LMFDB - Embedded newform 1156.2.a.g.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1156,2,Mod(1,1156)] 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("1156.1"); 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, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1156 = 2^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1156.a (trivial)

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:	yes
Analytic conductor:	$9.23070647366$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
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)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.84776$ of defining polynomial
Character	$\chi$	$=$	1156.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

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
$3$	1.84776	1.06680	0.533402	−	0.845862i	$-0.320913\pi$
$3$	1.84776	1.06680	0.533402	+	0.845862i	$0.320913\pi$
$4$	0	0
$5$	−1.84776	−0.826343	−0.413171	−	0.910653i	$-0.635579\pi$
$5$	−1.84776	−0.826343	−0.413171	+	0.910653i	$0.635579\pi$
$6$	0	0
$7$	0.765367	0.289281	0.144641	−	0.989484i	$-0.453797\pi$
$7$	0.765367	0.289281	0.144641	+	0.989484i	$0.453797\pi$
$8$	0	0
$9$	0.414214	0.138071
$10$	0	0
$11$	−3.37849	−1.01865	−0.509327	−	0.860573i	$-0.670106\pi$
$11$	−3.37849	−1.01865	−0.509327	+	0.860573i	$0.670106\pi$
$12$	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	$-0.595856\pi$
$19$	−2.58579	−0.593220	−0.296610	+	0.954999i	$0.595856\pi$
$20$	0	0
$21$	1.41421	0.308607
$22$	0	0
$23$	−7.07401	−1.47503	−0.737517	−	0.675329i	$-0.764002\pi$
$23$	−7.07401	−1.47503	−0.737517	+	0.675329i	$0.764002\pi$
$24$	0	0
$25$	−1.58579	−0.317157
$26$	0	0
$27$	−4.77791	−0.919509
$28$	0	0
$29$	4.01254	0.745111	0.372555	−	0.928010i	$-0.378482\pi$
$29$	4.01254	0.745111	0.372555	+	0.928010i	$0.378482\pi$
$30$	0	0
$31$	6.43996	1.15665	0.578326	−	0.815806i	$-0.303706\pi$
$31$	6.43996	1.15665	0.578326	+	0.815806i	$0.303706\pi$
$32$	0	0
$33$	−6.24264	−1.08670
$34$	0	0
$35$	−1.41421	−0.239046
$36$	0	0
$37$	6.62567	1.08925	0.544627	−	0.838679i	$-0.316671\pi$
$37$	6.62567	1.08925	0.544627	+	0.838679i	$0.316671\pi$
$38$	0	0
$39$	−12.6173	−2.02038
$40$	0	0
$41$	1.21371	0.189549	0.0947747	−	0.995499i	$-0.469787\pi$
$41$	1.21371	0.189549	0.0947747	+	0.995499i	$0.469787\pi$
$42$	0	0
$43$	4.24264	0.646997	0.323498	−	0.946229i	$-0.395141\pi$
$43$	4.24264	0.646997	0.323498	+	0.946229i	$0.395141\pi$
$44$	0	0
$45$	−0.765367	−0.114094
$46$	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	$-0.367209\pi$
$53$	5.89949	0.810358	0.405179	+	0.914237i	$0.367209\pi$
$54$	0	0
$55$	6.24264	0.841757
$56$	0	0
$57$	−4.77791	−0.632850
$58$	0	0
$59$	−13.8995	−1.80956	−0.904780	−	0.425879i	$-0.859965\pi$
$59$	−13.8995	−1.80956	−0.904780	+	0.425879i	$0.859965\pi$
$60$	0	0
$61$	1.84776	0.236581	0.118291	−	0.992979i	$-0.462259\pi$
$61$	1.84776	0.236581	0.118291	+	0.992979i	$0.462259\pi$
$62$	0	0
$63$	0.317025	0.0399414
$64$	0	0
$65$	12.6173	1.56498
$66$	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
$71$	−1.21371	−0.144041	−0.0720203	−	0.997403i	$-0.522945\pi$
$71$	−1.21371	−0.144041	−0.0720203	+	0.997403i	$0.522945\pi$
$72$	0	0
$73$	8.60474	1.00711	0.503555	−	0.863963i	$-0.332025\pi$
$73$	8.60474	1.00711	0.503555	+	0.863963i	$0.332025\pi$
$74$	0	0
$75$	−2.93015	−0.338345
$76$	0	0
$77$	−2.58579	−0.294678
$78$	0	0
$79$	11.4036	1.28300	0.641501	−	0.767122i	$-0.278312\pi$
$79$	11.4036	1.28300	0.641501	+	0.767122i	$0.278312\pi$
$80$	0	0
$81$	−10.0711	−1.11901
$82$	0	0
$83$	0.242641	0.0266333	0.0133166	−	0.999911i	$-0.495761\pi$
$83$	0.242641	0.0266333	0.0133166	+	0.999911i	$0.495761\pi$
$84$	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
$91$	−5.22625	−0.547860
$92$	0	0
$93$	11.8995	1.23392
$94$	0	0
$95$	4.77791	0.490203
$96$	0	0
$97$	0.951076	0.0965671	0.0482836	−	0.998834i	$-0.484625\pi$
$97$	0.951076	0.0965671	0.0482836	+	0.998834i	$0.484625\pi$
$98$	0	0
$99$	−1.39942	−0.140647
$100$	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
$105$	−2.61313	−0.255015
$106$	0	0
$107$	−3.82683	−0.369954	−0.184977	−	0.982743i	$-0.559221\pi$
$107$	−3.82683	−0.369954	−0.184977	+	0.982743i	$0.559221\pi$
$108$	0	0
$109$	−4.90923	−0.470219	−0.235109	−	0.971969i	$-0.575545\pi$
$109$	−4.90923	−0.470219	−0.235109	+	0.971969i	$0.575545\pi$
$110$	0	0
$111$	12.2426	1.16202
$112$	0	0
$113$	−1.39942	−0.131646	−0.0658231	−	0.997831i	$-0.520967\pi$
$113$	−1.39942	−0.131646	−0.0658231	+	0.997831i	$0.520967\pi$
$114$	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
$125$	12.1689	1.08842
$126$	0	0
$127$	−0.242641	−0.0215309	−0.0107654	−	0.999942i	$-0.503427\pi$
$127$	−0.242641	−0.0215309	−0.0107654	+	0.999942i	$0.503427\pi$
$128$	0	0
$129$	7.83938	0.690219
$130$	0	0
$131$	−10.9552	−0.957164	−0.478582	−	0.878043i	$-0.658849\pi$
$131$	−10.9552	−0.957164	−0.478582	+	0.878043i	$0.658849\pi$
$132$	0	0
$133$	−1.97908	−0.171608
$134$	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
$139$	17.5265	1.48658	0.743290	−	0.668970i	$-0.233264\pi$
$139$	17.5265	1.48658	0.743290	+	0.668970i	$0.233264\pi$
$140$	0	0
$141$	6.75699	0.569041
$142$	0	0
$143$	23.0698	1.92919
$144$	0	0
$145$	−7.41421	−0.615717
$146$	0	0
$147$	−11.8519	−0.977530
$148$	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	$-0.297751\pi$
$151$	14.5858	1.18697	0.593487	+	0.804843i	$0.297751\pi$
$152$	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
$159$	10.9008	0.864493
$160$	0	0
$161$	−5.41421	−0.426700
$162$	0	0
$163$	−7.25972	−0.568625	−0.284313	−	0.958732i	$-0.591765\pi$
$163$	−7.25972	−0.568625	−0.284313	+	0.958732i	$0.591765\pi$
$164$	0	0
$165$	11.5349	0.897990
$166$	0	0
$167$	−12.3003	−0.951823	−0.475911	−	0.879493i	$-0.657882\pi$
$167$	−12.3003	−0.951823	−0.475911	+	0.879493i	$0.657882\pi$
$168$	0	0
$169$	33.6274	2.58672
$170$	0	0
$171$	−1.07107	−0.0819066
$172$	0	0
$173$	−25.1033	−1.90857	−0.954283	−	0.298905i	$-0.903379\pi$
$173$	−25.1033	−1.90857	−0.954283	+	0.298905i	$0.903379\pi$
$174$	0	0
$175$	−1.21371	−0.0917477
$176$	0	0
$177$	−25.6829	−1.93045
$178$	0	0
$179$	−18.3848	−1.37414	−0.687071	−	0.726590i	$-0.741104\pi$
$179$	−18.3848	−1.37414	−0.687071	+	0.726590i	$0.741104\pi$
$180$	0	0
$181$	3.82683	0.284446	0.142223	−	0.989835i	$-0.454575\pi$
$181$	3.82683	0.284446	0.142223	+	0.989835i	$0.454575\pi$
$182$	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
$193$	−15.5474	−1.11913	−0.559565	−	0.828787i	$-0.689032\pi$
$193$	−15.5474	−1.11913	−0.559565	+	0.828787i	$0.689032\pi$
$194$	0	0
$195$	23.3137	1.66953
$196$	0	0
$197$	−12.5629	−0.895069	−0.447535	−	0.894267i	$-0.647698\pi$
$197$	−12.5629	−0.895069	−0.447535	+	0.894267i	$0.647698\pi$
$198$	0	0
$199$	1.66205	0.117820	0.0589098	−	0.998263i	$-0.481238\pi$
$199$	1.66205	0.117820	0.0589098	+	0.998263i	$0.481238\pi$
$200$	0	0
$201$	−20.9050	−1.47453
$202$	0	0
$203$	3.07107	0.215547
$204$	0	0
$205$	−2.24264	−0.156633
$206$	0	0
$207$	−2.93015	−0.203660
$208$	0	0
$209$	8.73606	0.604286
$210$	0	0
$211$	2.03347	0.139990	0.0699949	−	0.997547i	$-0.477702\pi$
$211$	2.03347	0.139990	0.0699949	+	0.997547i	$0.477702\pi$
$212$	0	0
$213$	−2.24264	−0.153663
$214$	0	0
$215$	−7.83938	−0.534641
$216$	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	$-0.695520\pi$
$223$	−17.2132	−1.15268	−0.576341	+	0.817210i	$0.695520\pi$
$224$	0	0
$225$	−0.656854	−0.0437903
$226$	0	0
$227$	19.4287	1.28953	0.644763	−	0.764383i	$-0.276956\pi$
$227$	19.4287	1.28953	0.644763	+	0.764383i	$0.276956\pi$
$228$	0	0
$229$	−3.75736	−0.248293	−0.124147	−	0.992264i	$-0.539619\pi$
$229$	−3.75736	−0.248293	−0.124147	+	0.992264i	$0.539619\pi$
$230$	0	0
$231$	−4.77791	−0.314363
$232$	0	0
$233$	−8.97616	−0.588048	−0.294024	−	0.955798i	$-0.594995\pi$
$233$	−8.97616	−0.588048	−0.294024	+	0.955798i	$0.594995\pi$
$234$	0	0
$235$	−6.75699	−0.440777
$236$	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
$241$	−19.8770	−1.28039	−0.640195	−	0.768212i	$-0.721147\pi$
$241$	−19.8770	−1.28039	−0.640195	+	0.768212i	$0.721147\pi$
$242$	0	0
$243$	−4.27518	−0.274253
$244$	0	0
$245$	11.8519	0.757191
$246$	0	0
$247$	17.6569	1.12348
$248$	0	0
$249$	0.448342	0.0284125
$250$	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	$-0.462612\pi$
$257$	3.75736	0.234378	0.117189	+	0.993110i	$0.462612\pi$
$258$	0	0
$259$	5.07107	0.315101
$260$	0	0
$261$	1.66205	0.102878
$262$	0	0
$263$	−13.2132	−0.814761	−0.407381	−	0.913259i	$-0.633558\pi$
$263$	−13.2132	−0.814761	−0.407381	+	0.913259i	$0.633558\pi$
$264$	0	0
$265$	−10.9008	−0.669634
$266$	0	0
$267$	−20.0083	−1.22449
$268$	0	0
$269$	−8.41904	−0.513318	−0.256659	−	0.966502i	$-0.582622\pi$
$269$	−8.41904	−0.513318	−0.256659	+	0.966502i	$0.582622\pi$
$270$	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
$275$	5.35757	0.323074
$276$	0	0
$277$	15.0991	0.907217	0.453608	−	0.891201i	$-0.350136\pi$
$277$	15.0991	0.907217	0.453608	+	0.891201i	$0.350136\pi$
$278$	0	0
$279$	2.66752	0.159700
$280$	0	0
$281$	−15.0711	−0.899065	−0.449532	−	0.893264i	$-0.648409\pi$
$281$	−15.0711	−0.899065	−0.449532	+	0.893264i	$0.648409\pi$
$282$	0	0
$283$	7.07401	0.420506	0.210253	−	0.977647i	$-0.432571\pi$
$283$	7.07401	0.420506	0.210253	+	0.977647i	$0.432571\pi$
$284$	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
$295$	25.6829	1.49532
$296$	0	0
$297$	16.1421	0.936662
$298$	0	0
$299$	48.3044	2.79351
$300$	0	0
$301$	3.24718	0.187164
$302$	0	0
$303$	−23.7038	−1.36175
$304$	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
$309$	7.39104	0.420461
$310$	0	0
$311$	12.7486	0.722907	0.361454	−	0.932390i	$-0.382281\pi$
$311$	12.7486	0.722907	0.361454	+	0.932390i	$0.382281\pi$
$312$	0	0
$313$	−22.7528	−1.28606	−0.643031	−	0.765840i	$-0.722323\pi$
$313$	−22.7528	−1.28606	−0.643031	+	0.765840i	$0.722323\pi$
$314$	0	0
$315$	−0.585786	−0.0330053
$316$	0	0
$317$	0.393949	0.0221264	0.0110632	−	0.999939i	$-0.496478\pi$
$317$	0.393949	0.0221264	0.0110632	+	0.999939i	$0.496478\pi$
$318$	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
$329$	2.79884	0.154305
$330$	0	0
$331$	28.7279	1.57903	0.789515	−	0.613732i	$-0.210332\pi$
$331$	28.7279	1.57903	0.789515	+	0.613732i	$0.210332\pi$
$332$	0	0
$333$	2.74444	0.150395
$334$	0	0
$335$	20.9050	1.14216
$336$	0	0
$337$	25.9999	1.41631	0.708154	−	0.706058i	$-0.249528\pi$
$337$	25.9999	1.41631	0.708154	+	0.706058i	$0.249528\pi$
$338$	0	0
$339$	−2.58579	−0.140441
$340$	0	0
$341$	−21.7574	−1.17823
$342$	0	0
$343$	−10.2668	−0.554355
$344$	0	0
$345$	24.1522	1.30031
$346$	0	0
$347$	7.78498	0.417920	0.208960	−	0.977924i	$-0.432992\pi$
$347$	7.78498	0.417920	0.208960	+	0.977924i	$0.432992\pi$
$348$	0	0
$349$	−28.0416	−1.50103	−0.750517	−	0.660851i	$-0.770195\pi$
$349$	−28.0416	−1.50103	−0.750517	+	0.660851i	$0.770195\pi$
$350$	0	0
$351$	32.6256	1.74143
$352$	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	$-0.203863\pi$
$359$	30.3848	1.60365	0.801824	+	0.597561i	$0.203863\pi$
$360$	0	0
$361$	−12.3137	−0.648090
$362$	0	0
$363$	0.765367	0.0401713
$364$	0	0
$365$	−15.8995	−0.832218
$366$	0	0
$367$	−3.82683	−0.199759	−0.0998796	−	0.995000i	$-0.531846\pi$
$367$	−3.82683	−0.199759	−0.0998796	+	0.995000i	$0.531846\pi$
$368$	0	0
$369$	0.502734	0.0261713
$370$	0	0
$371$	4.51528	0.234422
$372$	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
$377$	−27.3994	−1.41114
$378$	0	0
$379$	10.9552	0.562733	0.281366	−	0.959600i	$-0.409212\pi$
$379$	10.9552	0.562733	0.281366	+	0.959600i	$0.409212\pi$
$380$	0	0
$381$	−0.448342	−0.0229692
$382$	0	0
$383$	25.8995	1.32340	0.661701	−	0.749768i	$-0.269835\pi$
$383$	25.8995	1.32340	0.661701	+	0.749768i	$0.269835\pi$
$384$	0	0
$385$	4.77791	0.243505
$386$	0	0
$387$	1.75736	0.0893316
$388$	0	0
$389$	−2.10051	−0.106500	−0.0532499	−	0.998581i	$-0.516958\pi$
$389$	−2.10051	−0.106500	−0.0532499	+	0.998581i	$0.516958\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−20.2426	−1.02111
$394$	0	0
$395$	−21.0711	−1.06020
$396$	0	0
$397$	18.8715	0.947135	0.473568	−	0.880757i	$-0.342966\pi$
$397$	18.8715	0.947135	0.473568	+	0.880757i	$0.342966\pi$
$398$	0	0
$399$	−3.65685	−0.183072
$400$	0	0
$401$	9.23880	0.461363	0.230682	−	0.973029i	$-0.425904\pi$
$401$	9.23880	0.461363	0.230682	+	0.973029i	$0.425904\pi$
$402$	0	0
$403$	−43.9748	−2.19054
$404$	0	0
$405$	18.6089	0.924684
$406$	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
$411$	22.4357	1.10667
$412$	0	0
$413$	−10.6382	−0.523472
$414$	0	0
$415$	−0.448342	−0.0220082
$416$	0	0
$417$	32.3848	1.58589
$418$	0	0
$419$	−35.6327	−1.74077	−0.870385	−	0.492371i	$-0.836130\pi$
$419$	−35.6327	−1.74077	−0.870385	+	0.492371i	$0.836130\pi$
$420$	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
$431$	−20.5880	−0.991688	−0.495844	−	0.868412i	$-0.665141\pi$
$431$	−20.5880	−0.991688	−0.495844	+	0.868412i	$0.665141\pi$
$432$	0	0
$433$	8.24264	0.396116	0.198058	−	0.980190i	$-0.436537\pi$
$433$	8.24264	0.396116	0.198058	+	0.980190i	$0.436537\pi$
$434$	0	0
$435$	−13.6997	−0.656849
$436$	0	0
$437$	18.2919	0.875019
$438$	0	0
$439$	−14.8365	−0.708106	−0.354053	−	0.935225i	$-0.615197\pi$
$439$	−14.8365	−0.708106	−0.354053	+	0.935225i	$0.615197\pi$
$440$	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
$445$	20.0083	0.948486
$446$	0	0
$447$	−17.8435	−0.843970
$448$	0	0
$449$	21.2220	1.00153	0.500765	−	0.865583i	$-0.333052\pi$
$449$	21.2220	1.00153	0.500765	+	0.865583i	$0.333052\pi$
$450$	0	0
$451$	−4.10051	−0.193085
$452$	0	0
$453$	26.9510	1.26627
$454$	0	0
$455$	9.65685	0.452720
$456$	0	0
$457$	4.24264	0.198462	0.0992312	−	0.995064i	$-0.468362\pi$
$457$	4.24264	0.198462	0.0992312	+	0.995064i	$0.468362\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−7.75736	−0.361296	−0.180648	−	0.983548i	$-0.557820\pi$
$461$	−7.75736	−0.361296	−0.180648	+	0.983548i	$0.557820\pi$
$462$	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
$465$	−21.9874	−1.01964
$466$	0	0
$467$	−28.2426	−1.30691	−0.653457	−	0.756964i	$-0.726682\pi$
$467$	−28.2426	−1.30691	−0.653457	+	0.756964i	$0.726682\pi$
$468$	0	0
$469$	−8.65914	−0.399842
$470$	0	0
$471$	4.32957	0.199496
$472$	0	0
$473$	−14.3337	−0.659066
$474$	0	0
$475$	4.10051	0.188144
$476$	0	0
$477$	2.44365	0.111887
$478$	0	0
$479$	−8.23333	−0.376190	−0.188095	−	0.982151i	$-0.560231\pi$
$479$	−8.23333	−0.376190	−0.188095	+	0.982151i	$0.560231\pi$
$480$	0	0
$481$	−45.2429	−2.06290
$482$	0	0
$483$	−10.0042	−0.455205
$484$	0	0
$485$	−1.75736	−0.0797976
$486$	0	0
$487$	3.64113	0.164995	0.0824976	−	0.996591i	$-0.473710\pi$
$487$	3.64113	0.164995	0.0824976	+	0.996591i	$0.473710\pi$
$488$	0	0
$489$	−13.4142	−0.606612
$490$	0	0
$491$	−23.7574	−1.07215	−0.536077	−	0.844169i	$-0.680094\pi$
$491$	−23.7574	−1.07215	−0.536077	+	0.844169i	$0.680094\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	2.58579	0.116222
$496$	0	0
$497$	−0.928932	−0.0416683
$498$	0	0
$499$	−37.5348	−1.68029	−0.840145	−	0.542362i	$-0.817530\pi$
$499$	−37.5348	−1.68029	−0.840145	+	0.542362i	$0.817530\pi$
$500$	0	0
$501$	−22.7279	−1.01541
$502$	0	0
$503$	7.89377	0.351966	0.175983	−	0.984393i	$-0.443690\pi$
$503$	7.89377	0.351966	0.175983	+	0.984393i	$0.443690\pi$
$504$	0	0
$505$	23.7038	1.05481
$506$	0	0
$507$	62.1354	2.75953
$508$	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
$513$	12.3547	0.545471
$514$	0	0
$515$	−7.39104	−0.325688
$516$	0	0
$517$	−12.3547	−0.543357
$518$	0	0
$519$	−46.3848	−2.03607
$520$	0	0
$521$	−27.5307	−1.20614	−0.603070	−	0.797688i	$-0.706056\pi$
$521$	−27.5307	−1.20614	−0.603070	+	0.797688i	$0.706056\pi$
$522$	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
$533$	−8.28772	−0.358981
$534$	0	0
$535$	7.07107	0.305709
$536$	0	0
$537$	−33.9706	−1.46594
$538$	0	0
$539$	21.6704	0.933409
$540$	0	0
$541$	−7.89377	−0.339380	−0.169690	−	0.985498i	$-0.554277\pi$
$541$	−7.89377	−0.339380	−0.169690	+	0.985498i	$0.554277\pi$
$542$	0	0
$543$	7.07107	0.303449
$544$	0	0
$545$	9.07107	0.388562
$546$	0	0
$547$	24.5461	1.04952	0.524758	−	0.851251i	$-0.324156\pi$
$547$	24.5461	1.04952	0.524758	+	0.851251i	$0.324156\pi$
$548$	0	0
$549$	0.765367	0.0326651
$550$	0	0
$551$	−10.3756	−0.442015
$552$	0	0
$553$	8.72792	0.371149
$554$	0	0
$555$	−22.6215	−0.960227
$556$	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	$-0.802221\pi$
$563$	−38.5858	−1.62620	−0.813099	+	0.582126i	$0.802221\pi$
$564$	0	0
$565$	2.58579	0.108785
$566$	0	0
$567$	−7.70806	−0.323708
$568$	0	0
$569$	−27.5563	−1.15522	−0.577611	−	0.816312i	$-0.696015\pi$
$569$	−27.5563	−1.15522	−0.577611	+	0.816312i	$0.696015\pi$
$570$	0	0
$571$	−25.4747	−1.06608	−0.533041	−	0.846089i	$-0.678951\pi$
$571$	−25.4747	−1.06608	−0.533041	+	0.846089i	$0.678951\pi$
$572$	0	0
$573$	0.634051	0.0264878
$574$	0	0
$575$	11.2179	0.467818
$576$	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
$581$	0.185709	0.00770451
$582$	0	0
$583$	−19.9314	−0.825474
$584$	0	0
$585$	5.22625	0.216079
$586$	0	0
$587$	29.2132	1.20576	0.602879	−	0.797833i	$-0.294020\pi$
$587$	29.2132	1.20576	0.602879	+	0.797833i	$0.294020\pi$
$588$	0	0
$589$	−16.6524	−0.686149
$590$	0	0
$591$	−23.2132	−0.954864
$592$	0	0
$593$	−33.2132	−1.36390	−0.681951	−	0.731397i	$-0.738868\pi$
$593$	−33.2132	−1.36390	−0.681951	+	0.731397i	$0.738868\pi$
$594$	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
$601$	−5.09494	−0.207827	−0.103913	−	0.994586i	$-0.533136\pi$
$601$	−5.09494	−0.207827	−0.103913	+	0.994586i	$0.533136\pi$
$602$	0	0
$603$	−4.68629	−0.190841
$604$	0	0
$605$	−0.765367	−0.0311166
$606$	0	0
$607$	15.9189	0.646127	0.323063	−	0.946377i	$-0.395287\pi$
$607$	15.9189	0.646127	0.323063	+	0.946377i	$0.395287\pi$
$608$	0	0
$609$	5.67459	0.229946
$610$	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
$615$	−4.14386	−0.167097
$616$	0	0
$617$	−34.1020	−1.37289	−0.686446	−	0.727180i	$-0.740830\pi$
$617$	−34.1020	−1.37289	−0.686446	+	0.727180i	$0.740830\pi$
$618$	0	0
$619$	13.3827	0.537894	0.268947	−	0.963155i	$-0.413324\pi$
$619$	13.3827	0.537894	0.268947	+	0.963155i	$0.413324\pi$
$620$	0	0
$621$	33.7990	1.35631
$622$	0	0
$623$	−8.28772	−0.332040
$624$	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	$-0.593764\pi$
$631$	−14.5858	−0.580651	−0.290325	+	0.956928i	$0.593764\pi$
$632$	0	0
$633$	3.75736	0.149342
$634$	0	0
$635$	0.448342	0.0177919
$636$	0	0
$637$	43.7990	1.73538
$638$	0	0
$639$	−0.502734	−0.0198879
$640$	0	0
$641$	−46.4566	−1.83493	−0.917463	−	0.397821i	$-0.869766\pi$
$641$	−46.4566	−1.83493	−0.917463	+	0.397821i	$0.869766\pi$
$642$	0	0
$643$	5.54328	0.218606	0.109303	−	0.994009i	$-0.465138\pi$
$643$	5.54328	0.218606	0.109303	+	0.994009i	$0.465138\pi$
$644$	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
$649$	46.9593	1.84332
$650$	0	0
$651$	9.10748	0.356950
$652$	0	0
$653$	45.1885	1.76836	0.884181	−	0.467144i	$-0.154717\pi$
$653$	45.1885	1.76836	0.884181	+	0.467144i	$0.154717\pi$
$654$	0	0
$655$	20.2426	0.790945
$656$	0	0
$657$	3.56420	0.139053
$658$	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	$-0.420381\pi$
$661$	12.7279	0.495059	0.247529	+	0.968880i	$0.420381\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.65685	0.141807
$666$	0	0
$667$	−28.3848	−1.09906
$668$	0	0
$669$	−31.8059	−1.22969
$670$	0	0
$671$	−6.24264	−0.240994
$672$	0	0
$673$	31.0405	1.19652	0.598261	−	0.801301i	$-0.295858\pi$
$673$	31.0405	1.19652	0.598261	+	0.801301i	$0.295858\pi$
$674$	0	0
$675$	7.57675	0.291629
$676$	0	0
$677$	1.66205	0.0638778	0.0319389	−	0.999490i	$-0.489832\pi$
$677$	1.66205	0.0638778	0.0319389	+	0.999490i	$0.489832\pi$
$678$	0	0
$679$	0.727922	0.0279351
$680$	0	0
$681$	35.8995	1.37567
$682$	0	0
$683$	−4.27518	−0.163585	−0.0817925	−	0.996649i	$-0.526064\pi$
$683$	−4.27518	−0.163585	−0.0817925	+	0.996649i	$0.526064\pi$
$684$	0	0
$685$	−22.4357	−0.857226
$686$	0	0
$687$	−6.94269	−0.264880
$688$	0	0
$689$	−40.2843	−1.53471
$690$	0	0
$691$	7.07401	0.269108	0.134554	−	0.990906i	$-0.457040\pi$
$691$	7.07401	0.269108	0.134554	+	0.990906i	$0.457040\pi$
$692$	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
$703$	−17.1326	−0.646167
$704$	0	0
$705$	−12.4853	−0.470223
$706$	0	0
$707$	−9.81845	−0.369261
$708$	0	0
$709$	−3.37849	−0.126882	−0.0634410	−	0.997986i	$-0.520207\pi$
$709$	−3.37849	−0.126882	−0.0634410	+	0.997986i	$0.520207\pi$
$710$	0	0
$711$	4.72352	0.177146
$712$	0	0
$713$	−45.5563	−1.70610
$714$	0	0
$715$	−42.6274	−1.59418
$716$	0	0
$717$	41.8100	1.56142
$718$	0	0
$719$	−15.0222	−0.560233	−0.280116	−	0.959966i	$-0.590373\pi$
$719$	−15.0222	−0.560233	−0.280116	+	0.959966i	$0.590373\pi$
$720$	0	0
$721$	3.06147	0.114015
$722$	0	0
$723$	−36.7279	−1.36593
$724$	0	0
$725$	−6.36304	−0.236317
$726$	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	$-0.527829\pi$
$733$	−4.72792	−0.174630	−0.0873149	+	0.996181i	$0.527829\pi$
$734$	0	0
$735$	21.8995	0.807775
$736$	0	0
$737$	38.2233	1.40797
$738$	0	0
$739$	−16.9289	−0.622741	−0.311370	−	0.950289i	$-0.600788\pi$
$739$	−16.9289	−0.622741	−0.311370	+	0.950289i	$0.600788\pi$
$740$	0	0
$741$	32.6256	1.19853
$742$	0	0
$743$	28.1647	1.03326	0.516632	−	0.856208i	$-0.327186\pi$
$743$	28.1647	1.03326	0.516632	+	0.856208i	$0.327186\pi$
$744$	0	0
$745$	17.8435	0.653737
$746$	0	0
$747$	0.100505	0.00367729
$748$	0	0
$749$	−2.92893	−0.107021
$750$	0	0
$751$	−46.0852	−1.68167	−0.840836	−	0.541290i	$-0.817936\pi$
$751$	−46.0852	−1.68167	−0.840836	+	0.541290i	$0.817936\pi$
$752$	0	0
$753$	42.4441	1.54675
$754$	0	0
$755$	−26.9510	−0.980848
$756$	0	0
$757$	46.6690	1.69622	0.848108	−	0.529824i	$-0.177742\pi$
$757$	46.6690	1.69622	0.848108	+	0.529824i	$0.177742\pi$
$758$	0	0
$759$	44.1605	1.60292
$760$	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
$771$	6.94269	0.250035
$772$	0	0
$773$	20.9289	0.752761	0.376381	−	0.926465i	$-0.377168\pi$
$773$	20.9289	0.752761	0.376381	+	0.926465i	$0.377168\pi$
$774$	0	0
$775$	−10.2124	−0.366840
$776$	0	0
$777$	9.37011	0.336151
$778$	0	0
$779$	−3.13839	−0.112445
$780$	0	0
$781$	4.10051	0.146728
$782$	0	0
$783$	−19.1716	−0.685136
$784$	0	0
$785$	−4.32957	−0.154529
$786$	0	0
$787$	35.2931	1.25806	0.629032	−	0.777379i	$-0.283451\pi$
$787$	35.2931	1.25806	0.629032	+	0.777379i	$0.283451\pi$
$788$	0	0
$789$	−24.4148	−0.869190
$790$	0	0
$791$	−1.07107	−0.0380828
$792$	0	0
$793$	−12.6173	−0.448053
$794$	0	0
$795$	−20.1421	−0.714368
$796$	0	0
$797$	−48.5269	−1.71891	−0.859456	−	0.511210i	$-0.829197\pi$
$797$	−48.5269	−1.71891	−0.859456	+	0.511210i	$0.829197\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−4.48528	−0.158480
$802$	0	0
$803$	−29.0711	−1.02590
$804$	0	0
$805$	10.0042	0.352600
$806$	0	0
$807$	−15.5563	−0.547609
$808$	0	0
$809$	29.7724	1.04674	0.523371	−	0.852105i	$-0.324674\pi$
$809$	29.7724	1.04674	0.523371	+	0.852105i	$0.324674\pi$
$810$	0	0
$811$	23.8352	0.836966	0.418483	−	0.908225i	$-0.362562\pi$
$811$	23.8352	0.836966	0.418483	+	0.908225i	$0.362562\pi$
$812$	0	0
$813$	20.9050	0.733171
$814$	0	0
$815$	13.4142	0.469879
$816$	0	0
$817$	−10.9706	−0.383811
$818$	0	0
$819$	−2.16478	−0.0756437
$820$	0	0
$821$	5.91470	0.206424	0.103212	−	0.994659i	$-0.467088\pi$
$821$	5.91470	0.206424	0.103212	+	0.994659i	$0.467088\pi$
$822$	0	0
$823$	−30.2207	−1.05343	−0.526714	−	0.850042i	$-0.676576\pi$
$823$	−30.2207	−1.05343	−0.526714	+	0.850042i	$0.676576\pi$
$824$	0	0
$825$	9.89949	0.344656
$826$	0	0
$827$	3.64113	0.126614	0.0633072	−	0.997994i	$-0.479835\pi$
$827$	3.64113	0.126614	0.0633072	+	0.997994i	$0.479835\pi$
$828$	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
$839$	6.06854	0.209509	0.104755	−	0.994498i	$-0.466594\pi$
$839$	6.06854	0.209509	0.104755	+	0.994498i	$0.466594\pi$
$840$	0	0
$841$	−12.8995	−0.444810
$842$	0	0
$843$	−27.8477	−0.959126
$844$	0	0
$845$	−62.1354	−2.13752
$846$	0	0
$847$	0.317025	0.0108931
$848$	0	0
$849$	13.0711	0.448598
$850$	0	0
$851$	−46.8701	−1.60668
$852$	0	0
$853$	−21.6704	−0.741979	−0.370990	−	0.928637i	$-0.620982\pi$
$853$	−21.6704	−0.741979	−0.370990	+	0.928637i	$0.620982\pi$
$854$	0	0
$855$	1.97908	0.0676829
$856$	0	0
$857$	12.9343	0.441828	0.220914	−	0.975293i	$-0.429096\pi$
$857$	12.9343	0.441828	0.220914	+	0.975293i	$0.429096\pi$
$858$	0	0
$859$	−16.7279	−0.570749	−0.285375	−	0.958416i	$-0.592118\pi$
$859$	−16.7279	−0.570749	−0.285375	+	0.958416i	$0.592118\pi$
$860$	0	0
$861$	1.71644	0.0584962
$862$	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
$873$	0.393949	0.0133331
$874$	0	0
$875$	9.31371	0.314861
$876$	0	0
$877$	−8.60474	−0.290562	−0.145281	−	0.989390i	$-0.546409\pi$
$877$	−8.60474	−0.290562	−0.145281	+	0.989390i	$0.546409\pi$
$878$	0	0
$879$	−14.7821	−0.498587
$880$	0	0
$881$	−9.68714	−0.326368	−0.163184	−	0.986596i	$-0.552176\pi$
$881$	−9.68714	−0.326368	−0.163184	+	0.986596i	$0.552176\pi$
$882$	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
$887$	−35.4788	−1.19126	−0.595631	−	0.803258i	$-0.703098\pi$
$887$	−35.4788	−1.19126	−0.595631	+	0.803258i	$0.703098\pi$
$888$	0	0
$889$	−0.185709	−0.00622848
$890$	0	0
$891$	34.0250	1.13988
$892$	0	0
$893$	−9.45584	−0.316428
$894$	0	0
$895$	33.9706	1.13551
$896$	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
$907$	−7.59928	−0.252330	−0.126165	−	0.992009i	$-0.540267\pi$
$907$	−7.59928	−0.252330	−0.126165	+	0.992009i	$0.540267\pi$
$908$	0	0
$909$	−5.31371	−0.176245
$910$	0	0
$911$	3.03894	0.100684	0.0503422	−	0.998732i	$-0.483969\pi$
$911$	3.03894	0.100684	0.0503422	+	0.998732i	$0.483969\pi$
$912$	0	0
$913$	−0.819760	−0.0271301
$914$	0	0
$915$	−6.30864	−0.208557
$916$	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
$921$	11.7206	0.386207
$922$	0	0
$923$	8.28772	0.272794
$924$	0	0
$925$	−10.5069	−0.345465
$926$	0	0
$927$	1.65685	0.0544182
$928$	0	0
$929$	49.4412	1.62211	0.811056	−	0.584969i	$-0.198893\pi$
$929$	49.4412	1.62211	0.811056	+	0.584969i	$0.198893\pi$
$930$	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	$-0.624590\pi$
$937$	−23.3553	−0.762986	−0.381493	+	0.924372i	$0.624590\pi$
$938$	0	0
$939$	−42.0416	−1.37198
$940$	0	0
$941$	−1.66205	−0.0541813	−0.0270906	−	0.999633i	$-0.508624\pi$
$941$	−1.66205	−0.0541813	−0.0270906	+	0.999633i	$0.508624\pi$
$942$	0	0
$943$	−8.58579	−0.279592
$944$	0	0
$945$	6.75699	0.219805
$946$	0	0
$947$	9.05309	0.294186	0.147093	−	0.989123i	$-0.453008\pi$
$947$	9.05309	0.294186	0.147093	+	0.989123i	$0.453008\pi$
$948$	0	0
$949$	−58.7569	−1.90733
$950$	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
$955$	−0.634051	−0.0205174
$956$	0	0
$957$	−25.0489	−0.809715
$958$	0	0
$959$	9.29319	0.300093
$960$	0	0
$961$	10.4731	0.337842
$962$	0	0
$963$	−1.58513	−0.0510800
$964$	0	0
$965$	28.7279	0.924785
$966$	0	0
$967$	−33.0122	−1.06160	−0.530800	−	0.847497i	$-0.678109\pi$
$967$	−33.0122	−1.06160	−0.530800	+	0.847497i	$0.678109\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−37.2132	−1.19423	−0.597114	−	0.802156i	$-0.703686\pi$
$971$	−37.2132	−1.19423	−0.597114	+	0.802156i	$0.703686\pi$
$972$	0	0
$973$	13.4142	0.430040
$974$	0	0
$975$	20.0083	0.640779
$976$	0	0
$977$	38.3848	1.22804	0.614019	−	0.789291i	$-0.289552\pi$
$977$	38.3848	1.22804	0.614019	+	0.789291i	$0.289552\pi$
$978$	0	0
$979$	36.5838	1.16922
$980$	0	0
$981$	−2.03347	−0.0649236
$982$	0	0
$983$	−39.3282	−1.25437	−0.627187	−	0.778868i	$-0.715794\pi$
$983$	−39.3282	−1.25437	−0.627187	+	0.778868i	$0.715794\pi$
$984$	0	0
$985$	23.2132	0.739634
$986$	0	0
$987$	5.17157	0.164613
$988$	0	0
$989$	−30.0125	−0.954342
$990$	0	0
$991$	45.3742	1.44136	0.720680	−	0.693268i	$-0.243830\pi$
$991$	45.3742	1.44136	0.720680	+	0.693268i	$0.243830\pi$
$992$	0	0
$993$	53.0823	1.68452
$994$	0	0
$995$	−3.07107	−0.0973594
$996$	0	0
$997$	−30.9636	−0.980626	−0.490313	−	0.871546i	$-0.663118\pi$
$997$	−30.9636	−0.980626	−0.490313	+	0.871546i	$0.663118\pi$
$998$	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.a.g.1.4		4
4.3	odd	2		4624.2.a.bl.1.1		4
17.2	even	8		1156.2.e.f.905.4		8
17.3	odd	16		1156.2.h.c.757.1		4
17.4	even	4		1156.2.b.d.577.1		4
17.5	odd	16		68.2.h.a.25.1	✓	4
17.6	odd	16		1156.2.h.c.733.1		4
17.7	odd	16		68.2.h.a.49.1	yes	4
17.8	even	8		1156.2.e.f.829.1		8
17.9	even	8		1156.2.e.f.829.4		8
17.10	odd	16		1156.2.h.b.1001.1		4
17.11	odd	16		1156.2.h.a.733.1		4
17.12	odd	16		1156.2.h.b.977.1		4
17.13	even	4		1156.2.b.d.577.4		4
17.14	odd	16		1156.2.h.a.757.1		4
17.15	even	8		1156.2.e.f.905.1		8
17.16	even	2	inner	1156.2.a.g.1.1		4
51.5	even	16		612.2.w.a.433.1		4
51.41	even	16		612.2.w.a.253.1		4
68.7	even	16		272.2.v.c.49.1		4
68.39	even	16		272.2.v.c.161.1		4
68.67	odd	2		4624.2.a.bl.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.h.a.25.1	✓	4	17.5	odd	16
68.2.h.a.49.1	yes	4	17.7	odd	16
272.2.v.c.49.1		4	68.7	even	16
272.2.v.c.161.1		4	68.39	even	16
612.2.w.a.253.1		4	51.41	even	16
612.2.w.a.433.1		4	51.5	even	16
1156.2.a.g.1.1		4	17.16	even	2	inner
1156.2.a.g.1.4		4	1.1	even	1	trivial
1156.2.b.d.577.1		4	17.4	even	4
1156.2.b.d.577.4		4	17.13	even	4
1156.2.e.f.829.1		8	17.8	even	8
1156.2.e.f.829.4		8	17.9	even	8
1156.2.e.f.905.1		8	17.15	even	8
1156.2.e.f.905.4		8	17.2	even	8
1156.2.h.a.733.1		4	17.11	odd	16
1156.2.h.a.757.1		4	17.14	odd	16
1156.2.h.b.977.1		4	17.12	odd	16
1156.2.h.b.1001.1		4	17.10	odd	16
1156.2.h.c.733.1		4	17.6	odd	16
1156.2.h.c.757.1		4	17.3	odd	16
4624.2.a.bl.1.1		4	4.3	odd	2
4624.2.a.bl.1.4		4	68.67	odd	2

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.a (trivial)

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

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Representations

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Database

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