	42.9232	+	24.7817i	1.44941	+	0.836818i	0.998446	−	0.0557229i	$-0.0177464\pi$
$877$	42.9232	+	24.7817i	1.44941	+	0.836818i	0.450966	+	0.892541i	$0.351080\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−9.81413			−0.330646			−0.165323	−	0.986239i	$-0.552867\pi$
$881$	−9.81413			−0.330646			−0.165323	+	0.986239i	$0.552867\pi$
$882$		0			0
$883$	−6.16031			−0.207311			−0.103656	−	0.994613i	$-0.533054\pi$
$883$	−6.16031			−0.207311			−0.103656	+	0.994613i	$0.533054\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−17.2629	+	29.9002i	−0.579632	+	1.00395i	0.415890	+	0.909415i	$0.363470\pi$
$887$	−17.2629	+	29.9002i	−0.579632	+	1.00395i	−0.995521	+	0.0945365i	$0.969863\pi$
$888$		0			0
$889$	16.4915	−	9.52138i	0.553108	−	0.319337i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	6.21120	+	10.7581i	0.207850	+	0.360007i
$894$		0			0
$895$	0.0576474	+	0.0332827i	0.00192694	+	0.00111252i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			0.552979i			0.0184429i
$900$		0			0
$901$	47.0922			1.56887
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.325649	−	0.188014i	−0.0108249	−	0.00624978i
$906$		0			0
$907$	21.8265	+	37.8046i	0.724737	+	1.25528i	0.959082	+	0.283128i	$0.0913721\pi$
$907$	21.8265	+	37.8046i	0.724737	+	1.25528i	−0.234345	+	0.972153i	$0.575295\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−8.70581	−	15.0789i	−0.288436	−	0.499587i	0.685000	−	0.728543i	$-0.259802\pi$
$911$	−8.70581	−	15.0789i	−0.288436	−	0.499587i	−0.973437	+	0.228956i	$0.926469\pi$
$912$		0			0
$913$	17.4789	−	30.2743i	0.578467	−	1.00193i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	2.81704i		−	0.0930270i
$918$		0			0
$919$	−1.83009			−0.0603692			−0.0301846	−	0.999544i	$-0.509610\pi$
$919$	−1.83009			−0.0603692			−0.0301846	+	0.999544i	$0.509610\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−38.6359	+	15.2104i	−1.27172	+	0.500656i
$924$		0			0
$925$	−9.48620	+	5.47686i	−0.311904	+	0.180078i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.9913	+	11.5420i	−0.655892	+	0.378680i	−0.790710	−	0.612191i	$-0.790288\pi$
$929$	−19.9913	+	11.5420i	−0.655892	+	0.378680i	0.134818	+	0.990870i	$0.456955\pi$
$930$		0			0
$931$	−27.1696	−	15.6864i	−0.890447	−	0.514100i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.53235			−0.115520
$936$		0			0
$937$	−35.5716			−1.16207			−0.581036	−	0.813878i	$-0.697353\pi$
$937$	−35.5716			−1.16207			−0.581036	+	0.813878i	$0.697353\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.0588	+	6.96215i	0.393106	+	0.226960i	0.683505	−	0.729946i	$-0.260455\pi$
$941$	12.0588	+	6.96215i	0.393106	+	0.226960i	−0.290399	+	0.956906i	$0.593788\pi$
$942$		0			0
$943$	−2.40003	+	1.38566i	−0.0781559	+	0.0451233i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−31.0660	+	17.9360i	−1.00951	+	0.582841i	−0.911048	−	0.412299i	$-0.864726\pi$
$947$	−31.0660	+	17.9360i	−1.00951	+	0.582841i	−0.0984625	+	0.995141i	$0.531392\pi$
$948$		0			0
$949$	−25.4020	+	10.0004i	−0.824584	+	0.324627i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	10.2863			0.333205			0.166602	−	0.986024i	$-0.446720\pi$
$953$	10.2863			0.333205			0.166602	+	0.986024i	$0.446720\pi$
$954$		0			0
$955$			0.481061i			0.0155668i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−11.5961	+	20.0850i	−0.374457	+	0.648579i
$960$		0			0
$961$	−15.4952	−	26.8385i	−0.499845	−	0.865757i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.22904	+	2.12875i	0.0395640	+	0.0685269i
$966$		0			0
$967$	−39.4742	−	22.7905i	−1.26941	−	0.732892i	−0.294530	−	0.955642i	$-0.595163\pi$
$967$	−39.4742	−	22.7905i	−1.26941	−	0.732892i	−0.974876	+	0.222750i	$0.928497\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	48.0053			1.54056			0.770281	−	0.637704i	$-0.220116\pi$
$971$	48.0053			1.54056			0.770281	+	0.637704i	$0.220116\pi$
$972$		0			0
$973$		−	23.1951i		−	0.743599i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−42.7507	−	24.6821i	−1.36772	−	0.789652i	−0.377081	−	0.926180i	$-0.623072\pi$
$977$	−42.7507	−	24.6821i	−1.36772	−	0.789652i	−0.990636	+	0.136529i	$0.956405\pi$
$978$		0			0
$979$	−12.2702	−	21.2526i	−0.392157	−	0.679236i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	19.8114	−	11.4381i	0.631885	−	0.364819i	−0.149597	−	0.988747i	$-0.547798\pi$
$983$	19.8114	−	11.4381i	0.631885	−	0.364819i	0.781482	+	0.623928i	$0.214464\pi$
$984$		0			0
$985$	0.543886	−	0.942038i	0.0173296	−	0.0300158i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.17767			−0.132842
$990$		0			0
$991$	−18.5403			−0.588951			−0.294475	−	0.955659i	$-0.595145\pi$
$991$	−18.5403			−0.588951			−0.294475	+	0.955659i	$0.595145\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.45905	+	0.842383i	0.0462550	+	0.0267053i
$996$		0			0
$997$	−10.0331	−	17.3779i	−0.317753	−	0.550364i	0.662266	−	0.749269i	$-0.269595\pi$
$997$	−10.0331	−	17.3779i	−0.317753	−	0.550364i	−0.980019	+	0.198905i	$0.936261\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2808.2.cw.b.2521.19		80
3.2	odd	2		936.2.cw.b.337.20	yes	80
9.2	odd	6		936.2.cw.b.25.19	✓	80
9.7	even	3	inner	2808.2.cw.b.1585.22		80
13.12	even	2	inner	2808.2.cw.b.2521.22		80
39.38	odd	2		936.2.cw.b.337.19	yes	80
117.25	even	6	inner	2808.2.cw.b.1585.19		80
117.38	odd	6		936.2.cw.b.25.20	yes	80

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.cw.b.25.19	✓	80	9.2	odd	6
936.2.cw.b.25.20	yes	80	117.38	odd	6
936.2.cw.b.337.19	yes	80	39.38	odd	2
936.2.cw.b.337.20	yes	80	3.2	odd	2
2808.2.cw.b.1585.19		80	117.25	even	6	inner
2808.2.cw.b.1585.22		80	9.7	even	3	inner
2808.2.cw.b.2521.19		80	1.1	even	1	trivial
2808.2.cw.b.2521.22		80	13.12	even	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2808 = 2^{3} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2808.cw (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.107804 - 0.0622405i) q^{5} +(1.23873 - 0.715181i) q^{7} +(4.42238 - 2.55326i) q^{11} +(-1.32079 - 3.35493i) q^{13} +5.55695 q^{17} +6.33272i q^{19} +(0.310585 - 0.537949i) q^{23} +(-2.49225 - 4.31671i) q^{25} +(2.82232 + 4.88840i) q^{29} +(0.0848405 + 0.0489827i) q^{31} -0.178053 q^{35} -2.19755i q^{37} +(-3.86373 - 2.23073i) q^{41} +(-3.36274 - 5.82443i) q^{43} +(1.69881 - 0.980810i) q^{47} +(-2.47703 + 4.29034i) q^{49} +8.47447 q^{53} -0.635665 q^{55} +(-5.54284 - 3.20016i) q^{59} +(1.96107 + 3.39667i) q^{61} +(-0.0664265 + 0.443880i) q^{65} +(-10.5326 - 6.08102i) q^{67} -11.5162i q^{71} -7.57155i q^{73} +(3.65209 - 6.32560i) q^{77} +(6.67133 + 11.5551i) q^{79} +(5.92857 - 3.42286i) q^{83} +(-0.599059 - 0.345867i) q^{85} -4.80570i q^{89} +(-4.03548 - 3.21125i) q^{91} +(0.394152 - 0.682691i) q^{95} +(-1.15415 + 0.666350i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 6 q^{13} + 4 q^{17} - 10 q^{23} + 44 q^{25} + 52 q^{35} - 26 q^{43} + 48 q^{49} - 60 q^{53} - 16 q^{55} - 10 q^{61} + 26 q^{65} - 32 q^{77} + 6 q^{79} - 4 q^{91} - 8 q^{95}+O(q^{100}) \) 80 * q + 6 * q^13 + 4 * q^17 - 10 * q^23 + 44 * q^25 + 52 * q^35 - 26 * q^43 + 48 * q^49 - 60 * q^53 - 16 * q^55 - 10 * q^61 + 26 * q^65 - 32 * q^77 + 6 * q^79 - 4 * q^91 - 8 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more