LMFDB - Embedded newform 3744.2.j.a.2159.36

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3744,2,Mod(2159,3744)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3744, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3744.2159"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3744 = 2^{5} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3744.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2159.36
Character	$\chi$	$=$	3744.2159
Dual form			3744.2.j.a.2159.35

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.22260 q^{5} +0.534598i q^{7} +2.35523i q^{11} -1.00000i q^{13} -4.78791i q^{17} +4.25343 q^{19} -2.55292 q^{23} -0.0600300 q^{25} +8.32790 q^{29} -0.284086i q^{31} +1.18820i q^{35} -6.30002i q^{37} -4.33386i q^{41} +0.666197 q^{43} -1.10114 q^{47} +6.71421 q^{49} +1.42724 q^{53} +5.23475i q^{55} +3.21354i q^{59} +4.98877i q^{61} -2.22260i q^{65} -0.658322 q^{67} +12.3352 q^{71} +14.1390 q^{73} -1.25910 q^{77} -11.8382i q^{79} +14.7707i q^{83} -10.6416i q^{85} -1.91146i q^{89} +0.534598 q^{91} +9.45368 q^{95} -7.69682 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 32 q^{19} + 48 q^{25} + 32 q^{43} - 48 q^{49} - 32 q^{67} - 32 q^{97}+O(q^{100}) $ 48 * q - 32 * q^19 + 48 * q^25 + 32 * q^43 - 48 * q^49 - 32 * q^67 - 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$.

$n$	$703$	$2017$	$2081$	$2341$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.22260	0.993979	0.496989	−	0.867757i	$-0.334439\pi$
$5$	2.22260	0.993979	0.496989	+	0.867757i	$0.334439\pi$
$6$	0	0
$7$	0.534598i	0.202059i	0.994883	+	0.101029i	$0.0322136\pi$
$7$	0.534598i	0.202059i	−0.994883	+	0.101029i	$0.967786\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.35523i	0.710129i	0.934842	+	0.355065i	$0.115541\pi$
$11$	2.35523i	0.710129i	−0.934842	+	0.355065i	$0.884459\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.78791i	− 1.16124i	−0.814176	−	0.580619i	$-0.802811\pi$
$17$	− 4.78791i	− 1.16124i	0.814176	−	0.580619i	$-0.197189\pi$
$18$	0	0
$19$	4.25343	0.975803	0.487901	−	0.872899i	$-0.337763\pi$
$19$	4.25343	0.975803	0.487901	+	0.872899i	$0.337763\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.55292	−0.532320	−0.266160	−	0.963929i	$-0.585755\pi$
$23$	−2.55292	−0.532320	−0.266160	+	0.963929i	$0.585755\pi$
$24$	0	0
$25$	−0.0600300	−0.0120060
$26$	0	0
$27$	0	0
$28$	0	0
$29$	8.32790	1.54645	0.773226	−	0.634130i	$-0.218642\pi$
$29$	8.32790	1.54645	0.773226	+	0.634130i	$0.218642\pi$
$30$	0	0
$31$	− 0.284086i	− 0.0510234i	−0.999675	−	0.0255117i	$-0.991878\pi$
$31$	− 0.284086i	− 0.0510234i	0.999675	−	0.0255117i	$-0.00812150\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.18820i	0.200842i
$36$	0	0
$37$	− 6.30002i	− 1.03572i	−0.855467	−	0.517858i	$-0.826730\pi$
$37$	− 6.30002i	− 1.03572i	0.855467	−	0.517858i	$-0.173270\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 4.33386i	− 0.676835i	−0.940996	−	0.338417i	$-0.890108\pi$
$41$	− 4.33386i	− 0.676835i	0.940996	−	0.338417i	$-0.109892\pi$
$42$	0	0
$43$	0.666197	0.101594	0.0507970	−	0.998709i	$-0.483824\pi$
$43$	0.666197	0.101594	0.0507970	+	0.998709i	$0.483824\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.10114	−0.160618	−0.0803092	−	0.996770i	$-0.525591\pi$
$47$	−1.10114	−0.160618	−0.0803092	+	0.996770i	$0.525591\pi$
$48$	0	0
$49$	6.71421	0.959172
$50$	0	0
$51$	0	0
$52$	0	0
$53$	1.42724	0.196047	0.0980233	−	0.995184i	$-0.468748\pi$
$53$	1.42724	0.196047	0.0980233	+	0.995184i	$0.468748\pi$
$54$	0	0
$55$	5.23475i	0.705853i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.21354i	0.418368i	0.977876	+	0.209184i	$0.0670807\pi$
$59$	3.21354i	0.418368i	−0.977876	+	0.209184i	$0.932919\pi$
$60$	0	0
$61$	4.98877i	0.638747i	0.947629	+	0.319373i	$0.103472\pi$
$61$	4.98877i	0.638747i	−0.947629	+	0.319373i	$0.896528\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 2.22260i	− 0.275680i
$66$	0	0
$67$	−0.658322	−0.0804268	−0.0402134	−	0.999191i	$-0.512804\pi$
$67$	−0.658322	−0.0804268	−0.0402134	+	0.999191i	$0.512804\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.3352	1.46391	0.731957	−	0.681350i	$-0.238607\pi$
$71$	12.3352	1.46391	0.731957	+	0.681350i	$0.238607\pi$
$72$	0	0
$73$	14.1390	1.65485	0.827423	−	0.561579i	$-0.189806\pi$
$73$	14.1390	1.65485	0.827423	+	0.561579i	$0.189806\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.25910	−0.143488
$78$	0	0
$79$	− 11.8382i	− 1.33190i	−0.745998	−	0.665948i	$-0.768027\pi$
$79$	− 11.8382i	− 1.33190i	0.745998	−	0.665948i	$-0.231973\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	14.7707i	1.62130i	0.585531	+	0.810650i	$0.300886\pi$
$83$	14.7707i	1.62130i	−0.585531	+	0.810650i	$0.699114\pi$
$84$	0	0
$85$	− 10.6416i	− 1.15425i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1.91146i	− 0.202614i	−0.994855	−	0.101307i	$-0.967697\pi$
$89$	− 1.91146i	− 0.202614i	0.994855	−	0.101307i	$-0.0323025\pi$
$90$	0	0
$91$	0.534598	0.0560411
$92$	0	0
$93$	0	0
$94$	0	0
$95$	9.45368	0.969927
$96$	0	0
$97$	−7.69682	−0.781493	−0.390747	−	0.920498i	$-0.627783\pi$
$97$	−7.69682	−0.781493	−0.390747	+	0.920498i	$0.627783\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.87035	0.186106	0.0930532	−	0.995661i	$-0.470337\pi$
$101$	1.87035	0.186106	0.0930532	+	0.995661i	$0.470337\pi$
$102$	0	0
$103$	0.952030i	0.0938063i	0.998899	+	0.0469031i	$0.0149352\pi$
$103$	0.952030i	0.0938063i	−0.998899	+	0.0469031i	$0.985065\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.5895i	1.60377i	0.597481	+	0.801883i	$0.296168\pi$
$107$	16.5895i	1.60377i	−0.597481	+	0.801883i	$0.703832\pi$
$108$	0	0
$109$	− 9.32470i	− 0.893144i	−0.894748	−	0.446572i	$-0.852645\pi$
$109$	− 9.32470i	− 0.893144i	0.894748	−	0.446572i	$-0.147355\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.63306i	0.341770i	0.985291	+	0.170885i	$0.0546626\pi$
$113$	3.63306i	0.341770i	−0.985291	+	0.170885i	$0.945337\pi$
$114$	0	0
$115$	−5.67413	−0.529115
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.55960	0.234638
$120$	0	0
$121$	5.45288	0.495717
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.2464	−1.00591
$126$	0	0
$127$	− 13.2516i	− 1.17589i	−0.808902	−	0.587943i	$-0.799938\pi$
$127$	− 13.2516i	− 1.17589i	0.808902	−	0.587943i	$-0.200062\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.2828i	0.898409i	0.893429	+	0.449204i	$0.148292\pi$
$131$	10.2828i	0.898409i	−0.893429	+	0.449204i	$0.851708\pi$
$132$	0	0
$133$	2.27387i	0.197170i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.8206i	1.69339i	0.532081	+	0.846694i	$0.321410\pi$
$137$	19.8206i	1.69339i	−0.532081	+	0.846694i	$0.678590\pi$
$138$	0	0
$139$	21.6113	1.83304	0.916522	−	0.399983i	$-0.130984\pi$
$139$	21.6113	1.83304	0.916522	+	0.399983i	$0.130984\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.35523	0.196954
$144$	0	0
$145$	18.5096	1.53714
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.98437	−0.326413	−0.163206	−	0.986592i	$-0.552184\pi$
$149$	−3.98437	−0.326413	−0.163206	+	0.986592i	$0.552184\pi$
$150$	0	0
$151$	− 6.03988i	− 0.491518i	−0.969331	−	0.245759i	$-0.920963\pi$
$151$	− 6.03988i	− 0.491518i	0.969331	−	0.245759i	$-0.0790372\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 0.631411i	− 0.0507162i
$156$	0	0
$157$	9.13676i	0.729193i	0.931166	+	0.364597i	$0.118793\pi$
$157$	9.13676i	0.729193i	−0.931166	+	0.364597i	$0.881207\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1.36478i	− 0.107560i
$162$	0	0
$163$	−21.7664	−1.70487	−0.852437	−	0.522829i	$-0.824876\pi$
$163$	−21.7664	−1.70487	−0.852437	+	0.522829i	$0.824876\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.7137	−1.13858	−0.569291	−	0.822136i	$-0.692782\pi$
$167$	−14.7137	−1.13858	−0.569291	+	0.822136i	$0.692782\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.1757	1.07776	0.538881	−	0.842382i	$-0.318847\pi$
$173$	14.1757	1.07776	0.538881	+	0.842382i	$0.318847\pi$
$174$	0	0
$175$	− 0.0320919i	− 0.00242592i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 17.7742i	− 1.32851i	−0.747506	−	0.664255i	$-0.768749\pi$
$179$	− 17.7742i	− 1.32851i	0.747506	−	0.664255i	$-0.231251\pi$
$180$	0	0
$181$	− 15.3188i	− 1.13864i	−0.822117	−	0.569318i	$-0.807207\pi$
$181$	− 15.3188i	− 1.13864i	0.822117	−	0.569318i	$-0.192793\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 14.0024i	− 1.02948i
$186$	0	0
$187$	11.2766	0.824629
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.40080	−0.246073	−0.123037	−	0.992402i	$-0.539263\pi$
$191$	−3.40080	−0.246073	−0.123037	+	0.992402i	$0.539263\pi$
$192$	0	0
$193$	9.94428	0.715805	0.357902	−	0.933759i	$-0.383492\pi$
$193$	9.94428	0.715805	0.357902	+	0.933759i	$0.383492\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.736457	−0.0524704	−0.0262352	−	0.999656i	$-0.508352\pi$
$197$	−0.736457	−0.0524704	−0.0262352	+	0.999656i	$0.508352\pi$
$198$	0	0
$199$	− 14.0332i	− 0.994789i	−0.867524	−	0.497395i	$-0.834290\pi$
$199$	− 14.0332i	− 0.994789i	0.867524	−	0.497395i	$-0.165710\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.45208i	0.312475i
$204$	0	0
$205$	− 9.63245i	− 0.672760i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.0178i	0.692946i
$210$	0	0
$211$	1.55071	0.106756	0.0533778	−	0.998574i	$-0.483001\pi$
$211$	1.55071	0.106756	0.0533778	+	0.998574i	$0.483001\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.48069	0.100982
$216$	0	0
$217$	0.151872	0.0103097
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.78791	−0.322069
$222$	0	0
$223$	− 3.64245i	− 0.243917i	−0.992535	−	0.121958i	$-0.961083\pi$
$223$	− 3.64245i	− 0.243917i	0.992535	−	0.121958i	$-0.0389174\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 12.9635i	− 0.860417i	−0.902730	−	0.430208i	$-0.858440\pi$
$227$	− 12.9635i	− 0.860417i	0.902730	−	0.430208i	$-0.141560\pi$
$228$	0	0
$229$	− 3.85891i	− 0.255004i	−0.991838	−	0.127502i	$-0.959304\pi$
$229$	− 3.85891i	− 0.255004i	0.991838	−	0.127502i	$-0.0406959\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.6243i	1.15461i	0.816529	+	0.577304i	$0.195895\pi$
$233$	17.6243i	1.15461i	−0.816529	+	0.577304i	$0.804105\pi$
$234$	0	0
$235$	−2.44741	−0.159651
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.4649	−1.25908	−0.629540	−	0.776968i	$-0.716757\pi$
$239$	−19.4649	−1.25908	−0.629540	+	0.776968i	$0.716757\pi$
$240$	0	0
$241$	9.66859	0.622809	0.311404	−	0.950278i	$-0.399201\pi$
$241$	9.66859	0.622809	0.311404	+	0.950278i	$0.399201\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	14.9230	0.953397
$246$	0	0
$247$	− 4.25343i	− 0.270639i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.1660i	0.641670i	0.947135	+	0.320835i	$0.103964\pi$
$251$	10.1660i	0.641670i	−0.947135	+	0.320835i	$0.896036\pi$
$252$	0	0
$253$	− 6.01271i	− 0.378016i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.2601i	0.702386i	0.936303	+	0.351193i	$0.114224\pi$
$257$	11.2601i	0.702386i	−0.936303	+	0.351193i	$0.885776\pi$
$258$	0	0
$259$	3.36797	0.209276
$260$	0	0
$261$	0	0
$262$	0	0
$263$	29.3228	1.80812	0.904061	−	0.427404i	$-0.140572\pi$
$263$	29.3228	1.80812	0.904061	+	0.427404i	$0.140572\pi$
$264$	0	0
$265$	3.17219	0.194866
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.81250	0.110510	0.0552551	−	0.998472i	$-0.482403\pi$
$269$	1.81250	0.110510	0.0552551	+	0.998472i	$0.482403\pi$
$270$	0	0
$271$	− 28.5969i	− 1.73714i	−0.495566	−	0.868570i	$-0.665039\pi$
$271$	− 28.5969i	− 1.73714i	0.495566	−	0.868570i	$-0.334961\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 0.141385i	− 0.00852581i
$276$	0	0
$277$	18.6565i	1.12096i	0.828168	+	0.560480i	$0.189383\pi$
$277$	18.6565i	1.12096i	−0.828168	+	0.560480i	$0.810617\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	10.2308i	0.610317i	0.952302	+	0.305159i	$0.0987095\pi$
$281$	10.2308i	0.610317i	−0.952302	+	0.305159i	$0.901291\pi$
$282$	0	0
$283$	6.27100	0.372772	0.186386	−	0.982477i	$-0.440322\pi$
$283$	6.27100	0.372772	0.186386	+	0.982477i	$0.440322\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.31687	0.136761
$288$	0	0
$289$	−5.92404	−0.348473
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.3609	0.838971	0.419486	−	0.907762i	$-0.362210\pi$
$293$	14.3609	0.838971	0.419486	+	0.907762i	$0.362210\pi$
$294$	0	0
$295$	7.14243i	0.415849i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.55292i	0.147639i
$300$	0	0
$301$	0.356147i	0.0205280i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	11.0881i	0.634901i
$306$	0	0
$307$	−20.5030	−1.17017	−0.585083	−	0.810973i	$-0.698938\pi$
$307$	−20.5030	−1.17017	−0.585083	+	0.810973i	$0.698938\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	21.8586	1.23949	0.619744	−	0.784804i	$-0.287236\pi$
$311$	21.8586	1.23949	0.619744	+	0.784804i	$0.287236\pi$
$312$	0	0
$313$	−16.6056	−0.938605	−0.469302	−	0.883038i	$-0.655495\pi$
$313$	−16.6056	−0.938605	−0.469302	+	0.883038i	$0.655495\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.8968	−1.39834	−0.699171	−	0.714954i	$-0.746447\pi$
$317$	−24.8968	−1.39834	−0.699171	+	0.714954i	$0.746447\pi$
$318$	0	0
$319$	19.6141i	1.09818i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 20.3650i	− 1.13314i
$324$	0	0
$325$	0.0600300i	0.00332987i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 0.588669i	− 0.0324544i
$330$	0	0
$331$	29.5671	1.62515	0.812577	−	0.582854i	$-0.198064\pi$
$331$	29.5671	1.62515	0.812577	+	0.582854i	$0.198064\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.46319	−0.0799425
$336$	0	0
$337$	0.297033	0.0161804	0.00809022	−	0.999967i	$-0.497425\pi$
$337$	0.297033	0.0161804	0.00809022	+	0.999967i	$0.497425\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.669089	0.0362332
$342$	0	0
$343$	7.33158i	0.395868i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.16773i	0.384784i	0.981318	+	0.192392i	$0.0616245\pi$
$347$	7.16773i	0.384784i	−0.981318	+	0.192392i	$0.938375\pi$
$348$	0	0
$349$	− 3.32515i	− 0.177991i	−0.996032	−	0.0889955i	$-0.971634\pi$
$349$	− 3.32515i	− 0.177991i	0.996032	−	0.0889955i	$-0.0283657\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2.31174i	− 0.123041i	−0.998106	−	0.0615207i	$-0.980405\pi$
$353$	− 2.31174i	− 0.123041i	0.998106	−	0.0615207i	$-0.0195950\pi$
$354$	0	0
$355$	27.4162	1.45510
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.0147	1.58411	0.792057	−	0.610448i	$-0.209010\pi$
$359$	30.0147	1.58411	0.792057	+	0.610448i	$0.209010\pi$
$360$	0	0
$361$	−0.908366	−0.0478087
$362$	0	0
$363$	0	0
$364$	0	0
$365$	31.4254	1.64488
$366$	0	0
$367$	− 26.3530i	− 1.37562i	−0.725893	−	0.687808i	$-0.758573\pi$
$367$	− 26.3530i	− 1.37562i	0.725893	−	0.687808i	$-0.241427\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.763000i	0.0396130i
$372$	0	0
$373$	22.9710i	1.18939i	0.803951	+	0.594696i	$0.202727\pi$
$373$	22.9710i	1.18939i	−0.803951	+	0.594696i	$0.797273\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.32790i	− 0.428909i
$378$	0	0
$379$	16.2876	0.836637	0.418318	−	0.908301i	$-0.362620\pi$
$379$	16.2876	0.836637	0.418318	+	0.908301i	$0.362620\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.6718	−0.596404	−0.298202	−	0.954503i	$-0.596387\pi$
$383$	−11.6718	−0.596404	−0.298202	+	0.954503i	$0.596387\pi$
$384$	0	0
$385$	−2.79848	−0.142624
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.0270	0.508387	0.254193	−	0.967153i	$-0.418190\pi$
$389$	10.0270	0.508387	0.254193	+	0.967153i	$0.418190\pi$
$390$	0	0
$391$	12.2231i	0.618150i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 26.3115i	− 1.32388i
$396$	0	0
$397$	18.9924i	0.953201i	0.879120	+	0.476601i	$0.158131\pi$
$397$	18.9924i	0.953201i	−0.879120	+	0.476601i	$0.841869\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 23.9527i	− 1.19614i	−0.801444	−	0.598070i	$-0.795935\pi$
$401$	− 23.9527i	− 1.19614i	0.801444	−	0.598070i	$-0.204065\pi$
$402$	0	0
$403$	−0.284086	−0.0141513
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.8380	0.735492
$408$	0	0
$409$	−19.3336	−0.955985	−0.477993	−	0.878364i	$-0.658635\pi$
$409$	−19.3336	−0.955985	−0.477993	+	0.878364i	$0.658635\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1.71795	−0.0845349
$414$	0	0
$415$	32.8295i	1.61154i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.67129i	− 0.0816477i	−0.999166	−	0.0408239i	$-0.987002\pi$
$419$	− 1.67129i	− 0.0816477i	0.999166	−	0.0408239i	$-0.0129982\pi$
$420$	0	0
$421$	− 18.7410i	− 0.913380i	−0.889626	−	0.456690i	$-0.849035\pi$
$421$	− 18.7410i	− 0.913380i	0.889626	−	0.456690i	$-0.150965\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.287418i	0.0139418i
$426$	0	0
$427$	−2.66699	−0.129065
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.7512	−0.855043	−0.427522	−	0.904005i	$-0.640613\pi$
$431$	−17.7512	−0.855043	−0.427522	+	0.904005i	$0.640613\pi$
$432$	0	0
$433$	−35.1148	−1.68751	−0.843754	−	0.536729i	$-0.819660\pi$
$433$	−35.1148	−1.68751	−0.843754	+	0.536729i	$0.819660\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−10.8586	−0.519440
$438$	0	0
$439$	− 2.03379i	− 0.0970674i	−0.998822	−	0.0485337i	$-0.984545\pi$
$439$	− 2.03379i	− 0.0970674i	0.998822	−	0.0485337i	$-0.0154548\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.40321i	0.256714i	0.991728	+	0.128357i	$0.0409704\pi$
$443$	5.40321i	0.256714i	−0.991728	+	0.128357i	$0.959030\pi$
$444$	0	0
$445$	− 4.24842i	− 0.201394i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	23.2885i	1.09905i	0.835477	+	0.549525i	$0.185191\pi$
$449$	23.2885i	1.09905i	−0.835477	+	0.549525i	$0.814809\pi$
$450$	0	0
$451$	10.2072	0.480640
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.18820	0.0557036
$456$	0	0
$457$	21.0679	0.985515	0.492758	−	0.870167i	$-0.335989\pi$
$457$	21.0679	0.985515	0.492758	+	0.870167i	$0.335989\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.7196	−1.01158	−0.505792	−	0.862655i	$-0.668800\pi$
$461$	−21.7196	−1.01158	−0.505792	+	0.862655i	$0.668800\pi$
$462$	0	0
$463$	31.7599i	1.47601i	0.674798	+	0.738003i	$0.264231\pi$
$463$	31.7599i	1.47601i	−0.674798	+	0.738003i	$0.735769\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 40.1071i	− 1.85594i	−0.372660	−	0.927968i	$-0.621554\pi$
$467$	− 40.1071i	− 1.85594i	0.372660	−	0.927968i	$-0.378446\pi$
$468$	0	0
$469$	− 0.351937i	− 0.0162510i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.56905i	0.0721449i
$474$	0	0
$475$	−0.255333	−0.0117155
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−40.3160	−1.84208	−0.921042	−	0.389463i	$-0.872661\pi$
$479$	−40.3160	−1.84208	−0.921042	+	0.389463i	$0.872661\pi$
$480$	0	0
$481$	−6.30002	−0.287256
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−17.1070	−0.776788
$486$	0	0
$487$	− 6.50924i	− 0.294962i	−0.989065	−	0.147481i	$-0.952883\pi$
$487$	− 6.50924i	− 0.294962i	0.989065	−	0.147481i	$-0.0471165\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 30.1527i	− 1.36077i	−0.732854	−	0.680386i	$-0.761812\pi$
$491$	− 30.1527i	− 1.36077i	0.732854	−	0.680386i	$-0.238188\pi$
$492$	0	0
$493$	− 39.8732i	− 1.79580i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.59435i	0.295797i
$498$	0	0
$499$	25.4542	1.13949	0.569744	−	0.821822i	$-0.307043\pi$
$499$	25.4542	1.13949	0.569744	+	0.821822i	$0.307043\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−28.7136	−1.28028	−0.640138	−	0.768260i	$-0.721123\pi$
$503$	−28.7136	−1.28028	−0.640138	+	0.768260i	$0.721123\pi$
$504$	0	0
$505$	4.15704	0.184986
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11.0695	−0.490645	−0.245323	−	0.969441i	$-0.578894\pi$
$509$	−11.0695	−0.490645	−0.245323	+	0.969441i	$0.578894\pi$
$510$	0	0
$511$	7.55869i	0.334377i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.11599i	0.0932414i
$516$	0	0
$517$	− 2.59345i	− 0.114060i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13.6152i	0.596493i	0.954489	+	0.298246i	$0.0964017\pi$
$521$	13.6152i	0.596493i	−0.954489	+	0.298246i	$0.903598\pi$
$522$	0	0
$523$	−30.2130	−1.32112	−0.660561	−	0.750772i	$-0.729682\pi$
$523$	−30.2130	−1.32112	−0.660561	+	0.750772i	$0.729682\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.36018	−0.0592503
$528$	0	0
$529$	−16.4826	−0.716635
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.33386	−0.187720
$534$	0	0
$535$	36.8719i	1.59411i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	15.8135i	0.681136i
$540$	0	0
$541$	− 21.4574i	− 0.922527i	−0.887263	−	0.461264i	$-0.847396\pi$
$541$	− 21.4574i	− 0.922527i	0.887263	−	0.461264i	$-0.152604\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 20.7251i	− 0.887766i
$546$	0	0
$547$	5.45515	0.233245	0.116623	−	0.993176i	$-0.462793\pi$
$547$	5.45515	0.233245	0.116623	+	0.993176i	$0.462793\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	35.4221	1.50903
$552$	0	0
$553$	6.32865	0.269122
$554$	0	0
$555$	0	0
$556$	0	0
$557$	35.6007	1.50845	0.754225	−	0.656616i	$-0.228013\pi$
$557$	35.6007	1.50845	0.754225	+	0.656616i	$0.228013\pi$
$558$	0	0
$559$	− 0.666197i	− 0.0281771i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17.7813i	0.749392i	0.927148	+	0.374696i	$0.122253\pi$
$563$	17.7813i	0.749392i	−0.927148	+	0.374696i	$0.877747\pi$
$564$	0	0
$565$	8.07486i	0.339712i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 7.00142i	− 0.293515i	−0.989173	−	0.146757i	$-0.953116\pi$
$569$	− 7.00142i	− 0.293515i	0.989173	−	0.146757i	$-0.0468837\pi$
$570$	0	0
$571$	−32.3791	−1.35502	−0.677511	−	0.735513i	$-0.736941\pi$
$571$	−32.3791	−1.35502	−0.677511	+	0.735513i	$0.736941\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.153252	0.00639104
$576$	0	0
$577$	43.8685	1.82627	0.913135	−	0.407657i	$-0.133654\pi$
$577$	43.8685	1.82627	0.913135	+	0.407657i	$0.133654\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7.89641	−0.327598
$582$	0	0
$583$	3.36148i	0.139218i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.5529i	0.765758i	0.923798	+	0.382879i	$0.125067\pi$
$587$	18.5529i	0.765758i	−0.923798	+	0.382879i	$0.874933\pi$
$588$	0	0
$589$	− 1.20834i	− 0.0497888i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 23.1700i	− 0.951479i	−0.879586	−	0.475739i	$-0.842181\pi$
$593$	− 23.1700i	− 0.951479i	0.879586	−	0.475739i	$-0.157819\pi$
$594$	0	0
$595$	5.68899	0.233226
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−41.7546	−1.70605	−0.853024	−	0.521871i	$-0.825234\pi$
$599$	−41.7546	−1.70605	−0.853024	+	0.521871i	$0.825234\pi$
$600$	0	0
$601$	−44.9068	−1.83179	−0.915894	−	0.401420i	$-0.868517\pi$
$601$	−44.9068	−1.83179	−0.915894	+	0.401420i	$0.868517\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	12.1196	0.492732
$606$	0	0
$607$	− 48.1964i	− 1.95623i	−0.208066	−	0.978115i	$-0.566717\pi$
$607$	− 48.1964i	− 1.95623i	0.208066	−	0.978115i	$-0.433283\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.10114i	0.0445475i
$612$	0	0
$613$	27.5841i	1.11411i	0.830475	+	0.557056i	$0.188069\pi$
$613$	27.5841i	1.11411i	−0.830475	+	0.557056i	$0.811931\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 23.8625i	− 0.960668i	−0.877086	−	0.480334i	$-0.840515\pi$
$617$	− 23.8625i	− 0.960668i	0.877086	−	0.480334i	$-0.159485\pi$
$618$	0	0
$619$	20.4340	0.821312	0.410656	−	0.911790i	$-0.365300\pi$
$619$	20.4340	0.821312	0.410656	+	0.911790i	$0.365300\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.02186	0.0409401
$624$	0	0
$625$	−24.6962	−0.987850
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−30.1639	−1.20271
$630$	0	0
$631$	43.7874i	1.74315i	0.490265	+	0.871574i	$0.336900\pi$
$631$	43.7874i	1.74315i	−0.490265	+	0.871574i	$0.663100\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 29.4530i	− 1.16881i
$636$	0	0
$637$	− 6.71421i	− 0.266026i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26.4737i	1.04565i	0.852441	+	0.522824i	$0.175122\pi$
$641$	26.4737i	1.04565i	−0.852441	+	0.522824i	$0.824878\pi$
$642$	0	0
$643$	−30.3343	−1.19627	−0.598134	−	0.801396i	$-0.704091\pi$
$643$	−30.3343	−1.19627	−0.598134	+	0.801396i	$0.704091\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.4455	−0.449971	−0.224985	−	0.974362i	$-0.572233\pi$
$647$	−11.4455	−0.449971	−0.224985	+	0.974362i	$0.572233\pi$
$648$	0	0
$649$	−7.56864	−0.297095
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.80591	0.227203	0.113601	−	0.993526i	$-0.463761\pi$
$653$	5.80591	0.227203	0.113601	+	0.993526i	$0.463761\pi$
$654$	0	0
$655$	22.8545i	0.892999i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.6774i	0.571752i	0.958267	+	0.285876i	$0.0922845\pi$
$659$	14.6774i	0.571752i	−0.958267	+	0.285876i	$0.907715\pi$
$660$	0	0
$661$	− 3.77769i	− 0.146935i	−0.997298	−	0.0734676i	$-0.976593\pi$
$661$	− 3.77769i	− 0.146935i	0.997298	−	0.0734676i	$-0.0234065\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.05392i	0.195983i
$666$	0	0
$667$	−21.2605	−0.823208
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.7497	−0.453593
$672$	0	0
$673$	−37.6779	−1.45238	−0.726188	−	0.687496i	$-0.758710\pi$
$673$	−37.6779	−1.45238	−0.726188	+	0.687496i	$0.758710\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.4215	1.32292	0.661462	−	0.749979i	$-0.269936\pi$
$677$	34.4215	1.32292	0.661462	+	0.749979i	$0.269936\pi$
$678$	0	0
$679$	− 4.11470i	− 0.157908i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.2294i	0.582739i	0.956611	+	0.291369i	$0.0941108\pi$
$683$	15.2294i	0.582739i	−0.956611	+	0.291369i	$0.905889\pi$
$684$	0	0
$685$	44.0533i	1.68319i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.42724i	− 0.0543736i
$690$	0	0
$691$	−0.308783	−0.0117467	−0.00587334	−	0.999983i	$-0.501870\pi$
$691$	−0.308783	−0.0117467	−0.00587334	+	0.999983i	$0.501870\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	48.0333	1.82201
$696$	0	0
$697$	−20.7501	−0.785966
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.0310215	−0.00117166	−0.000585832	−	1.00000i	$-0.500186\pi$
$701$	−0.0310215	−0.00117166	−0.000585832		1.00000i	$0.500186\pi$
$702$	0	0
$703$	− 26.7967i	− 1.01065i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.999883i	0.0376045i
$708$	0	0
$709$	23.9935i	0.901094i	0.892753	+	0.450547i	$0.148771\pi$
$709$	23.9935i	0.901094i	−0.892753	+	0.450547i	$0.851229\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0.725249i	0.0271608i
$714$	0	0
$715$	5.23475	0.195768
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−28.8021	−1.07414	−0.537069	−	0.843538i	$-0.680468\pi$
$719$	−28.8021	−1.07414	−0.537069	+	0.843538i	$0.680468\pi$
$720$	0	0
$721$	−0.508953	−0.0189544
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.499924	−0.0185667
$726$	0	0
$727$	11.5476i	0.428277i	0.976803	+	0.214138i	$0.0686943\pi$
$727$	11.5476i	0.428277i	−0.976803	+	0.214138i	$0.931306\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 3.18969i	− 0.117975i
$732$	0	0
$733$	34.4289i	1.27166i	0.771830	+	0.635830i	$0.219342\pi$
$733$	34.4289i	1.27166i	−0.771830	+	0.635830i	$0.780658\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.55050i	− 0.0571134i
$738$	0	0
$739$	−47.2448	−1.73793	−0.868964	−	0.494876i	$-0.835214\pi$
$739$	−47.2448	−1.73793	−0.868964	+	0.494876i	$0.835214\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−40.8575	−1.49892	−0.749459	−	0.662051i	$-0.769686\pi$
$743$	−40.8575	−1.49892	−0.749459	+	0.662051i	$0.769686\pi$
$744$	0	0
$745$	−8.85568	−0.324447
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8.86870	−0.324055
$750$	0	0
$751$	− 54.4523i	− 1.98699i	−0.113859	−	0.993497i	$-0.536321\pi$
$751$	− 54.4523i	− 1.98699i	0.113859	−	0.993497i	$-0.463679\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 13.4243i	− 0.488559i
$756$	0	0
$757$	10.5247i	0.382527i	0.981539	+	0.191263i	$0.0612585\pi$
$757$	10.5247i	0.382527i	−0.981539	+	0.191263i	$0.938742\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.22084i	0.0442555i	0.999755	+	0.0221278i	$0.00704406\pi$
$761$	1.22084i	0.0442555i	−0.999755	+	0.0221278i	$0.992956\pi$
$762$	0	0
$763$	4.98496	0.180468
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.21354	0.116034
$768$	0	0
$769$	−6.42116	−0.231553	−0.115777	−	0.993275i	$-0.536936\pi$
$769$	−6.42116	−0.231553	−0.115777	+	0.993275i	$0.536936\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−43.3705	−1.55993	−0.779964	−	0.625825i	$-0.784762\pi$
$773$	−43.3705	−1.55993	−0.779964	+	0.625825i	$0.784762\pi$
$774$	0	0
$775$	0.0170537i	0 0.000612587i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 18.4337i	− 0.660457i
$780$	0	0
$781$	29.0522i	1.03957i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	20.3074i	0.724803i
$786$	0	0
$787$	−4.57163	−0.162961	−0.0814804	−	0.996675i	$-0.525965\pi$
$787$	−4.57163	−0.162961	−0.0814804	+	0.996675i	$0.525965\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.94223	−0.0690576
$792$	0	0
$793$	4.98877	0.177157
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−45.1833	−1.60047	−0.800237	−	0.599684i	$-0.795293\pi$
$797$	−45.1833	−1.60047	−0.800237	+	0.599684i	$0.795293\pi$
$798$	0	0
$799$	5.27217i	0.186516i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	33.3007i	1.17515i
$804$	0	0
$805$	− 3.03338i	− 0.106912i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4.51903i	− 0.158881i	−0.996840	−	0.0794403i	$-0.974687\pi$
$809$	− 4.51903i	− 0.158881i	0.996840	−	0.0794403i	$-0.0253133\pi$
$810$	0	0
$811$	−51.7819	−1.81831	−0.909154	−	0.416460i	$-0.863271\pi$
$811$	−51.7819	−1.81831	−0.909154	+	0.416460i	$0.863271\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−48.3781	−1.69461
$816$	0	0
$817$	2.83362	0.0991358
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−46.7224	−1.63062	−0.815312	−	0.579022i	$-0.803434\pi$
$821$	−46.7224	−1.63062	−0.815312	+	0.579022i	$0.803434\pi$
$822$	0	0
$823$	0.605003i	0.0210891i	0.999944	+	0.0105445i	$0.00335649\pi$
$823$	0.605003i	0.0210891i	−0.999944	+	0.0105445i	$0.996644\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 7.96743i	− 0.277055i	−0.990359	−	0.138527i	$-0.955763\pi$
$827$	− 7.96743i	− 0.277055i	0.990359	−	0.138527i	$-0.0442369\pi$
$828$	0	0
$829$	11.5924i	0.402622i	0.979527	+	0.201311i	$0.0645202\pi$
$829$	11.5924i	0.402622i	−0.979527	+	0.201311i	$0.935480\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 32.1470i	− 1.11383i
$834$	0	0
$835$	−32.7028	−1.13173
$836$	0	0
$837$	0	0
$838$	0	0
$839$	29.8351	1.03002	0.515012	−	0.857183i	$-0.327788\pi$
$839$	29.8351	1.03002	0.515012	+	0.857183i	$0.327788\pi$
$840$	0	0
$841$	40.3539	1.39152
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.22260	−0.0764599
$846$	0	0
$847$	2.91510i	0.100164i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	16.0834i	0.551333i
$852$	0	0
$853$	− 27.9548i	− 0.957155i	−0.878045	−	0.478578i	$-0.841152\pi$
$853$	− 27.9548i	− 0.957155i	0.878045	−	0.478578i	$-0.158848\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.72159i	0.195446i	0.995214	+	0.0977229i	$0.0311559\pi$
$857$	5.72159i	0.195446i	−0.995214	+	0.0977229i	$0.968844\pi$
$858$	0	0
$859$	−18.9201	−0.645544	−0.322772	−	0.946477i	$-0.604615\pi$
$859$	−18.9201	−0.645544	−0.322772	+	0.946477i	$0.604615\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14.3411	−0.488176	−0.244088	−	0.969753i	$-0.578489\pi$
$863$	−14.3411	−0.488176	−0.244088	+	0.969753i	$0.578489\pi$
$864$	0	0
$865$	31.5071	1.07127
$866$	0	0
$867$	0	0
$868$	0	0
$869$	27.8816	0.945819
$870$	0	0
$871$	0.658322i	0.0223064i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 6.01232i	− 0.203254i
$876$	0	0
$877$	24.2543i	0.819011i	0.912308	+	0.409505i	$0.134299\pi$
$877$	24.2543i	0.819011i	−0.912308	+	0.409505i	$0.865701\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 58.3091i	− 1.96448i	−0.187621	−	0.982241i	$-0.560078\pi$
$881$	− 58.3091i	− 1.96448i	0.187621	−	0.982241i	$-0.439922\pi$
$882$	0	0
$883$	−39.8809	−1.34210	−0.671050	−	0.741412i	$-0.734156\pi$
$883$	−39.8809	−1.34210	−0.671050	+	0.741412i	$0.734156\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	30.1897	1.01367	0.506836	−	0.862043i	$-0.330815\pi$
$887$	30.1897	1.01367	0.506836	+	0.862043i	$0.330815\pi$
$888$	0	0
$889$	7.08426	0.237598
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.68364	−0.156732
$894$	0	0
$895$	− 39.5051i	− 1.32051i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2.36584i	− 0.0789052i
$900$	0	0
$901$	− 6.83350i	− 0.227657i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 34.0476i	− 1.13178i
$906$	0	0
$907$	−20.4434	−0.678810	−0.339405	−	0.940640i	$-0.610226\pi$
$907$	−20.4434	−0.678810	−0.339405	+	0.940640i	$0.610226\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.47336	0.148209	0.0741045	−	0.997250i	$-0.476390\pi$
$911$	4.47336	0.148209	0.0741045	+	0.997250i	$0.476390\pi$
$912$	0	0
$913$	−34.7885	−1.15133
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.49714	−0.181532
$918$	0	0
$919$	49.2114i	1.62333i	0.584121	+	0.811667i	$0.301439\pi$
$919$	49.2114i	1.62333i	−0.584121	+	0.811667i	$0.698561\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 12.3352i	− 0.406017i
$924$	0	0
$925$	0.378190i	0.0124348i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.4660i	0.343379i	0.985151	+	0.171690i	$0.0549226\pi$
$929$	10.4660i	0.343379i	−0.985151	+	0.171690i	$0.945077\pi$
$930$	0	0
$931$	28.5584	0.935963
$932$	0	0
$933$	0	0
$934$	0	0
$935$	25.0635	0.819663
$936$	0	0
$937$	56.2534	1.83772	0.918859	−	0.394585i	$-0.129112\pi$
$937$	56.2534	1.83772	0.918859	+	0.394585i	$0.129112\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.89072	−0.126834	−0.0634170	−	0.997987i	$-0.520200\pi$
$941$	−3.89072	−0.126834	−0.0634170	+	0.997987i	$0.520200\pi$
$942$	0	0
$943$	11.0640i	0.360293i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 22.4095i	− 0.728211i	−0.931358	−	0.364105i	$-0.881375\pi$
$947$	− 22.4095i	− 0.728211i	0.931358	−	0.364105i	$-0.118625\pi$
$948$	0	0
$949$	− 14.1390i	− 0.458972i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 52.6242i	− 1.70466i	−0.523001	−	0.852332i	$-0.675187\pi$
$953$	− 52.6242i	− 1.70466i	0.523001	−	0.852332i	$-0.324813\pi$
$954$	0	0
$955$	−7.55863	−0.244592
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.5960	−0.342164
$960$	0	0
$961$	30.9193	0.997397
$962$	0	0
$963$	0	0
$964$	0	0
$965$	22.1022	0.711495
$966$	0	0
$967$	54.9989i	1.76865i	0.466876	+	0.884323i	$0.345379\pi$
$967$	54.9989i	1.76865i	−0.466876	+	0.884323i	$0.654621\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38.5659i	1.23764i	0.785533	+	0.618820i	$0.212389\pi$
$971$	38.5659i	1.23764i	−0.785533	+	0.618820i	$0.787611\pi$
$972$	0	0
$973$	11.5533i	0.370383i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.8882i	0.348345i	0.984715	+	0.174172i	$0.0557250\pi$
$977$	10.8882i	0.348345i	−0.984715	+	0.174172i	$0.944275\pi$
$978$	0	0
$979$	4.50193	0.143882
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27.2804	0.870111	0.435055	−	0.900404i	$-0.356729\pi$
$983$	27.2804	0.870111	0.435055	+	0.900404i	$0.356729\pi$
$984$	0	0
$985$	−1.63685	−0.0521545
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.70075	−0.0540806
$990$	0	0
$991$	− 25.6230i	− 0.813941i	−0.913441	−	0.406971i	$-0.866585\pi$
$991$	− 25.6230i	− 0.813941i	0.913441	−	0.406971i	$-0.133415\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 31.1903i	− 0.988799i
$996$	0	0
$997$	48.5538i	1.53771i	0.639421	+	0.768856i	$0.279174\pi$
$997$	48.5538i	1.53771i	−0.639421	+	0.768856i	$0.720826\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.j.a.2159.36		48
3.2	odd	2	inner	3744.2.j.a.2159.14		48
4.3	odd	2		936.2.j.a.755.11	✓	48
8.3	odd	2	inner	3744.2.j.a.2159.13		48
8.5	even	2		936.2.j.a.755.37	yes	48
12.11	even	2		936.2.j.a.755.38	yes	48
24.5	odd	2		936.2.j.a.755.12	yes	48
24.11	even	2	inner	3744.2.j.a.2159.35		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.j.a.755.11	✓	48	4.3	odd	2
936.2.j.a.755.12	yes	48	24.5	odd	2
936.2.j.a.755.37	yes	48	8.5	even	2
936.2.j.a.755.38	yes	48	12.11	even	2
3744.2.j.a.2159.13		48	8.3	odd	2	inner
3744.2.j.a.2159.14		48	3.2	odd	2	inner
3744.2.j.a.2159.35		48	24.11	even	2	inner
3744.2.j.a.2159.36		48	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3744 = 2^{5} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3744.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.22260 q^{5} +0.534598i q^{7} +2.35523i q^{11} -1.00000i q^{13} -4.78791i q^{17} +4.25343 q^{19} -2.55292 q^{23} -0.0600300 q^{25} +8.32790 q^{29} -0.284086i q^{31} +1.18820i q^{35} -6.30002i q^{37} -4.33386i q^{41} +0.666197 q^{43} -1.10114 q^{47} +6.71421 q^{49} +1.42724 q^{53} +5.23475i q^{55} +3.21354i q^{59} +4.98877i q^{61} -2.22260i q^{65} -0.658322 q^{67} +12.3352 q^{71} +14.1390 q^{73} -1.25910 q^{77} -11.8382i q^{79} +14.7707i q^{83} -10.6416i q^{85} -1.91146i q^{89} +0.534598 q^{91} +9.45368 q^{95} -7.69682 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 32 q^{19} + 48 q^{25} + 32 q^{43} - 48 q^{49} - 32 q^{67} - 32 q^{97}+O(q^{100}) \) 48 * q - 32 * q^19 + 48 * q^25 + 32 * q^43 - 48 * q^49 - 32 * q^67 - 32 * q^97

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