LMFDB - Embedded newform 1156.2.b.d.577.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1156,2,Mod(577,1156)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1156.577"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1156, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1156 = 2^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1156.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$9.23070647366$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.2048.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 2 $ x^4 + 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 68)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.4
Root			$1.84776i$ of defining polynomial
Character	$\chi$	$=$	1156.577
Dual form			1156.2.b.d.577.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.84776i q^{3} -1.84776i q^{5} -0.765367i q^{7} -0.414214 q^{9} +3.37849i q^{11} -6.82843 q^{13} +3.41421 q^{15} +2.58579 q^{19} +1.41421 q^{21} +7.07401i q^{23} +1.58579 q^{25} +4.77791i q^{27} +4.01254i q^{29} +6.43996i q^{31} -6.24264 q^{33} -1.41421 q^{35} +6.62567i q^{37} -12.6173i q^{39} -1.21371i q^{41} -4.24264 q^{43} +0.765367i q^{45} +3.65685 q^{47} +6.41421 q^{49} -5.89949 q^{53} +6.24264 q^{55} +4.77791i q^{57} +13.8995 q^{59} -1.84776i q^{61} +0.317025i q^{63} +12.6173i q^{65} -11.3137 q^{67} -13.0711 q^{69} -1.21371i q^{71} +8.60474i q^{73} +2.93015i q^{75} +2.58579 q^{77} -11.4036i q^{79} -10.0711 q^{81} -0.242641 q^{83} -7.41421 q^{87} -10.8284 q^{89} +5.22625i q^{91} -11.8995 q^{93} -4.77791i q^{95} +0.951076i q^{97} -1.39942i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9} - 16 q^{13} + 8 q^{15} + 16 q^{19} + 12 q^{25} - 8 q^{33} - 8 q^{47} + 20 q^{49} + 16 q^{53} + 8 q^{55} + 16 q^{59} - 24 q^{69} + 16 q^{77} - 12 q^{81} + 16 q^{83} - 24 q^{87} - 32 q^{89}+ \cdots - 8 q^{93}+O(q^{100}) $ 4 * q + 4 * q^9 - 16 * q^13 + 8 * q^15 + 16 * q^19 + 12 * q^25 - 8 * q^33 - 8 * q^47 + 20 * q^49 + 16 * q^53 + 8 * q^55 + 16 * q^59 - 24 * q^69 + 16 * q^77 - 12 * q^81 + 16 * q^83 - 24 * q^87 - 32 * q^89 - 8 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times$.

$n$	$579$	$581$
$\chi(n)$	$1$	$-1$

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
$3$	1.84776i	1.06680i	0.845862	+	0.533402i	$0.179087\pi$
$3$	1.84776i	1.06680i	−0.845862	+	0.533402i	$0.820913\pi$
$4$	0	0
$5$	− 1.84776i	− 0.826343i	−0.910653	−	0.413171i	$-0.864421\pi$
$5$	− 1.84776i	− 0.826343i	0.910653	−	0.413171i	$-0.135579\pi$
$6$	0	0
$7$	− 0.765367i	− 0.289281i	−0.989484	−	0.144641i	$-0.953797\pi$
$7$	− 0.765367i	− 0.289281i	0.989484	−	0.144641i	$-0.0462026\pi$
$8$	0	0
$9$	−0.414214	−0.138071
$10$	0	0
$11$	3.37849i	1.01865i	0.860573	+	0.509327i	$0.170106\pi$
$11$	3.37849i	1.01865i	−0.860573	+	0.509327i	$0.829894\pi$
$12$	0	0
$13$	−6.82843	−1.89386	−0.946932	−	0.321433i	$-0.895836\pi$
$13$	−6.82843	−1.89386	−0.946932	+	0.321433i	$0.895836\pi$
$14$	0	0
$15$	3.41421	0.881546
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	2.58579	0.593220	0.296610	−	0.954999i	$-0.404144\pi$
$19$	2.58579	0.593220	0.296610	+	0.954999i	$0.404144\pi$
$20$	0	0
$21$	1.41421	0.308607
$22$	0	0
$23$	7.07401i	1.47503i	0.675329	+	0.737517i	$0.264002\pi$
$23$	7.07401i	1.47503i	−0.675329	+	0.737517i	$0.735998\pi$
$24$	0	0
$25$	1.58579	0.317157
$26$	0	0
$27$	4.77791i	0.919509i
$28$	0	0
$29$	4.01254i	0.745111i	0.928010	+	0.372555i	$0.121518\pi$
$29$	4.01254i	0.745111i	−0.928010	+	0.372555i	$0.878482\pi$
$30$	0	0
$31$	6.43996i	1.15665i	0.815806	+	0.578326i	$0.196294\pi$
$31$	6.43996i	1.15665i	−0.815806	+	0.578326i	$0.803706\pi$
$32$	0	0
$33$	−6.24264	−1.08670
$34$	0	0
$35$	−1.41421	−0.239046
$36$	0	0
$37$	6.62567i	1.08925i	0.838679	+	0.544627i	$0.183329\pi$
$37$	6.62567i	1.08925i	−0.838679	+	0.544627i	$0.816671\pi$
$38$	0	0
$39$	− 12.6173i	− 2.02038i
$40$	0	0
$41$	− 1.21371i	− 0.189549i	−0.995499	−	0.0947747i	$-0.969787\pi$
$41$	− 1.21371i	− 0.189549i	0.995499	−	0.0947747i	$-0.0302131\pi$
$42$	0	0
$43$	−4.24264	−0.646997	−0.323498	−	0.946229i	$-0.604859\pi$
$43$	−4.24264	−0.646997	−0.323498	+	0.946229i	$0.604859\pi$
$44$	0	0
$45$	0.765367i	0.114094i
$46$	0	0
$47$	3.65685	0.533407	0.266704	−	0.963779i	$-0.414066\pi$
$47$	3.65685	0.533407	0.266704	+	0.963779i	$0.414066\pi$
$48$	0	0
$49$	6.41421	0.916316
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.89949	−0.810358	−0.405179	−	0.914237i	$-0.632791\pi$
$53$	−5.89949	−0.810358	−0.405179	+	0.914237i	$0.632791\pi$
$54$	0	0
$55$	6.24264	0.841757
$56$	0	0
$57$	4.77791i	0.632850i
$58$	0	0
$59$	13.8995	1.80956	0.904780	−	0.425879i	$-0.140035\pi$
$59$	13.8995	1.80956	0.904780	+	0.425879i	$0.140035\pi$
$60$	0	0
$61$	− 1.84776i	− 0.236581i	−0.992979	−	0.118291i	$-0.962259\pi$
$61$	− 1.84776i	− 0.236581i	0.992979	−	0.118291i	$-0.0377415\pi$
$62$	0	0
$63$	0.317025i	0.0399414i
$64$	0	0
$65$	12.6173i	1.56498i
$66$	0	0
$67$	−11.3137	−1.38219	−0.691095	−	0.722764i	$-0.742871\pi$
$67$	−11.3137	−1.38219	−0.691095	+	0.722764i	$0.742871\pi$
$68$	0	0
$69$	−13.0711	−1.57357
$70$	0	0
$71$	− 1.21371i	− 0.144041i	−0.997403	−	0.0720203i	$-0.977055\pi$
$71$	− 1.21371i	− 0.144041i	0.997403	−	0.0720203i	$-0.0229447\pi$
$72$	0	0
$73$	8.60474i	1.00711i	0.863963	+	0.503555i	$0.167975\pi$
$73$	8.60474i	1.00711i	−0.863963	+	0.503555i	$0.832025\pi$
$74$	0	0
$75$	2.93015i	0.338345i
$76$	0	0
$77$	2.58579	0.294678
$78$	0	0
$79$	− 11.4036i	− 1.28300i	−0.767122	−	0.641501i	$-0.778312\pi$
$79$	− 11.4036i	− 1.28300i	0.767122	−	0.641501i	$-0.221688\pi$
$80$	0	0
$81$	−10.0711	−1.11901
$82$	0	0
$83$	−0.242641	−0.0266333	−0.0133166	−	0.999911i	$-0.504239\pi$
$83$	−0.242641	−0.0266333	−0.0133166	+	0.999911i	$0.504239\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−7.41421	−0.794887
$88$	0	0
$89$	−10.8284	−1.14781	−0.573905	−	0.818922i	$-0.694572\pi$
$89$	−10.8284	−1.14781	−0.573905	+	0.818922i	$0.694572\pi$
$90$	0	0
$91$	5.22625i	0.547860i
$92$	0	0
$93$	−11.8995	−1.23392
$94$	0	0
$95$	− 4.77791i	− 0.490203i
$96$	0	0
$97$	0.951076i	0.0965671i	0.998834	+	0.0482836i	$0.0153751\pi$
$97$	0.951076i	0.0965671i	−0.998834	+	0.0482836i	$0.984625\pi$
$98$	0	0
$99$	− 1.39942i	− 0.140647i
$100$	0	0
$101$	−12.8284	−1.27648	−0.638238	−	0.769839i	$-0.720336\pi$
$101$	−12.8284	−1.27648	−0.638238	+	0.769839i	$0.720336\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	− 2.61313i	− 0.255015i
$106$	0	0
$107$	− 3.82683i	− 0.369954i	−0.982743	−	0.184977i	$-0.940779\pi$
$107$	− 3.82683i	− 0.369954i	0.982743	−	0.184977i	$-0.0592211\pi$
$108$	0	0
$109$	4.90923i	0.470219i	0.971969	+	0.235109i	$0.0755448\pi$
$109$	4.90923i	0.470219i	−0.971969	+	0.235109i	$0.924455\pi$
$110$	0	0
$111$	−12.2426	−1.16202
$112$	0	0
$113$	1.39942i	0.131646i	0.997831	+	0.0658231i	$0.0209673\pi$
$113$	1.39942i	0.131646i	−0.997831	+	0.0658231i	$0.979033\pi$
$114$	0	0
$115$	13.0711	1.21888
$116$	0	0
$117$	2.82843	0.261488
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−0.414214	−0.0376558
$122$	0	0
$123$	2.24264	0.202212
$124$	0	0
$125$	− 12.1689i	− 1.08842i
$126$	0	0
$127$	0.242641	0.0215309	0.0107654	−	0.999942i	$-0.496573\pi$
$127$	0.242641	0.0215309	0.0107654	+	0.999942i	$0.496573\pi$
$128$	0	0
$129$	− 7.83938i	− 0.690219i
$130$	0	0
$131$	− 10.9552i	− 0.957164i	−0.878043	−	0.478582i	$-0.841151\pi$
$131$	− 10.9552i	− 0.957164i	0.878043	−	0.478582i	$-0.158849\pi$
$132$	0	0
$133$	− 1.97908i	− 0.171608i
$134$	0	0
$135$	8.82843	0.759830
$136$	0	0
$137$	12.1421	1.03737	0.518686	−	0.854965i	$-0.326421\pi$
$137$	12.1421	1.03737	0.518686	+	0.854965i	$0.326421\pi$
$138$	0	0
$139$	17.5265i	1.48658i	0.668970	+	0.743290i	$0.266736\pi$
$139$	17.5265i	1.48658i	−0.668970	+	0.743290i	$0.733264\pi$
$140$	0	0
$141$	6.75699i	0.569041i
$142$	0	0
$143$	− 23.0698i	− 1.92919i
$144$	0	0
$145$	7.41421	0.615717
$146$	0	0
$147$	11.8519i	0.977530i
$148$	0	0
$149$	−9.65685	−0.791120	−0.395560	−	0.918440i	$-0.629450\pi$
$149$	−9.65685	−0.791120	−0.395560	+	0.918440i	$0.629450\pi$
$150$	0	0
$151$	−14.5858	−1.18697	−0.593487	−	0.804843i	$-0.702249\pi$
$151$	−14.5858	−1.18697	−0.593487	+	0.804843i	$0.702249\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11.8995	0.955790
$156$	0	0
$157$	2.34315	0.187003	0.0935017	−	0.995619i	$-0.470194\pi$
$157$	2.34315	0.187003	0.0935017	+	0.995619i	$0.470194\pi$
$158$	0	0
$159$	− 10.9008i	− 0.864493i
$160$	0	0
$161$	5.41421	0.426700
$162$	0	0
$163$	7.25972i	0.568625i	0.958732	+	0.284313i	$0.0917653\pi$
$163$	7.25972i	0.568625i	−0.958732	+	0.284313i	$0.908235\pi$
$164$	0	0
$165$	11.5349i	0.897990i
$166$	0	0
$167$	− 12.3003i	− 0.951823i	−0.879493	−	0.475911i	$-0.842118\pi$
$167$	− 12.3003i	− 0.951823i	0.879493	−	0.475911i	$-0.157882\pi$
$168$	0	0
$169$	33.6274	2.58672
$170$	0	0
$171$	−1.07107	−0.0819066
$172$	0	0
$173$	− 25.1033i	− 1.90857i	−0.298905	−	0.954283i	$-0.596621\pi$
$173$	− 25.1033i	− 1.90857i	0.298905	−	0.954283i	$-0.403379\pi$
$174$	0	0
$175$	− 1.21371i	− 0.0917477i
$176$	0	0
$177$	25.6829i	1.93045i
$178$	0	0
$179$	18.3848	1.37414	0.687071	−	0.726590i	$-0.258896\pi$
$179$	18.3848	1.37414	0.687071	+	0.726590i	$0.258896\pi$
$180$	0	0
$181$	− 3.82683i	− 0.284446i	−0.989835	−	0.142223i	$-0.954575\pi$
$181$	− 3.82683i	− 0.284446i	0.989835	−	0.142223i	$-0.0454251\pi$
$182$	0	0
$183$	3.41421	0.252386
$184$	0	0
$185$	12.2426	0.900097
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.65685	0.265997
$190$	0	0
$191$	0.343146	0.0248292	0.0124146	−	0.999923i	$-0.496048\pi$
$191$	0.343146	0.0248292	0.0124146	+	0.999923i	$0.496048\pi$
$192$	0	0
$193$	15.5474i	1.11913i	0.828787	+	0.559565i	$0.189032\pi$
$193$	15.5474i	1.11913i	−0.828787	+	0.559565i	$0.810968\pi$
$194$	0	0
$195$	−23.3137	−1.66953
$196$	0	0
$197$	12.5629i	0.895069i	0.894267	+	0.447535i	$0.147698\pi$
$197$	12.5629i	0.895069i	−0.894267	+	0.447535i	$0.852302\pi$
$198$	0	0
$199$	1.66205i	0.117820i	0.998263	+	0.0589098i	$0.0187624\pi$
$199$	1.66205i	0.117820i	−0.998263	+	0.0589098i	$0.981238\pi$
$200$	0	0
$201$	− 20.9050i	− 1.47453i
$202$	0	0
$203$	3.07107	0.215547
$204$	0	0
$205$	−2.24264	−0.156633
$206$	0	0
$207$	− 2.93015i	− 0.203660i
$208$	0	0
$209$	8.73606i	0.604286i
$210$	0	0
$211$	− 2.03347i	− 0.139990i	−0.997547	−	0.0699949i	$-0.977702\pi$
$211$	− 2.03347i	− 0.139990i	0.997547	−	0.0699949i	$-0.0222983\pi$
$212$	0	0
$213$	2.24264	0.153663
$214$	0	0
$215$	7.83938i	0.534641i
$216$	0	0
$217$	4.92893	0.334598
$218$	0	0
$219$	−15.8995	−1.07439
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.2132	1.15268	0.576341	−	0.817210i	$-0.304480\pi$
$223$	17.2132	1.15268	0.576341	+	0.817210i	$0.304480\pi$
$224$	0	0
$225$	−0.656854	−0.0437903
$226$	0	0
$227$	− 19.4287i	− 1.28953i	−0.764383	−	0.644763i	$-0.776956\pi$
$227$	− 19.4287i	− 1.28953i	0.764383	−	0.644763i	$-0.223044\pi$
$228$	0	0
$229$	3.75736	0.248293	0.124147	−	0.992264i	$-0.460381\pi$
$229$	3.75736	0.248293	0.124147	+	0.992264i	$0.460381\pi$
$230$	0	0
$231$	4.77791i	0.314363i
$232$	0	0
$233$	− 8.97616i	− 0.588048i	−0.955798	−	0.294024i	$-0.905005\pi$
$233$	− 8.97616i	− 0.588048i	0.955798	−	0.294024i	$-0.0949946\pi$
$234$	0	0
$235$	− 6.75699i	− 0.440777i
$236$	0	0
$237$	21.0711	1.36871
$238$	0	0
$239$	22.6274	1.46365	0.731823	−	0.681495i	$-0.238670\pi$
$239$	22.6274	1.46365	0.731823	+	0.681495i	$0.238670\pi$
$240$	0	0
$241$	− 19.8770i	− 1.28039i	−0.768212	−	0.640195i	$-0.778853\pi$
$241$	− 19.8770i	− 1.28039i	0.768212	−	0.640195i	$-0.221147\pi$
$242$	0	0
$243$	− 4.27518i	− 0.274253i
$244$	0	0
$245$	− 11.8519i	− 0.757191i
$246$	0	0
$247$	−17.6569	−1.12348
$248$	0	0
$249$	− 0.448342i	− 0.0284125i
$250$	0	0
$251$	22.9706	1.44989	0.724945	−	0.688807i	$-0.241865\pi$
$251$	22.9706	1.44989	0.724945	+	0.688807i	$0.241865\pi$
$252$	0	0
$253$	−23.8995	−1.50255
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−3.75736	−0.234378	−0.117189	−	0.993110i	$-0.537388\pi$
$257$	−3.75736	−0.234378	−0.117189	+	0.993110i	$0.537388\pi$
$258$	0	0
$259$	5.07107	0.315101
$260$	0	0
$261$	− 1.66205i	− 0.102878i
$262$	0	0
$263$	13.2132	0.814761	0.407381	−	0.913259i	$-0.366442\pi$
$263$	13.2132	0.814761	0.407381	+	0.913259i	$0.366442\pi$
$264$	0	0
$265$	10.9008i	0.669634i
$266$	0	0
$267$	− 20.0083i	− 1.22449i
$268$	0	0
$269$	− 8.41904i	− 0.513318i	−0.966502	−	0.256659i	$-0.917378\pi$
$269$	− 8.41904i	− 0.513318i	0.966502	−	0.256659i	$-0.0826217\pi$
$270$	0	0
$271$	11.3137	0.687259	0.343629	−	0.939105i	$-0.388344\pi$
$271$	11.3137	0.687259	0.343629	+	0.939105i	$0.388344\pi$
$272$	0	0
$273$	−9.65685	−0.584459
$274$	0	0
$275$	5.35757i	0.323074i
$276$	0	0
$277$	15.0991i	0.907217i	0.891201	+	0.453608i	$0.149864\pi$
$277$	15.0991i	0.907217i	−0.891201	+	0.453608i	$0.850136\pi$
$278$	0	0
$279$	− 2.66752i	− 0.159700i
$280$	0	0
$281$	15.0711	0.899065	0.449532	−	0.893264i	$-0.351591\pi$
$281$	15.0711	0.899065	0.449532	+	0.893264i	$0.351591\pi$
$282$	0	0
$283$	− 7.07401i	− 0.420506i	−0.977647	−	0.210253i	$-0.932571\pi$
$283$	− 7.07401i	− 0.420506i	0.977647	−	0.210253i	$-0.0674288\pi$
$284$	0	0
$285$	8.82843	0.522951
$286$	0	0
$287$	−0.928932	−0.0548331
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−1.75736	−0.103018
$292$	0	0
$293$	−8.00000	−0.467365	−0.233682	−	0.972313i	$-0.575078\pi$
$293$	−8.00000	−0.467365	−0.233682	+	0.972313i	$0.575078\pi$
$294$	0	0
$295$	− 25.6829i	− 1.49532i
$296$	0	0
$297$	−16.1421	−0.936662
$298$	0	0
$299$	− 48.3044i	− 2.79351i
$300$	0	0
$301$	3.24718i	0.187164i
$302$	0	0
$303$	− 23.7038i	− 1.36175i
$304$	0	0
$305$	−3.41421	−0.195497
$306$	0	0
$307$	6.34315	0.362022	0.181011	−	0.983481i	$-0.442063\pi$
$307$	6.34315	0.362022	0.181011	+	0.983481i	$0.442063\pi$
$308$	0	0
$309$	7.39104i	0.420461i
$310$	0	0
$311$	12.7486i	0.722907i	0.932390	+	0.361454i	$0.117719\pi$
$311$	12.7486i	0.722907i	−0.932390	+	0.361454i	$0.882281\pi$
$312$	0	0
$313$	22.7528i	1.28606i	0.765840	+	0.643031i	$0.222323\pi$
$313$	22.7528i	1.28606i	−0.765840	+	0.643031i	$0.777677\pi$
$314$	0	0
$315$	0.585786	0.0330053
$316$	0	0
$317$	− 0.393949i	− 0.0221264i	−0.999939	−	0.0110632i	$-0.996478\pi$
$317$	− 0.393949i	− 0.0221264i	0.999939	−	0.0110632i	$-0.00352159\pi$
$318$	0	0
$319$	−13.5563	−0.759010
$320$	0	0
$321$	7.07107	0.394669
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−10.8284	−0.600653
$326$	0	0
$327$	−9.07107	−0.501631
$328$	0	0
$329$	− 2.79884i	− 0.154305i
$330$	0	0
$331$	−28.7279	−1.57903	−0.789515	−	0.613732i	$-0.789668\pi$
$331$	−28.7279	−1.57903	−0.789515	+	0.613732i	$0.789668\pi$
$332$	0	0
$333$	− 2.74444i	− 0.150395i
$334$	0	0
$335$	20.9050i	1.14216i
$336$	0	0
$337$	25.9999i	1.41631i	0.706058	+	0.708154i	$0.250472\pi$
$337$	25.9999i	1.41631i	−0.706058	+	0.708154i	$0.749528\pi$
$338$	0	0
$339$	−2.58579	−0.140441
$340$	0	0
$341$	−21.7574	−1.17823
$342$	0	0
$343$	− 10.2668i	− 0.554355i
$344$	0	0
$345$	24.1522i	1.30031i
$346$	0	0
$347$	− 7.78498i	− 0.417920i	−0.977924	−	0.208960i	$-0.932992\pi$
$347$	− 7.78498i	− 0.417920i	0.977924	−	0.208960i	$-0.0670078\pi$
$348$	0	0
$349$	28.0416	1.50103	0.750517	−	0.660851i	$-0.229805\pi$
$349$	28.0416	1.50103	0.750517	+	0.660851i	$0.229805\pi$
$350$	0	0
$351$	− 32.6256i	− 1.74143i
$352$	0	0
$353$	15.3137	0.815066	0.407533	−	0.913190i	$-0.366389\pi$
$353$	15.3137	0.815066	0.407533	+	0.913190i	$0.366389\pi$
$354$	0	0
$355$	−2.24264	−0.119027
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.3848	−1.60365	−0.801824	−	0.597561i	$-0.796137\pi$
$359$	−30.3848	−1.60365	−0.801824	+	0.597561i	$0.796137\pi$
$360$	0	0
$361$	−12.3137	−0.648090
$362$	0	0
$363$	− 0.765367i	− 0.0401713i
$364$	0	0
$365$	15.8995	0.832218
$366$	0	0
$367$	3.82683i	0.199759i	0.995000	+	0.0998796i	$0.0318458\pi$
$367$	3.82683i	0.199759i	−0.995000	+	0.0998796i	$0.968154\pi$
$368$	0	0
$369$	0.502734i	0.0261713i
$370$	0	0
$371$	4.51528i	0.234422i
$372$	0	0
$373$	−18.4853	−0.957132	−0.478566	−	0.878052i	$-0.658843\pi$
$373$	−18.4853	−0.957132	−0.478566	+	0.878052i	$0.658843\pi$
$374$	0	0
$375$	22.4853	1.16113
$376$	0	0
$377$	− 27.3994i	− 1.41114i
$378$	0	0
$379$	10.9552i	0.562733i	0.959600	+	0.281366i	$0.0907876\pi$
$379$	10.9552i	0.562733i	−0.959600	+	0.281366i	$0.909212\pi$
$380$	0	0
$381$	0.448342i	0.0229692i
$382$	0	0
$383$	−25.8995	−1.32340	−0.661701	−	0.749768i	$-0.730165\pi$
$383$	−25.8995	−1.32340	−0.661701	+	0.749768i	$0.730165\pi$
$384$	0	0
$385$	− 4.77791i	− 0.243505i
$386$	0	0
$387$	1.75736	0.0893316
$388$	0	0
$389$	2.10051	0.106500	0.0532499	−	0.998581i	$-0.483042\pi$
$389$	2.10051	0.106500	0.0532499	+	0.998581i	$0.483042\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	20.2426	1.02111
$394$	0	0
$395$	−21.0711	−1.06020
$396$	0	0
$397$	− 18.8715i	− 0.947135i	−0.880757	−	0.473568i	$-0.842966\pi$
$397$	− 18.8715i	− 0.947135i	0.880757	−	0.473568i	$-0.157034\pi$
$398$	0	0
$399$	3.65685	0.183072
$400$	0	0
$401$	− 9.23880i	− 0.461363i	−0.973029	−	0.230682i	$-0.925904\pi$
$401$	− 9.23880i	− 0.461363i	0.973029	−	0.230682i	$-0.0740956\pi$
$402$	0	0
$403$	− 43.9748i	− 2.19054i
$404$	0	0
$405$	18.6089i	0.924684i
$406$	0	0
$407$	−22.3848	−1.10957
$408$	0	0
$409$	−11.6569	−0.576394	−0.288197	−	0.957571i	$-0.593056\pi$
$409$	−11.6569	−0.576394	−0.288197	+	0.957571i	$0.593056\pi$
$410$	0	0
$411$	22.4357i	1.10667i
$412$	0	0
$413$	− 10.6382i	− 0.523472i
$414$	0	0
$415$	0.448342i	0.0220082i
$416$	0	0
$417$	−32.3848	−1.58589
$418$	0	0
$419$	35.6327i	1.74077i	0.492371	+	0.870385i	$0.336130\pi$
$419$	35.6327i	1.74077i	−0.492371	+	0.870385i	$0.663870\pi$
$420$	0	0
$421$	−1.17157	−0.0570990	−0.0285495	−	0.999592i	$-0.509089\pi$
$421$	−1.17157	−0.0570990	−0.0285495	+	0.999592i	$0.509089\pi$
$422$	0	0
$423$	−1.51472	−0.0736481
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.41421	−0.0684386
$428$	0	0
$429$	42.6274	2.05807
$430$	0	0
$431$	20.5880i	0.991688i	0.868412	+	0.495844i	$0.165141\pi$
$431$	20.5880i	0.991688i	−0.868412	+	0.495844i	$0.834859\pi$
$432$	0	0
$433$	−8.24264	−0.396116	−0.198058	−	0.980190i	$-0.563463\pi$
$433$	−8.24264	−0.396116	−0.198058	+	0.980190i	$0.563463\pi$
$434$	0	0
$435$	13.6997i	0.656849i
$436$	0	0
$437$	18.2919i	0.875019i
$438$	0	0
$439$	− 14.8365i	− 0.708106i	−0.935225	−	0.354053i	$-0.884803\pi$
$439$	− 14.8365i	− 0.708106i	0.935225	−	0.354053i	$-0.115197\pi$
$440$	0	0
$441$	−2.65685	−0.126517
$442$	0	0
$443$	−21.6569	−1.02895	−0.514474	−	0.857506i	$-0.672013\pi$
$443$	−21.6569	−1.02895	−0.514474	+	0.857506i	$0.672013\pi$
$444$	0	0
$445$	20.0083i	0.948486i
$446$	0	0
$447$	− 17.8435i	− 0.843970i
$448$	0	0
$449$	− 21.2220i	− 1.00153i	−0.865583	−	0.500765i	$-0.833052\pi$
$449$	− 21.2220i	− 1.00153i	0.865583	−	0.500765i	$-0.166948\pi$
$450$	0	0
$451$	4.10051	0.193085
$452$	0	0
$453$	− 26.9510i	− 1.26627i
$454$	0	0
$455$	9.65685	0.452720
$456$	0	0
$457$	−4.24264	−0.198462	−0.0992312	−	0.995064i	$-0.531638\pi$
$457$	−4.24264	−0.198462	−0.0992312	+	0.995064i	$0.531638\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.75736	0.361296	0.180648	−	0.983548i	$-0.442180\pi$
$461$	7.75736	0.361296	0.180648	+	0.983548i	$0.442180\pi$
$462$	0	0
$463$	7.65685	0.355844	0.177922	−	0.984045i	$-0.443062\pi$
$463$	7.65685	0.355844	0.177922	+	0.984045i	$0.443062\pi$
$464$	0	0
$465$	21.9874i	1.01964i
$466$	0	0
$467$	28.2426	1.30691	0.653457	−	0.756964i	$-0.273318\pi$
$467$	28.2426	1.30691	0.653457	+	0.756964i	$0.273318\pi$
$468$	0	0
$469$	8.65914i	0.399842i
$470$	0	0
$471$	4.32957i	0.199496i
$472$	0	0
$473$	− 14.3337i	− 0.659066i
$474$	0	0
$475$	4.10051	0.188144
$476$	0	0
$477$	2.44365	0.111887
$478$	0	0
$479$	− 8.23333i	− 0.376190i	−0.982151	−	0.188095i	$-0.939769\pi$
$479$	− 8.23333i	− 0.376190i	0.982151	−	0.188095i	$-0.0602313\pi$
$480$	0	0
$481$	− 45.2429i	− 2.06290i
$482$	0	0
$483$	10.0042i	0.455205i
$484$	0	0
$485$	1.75736	0.0797976
$486$	0	0
$487$	− 3.64113i	− 0.164995i	−0.996591	−	0.0824976i	$-0.973710\pi$
$487$	− 3.64113i	− 0.164995i	0.996591	−	0.0824976i	$-0.0262897\pi$
$488$	0	0
$489$	−13.4142	−0.606612
$490$	0	0
$491$	23.7574	1.07215	0.536077	−	0.844169i	$-0.319906\pi$
$491$	23.7574	1.07215	0.536077	+	0.844169i	$0.319906\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−2.58579	−0.116222
$496$	0	0
$497$	−0.928932	−0.0416683
$498$	0	0
$499$	37.5348i	1.68029i	0.542362	+	0.840145i	$0.317530\pi$
$499$	37.5348i	1.68029i	−0.542362	+	0.840145i	$0.682470\pi$
$500$	0	0
$501$	22.7279	1.01541
$502$	0	0
$503$	− 7.89377i	− 0.351966i	−0.984393	−	0.175983i	$-0.943690\pi$
$503$	− 7.89377i	− 0.351966i	0.984393	−	0.175983i	$-0.0563104\pi$
$504$	0	0
$505$	23.7038i	1.05481i
$506$	0	0
$507$	62.1354i	2.75953i
$508$	0	0
$509$	−6.68629	−0.296365	−0.148182	−	0.988960i	$-0.547342\pi$
$509$	−6.68629	−0.296365	−0.148182	+	0.988960i	$0.547342\pi$
$510$	0	0
$511$	6.58579	0.291338
$512$	0	0
$513$	12.3547i	0.545471i
$514$	0	0
$515$	− 7.39104i	− 0.325688i
$516$	0	0
$517$	12.3547i	0.543357i
$518$	0	0
$519$	46.3848	2.03607
$520$	0	0
$521$	27.5307i	1.20614i	0.797688	+	0.603070i	$0.206056\pi$
$521$	27.5307i	1.20614i	−0.797688	+	0.603070i	$0.793944\pi$
$522$	0	0
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	0	0
$525$	2.24264	0.0978769
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−27.0416	−1.17572
$530$	0	0
$531$	−5.75736	−0.249848
$532$	0	0
$533$	8.28772i	0.358981i
$534$	0	0
$535$	−7.07107	−0.305709
$536$	0	0
$537$	33.9706i	1.46594i
$538$	0	0
$539$	21.6704i	0.933409i
$540$	0	0
$541$	− 7.89377i	− 0.339380i	−0.985498	−	0.169690i	$-0.945723\pi$
$541$	− 7.89377i	− 0.339380i	0.985498	−	0.169690i	$-0.0542766\pi$
$542$	0	0
$543$	7.07107	0.303449
$544$	0	0
$545$	9.07107	0.388562
$546$	0	0
$547$	24.5461i	1.04952i	0.851251	+	0.524758i	$0.175844\pi$
$547$	24.5461i	1.04952i	−0.851251	+	0.524758i	$0.824156\pi$
$548$	0	0
$549$	0.765367i	0.0326651i
$550$	0	0
$551$	10.3756i	0.442015i
$552$	0	0
$553$	−8.72792	−0.371149
$554$	0	0
$555$	22.6215i	0.960227i
$556$	0	0
$557$	−22.8284	−0.967272	−0.483636	−	0.875269i	$-0.660684\pi$
$557$	−22.8284	−0.967272	−0.483636	+	0.875269i	$0.660684\pi$
$558$	0	0
$559$	28.9706	1.22532
$560$	0	0
$561$	0	0
$562$	0	0
$563$	38.5858	1.62620	0.813099	−	0.582126i	$-0.197779\pi$
$563$	38.5858	1.62620	0.813099	+	0.582126i	$0.197779\pi$
$564$	0	0
$565$	2.58579	0.108785
$566$	0	0
$567$	7.70806i	0.323708i
$568$	0	0
$569$	27.5563	1.15522	0.577611	−	0.816312i	$-0.303985\pi$
$569$	27.5563	1.15522	0.577611	+	0.816312i	$0.303985\pi$
$570$	0	0
$571$	25.4747i	1.06608i	0.846089	+	0.533041i	$0.178951\pi$
$571$	25.4747i	1.06608i	−0.846089	+	0.533041i	$0.821049\pi$
$572$	0	0
$573$	0.634051i	0.0264878i
$574$	0	0
$575$	11.2179i	0.467818i
$576$	0	0
$577$	−1.79899	−0.0748929	−0.0374465	−	0.999299i	$-0.511922\pi$
$577$	−1.79899	−0.0748929	−0.0374465	+	0.999299i	$0.511922\pi$
$578$	0	0
$579$	−28.7279	−1.19389
$580$	0	0
$581$	0.185709i	0.00770451i
$582$	0	0
$583$	− 19.9314i	− 0.825474i
$584$	0	0
$585$	− 5.22625i	− 0.216079i
$586$	0	0
$587$	−29.2132	−1.20576	−0.602879	−	0.797833i	$-0.705980\pi$
$587$	−29.2132	−1.20576	−0.602879	+	0.797833i	$0.705980\pi$
$588$	0	0
$589$	16.6524i	0.686149i
$590$	0	0
$591$	−23.2132	−0.954864
$592$	0	0
$593$	33.2132	1.36390	0.681951	−	0.731397i	$-0.261132\pi$
$593$	33.2132	1.36390	0.681951	+	0.731397i	$0.261132\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−3.07107	−0.125690
$598$	0	0
$599$	37.3137	1.52460	0.762298	−	0.647226i	$-0.224071\pi$
$599$	37.3137	1.52460	0.762298	+	0.647226i	$0.224071\pi$
$600$	0	0
$601$	5.09494i	0.207827i	0.994586	+	0.103913i	$0.0331365\pi$
$601$	5.09494i	0.207827i	−0.994586	+	0.103913i	$0.966864\pi$
$602$	0	0
$603$	4.68629	0.190841
$604$	0	0
$605$	0.765367i	0.0311166i
$606$	0	0
$607$	15.9189i	0.646127i	0.946377	+	0.323063i	$0.104713\pi$
$607$	15.9189i	0.646127i	−0.946377	+	0.323063i	$0.895287\pi$
$608$	0	0
$609$	5.67459i	0.229946i
$610$	0	0
$611$	−24.9706	−1.01020
$612$	0	0
$613$	0.343146	0.0138595	0.00692976	−	0.999976i	$-0.497794\pi$
$613$	0.343146	0.0138595	0.00692976	+	0.999976i	$0.497794\pi$
$614$	0	0
$615$	− 4.14386i	− 0.167097i
$616$	0	0
$617$	− 34.1020i	− 1.37289i	−0.727180	−	0.686446i	$-0.759170\pi$
$617$	− 34.1020i	− 1.37289i	0.727180	−	0.686446i	$-0.240830\pi$
$618$	0	0
$619$	− 13.3827i	− 0.537894i	−0.963155	−	0.268947i	$-0.913324\pi$
$619$	− 13.3827i	− 0.537894i	0.963155	−	0.268947i	$-0.0866757\pi$
$620$	0	0
$621$	−33.7990	−1.35631
$622$	0	0
$623$	8.28772i	0.332040i
$624$	0	0
$625$	−14.5563	−0.582254
$626$	0	0
$627$	−16.1421	−0.644655
$628$	0	0
$629$	0	0
$630$	0	0
$631$	14.5858	0.580651	0.290325	−	0.956928i	$-0.406236\pi$
$631$	14.5858	0.580651	0.290325	+	0.956928i	$0.406236\pi$
$632$	0	0
$633$	3.75736	0.149342
$634$	0	0
$635$	− 0.448342i	− 0.0177919i
$636$	0	0
$637$	−43.7990	−1.73538
$638$	0	0
$639$	0.502734i	0.0198879i
$640$	0	0
$641$	− 46.4566i	− 1.83493i	−0.397821	−	0.917463i	$-0.630234\pi$
$641$	− 46.4566i	− 1.83493i	0.397821	−	0.917463i	$-0.369766\pi$
$642$	0	0
$643$	5.54328i	0.218606i	0.994009	+	0.109303i	$0.0348618\pi$
$643$	5.54328i	0.218606i	−0.994009	+	0.109303i	$0.965138\pi$
$644$	0	0
$645$	−14.4853	−0.570357
$646$	0	0
$647$	32.9706	1.29621	0.648103	−	0.761552i	$-0.275563\pi$
$647$	32.9706	1.29621	0.648103	+	0.761552i	$0.275563\pi$
$648$	0	0
$649$	46.9593i	1.84332i
$650$	0	0
$651$	9.10748i	0.356950i
$652$	0	0
$653$	− 45.1885i	− 1.76836i	−0.467144	−	0.884181i	$-0.654717\pi$
$653$	− 45.1885i	− 1.76836i	0.467144	−	0.884181i	$-0.345283\pi$
$654$	0	0
$655$	−20.2426	−0.790945
$656$	0	0
$657$	− 3.56420i	− 0.139053i
$658$	0	0
$659$	−2.68629	−0.104643	−0.0523215	−	0.998630i	$-0.516662\pi$
$659$	−2.68629	−0.104643	−0.0523215	+	0.998630i	$0.516662\pi$
$660$	0	0
$661$	−12.7279	−0.495059	−0.247529	−	0.968880i	$-0.579619\pi$
$661$	−12.7279	−0.495059	−0.247529	+	0.968880i	$0.579619\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−3.65685	−0.141807
$666$	0	0
$667$	−28.3848	−1.09906
$668$	0	0
$669$	31.8059i	1.22969i
$670$	0	0
$671$	6.24264	0.240994
$672$	0	0
$673$	− 31.0405i	− 1.19652i	−0.801301	−	0.598261i	$-0.795858\pi$
$673$	− 31.0405i	− 1.19652i	0.801301	−	0.598261i	$-0.204142\pi$
$674$	0	0
$675$	7.57675i	0.291629i
$676$	0	0
$677$	1.66205i	0.0638778i	0.999490	+	0.0319389i	$0.0101682\pi$
$677$	1.66205i	0.0638778i	−0.999490	+	0.0319389i	$0.989832\pi$
$678$	0	0
$679$	0.727922	0.0279351
$680$	0	0
$681$	35.8995	1.37567
$682$	0	0
$683$	− 4.27518i	− 0.163585i	−0.996649	−	0.0817925i	$-0.973936\pi$
$683$	− 4.27518i	− 0.163585i	0.996649	−	0.0817925i	$-0.0260645\pi$
$684$	0	0
$685$	− 22.4357i	− 0.857226i
$686$	0	0
$687$	6.94269i	0.264880i
$688$	0	0
$689$	40.2843	1.53471
$690$	0	0
$691$	− 7.07401i	− 0.269108i	−0.990906	−	0.134554i	$-0.957040\pi$
$691$	− 7.07401i	− 0.269108i	0.990906	−	0.134554i	$-0.0429602\pi$
$692$	0	0
$693$	−1.07107	−0.0406865
$694$	0	0
$695$	32.3848	1.22842
$696$	0	0
$697$	0	0
$698$	0	0
$699$	16.5858	0.627332
$700$	0	0
$701$	−19.5147	−0.737061	−0.368530	−	0.929616i	$-0.620139\pi$
$701$	−19.5147	−0.737061	−0.368530	+	0.929616i	$0.620139\pi$
$702$	0	0
$703$	17.1326i	0.646167i
$704$	0	0
$705$	12.4853	0.470223
$706$	0	0
$707$	9.81845i	0.369261i
$708$	0	0
$709$	− 3.37849i	− 0.126882i	−0.997986	−	0.0634410i	$-0.979793\pi$
$709$	− 3.37849i	− 0.126882i	0.997986	−	0.0634410i	$-0.0202075\pi$
$710$	0	0
$711$	4.72352i	0.177146i
$712$	0	0
$713$	−45.5563	−1.70610
$714$	0	0
$715$	−42.6274	−1.59418
$716$	0	0
$717$	41.8100i	1.56142i
$718$	0	0
$719$	− 15.0222i	− 0.560233i	−0.959966	−	0.280116i	$-0.909627\pi$
$719$	− 15.0222i	− 0.560233i	0.959966	−	0.280116i	$-0.0903730\pi$
$720$	0	0
$721$	− 3.06147i	− 0.114015i
$722$	0	0
$723$	36.7279	1.36593
$724$	0	0
$725$	6.36304i	0.236317i
$726$	0	0
$727$	19.6569	0.729032	0.364516	−	0.931197i	$-0.381234\pi$
$727$	19.6569	0.729032	0.364516	+	0.931197i	$0.381234\pi$
$728$	0	0
$729$	−22.3137	−0.826434
$730$	0	0
$731$	0	0
$732$	0	0
$733$	4.72792	0.174630	0.0873149	−	0.996181i	$-0.472171\pi$
$733$	4.72792	0.174630	0.0873149	+	0.996181i	$0.472171\pi$
$734$	0	0
$735$	21.8995	0.807775
$736$	0	0
$737$	− 38.2233i	− 1.40797i
$738$	0	0
$739$	16.9289	0.622741	0.311370	−	0.950289i	$-0.399212\pi$
$739$	16.9289	0.622741	0.311370	+	0.950289i	$0.399212\pi$
$740$	0	0
$741$	− 32.6256i	− 1.19853i
$742$	0	0
$743$	28.1647i	1.03326i	0.856208	+	0.516632i	$0.172814\pi$
$743$	28.1647i	1.03326i	−0.856208	+	0.516632i	$0.827186\pi$
$744$	0	0
$745$	17.8435i	0.653737i
$746$	0	0
$747$	0.100505	0.00367729
$748$	0	0
$749$	−2.92893	−0.107021
$750$	0	0
$751$	− 46.0852i	− 1.68167i	−0.541290	−	0.840836i	$-0.682064\pi$
$751$	− 46.0852i	− 1.68167i	0.541290	−	0.840836i	$-0.317936\pi$
$752$	0	0
$753$	42.4441i	1.54675i
$754$	0	0
$755$	26.9510i	0.980848i
$756$	0	0
$757$	−46.6690	−1.69622	−0.848108	−	0.529824i	$-0.822258\pi$
$757$	−46.6690	−1.69622	−0.848108	+	0.529824i	$0.822258\pi$
$758$	0	0
$759$	− 44.1605i	− 1.60292i
$760$	0	0
$761$	−17.8579	−0.647347	−0.323674	−	0.946169i	$-0.604918\pi$
$761$	−17.8579	−0.647347	−0.323674	+	0.946169i	$0.604918\pi$
$762$	0	0
$763$	3.75736	0.136026
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−94.9117	−3.42706
$768$	0	0
$769$	3.79899	0.136995	0.0684975	−	0.997651i	$-0.478179\pi$
$769$	3.79899	0.136995	0.0684975	+	0.997651i	$0.478179\pi$
$770$	0	0
$771$	− 6.94269i	− 0.250035i
$772$	0	0
$773$	−20.9289	−0.752761	−0.376381	−	0.926465i	$-0.622832\pi$
$773$	−20.9289	−0.752761	−0.376381	+	0.926465i	$0.622832\pi$
$774$	0	0
$775$	10.2124i	0.366840i
$776$	0	0
$777$	9.37011i	0.336151i
$778$	0	0
$779$	− 3.13839i	− 0.112445i
$780$	0	0
$781$	4.10051	0.146728
$782$	0	0
$783$	−19.1716	−0.685136
$784$	0	0
$785$	− 4.32957i	− 0.154529i
$786$	0	0
$787$	35.2931i	1.25806i	0.777379	+	0.629032i	$0.216549\pi$
$787$	35.2931i	1.25806i	−0.777379	+	0.629032i	$0.783451\pi$
$788$	0	0
$789$	24.4148i	0.869190i
$790$	0	0
$791$	1.07107	0.0380828
$792$	0	0
$793$	12.6173i	0.448053i
$794$	0	0
$795$	−20.1421	−0.714368
$796$	0	0
$797$	48.5269	1.71891	0.859456	−	0.511210i	$-0.170803\pi$
$797$	48.5269	1.71891	0.859456	+	0.511210i	$0.170803\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	4.48528	0.158480
$802$	0	0
$803$	−29.0711	−1.02590
$804$	0	0
$805$	− 10.0042i	− 0.352600i
$806$	0	0
$807$	15.5563	0.547609
$808$	0	0
$809$	− 29.7724i	− 1.04674i	−0.852105	−	0.523371i	$-0.824674\pi$
$809$	− 29.7724i	− 1.04674i	0.852105	−	0.523371i	$-0.175326\pi$
$810$	0	0
$811$	23.8352i	0.836966i	0.908225	+	0.418483i	$0.137438\pi$
$811$	23.8352i	0.836966i	−0.908225	+	0.418483i	$0.862562\pi$
$812$	0	0
$813$	20.9050i	0.733171i
$814$	0	0
$815$	13.4142	0.469879
$816$	0	0
$817$	−10.9706	−0.383811
$818$	0	0
$819$	− 2.16478i	− 0.0756437i
$820$	0	0
$821$	5.91470i	0.206424i	0.994659	+	0.103212i	$0.0329121\pi$
$821$	5.91470i	0.206424i	−0.994659	+	0.103212i	$0.967088\pi$
$822$	0	0
$823$	30.2207i	1.05343i	0.850042	+	0.526714i	$0.176576\pi$
$823$	30.2207i	1.05343i	−0.850042	+	0.526714i	$0.823424\pi$
$824$	0	0
$825$	−9.89949	−0.344656
$826$	0	0
$827$	− 3.64113i	− 0.126614i	−0.997994	−	0.0633072i	$-0.979835\pi$
$827$	− 3.64113i	− 0.126614i	0.997994	−	0.0633072i	$-0.0201648\pi$
$828$	0	0
$829$	29.6569	1.03003	0.515013	−	0.857183i	$-0.327787\pi$
$829$	29.6569	1.03003	0.515013	+	0.857183i	$0.327787\pi$
$830$	0	0
$831$	−27.8995	−0.967823
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−22.7279	−0.786532
$836$	0	0
$837$	−30.7696	−1.06355
$838$	0	0
$839$	− 6.06854i	− 0.209509i	−0.994498	−	0.104755i	$-0.966594\pi$
$839$	− 6.06854i	− 0.209509i	0.994498	−	0.104755i	$-0.0334057\pi$
$840$	0	0
$841$	12.8995	0.444810
$842$	0	0
$843$	27.8477i	0.959126i
$844$	0	0
$845$	− 62.1354i	− 2.13752i
$846$	0	0
$847$	0.317025i	0.0108931i
$848$	0	0
$849$	13.0711	0.448598
$850$	0	0
$851$	−46.8701	−1.60668
$852$	0	0
$853$	− 21.6704i	− 0.741979i	−0.928637	−	0.370990i	$-0.879018\pi$
$853$	− 21.6704i	− 0.741979i	0.928637	−	0.370990i	$-0.120982\pi$
$854$	0	0
$855$	1.97908i	0.0676829i
$856$	0	0
$857$	− 12.9343i	− 0.441828i	−0.975293	−	0.220914i	$-0.929096\pi$
$857$	− 12.9343i	− 0.441828i	0.975293	−	0.220914i	$-0.0709040\pi$
$858$	0	0
$859$	16.7279	0.570749	0.285375	−	0.958416i	$-0.407882\pi$
$859$	16.7279	0.570749	0.285375	+	0.958416i	$0.407882\pi$
$860$	0	0
$861$	− 1.71644i	− 0.0584962i
$862$	0	0
$863$	14.0000	0.476566	0.238283	−	0.971196i	$-0.423415\pi$
$863$	14.0000	0.476566	0.238283	+	0.971196i	$0.423415\pi$
$864$	0	0
$865$	−46.3848	−1.57713
$866$	0	0
$867$	0	0
$868$	0	0
$869$	38.5269	1.30694
$870$	0	0
$871$	77.2548	2.61768
$872$	0	0
$873$	− 0.393949i	− 0.0133331i
$874$	0	0
$875$	−9.31371	−0.314861
$876$	0	0
$877$	8.60474i	0.290562i	0.989390	+	0.145281i	$0.0464086\pi$
$877$	8.60474i	0.290562i	−0.989390	+	0.145281i	$0.953591\pi$
$878$	0	0
$879$	− 14.7821i	− 0.498587i
$880$	0	0
$881$	− 9.68714i	− 0.326368i	−0.986596	−	0.163184i	$-0.947824\pi$
$881$	− 9.68714i	− 0.326368i	0.986596	−	0.163184i	$-0.0521764\pi$
$882$	0	0
$883$	−14.3431	−0.482685	−0.241343	−	0.970440i	$-0.577588\pi$
$883$	−14.3431	−0.482685	−0.241343	+	0.970440i	$0.577588\pi$
$884$	0	0
$885$	47.4558	1.59521
$886$	0	0
$887$	− 35.4788i	− 1.19126i	−0.803258	−	0.595631i	$-0.796902\pi$
$887$	− 35.4788i	− 1.19126i	0.803258	−	0.595631i	$-0.203098\pi$
$888$	0	0
$889$	− 0.185709i	− 0.00622848i
$890$	0	0
$891$	− 34.0250i	− 1.13988i
$892$	0	0
$893$	9.45584	0.316428
$894$	0	0
$895$	− 33.9706i	− 1.13551i
$896$	0	0
$897$	89.2548	2.98013
$898$	0	0
$899$	−25.8406	−0.861833
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−6.00000	−0.199667
$904$	0	0
$905$	−7.07107	−0.235050
$906$	0	0
$907$	7.59928i	0.252330i	0.992009	+	0.126165i	$0.0402668\pi$
$907$	7.59928i	0.252330i	−0.992009	+	0.126165i	$0.959733\pi$
$908$	0	0
$909$	5.31371	0.176245
$910$	0	0
$911$	− 3.03894i	− 0.100684i	−0.998732	−	0.0503422i	$-0.983969\pi$
$911$	− 3.03894i	− 0.100684i	0.998732	−	0.0503422i	$-0.0160312\pi$
$912$	0	0
$913$	− 0.819760i	− 0.0271301i
$914$	0	0
$915$	− 6.30864i	− 0.208557i
$916$	0	0
$917$	−8.38478	−0.276890
$918$	0	0
$919$	−14.3431	−0.473137	−0.236568	−	0.971615i	$-0.576023\pi$
$919$	−14.3431	−0.473137	−0.236568	+	0.971615i	$0.576023\pi$
$920$	0	0
$921$	11.7206i	0.386207i
$922$	0	0
$923$	8.28772i	0.272794i
$924$	0	0
$925$	10.5069i	0.345465i
$926$	0	0
$927$	−1.65685	−0.0544182
$928$	0	0
$929$	− 49.4412i	− 1.62211i	−0.584969	−	0.811056i	$-0.698893\pi$
$929$	− 49.4412i	− 1.62211i	0.584969	−	0.811056i	$-0.301107\pi$
$930$	0	0
$931$	16.5858	0.543577
$932$	0	0
$933$	−23.5563	−0.771200
$934$	0	0
$935$	0	0
$936$	0	0
$937$	23.3553	0.762986	0.381493	−	0.924372i	$-0.375410\pi$
$937$	23.3553	0.762986	0.381493	+	0.924372i	$0.375410\pi$
$938$	0	0
$939$	−42.0416	−1.37198
$940$	0	0
$941$	1.66205i	0.0541813i	0.999633	+	0.0270906i	$0.00862427\pi$
$941$	1.66205i	0.0541813i	−0.999633	+	0.0270906i	$0.991376\pi$
$942$	0	0
$943$	8.58579	0.279592
$944$	0	0
$945$	− 6.75699i	− 0.219805i
$946$	0	0
$947$	9.05309i	0.294186i	0.989123	+	0.147093i	$0.0469916\pi$
$947$	9.05309i	0.294186i	−0.989123	+	0.147093i	$0.953008\pi$
$948$	0	0
$949$	− 58.7569i	− 1.90733i
$950$	0	0
$951$	0.727922	0.0236045
$952$	0	0
$953$	12.1421	0.393322	0.196661	−	0.980472i	$-0.436990\pi$
$953$	12.1421	0.393322	0.196661	+	0.980472i	$0.436990\pi$
$954$	0	0
$955$	− 0.634051i	− 0.0205174i
$956$	0	0
$957$	− 25.0489i	− 0.809715i
$958$	0	0
$959$	− 9.29319i	− 0.300093i
$960$	0	0
$961$	−10.4731	−0.337842
$962$	0	0
$963$	1.58513i	0.0510800i
$964$	0	0
$965$	28.7279	0.924785
$966$	0	0
$967$	33.0122	1.06160	0.530800	−	0.847497i	$-0.321891\pi$
$967$	33.0122	1.06160	0.530800	+	0.847497i	$0.321891\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	37.2132	1.19423	0.597114	−	0.802156i	$-0.296314\pi$
$971$	37.2132	1.19423	0.597114	+	0.802156i	$0.296314\pi$
$972$	0	0
$973$	13.4142	0.430040
$974$	0	0
$975$	− 20.0083i	− 0.640779i
$976$	0	0
$977$	−38.3848	−1.22804	−0.614019	−	0.789291i	$-0.710448\pi$
$977$	−38.3848	−1.22804	−0.614019	+	0.789291i	$0.710448\pi$
$978$	0	0
$979$	− 36.5838i	− 1.16922i
$980$	0	0
$981$	− 2.03347i	− 0.0649236i
$982$	0	0
$983$	− 39.3282i	− 1.25437i	−0.778868	−	0.627187i	$-0.784206\pi$
$983$	− 39.3282i	− 1.25437i	0.778868	−	0.627187i	$-0.215794\pi$
$984$	0	0
$985$	23.2132	0.739634
$986$	0	0
$987$	5.17157	0.164613
$988$	0	0
$989$	− 30.0125i	− 0.954342i
$990$	0	0
$991$	45.3742i	1.44136i	0.693268	+	0.720680i	$0.256170\pi$
$991$	45.3742i	1.44136i	−0.693268	+	0.720680i	$0.743830\pi$
$992$	0	0
$993$	− 53.0823i	− 1.68452i
$994$	0	0
$995$	3.07107	0.0973594
$996$	0	0
$997$	30.9636i	0.980626i	0.871546	+	0.490313i	$0.163118\pi$
$997$	30.9636i	0.980626i	−0.871546	+	0.490313i	$0.836882\pi$
$998$	0	0
$999$	−31.6569	−1.00158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1156.2.b.d.577.4		4
17.2	even	8		1156.2.e.f.829.4		8
17.3	odd	16		68.2.h.a.25.1	✓	4
17.4	even	4		1156.2.a.g.1.4		4
17.5	odd	16		1156.2.h.a.757.1		4
17.6	odd	16		1156.2.h.b.1001.1		4
17.7	odd	16		1156.2.h.c.733.1		4
17.8	even	8		1156.2.e.f.905.4		8
17.9	even	8		1156.2.e.f.905.1		8
17.10	odd	16		1156.2.h.a.733.1		4
17.11	odd	16		68.2.h.a.49.1	yes	4
17.12	odd	16		1156.2.h.c.757.1		4
17.13	even	4		1156.2.a.g.1.1		4
17.14	odd	16		1156.2.h.b.977.1		4
17.15	even	8		1156.2.e.f.829.1		8
17.16	even	2	inner	1156.2.b.d.577.1		4
51.11	even	16		612.2.w.a.253.1		4
51.20	even	16		612.2.w.a.433.1		4
68.3	even	16		272.2.v.c.161.1		4
68.11	even	16		272.2.v.c.49.1		4
68.47	odd	4		4624.2.a.bl.1.4		4
68.55	odd	4		4624.2.a.bl.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.h.a.25.1	✓	4	17.3	odd	16
68.2.h.a.49.1	yes	4	17.11	odd	16
272.2.v.c.49.1		4	68.11	even	16
272.2.v.c.161.1		4	68.3	even	16
612.2.w.a.253.1		4	51.11	even	16
612.2.w.a.433.1		4	51.20	even	16
1156.2.a.g.1.1		4	17.13	even	4
1156.2.a.g.1.4		4	17.4	even	4
1156.2.b.d.577.1		4	17.16	even	2	inner
1156.2.b.d.577.4		4	1.1	even	1	trivial
1156.2.e.f.829.1		8	17.15	even	8
1156.2.e.f.829.4		8	17.2	even	8
1156.2.e.f.905.1		8	17.9	even	8
1156.2.e.f.905.4		8	17.8	even	8
1156.2.h.a.733.1		4	17.10	odd	16
1156.2.h.a.757.1		4	17.5	odd	16
1156.2.h.b.977.1		4	17.14	odd	16
1156.2.h.b.1001.1		4	17.6	odd	16
1156.2.h.c.733.1		4	17.7	odd	16
1156.2.h.c.757.1		4	17.12	odd	16
4624.2.a.bl.1.1		4	68.55	odd	4
4624.2.a.bl.1.4		4	68.47	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1156 = 2^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1156.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.84776i q^{3} -1.84776i q^{5} -0.765367i q^{7} -0.414214 q^{9} +3.37849i q^{11} -6.82843 q^{13} +3.41421 q^{15} +2.58579 q^{19} +1.41421 q^{21} +7.07401i q^{23} +1.58579 q^{25} +4.77791i q^{27} +4.01254i q^{29} +6.43996i q^{31} -6.24264 q^{33} -1.41421 q^{35} +6.62567i q^{37} -12.6173i q^{39} -1.21371i q^{41} -4.24264 q^{43} +0.765367i q^{45} +3.65685 q^{47} +6.41421 q^{49} -5.89949 q^{53} +6.24264 q^{55} +4.77791i q^{57} +13.8995 q^{59} -1.84776i q^{61} +0.317025i q^{63} +12.6173i q^{65} -11.3137 q^{67} -13.0711 q^{69} -1.21371i q^{71} +8.60474i q^{73} +2.93015i q^{75} +2.58579 q^{77} -11.4036i q^{79} -10.0711 q^{81} -0.242641 q^{83} -7.41421 q^{87} -10.8284 q^{89} +5.22625i q^{91} -11.8995 q^{93} -4.77791i q^{95} +0.951076i q^{97} -1.39942i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9} - 16 q^{13} + 8 q^{15} + 16 q^{19} + 12 q^{25} - 8 q^{33} - 8 q^{47} + 20 q^{49} + 16 q^{53} + 8 q^{55} + 16 q^{59} - 24 q^{69} + 16 q^{77} - 12 q^{81} + 16 q^{83} - 24 q^{87} - 32 q^{89}+ \cdots - 8 q^{93}+O(q^{100}) \) 4 * q + 4 * q^9 - 16 * q^13 + 8 * q^15 + 16 * q^19 + 12 * q^25 - 8 * q^33 - 8 * q^47 + 20 * q^49 + 16 * q^53 + 8 * q^55 + 16 * q^59 - 24 * q^69 + 16 * q^77 - 12 * q^81 + 16 * q^83 - 24 * q^87 - 32 * q^89 - 8 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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