LMFDB - Embedded newform 2880.2.bc.a.593.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(593,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2, 3]))

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

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

chi := DirichletCharacter("2880.593");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bc (of order $4$, 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:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$96$
Relative dimension:	$48$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			593.20
Character	$\chi$	$=$	2880.593
Dual form			2880.2.bc.a.1457.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.470363 - 2.18604i) q^{5} +(0.177389 + 0.177389i) q^{7} +O(q^{10})$	$q+(-0.470363 - 2.18604i) q^{5} +(0.177389 + 0.177389i) q^{7} +(1.34414 + 1.34414i) q^{11} -1.40679i q^{13} +(1.03869 + 1.03869i) q^{17} +(0.205281 - 0.205281i) q^{19} +(5.60286 + 5.60286i) q^{23} +(-4.55752 + 2.05646i) q^{25} +(5.31184 + 5.31184i) q^{29} +4.12332 q^{31} +(0.304341 - 0.471215i) q^{35} -3.60144i q^{37} -8.78009 q^{41} -3.50742 q^{43} +(6.65257 + 6.65257i) q^{47} -6.93707i q^{49} -5.30796i q^{53} +(2.30610 - 3.57056i) q^{55} +(9.02817 - 9.02817i) q^{59} +(-0.150056 - 0.150056i) q^{61} +(-3.07530 + 0.661702i) q^{65} +10.1797 q^{67} +6.76393 q^{71} +(1.14818 - 1.14818i) q^{73} +0.476869i q^{77} -5.19176i q^{79} -5.08514 q^{83} +(1.78205 - 2.75918i) q^{85} +8.79477i q^{89} +(0.249549 - 0.249549i) q^{91} +(-0.545309 - 0.352195i) q^{95} +(6.63701 - 6.63701i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 96 q+O(q^{10}) $ 96 * q	$ 96 q - 16 q^{19} - 64 q^{43} + 32 q^{61}+O(q^{100}) $ 96 * q - 16 * q^19 - 64 * q^43 + 32 * q^61

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.470363	−	2.18604i	−0.210353	−	0.977626i
$5$	−0.470363	−	2.18604i	−0.210353	−	0.977626i
$6$		0			0
$7$	0.177389	+	0.177389i	0.0670466	+	0.0670466i	0.739835	−	0.672788i	$-0.234904\pi$
$7$	0.177389	+	0.177389i	0.0670466	+	0.0670466i	−0.672788	+	0.739835i	$0.734904\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.34414	+	1.34414i	0.405272	+	0.405272i	0.880086	−	0.474814i	$-0.157485\pi$
$11$	1.34414	+	1.34414i	0.405272	+	0.405272i	−0.474814	+	0.880086i	$0.657485\pi$
$12$		0			0
$13$		−	1.40679i		−	0.390173i	−0.980786	−	0.195087i	$-0.937501\pi$
$13$		−	1.40679i		−	0.390173i	0.980786	−	0.195087i	$-0.0624988\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.03869	+	1.03869i	0.251919	+	0.251919i	0.821757	−	0.569838i	$-0.192994\pi$
$17$	1.03869	+	1.03869i	0.251919	+	0.251919i	−0.569838	+	0.821757i	$0.692994\pi$
$18$		0			0
$19$	0.205281	−	0.205281i	0.0470947	−	0.0470947i	−0.683167	−	0.730262i	$-0.739398\pi$
$19$	0.205281	−	0.205281i	0.0470947	−	0.0470947i	0.730262	+	0.683167i	$0.239398\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	5.60286	+	5.60286i	1.16828	+	1.16828i	0.982613	+	0.185664i	$0.0594437\pi$
$23$	5.60286	+	5.60286i	1.16828	+	1.16828i	0.185664	+	0.982613i	$0.440556\pi$
$24$		0			0
$25$	−4.55752	+	2.05646i	−0.911503	+	0.411293i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.31184	+	5.31184i	0.986384	+	0.986384i	0.999909	−	0.0135250i	$-0.00430527\pi$
$29$	5.31184	+	5.31184i	0.986384	+	0.986384i	−0.0135250	+	0.999909i	$0.504305\pi$
$30$		0			0
$31$	4.12332			0.740571			0.370285	−	0.928918i	$-0.379260\pi$
$31$	4.12332			0.740571			0.370285	+	0.928918i	$0.379260\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.304341	−	0.471215i	0.0514430	−	0.0796499i
$36$		0			0
$37$		−	3.60144i		−	0.592073i	−0.955177	−	0.296036i	$-0.904335\pi$
$37$		−	3.60144i		−	0.592073i	0.955177	−	0.296036i	$-0.0956650\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−8.78009			−1.37122			−0.685610	−	0.727969i	$-0.740465\pi$
$41$	−8.78009			−1.37122			−0.685610	+	0.727969i	$0.740465\pi$
$42$		0			0
$43$	−3.50742			−0.534876			−0.267438	−	0.963575i	$-0.586177\pi$
$43$	−3.50742			−0.534876			−0.267438	+	0.963575i	$0.586177\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.65257	+	6.65257i	0.970378	+	0.970378i	0.999574	−	0.0291960i	$-0.00929471\pi$
$47$	6.65257	+	6.65257i	0.970378	+	0.970378i	−0.0291960	+	0.999574i	$0.509295\pi$
$48$		0			0
$49$		−	6.93707i		−	0.991010i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	5.30796i		−	0.729105i	−0.931183	−	0.364552i	$-0.881222\pi$
$53$		−	5.30796i		−	0.729105i	0.931183	−	0.364552i	$-0.118778\pi$
$54$		0			0
$55$	2.30610	−	3.57056i	0.310954	−	0.481455i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	9.02817	−	9.02817i	1.17537	−	1.17537i	0.194457	−	0.980911i	$-0.437705\pi$
$59$	9.02817	−	9.02817i	1.17537	−	1.17537i	0.980911	−	0.194457i	$-0.0622945\pi$
$60$		0			0
$61$	−0.150056	−	0.150056i	−0.0192127	−	0.0192127i	0.697435	−	0.716648i	$-0.254325\pi$
$61$	−0.150056	−	0.150056i	−0.0192127	−	0.0192127i	−0.716648	+	0.697435i	$0.754325\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.07530	+	0.661702i	−0.381444	+	0.0820741i
$66$		0			0
$67$	10.1797			1.24364			0.621822	−	0.783159i	$-0.286393\pi$
$67$	10.1797			1.24364			0.621822	+	0.783159i	$0.286393\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	6.76393			0.802731			0.401365	−	0.915918i	$-0.368536\pi$
$71$	6.76393			0.802731			0.401365	+	0.915918i	$0.368536\pi$
$72$		0			0
$73$	1.14818	−	1.14818i	0.134384	−	0.134384i	−0.636715	−	0.771099i	$-0.719707\pi$
$73$	1.14818	−	1.14818i	0.134384	−	0.134384i	0.771099	+	0.636715i	$0.219707\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$			0.476869i			0.0543443i
$78$		0			0
$79$		−	5.19176i		−	0.584119i	−0.956400	−	0.292059i	$-0.905660\pi$
$79$		−	5.19176i		−	0.584119i	0.956400	−	0.292059i	$-0.0943405\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−5.08514			−0.558167			−0.279084	−	0.960267i	$-0.590031\pi$
$83$	−5.08514			−0.558167			−0.279084	+	0.960267i	$0.590031\pi$
$84$		0			0
$85$	1.78205	−	2.75918i	0.193291	−	0.299275i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			8.79477i			0.932244i	0.884721	+	0.466122i	$0.154349\pi$
$89$			8.79477i			0.932244i	−0.884721	+	0.466122i	$0.845651\pi$
$90$		0			0
$91$	0.249549	−	0.249549i	0.0261598	−	0.0261598i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.545309	−	0.352195i	−0.0559475	−	0.0361345i
$96$		0			0
$97$	6.63701	−	6.63701i	0.673886	−	0.673886i	−0.284723	−	0.958610i	$-0.591902\pi$
$97$	6.63701	−	6.63701i	0.673886	−	0.673886i	0.958610	+	0.284723i	$0.0919017\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−2.21659	−	2.21659i	−0.220559	−	0.220559i	0.588175	−	0.808734i	$-0.299847\pi$
$101$	−2.21659	−	2.21659i	−0.220559	−	0.220559i	−0.808734	+	0.588175i	$0.799847\pi$
$102$		0			0
$103$	−7.63400	+	7.63400i	−0.752201	+	0.752201i	−0.974890	−	0.222689i	$-0.928517\pi$
$103$	−7.63400	+	7.63400i	−0.752201	+	0.752201i	0.222689	+	0.974890i	$0.428517\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.01575			0.581564			0.290782	−	0.956789i	$-0.406084\pi$
$107$	6.01575			0.581564			0.290782	+	0.956789i	$0.406084\pi$
$108$		0			0
$109$	6.71925	−	6.71925i	0.643588	−	0.643588i	−0.307848	−	0.951436i	$-0.599609\pi$
$109$	6.71925	−	6.71925i	0.643588	−	0.643588i	0.951436	+	0.307848i	$0.0996088\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−11.1195	+	11.1195i	−1.04603	+	1.04603i	−0.0471464	+	0.998888i	$0.515013\pi$
$113$	−11.1195	+	11.1195i	−1.04603	+	1.04603i	−0.998888	+	0.0471464i	$0.984987\pi$
$114$		0			0
$115$	9.61269	−	14.8834i	0.896387	−	1.38789i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			0.368504i			0.0337807i
$120$		0			0
$121$		−	7.38660i		−	0.671509i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.63919	+	8.99562i	0.593827	+	0.804592i
$126$		0			0
$127$	4.96966	−	4.96966i	0.440986	−	0.440986i	−0.451357	−	0.892343i	$-0.649060\pi$
$127$	4.96966	−	4.96966i	0.440986	−	0.440986i	0.892343	+	0.451357i	$0.149060\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.790895	+	0.790895i	−0.0691008	+	0.0691008i	−0.740813	−	0.671712i	$-0.765559\pi$
$131$	−0.790895	+	0.790895i	−0.0691008	+	0.0691008i	0.671712	+	0.740813i	$0.265559\pi$
$132$		0			0
$133$	0.0728291			0.00631508
$134$		0			0
$135$		0			0
$136$		0			0
$137$	10.5573	−	10.5573i	0.901975	−	0.901975i	−0.0936320	−	0.995607i	$-0.529848\pi$
$137$	10.5573	−	10.5573i	0.901975	−	0.901975i	0.995607	+	0.0936320i	$0.0298477\pi$
$138$		0			0
$139$	−0.483599	−	0.483599i	−0.0410184	−	0.0410184i	0.686300	−	0.727318i	$-0.259234\pi$
$139$	−0.483599	−	0.483599i	−0.0410184	−	0.0410184i	−0.727318	+	0.686300i	$0.759234\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.89092	−	1.89092i	0.158126	−	0.158126i
$144$		0			0
$145$	9.11338	−	14.1104i	0.756825	−	1.17180i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	14.4866	−	14.4866i	1.18679	−	1.18679i	0.208842	−	0.977949i	$-0.433030\pi$
$149$	14.4866	−	14.4866i	1.18679	−	1.18679i	0.977949	−	0.208842i	$-0.0669696\pi$
$150$		0			0
$151$			13.5412i			1.10196i	0.834517	+	0.550982i	$0.185747\pi$
$151$			13.5412i			1.10196i	−0.834517	+	0.550982i	$0.814253\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.93946	−	9.01374i	−0.155781	−	0.724001i
$156$		0			0
$157$	0.663466			0.0529503			0.0264752	−	0.999649i	$-0.491572\pi$
$157$	0.663466			0.0529503			0.0264752	+	0.999649i	$0.491572\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1.98777i			0.156658i
$162$		0			0
$163$		−	5.91663i		−	0.463426i	−0.972784	−	0.231713i	$-0.925567\pi$
$163$		−	5.91663i		−	0.463426i	0.972784	−	0.231713i	$-0.0744330\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−11.3597	+	11.3597i	−0.879038	+	0.879038i	−0.993435	−	0.114398i	$-0.963506\pi$
$167$	−11.3597	+	11.3597i	−0.879038	+	0.879038i	0.114398	+	0.993435i	$0.463506\pi$
$168$		0			0
$169$	11.0209			0.847765
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.250616			−0.0190540			−0.00952698	−	0.999955i	$-0.503033\pi$
$173$	−0.250616			−0.0190540			−0.00952698	+	0.999955i	$0.503033\pi$
$174$		0			0
$175$	−1.17325	−	0.443659i	−0.0886890	−	0.0335374i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−15.8015	−	15.8015i	−1.18106	−	1.18106i	−0.979470	−	0.201591i	$-0.935389\pi$
$179$	−15.8015	−	15.8015i	−1.18106	−	1.18106i	−0.201591	−	0.979470i	$-0.564611\pi$
$180$		0			0
$181$	15.2721	−	15.2721i	1.13517	−	1.13517i	0.145860	−	0.989305i	$-0.453405\pi$
$181$	15.2721	−	15.2721i	1.13517	−	1.13517i	0.989305	−	0.145860i	$-0.0465950\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−7.87288	+	1.69398i	−0.578825	+	0.124544i
$186$		0			0
$187$			2.79228i			0.204192i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	1.05725i		−	0.0764996i	−0.999268	−	0.0382498i	$-0.987822\pi$
$191$		−	1.05725i		−	0.0764996i	0.999268	−	0.0382498i	$-0.0121783\pi$
$192$		0			0
$193$	−13.2076	−	13.2076i	−0.950707	−	0.950707i	0.0481339	−	0.998841i	$-0.484673\pi$
$193$	−13.2076	−	13.2076i	−0.950707	−	0.950707i	−0.998841	+	0.0481339i	$0.984673\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	24.5068			1.74604			0.873018	−	0.487688i	$-0.162160\pi$
$197$	24.5068			1.74604			0.873018	+	0.487688i	$0.162160\pi$
$198$		0			0
$199$	−18.2205			−1.29161			−0.645807	−	0.763500i	$-0.723479\pi$
$199$	−18.2205			−1.29161			−0.645807	+	0.763500i	$0.723479\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			1.88452i			0.132267i
$204$		0			0
$205$	4.12983	+	19.1936i	0.288440	+	1.34054i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.551851			0.0381724
$210$		0			0
$211$	1.37607	+	1.37607i	0.0947327	+	0.0947327i	0.752885	−	0.658152i	$-0.228662\pi$
$211$	1.37607	+	1.37607i	0.0947327	+	0.0947327i	−0.658152	+	0.752885i	$0.728662\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.64976	+	7.66734i	0.112513	+	0.522908i
$216$		0			0
$217$	0.731431	+	0.731431i	0.0496528	+	0.0496528i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.46122	−	1.46122i	0.0982923	−	0.0982923i
$222$		0			0
$223$	−17.6426	−	17.6426i	−1.18143	−	1.18143i	−0.979373	−	0.202061i	$-0.935236\pi$
$223$	−17.6426	−	17.6426i	−1.18143	−	1.18143i	−0.202061	−	0.979373i	$-0.564764\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			23.7322i			1.57516i	0.616212	+	0.787581i	$0.288667\pi$
$227$			23.7322i			1.57516i	−0.616212	+	0.787581i	$0.711333\pi$
$228$		0			0
$229$	9.07143	+	9.07143i	0.599457	+	0.599457i	0.940168	−	0.340711i	$-0.110668\pi$
$229$	9.07143	+	9.07143i	0.599457	+	0.599457i	−0.340711	+	0.940168i	$0.610668\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.6733	+	12.6733i	0.830254	+	0.830254i	0.987551	−	0.157297i	$-0.0502781\pi$
$233$	12.6733	+	12.6733i	0.830254	+	0.830254i	−0.157297	+	0.987551i	$0.550278\pi$
$234$		0			0
$235$	11.4136	−	17.6719i	0.744544	−	1.15279i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	24.2418			1.56807			0.784035	−	0.620716i	$-0.213158\pi$
$239$	24.2418			1.56807			0.784035	+	0.620716i	$0.213158\pi$
$240$		0			0
$241$	−17.3074			−1.11487			−0.557433	−	0.830222i	$-0.688214\pi$
$241$	−17.3074			−1.11487			−0.557433	+	0.830222i	$0.688214\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−15.1647	+	3.26294i	−0.968836	+	0.208462i
$246$		0			0
$247$	−0.288788	−	0.288788i	−0.0183751	−	0.0183751i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−4.86892	−	4.86892i	−0.307324	−	0.307324i	0.536547	−	0.843871i	$-0.319728\pi$
$251$	−4.86892	−	4.86892i	−0.307324	−	0.307324i	−0.843871	+	0.536547i	$0.819728\pi$
$252$		0			0
$253$			15.0620i			0.946941i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2.99023	−	2.99023i	−0.186525	−	0.186525i	0.607667	−	0.794192i	$-0.292106\pi$
$257$	−2.99023	−	2.99023i	−0.186525	−	0.186525i	−0.794192	+	0.607667i	$0.792106\pi$
$258$		0			0
$259$	0.638854	−	0.638854i	0.0396965	−	0.0396965i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	20.9527	+	20.9527i	1.29200	+	1.29200i	0.933551	+	0.358446i	$0.116693\pi$
$263$	20.9527	+	20.9527i	1.29200	+	1.29200i	0.358446	+	0.933551i	$0.383307\pi$
$264$		0			0
$265$	−11.6034	+	2.49667i	−0.712792	+	0.153369i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.6508	+	14.6508i	0.893278	+	0.893278i	0.994830	−	0.101552i	$-0.0323810\pi$
$269$	14.6508	+	14.6508i	0.893278	+	0.893278i	−0.101552	+	0.994830i	$0.532381\pi$
$270$		0			0
$271$	24.9661			1.51658			0.758290	−	0.651917i	$-0.226035\pi$
$271$	24.9661			1.51658			0.758290	+	0.651917i	$0.226035\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−8.89009	−	3.36176i	−0.536092	−	0.202722i
$276$		0			0
$277$			27.7345i			1.66640i	0.552968	+	0.833202i	$0.313495\pi$
$277$			27.7345i			1.66640i	−0.552968	+	0.833202i	$0.686505\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	11.9673			0.713911			0.356955	−	0.934121i	$-0.383815\pi$
$281$	11.9673			0.713911			0.356955	+	0.934121i	$0.383815\pi$
$282$		0			0
$283$	16.8585			1.00214			0.501068	−	0.865408i	$-0.332941\pi$
$283$	16.8585			1.00214			0.501068	+	0.865408i	$0.332941\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1.55749	−	1.55749i	−0.0919357	−	0.0919357i
$288$		0			0
$289$		−	14.8422i		−	0.873073i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	10.0720i		−	0.588411i	−0.955742	−	0.294206i	$-0.904945\pi$
$293$		−	10.0720i		−	0.588411i	0.955742	−	0.294206i	$-0.0950551\pi$
$294$		0			0
$295$	−23.9824	−	15.4894i	−1.39631	−	0.901828i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.88205	−	7.88205i	0.455831	−	0.455831i
$300$		0			0
$301$	−0.622176	−	0.622176i	−0.0358616	−	0.0358616i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.257448	+	0.398610i	−0.0147414	+	0.0228243i
$306$		0			0
$307$	−2.33633			−0.133341			−0.0666707	−	0.997775i	$-0.521238\pi$
$307$	−2.33633			−0.133341			−0.0666707	+	0.997775i	$0.521238\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−30.4952			−1.72922			−0.864612	−	0.502440i	$-0.832436\pi$
$311$	−30.4952			−1.72922			−0.864612	+	0.502440i	$0.832436\pi$
$312$		0			0
$313$	−8.25345	+	8.25345i	−0.466512	+	0.466512i	−0.900783	−	0.434270i	$-0.857006\pi$
$313$	−8.25345	+	8.25345i	−0.466512	+	0.466512i	0.434270	+	0.900783i	$0.357006\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	7.23096i		−	0.406131i	−0.979165	−	0.203066i	$-0.934910\pi$
$317$		−	7.23096i		−	0.406131i	0.979165	−	0.203066i	$-0.0650904\pi$
$318$		0			0
$319$			14.2797i			0.799508i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.426447			0.0237282
$324$		0			0
$325$	2.89301	+	6.41147i	0.160475	+	0.355644i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			2.36018i			0.130121i
$330$		0			0
$331$	6.17028	−	6.17028i	0.339149	−	0.339149i	−0.516898	−	0.856047i	$-0.672913\pi$
$331$	6.17028	−	6.17028i	0.339149	−	0.339149i	0.856047	+	0.516898i	$0.172913\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.78814	−	22.2531i	−0.261604	−	1.21582i
$336$		0			0
$337$	−3.81736	+	3.81736i	−0.207945	+	0.207945i	−0.803393	−	0.595449i	$-0.796974\pi$
$337$	−3.81736	+	3.81736i	−0.207945	+	0.207945i	0.595449	+	0.803393i	$0.296974\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5.54231	+	5.54231i	0.300133	+	0.300133i
$342$		0			0
$343$	2.47228	−	2.47228i	0.133490	−	0.133490i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.76593			0.0948002			0.0474001	−	0.998876i	$-0.484906\pi$
$347$	1.76593			0.0948002			0.0474001	+	0.998876i	$0.484906\pi$
$348$		0			0
$349$	−9.53222	+	9.53222i	−0.510248	+	0.510248i	−0.914602	−	0.404354i	$-0.867496\pi$
$349$	−9.53222	+	9.53222i	−0.510248	+	0.510248i	0.404354	+	0.914602i	$0.367496\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−20.3690	+	20.3690i	−1.08413	+	1.08413i	−0.0880116	+	0.996119i	$0.528051\pi$
$353$	−20.3690	+	20.3690i	−1.08413	+	1.08413i	−0.996119	+	0.0880116i	$0.971949\pi$
$354$		0			0
$355$	−3.18150	−	14.7862i	−0.168857	−	0.784770i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			12.8277i			0.677018i	0.940963	+	0.338509i	$0.109923\pi$
$359$			12.8277i			0.677018i	−0.940963	+	0.338509i	$0.890077\pi$
$360$		0			0
$361$			18.9157i			0.995564i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.05002	−	1.96990i	−0.159645	−	0.103109i
$366$		0			0
$367$	−11.3228	+	11.3228i	−0.591045	+	0.591045i	−0.937914	−	0.346869i	$-0.887245\pi$
$367$	−11.3228	+	11.3228i	−0.591045	+	0.591045i	0.346869	+	0.937914i	$0.387245\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.941573	−	0.941573i	0.0488840	−	0.0488840i
$372$		0			0
$373$	−31.7028			−1.64151			−0.820753	−	0.571283i	$-0.806446\pi$
$373$	−31.7028			−1.64151			−0.820753	+	0.571283i	$0.806446\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	7.47264	−	7.47264i	0.384861	−	0.384861i
$378$		0			0
$379$	23.8161	+	23.8161i	1.22335	+	1.22335i	0.966433	+	0.256919i	$0.0827074\pi$
$379$	23.8161	+	23.8161i	1.22335	+	1.22335i	0.256919	+	0.966433i	$0.417293\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−5.85262	+	5.85262i	−0.299055	+	0.299055i	−0.840644	−	0.541589i	$-0.817823\pi$
$383$	−5.85262	+	5.85262i	−0.299055	+	0.299055i	0.541589	+	0.840644i	$0.317823\pi$
$384$		0			0
$385$	1.04245	−	0.224302i	0.0531283	−	0.0114315i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	14.9797	−	14.9797i	0.759502	−	0.759502i	−0.216730	−	0.976232i	$-0.569539\pi$
$389$	14.9797	−	14.9797i	0.759502	−	0.759502i	0.976232	+	0.216730i	$0.0695391\pi$
$390$		0			0
$391$			11.6393i			0.588624i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−11.3494	+	2.44201i	−0.571049	+	0.122871i
$396$		0			0
$397$	−12.6966			−0.637223			−0.318612	−	0.947885i	$-0.603217\pi$
$397$	−12.6966			−0.637223			−0.318612	+	0.947885i	$0.603217\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	13.5414i		−	0.676223i	−0.941106	−	0.338112i	$-0.890212\pi$
$401$		−	13.5414i		−	0.676223i	0.941106	−	0.338112i	$-0.109788\pi$
$402$		0			0
$403$		−	5.80065i		−	0.288951i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.84082	−	4.84082i	0.239951	−	0.239951i
$408$		0			0
$409$	21.7152			1.07375			0.536875	−	0.843662i	$-0.319605\pi$
$409$	21.7152			1.07375			0.536875	+	0.843662i	$0.319605\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.20299			0.157609
$414$		0			0
$415$	2.39186	+	11.1163i	0.117412	+	0.545678i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−15.9250	−	15.9250i	−0.777986	−	0.777986i	0.201502	−	0.979488i	$-0.435418\pi$
$419$	−15.9250	−	15.9250i	−0.777986	−	0.777986i	−0.979488	+	0.201502i	$0.935418\pi$
$420$		0			0
$421$	12.3895	−	12.3895i	0.603827	−	0.603827i	−0.337499	−	0.941326i	$-0.609581\pi$
$421$	12.3895	−	12.3895i	0.603827	−	0.603827i	0.941326	+	0.337499i	$0.109581\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.86988	−	2.59782i	−0.333238	−	0.126013i
$426$		0			0
$427$		−	0.0532366i		−	0.00257630i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			15.4990i			0.746560i	0.927719	+	0.373280i	$0.121767\pi$
$431$			15.4990i			0.746560i	−0.927719	+	0.373280i	$0.878233\pi$
$432$		0			0
$433$	−0.432991	−	0.432991i	−0.0208082	−	0.0208082i	0.696626	−	0.717434i	$-0.254684\pi$
$433$	−0.432991	−	0.432991i	−0.0208082	−	0.0208082i	−0.717434	+	0.696626i	$0.754684\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.30032			0.110039
$438$		0			0
$439$	12.3191			0.587957			0.293978	−	0.955812i	$-0.405021\pi$
$439$	12.3191			0.587957			0.293978	+	0.955812i	$0.405021\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			16.1150i			0.765646i	0.923822	+	0.382823i	$0.125048\pi$
$443$			16.1150i			0.765646i	−0.923822	+	0.382823i	$0.874952\pi$
$444$		0			0
$445$	19.2257	−	4.13674i	0.911385	−	0.196100i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−15.6680			−0.739421			−0.369710	−	0.929147i	$-0.620543\pi$
$449$	−15.6680			−0.739421			−0.369710	+	0.929147i	$0.620543\pi$
$450$		0			0
$451$	−11.8016	−	11.8016i	−0.555718	−	0.555718i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.662901	−	0.428144i	−0.0310773	−	0.0200717i
$456$		0			0
$457$	−14.3003	−	14.3003i	−0.668942	−	0.668942i	0.288529	−	0.957471i	$-0.406834\pi$
$457$	−14.3003	−	14.3003i	−0.668942	−	0.668942i	−0.957471	+	0.288529i	$0.906834\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	14.5085	−	14.5085i	0.675727	−	0.675727i	−0.283304	−	0.959030i	$-0.591430\pi$
$461$	14.5085	−	14.5085i	0.675727	−	0.675727i	0.959030	+	0.283304i	$0.0914304\pi$
$462$		0			0
$463$	1.97616	+	1.97616i	0.0918399	+	0.0918399i	0.751534	−	0.659694i	$-0.229314\pi$
$463$	1.97616	+	1.97616i	0.0918399	+	0.0918399i	−0.659694	+	0.751534i	$0.729314\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	25.0244i		−	1.15799i	−0.815331	−	0.578995i	$-0.803445\pi$
$467$		−	25.0244i		−	1.15799i	0.815331	−	0.578995i	$-0.196555\pi$
$468$		0			0
$469$	1.80576	+	1.80576i	0.0833821	+	0.0833821i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.71444	−	4.71444i	−0.216770	−	0.216770i
$474$		0			0
$475$	−0.513419	+	1.35773i	−0.0235573	+	0.0622967i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−24.2643			−1.10867			−0.554333	−	0.832295i	$-0.687027\pi$
$479$	−24.2643			−1.10867			−0.554333	+	0.832295i	$0.687027\pi$
$480$		0			0
$481$	−5.06647			−0.231011
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−17.6306	−	11.3869i	−0.800562	−	0.517055i
$486$		0			0
$487$	−7.40444	−	7.40444i	−0.335527	−	0.335527i	0.519154	−	0.854681i	$-0.326247\pi$
$487$	−7.40444	−	7.40444i	−0.335527	−	0.335527i	−0.854681	+	0.519154i	$0.826247\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−27.8155	−	27.8155i	−1.25530	−	1.25530i	−0.953314	−	0.301982i	$-0.902352\pi$
$491$	−27.8155	−	27.8155i	−1.25530	−	1.25530i	−0.301982	−	0.953314i	$-0.597648\pi$
$492$		0			0
$493$			11.0347i			0.496978i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.19984	+	1.19984i	0.0538204	+	0.0538204i
$498$		0			0
$499$	−24.0617	+	24.0617i	−1.07715	+	1.07715i	−0.0803846	+	0.996764i	$0.525615\pi$
$499$	−24.0617	+	24.0617i	−1.07715	+	1.07715i	−0.996764	+	0.0803846i	$0.974385\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	20.5076	+	20.5076i	0.914389	+	0.914389i	0.996614	−	0.0822248i	$-0.0262026\pi$
$503$	20.5076	+	20.5076i	0.914389	+	0.914389i	−0.0822248	+	0.996614i	$0.526203\pi$
$504$		0			0
$505$	−3.80294	+	5.88815i	−0.169229	+	0.262019i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9.52950	−	9.52950i	−0.422388	−	0.422388i	0.463637	−	0.886025i	$-0.346544\pi$
$509$	−9.52950	−	9.52950i	−0.422388	−	0.422388i	−0.886025	+	0.463637i	$0.846544\pi$
$510$		0			0
$511$	0.407348			0.0180200
$512$		0			0
$513$		0			0
$514$		0			0
$515$	20.2790	+	13.0975i	0.893598	+	0.577143i
$516$		0			0
$517$			17.8839i			0.786534i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−17.8874			−0.783659			−0.391830	−	0.920038i	$-0.628158\pi$
$521$	−17.8874			−0.783659			−0.391830	+	0.920038i	$0.628158\pi$
$522$		0			0
$523$	−40.3081			−1.76255			−0.881274	−	0.472605i	$-0.843314\pi$
$523$	−40.3081			−1.76255			−0.881274	+	0.472605i	$0.843314\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.28286	+	4.28286i	0.186564	+	0.186564i
$528$		0			0
$529$			39.7841i			1.72974i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			12.3518i			0.535014i
$534$		0			0
$535$	−2.82959	−	13.1507i	−0.122334	−	0.568552i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	9.32436	−	9.32436i	0.401629	−	0.401629i
$540$		0			0
$541$	−28.8625	−	28.8625i	−1.24090	−	1.24090i	−0.959630	−	0.281266i	$-0.909246\pi$
$541$	−28.8625	−	28.8625i	−1.24090	−	1.24090i	−0.281266	−	0.959630i	$-0.590754\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−17.8490	−	11.5280i	−0.764568	−	0.493807i
$546$		0			0
$547$	37.5936			1.60739			0.803693	−	0.595044i	$-0.202865\pi$
$547$	37.5936			1.60739			0.803693	+	0.595044i	$0.202865\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	2.18084			0.0929069
$552$		0			0
$553$	0.920959	−	0.920959i	0.0391632	−	0.0391632i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	0.325753i		−	0.0138026i	−0.999976	−	0.00690129i	$-0.997803\pi$
$557$		−	0.325753i		−	0.0138026i	0.999976	−	0.00690129i	$-0.00219677\pi$
$558$		0			0
$559$			4.93420i			0.208694i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−44.3309			−1.86833			−0.934163	−	0.356846i	$-0.883852\pi$
$563$	−44.3309			−1.86833			−0.934163	+	0.356846i	$0.883852\pi$
$564$		0			0
$565$	29.5378	+	19.0774i	1.24267	+	0.802594i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	8.80728i		−	0.369220i	−0.982812	−	0.184610i	$-0.940898\pi$
$569$		−	8.80728i		−	0.369220i	0.982812	−	0.184610i	$-0.0591022\pi$
$570$		0			0
$571$	−7.90983	+	7.90983i	−0.331016	+	0.331016i	−0.852972	−	0.521956i	$-0.825202\pi$
$571$	−7.90983	+	7.90983i	−0.331016	+	0.331016i	0.521956	+	0.852972i	$0.325202\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−37.0572	−	14.0131i	−1.54539	−	0.584385i
$576$		0			0
$577$	−19.3021	+	19.3021i	−0.803557	+	0.803557i	−0.983650	−	0.180093i	$-0.942360\pi$
$577$	−19.3021	+	19.3021i	−0.803557	+	0.803557i	0.180093	+	0.983650i	$0.442360\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.902047	−	0.902047i	−0.0374232	−	0.0374232i
$582$		0			0
$583$	7.13462	−	7.13462i	0.295486	−	0.295486i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	28.2797			1.16723			0.583613	−	0.812032i	$-0.301638\pi$
$587$	28.2797			1.16723			0.583613	+	0.812032i	$0.301638\pi$
$588$		0			0
$589$	0.846441	−	0.846441i	0.0348770	−	0.0348770i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18.9784	+	18.9784i	−0.779351	+	0.779351i	−0.979720	−	0.200369i	$-0.935786\pi$
$593$	−18.9784	+	18.9784i	−0.779351	+	0.779351i	0.200369	+	0.979720i	$0.435786\pi$
$594$		0			0
$595$	0.805563	−	0.173331i	0.0330249	−	0.00710586i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	15.2909i		−	0.624769i	−0.949956	−	0.312384i	$-0.898872\pi$
$599$		−	15.2909i		−	0.624769i	0.949956	−	0.312384i	$-0.101128\pi$
$600$		0			0
$601$		−	8.54104i		−	0.348396i	−0.984711	−	0.174198i	$-0.944267\pi$
$601$		−	8.54104i		−	0.348396i	0.984711	−	0.174198i	$-0.0557333\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−16.1474	+	3.47438i	−0.656484	+	0.141254i
$606$		0			0
$607$	−26.6010	+	26.6010i	−1.07970	+	1.07970i	−0.0831652	+	0.996536i	$0.526503\pi$
$607$	−26.6010	+	26.6010i	−1.07970	+	1.07970i	−0.996536	+	0.0831652i	$0.973497\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.35878	−	9.35878i	0.378616	−	0.378616i
$612$		0			0
$613$	−6.11869			−0.247131			−0.123566	−	0.992336i	$-0.539433\pi$
$613$	−6.11869			−0.247131			−0.123566	+	0.992336i	$0.539433\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.179176	−	0.179176i	0.00721337	−	0.00721337i	−0.703491	−	0.710704i	$-0.748376\pi$
$617$	0.179176	−	0.179176i	0.00721337	−	0.00721337i	0.710704	+	0.703491i	$0.248376\pi$
$618$		0			0
$619$	−8.82114	−	8.82114i	−0.354552	−	0.354552i	0.507248	−	0.861800i	$-0.330663\pi$
$619$	−8.82114	−	8.82114i	−0.354552	−	0.354552i	−0.861800	+	0.507248i	$0.830663\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.56009	+	1.56009i	−0.0625038	+	0.0625038i
$624$		0			0
$625$	16.5419	−	18.7447i	0.661677	−	0.749789i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3.74078	−	3.74078i	0.149155	−	0.149155i
$630$		0			0
$631$			11.9376i			0.475228i	0.971360	+	0.237614i	$0.0763654\pi$
$631$			11.9376i			0.475228i	−0.971360	+	0.237614i	$0.923635\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13.2014	−	8.52632i	−0.523882	−	0.338357i
$636$		0			0
$637$	−9.75900			−0.386666
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	18.1107i		−	0.715330i	−0.933850	−	0.357665i	$-0.883573\pi$
$641$		−	18.1107i		−	0.715330i	0.933850	−	0.357665i	$-0.116427\pi$
$642$		0			0
$643$			25.0110i			0.986336i	0.869934	+	0.493168i	$0.164161\pi$
$643$			25.0110i			0.986336i	−0.869934	+	0.493168i	$0.835839\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	21.0736	−	21.0736i	0.828490	−	0.828490i	−0.158818	−	0.987308i	$-0.550768\pi$
$647$	21.0736	−	21.0736i	0.828490	−	0.828490i	0.987308	+	0.158818i	$0.0507683\pi$
$648$		0			0
$649$	24.2702			0.952688
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.05500			−0.119551			−0.0597757	−	0.998212i	$-0.519039\pi$
$653$	−3.05500			−0.119551			−0.0597757	+	0.998212i	$0.519039\pi$
$654$		0			0
$655$	2.10093	+	1.35692i	0.0820903	+	0.0530192i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−6.67289	−	6.67289i	−0.259939	−	0.259939i	0.565090	−	0.825029i	$-0.308841\pi$
$659$	−6.67289	−	6.67289i	−0.259939	−	0.259939i	−0.825029	+	0.565090i	$0.808841\pi$
$660$		0			0
$661$	19.4725	−	19.4725i	0.757391	−	0.757391i	−0.218456	−	0.975847i	$-0.570102\pi$
$661$	19.4725	−	19.4725i	0.757391	−	0.757391i	0.975847	+	0.218456i	$0.0701020\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.0342561	−	0.159207i	−0.00132840	−	0.00617379i
$666$		0			0
$667$			59.5230i			2.30474i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	0.403392i		−	0.0155728i
$672$		0			0
$673$	28.5244	+	28.5244i	1.09954	+	1.09954i	0.994465	+	0.105072i	$0.0335073\pi$
$673$	28.5244	+	28.5244i	1.09954	+	1.09954i	0.105072	+	0.994465i	$0.466493\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−39.9514			−1.53546			−0.767728	−	0.640776i	$-0.778613\pi$
$677$	−39.9514			−1.53546			−0.767728	+	0.640776i	$0.778613\pi$
$678$		0			0
$679$	2.35466			0.0903636
$680$		0			0
$681$		0			0
$682$		0			0
$683$			38.8034i			1.48477i	0.669973	+	0.742386i	$0.266306\pi$
$683$			38.8034i			1.48477i	−0.669973	+	0.742386i	$0.733694\pi$
$684$		0			0
$685$	−28.0445	−	18.1130i	−1.07153	−	0.692061i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−7.46719			−0.284477
$690$		0			0
$691$	−30.7324	−	30.7324i	−1.16911	−	1.16911i	−0.982418	−	0.186697i	$-0.940222\pi$
$691$	−30.7324	−	30.7324i	−1.16911	−	1.16911i	−0.186697	−	0.982418i	$-0.559778\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.829699	+	1.28463i	−0.0314723	+	0.0487289i
$696$		0			0
$697$	−9.11980	−	9.11980i	−0.345437	−	0.345437i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	15.4043	−	15.4043i	0.581812	−	0.581812i	−0.353589	−	0.935401i	$-0.615039\pi$
$701$	15.4043	−	15.4043i	0.581812	−	0.581812i	0.935401	+	0.353589i	$0.115039\pi$
$702$		0			0
$703$	−0.739307	−	0.739307i	−0.0278835	−	0.0278835i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	0.786395i		−	0.0295754i
$708$		0			0
$709$	−2.80368	−	2.80368i	−0.105294	−	0.105294i	0.652497	−	0.757791i	$-0.273721\pi$
$709$	−2.80368	−	2.80368i	−0.105294	−	0.105294i	−0.757791	+	0.652497i	$0.773721\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	23.1024	+	23.1024i	0.865192	+	0.865192i
$714$		0			0
$715$	−5.02303	−	3.24420i	−0.187851	−	0.121326i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	42.0497			1.56819			0.784094	−	0.620642i	$-0.213128\pi$
$719$	42.0497			1.56819			0.784094	+	0.620642i	$0.213128\pi$
$720$		0			0
$721$	−2.70837			−0.100865
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−35.1324	−	13.2852i	−1.30478	−	0.493400i
$726$		0			0
$727$	35.5338	+	35.5338i	1.31787	+	1.31787i	0.915456	+	0.402419i	$0.131830\pi$
$727$	35.5338	+	35.5338i	1.31787	+	1.31787i	0.402419	+	0.915456i	$0.368170\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.64312	−	3.64312i	−0.134746	−	0.134746i
$732$		0			0
$733$		−	7.47504i		−	0.276097i	−0.990425	−	0.138048i	$-0.955917\pi$
$733$		−	7.47504i		−	0.276097i	0.990425	−	0.138048i	$-0.0440830\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13.6828	+	13.6828i	0.504014	+	0.504014i
$738$		0			0
$739$	−24.9409	+	24.9409i	−0.917465	+	0.917465i	−0.996844	−	0.0793792i	$-0.974706\pi$
$739$	−24.9409	+	24.9409i	−0.917465	+	0.917465i	0.0793792	+	0.996844i	$0.474706\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.0898	−	12.0898i	−0.443531	−	0.443531i	0.449666	−	0.893197i	$-0.351543\pi$
$743$	−12.0898	−	12.0898i	−0.443531	−	0.443531i	−0.893197	+	0.449666i	$0.851543\pi$
$744$		0			0
$745$	−38.4823	−	24.8543i	−1.40988	−	0.910593i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.06713	+	1.06713i	0.0389919	+	0.0389919i
$750$		0			0
$751$	12.2060			0.445404			0.222702	−	0.974887i	$-0.428512\pi$
$751$	12.2060			0.445404			0.222702	+	0.974887i	$0.428512\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	29.6015	−	6.36926i	1.07731	−	0.231801i
$756$		0			0
$757$		−	7.64467i		−	0.277850i	−0.990303	−	0.138925i	$-0.955635\pi$
$757$		−	7.64467i		−	0.277850i	0.990303	−	0.138925i	$-0.0443648\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−35.2785			−1.27885			−0.639423	−	0.768855i	$-0.720827\pi$
$761$	−35.2785			−1.27885			−0.639423	+	0.768855i	$0.720827\pi$
$762$		0			0
$763$	2.38384			0.0863008
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−12.7007	−	12.7007i	−0.458597	−	0.458597i
$768$		0			0
$769$			21.2930i			0.767846i	0.923365	+	0.383923i	$0.125427\pi$
$769$			21.2930i			0.767846i	−0.923365	+	0.383923i	$0.874573\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			54.9059i			1.97483i	0.158161	+	0.987413i	$0.449443\pi$
$773$			54.9059i			1.97483i	−0.158161	+	0.987413i	$0.550557\pi$
$774$		0			0
$775$	−18.7921	+	8.47946i	−0.675033	+	0.304591i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−1.80239	+	1.80239i	−0.0645772	+	0.0645772i
$780$		0			0
$781$	9.09164	+	9.09164i	0.325324	+	0.325324i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.312070	−	1.45036i	−0.0111383	−	0.0517656i
$786$		0			0
$787$	−14.8192			−0.528246			−0.264123	−	0.964489i	$-0.585082\pi$
$787$	−14.8192			−0.528246			−0.264123	+	0.964489i	$0.585082\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.94495			−0.140266
$792$		0			0
$793$	−0.211098	+	0.211098i	−0.00749630	+	0.00749630i
$794$		0			0
$795$		0			0
$796$		0			0
$797$			18.8133i			0.666403i	0.942856	+	0.333201i	$0.108129\pi$
$797$			18.8133i			0.666403i	−0.942856	+	0.333201i	$0.891871\pi$
$798$		0			0
$799$			13.8199i			0.488914i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.08661			0.108924
$804$		0			0
$805$	4.34534	−	0.934973i	0.153153	−	0.0329535i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			5.64088i			0.198323i	0.995071	+	0.0991615i	$0.0316160\pi$
$809$			5.64088i			0.198323i	−0.995071	+	0.0991615i	$0.968384\pi$
$810$		0			0
$811$	−10.9683	+	10.9683i	−0.385147	+	0.385147i	−0.872953	−	0.487805i	$-0.837798\pi$
$811$	−10.9683	+	10.9683i	−0.385147	+	0.385147i	0.487805	+	0.872953i	$0.337798\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−12.9340	+	2.78296i	−0.453057	+	0.0974830i
$816$		0			0
$817$	−0.720006	+	0.720006i	−0.0251898	+	0.0251898i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	11.2050	+	11.2050i	0.391058	+	0.391058i	0.875064	−	0.484006i	$-0.160819\pi$
$821$	11.2050	+	11.2050i	0.391058	+	0.391058i	−0.484006	+	0.875064i	$0.660819\pi$
$822$		0			0
$823$	−1.58909	+	1.58909i	−0.0553921	+	0.0553921i	−0.734260	−	0.678868i	$-0.762471\pi$
$823$	−1.58909	+	1.58909i	−0.0553921	+	0.0553921i	0.678868	+	0.734260i	$0.262471\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−4.33443			−0.150723			−0.0753614	−	0.997156i	$-0.524011\pi$
$827$	−4.33443			−0.150723			−0.0753614	+	0.997156i	$0.524011\pi$
$828$		0			0
$829$	2.16612	−	2.16612i	0.0752326	−	0.0752326i	−0.668489	−	0.743722i	$-0.733059\pi$
$829$	2.16612	−	2.16612i	0.0752326	−	0.0752326i	0.743722	+	0.668489i	$0.233059\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	7.20547	−	7.20547i	0.249655	−	0.249655i
$834$		0			0
$835$	30.1758	+	19.4895i	1.04428	+	0.674462i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	20.6713i		−	0.713651i	−0.934171	−	0.356826i	$-0.883859\pi$
$839$		−	20.6713i		−	0.713651i	0.934171	−	0.356826i	$-0.116141\pi$
$840$		0			0
$841$			27.4312i			0.945905i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−5.18384	−	24.0922i	−0.178330	−	0.828796i
$846$		0			0
$847$	1.31030	−	1.31030i	0.0450224	−	0.0450224i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	20.1784	−	20.1784i	0.691705	−	0.691705i
$852$		0			0
$853$	−37.3727			−1.27962			−0.639808	−	0.768535i	$-0.720986\pi$
$853$	−37.3727			−1.27962			−0.639808	+	0.768535i	$0.720986\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.707244	+	0.707244i	−0.0241590	+	0.0241590i	−0.719083	−	0.694924i	$-0.755438\pi$
$857$	−0.707244	+	0.707244i	−0.0241590	+	0.0241590i	0.694924	+	0.719083i	$0.255438\pi$
$858$		0			0
$859$	7.21858	+	7.21858i	0.246295	+	0.246295i	0.819448	−	0.573153i	$-0.194280\pi$
$859$	7.21858	+	7.21858i	0.246295	+	0.246295i	−0.573153	+	0.819448i	$0.694280\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−2.00403	+	2.00403i	−0.0682181	+	0.0682181i	−0.740393	−	0.672175i	$-0.765360\pi$
$863$	−2.00403	+	2.00403i	−0.0682181	+	0.0682181i	0.672175	+	0.740393i	$0.265360\pi$
$864$		0			0
$865$	0.117880	+	0.547855i	0.00400805	+	0.0186276i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6.97843	−	6.97843i	0.236727	−	0.236727i
$870$		0			0
$871$		−	14.3206i		−	0.485237i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.418003	+	2.77344i	−0.0141311	+	0.0937593i
$876$		0			0
$877$	−19.2570			−0.650263			−0.325131	−	0.945669i	$-0.605409\pi$
$877$	−19.2570			−0.650263			−0.325131	+	0.945669i	$0.605409\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	46.9929i		−	1.58323i	−0.611019	−	0.791616i	$-0.709240\pi$
$881$		−	46.9929i		−	1.58323i	0.611019	−	0.791616i	$-0.290760\pi$
$882$		0			0
$883$			30.4104i			1.02339i	0.859166	+	0.511696i	$0.170983\pi$
$883$			30.4104i			1.02339i	−0.859166	+	0.511696i	$0.829017\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	13.3684	−	13.3684i	0.448868	−	0.448868i	−0.446110	−	0.894978i	$-0.647191\pi$
$887$	13.3684	−	13.3684i	0.448868	−	0.448868i	0.894978	+	0.446110i	$0.147191\pi$
$888$		0			0
$889$	1.76312			0.0591333
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.73130			0.0913993
$894$		0			0
$895$	−27.1102	+	41.9751i	−0.906195	+	1.40307i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	21.9024	+	21.9024i	0.730487	+	0.730487i
$900$		0			0
$901$	5.51333	−	5.51333i	0.183676	−	0.183676i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−40.5688	−	26.2019i	−1.34855	−	0.870982i
$906$		0			0
$907$		−	31.1982i		−	1.03592i	−0.855405	−	0.517960i	$-0.826692\pi$
$907$		−	31.1982i		−	1.03592i	0.855405	−	0.517960i	$-0.173308\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			10.8789i			0.360435i	0.983627	+	0.180218i	$0.0576802\pi$
$911$			10.8789i			0.360435i	−0.983627	+	0.180218i	$0.942320\pi$
$912$		0			0
$913$	−6.83512	−	6.83512i	−0.226210	−	0.226210i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−0.280592			−0.00926595
$918$		0			0
$919$	46.5372			1.53512			0.767560	−	0.640977i	$-0.221471\pi$
$919$	46.5372			1.53512			0.767560	+	0.640977i	$0.221471\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	9.51543i		−	0.313204i
$924$		0			0
$925$	7.40622	+	16.4136i	0.243515	+	0.539676i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	50.4820			1.65626			0.828131	−	0.560535i	$-0.189404\pi$
$929$	50.4820			1.65626			0.828131	+	0.560535i	$0.189404\pi$
$930$		0			0
$931$	−1.42405	−	1.42405i	−0.0466713	−	0.0466713i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	6.10403	−	1.31339i	0.199623	−	0.0429523i
$936$		0			0
$937$	4.09140	+	4.09140i	0.133660	+	0.133660i	0.770772	−	0.637111i	$-0.219871\pi$
$937$	4.09140	+	4.09140i	0.133660	+	0.133660i	−0.637111	+	0.770772i	$0.719871\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−39.9686	+	39.9686i	−1.30294	+	1.30294i	−0.376537	+	0.926402i	$0.622885\pi$
$941$	−39.9686	+	39.9686i	−1.30294	+	1.30294i	−0.926402	+	0.376537i	$0.877115\pi$
$942$		0			0
$943$	−49.1937	−	49.1937i	−1.60197	−	1.60197i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			7.83697i			0.254667i	0.991860	+	0.127334i	$0.0406419\pi$
$947$			7.83697i			0.254667i	−0.991860	+	0.127334i	$0.959358\pi$
$948$		0			0
$949$	−1.61525	−	1.61525i	−0.0524331	−	0.0524331i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	28.4387	+	28.4387i	0.921220	+	0.921220i	0.997116	−	0.0758955i	$-0.0241815\pi$
$953$	28.4387	+	28.4387i	0.921220	+	0.921220i	−0.0758955	+	0.997116i	$0.524182\pi$
$954$		0			0
$955$	−2.31118	+	0.497289i	−0.0747880	+	0.0160919i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3.74551			0.120949
$960$		0			0
$961$	−13.9982			−0.451555
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−22.6600	+	35.0848i	−0.729452	+	1.12942i
$966$		0			0
$967$	−21.6207	−	21.6207i	−0.695275	−	0.695275i	0.268113	−	0.963388i	$-0.413600\pi$
$967$	−21.6207	−	21.6207i	−0.695275	−	0.695275i	−0.963388	+	0.268113i	$0.913600\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	28.0929	+	28.0929i	0.901543	+	0.901543i	0.995570	−	0.0940266i	$-0.0299739\pi$
$971$	28.0929	+	28.0929i	0.901543	+	0.901543i	−0.0940266	+	0.995570i	$0.529974\pi$
$972$		0			0
$973$		−	0.171570i		−	0.00550029i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.3100	+	11.3100i	0.361838	+	0.361838i	0.864489	−	0.502651i	$-0.167642\pi$
$977$	11.3100	+	11.3100i	0.361838	+	0.361838i	−0.502651	+	0.864489i	$0.667642\pi$
$978$		0			0
$979$	−11.8214	+	11.8214i	−0.377812	+	0.377812i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−3.65368	−	3.65368i	−0.116534	−	0.116534i	0.646435	−	0.762969i	$-0.276259\pi$
$983$	−3.65368	−	3.65368i	−0.116534	−	0.116534i	−0.762969	+	0.646435i	$0.776259\pi$
$984$		0			0
$985$	−11.5271	−	53.5727i	−0.367284	−	1.70697i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−19.6516	−	19.6516i	−0.624884	−	0.624884i
$990$		0			0
$991$	40.2087			1.27727			0.638635	−	0.769509i	$-0.279499\pi$
$991$	40.2087			1.27727			0.638635	+	0.769509i	$0.279499\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	8.57024	+	39.8306i	0.271695	+	1.26272i
$996$		0			0
$997$			5.66421i			0.179387i	0.995969	+	0.0896937i	$0.0285888\pi$
$997$			5.66421i			0.179387i	−0.995969	+	0.0896937i	$0.971411\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bc.a.593.20		96
3.2	odd	2	inner	2880.2.bc.a.593.29		96
4.3	odd	2		720.2.bc.a.413.8	yes	96
5.2	odd	4		2880.2.bg.a.17.5		96
12.11	even	2		720.2.bc.a.413.41	yes	96
15.2	even	4		2880.2.bg.a.17.44		96
16.5	even	4		2880.2.bg.a.2033.44		96
16.11	odd	4		720.2.bg.a.53.17	yes	96
20.7	even	4		720.2.bg.a.557.32	yes	96
48.5	odd	4		2880.2.bg.a.2033.5		96
48.11	even	4		720.2.bg.a.53.32	yes	96
60.47	odd	4		720.2.bg.a.557.17	yes	96
80.27	even	4		720.2.bc.a.197.41	yes	96
80.37	odd	4	inner	2880.2.bc.a.1457.29		96
240.107	odd	4		720.2.bc.a.197.8	✓	96
240.197	even	4	inner	2880.2.bc.a.1457.20		96

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.bc.a.197.8	✓	96	240.107	odd	4
720.2.bc.a.197.41	yes	96	80.27	even	4
720.2.bc.a.413.8	yes	96	4.3	odd	2
720.2.bc.a.413.41	yes	96	12.11	even	2
720.2.bg.a.53.17	yes	96	16.11	odd	4
720.2.bg.a.53.32	yes	96	48.11	even	4
720.2.bg.a.557.17	yes	96	60.47	odd	4
720.2.bg.a.557.32	yes	96	20.7	even	4
2880.2.bc.a.593.20		96	1.1	even	1	trivial
2880.2.bc.a.593.29		96	3.2	odd	2	inner
2880.2.bc.a.1457.20		96	240.197	even	4	inner
2880.2.bc.a.1457.29		96	80.37	odd	4	inner
2880.2.bg.a.17.5		96	5.2	odd	4
2880.2.bg.a.17.44		96	15.2	even	4
2880.2.bg.a.2033.5		96	48.5	odd	4
2880.2.bg.a.2033.44		96	16.5	even	4

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:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.bc (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.470363 - 2.18604i) q^{5} +(0.177389 + 0.177389i) q^{7} +O(q^{10})\)	\(q+(-0.470363 - 2.18604i) q^{5} +(0.177389 + 0.177389i) q^{7} +(1.34414 + 1.34414i) q^{11} -1.40679i q^{13} +(1.03869 + 1.03869i) q^{17} +(0.205281 - 0.205281i) q^{19} +(5.60286 + 5.60286i) q^{23} +(-4.55752 + 2.05646i) q^{25} +(5.31184 + 5.31184i) q^{29} +4.12332 q^{31} +(0.304341 - 0.471215i) q^{35} -3.60144i q^{37} -8.78009 q^{41} -3.50742 q^{43} +(6.65257 + 6.65257i) q^{47} -6.93707i q^{49} -5.30796i q^{53} +(2.30610 - 3.57056i) q^{55} +(9.02817 - 9.02817i) q^{59} +(-0.150056 - 0.150056i) q^{61} +(-3.07530 + 0.661702i) q^{65} +10.1797 q^{67} +6.76393 q^{71} +(1.14818 - 1.14818i) q^{73} +0.476869i q^{77} -5.19176i q^{79} -5.08514 q^{83} +(1.78205 - 2.75918i) q^{85} +8.79477i q^{89} +(0.249549 - 0.249549i) q^{91} +(-0.545309 - 0.352195i) q^{95} +(6.63701 - 6.63701i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q+O(q^{10}) \) 96 * q	\( 96 q - 16 q^{19} - 64 q^{43} + 32 q^{61}+O(q^{100}) \) 96 * q - 16 * q^19 - 64 * q^43 + 32 * q^61

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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