LMFDB - Embedded newform 5328.2.e.g.2591.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5328,2,Mod(2591,5328)] 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("5328.2591"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5328, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5328 = 2^{4} \cdot 3^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5328.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,0,0,16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$42.5442941969$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.9
Character	$\chi$	$=$	5328.2591
Dual form			5328.2.e.g.2591.16

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.827995i q^{5} -4.12813i q^{7} +5.90923 q^{11} -1.64385 q^{13} +6.56445i q^{17} -2.04102i q^{19} +6.81174 q^{23} +4.31442 q^{25} +9.84801i q^{29} +10.7545i q^{31} -3.41807 q^{35} +1.00000 q^{37} +3.00378i q^{41} +11.5653i q^{43} -4.55822 q^{47} -10.0414 q^{49} -3.86226i q^{53} -4.89282i q^{55} +3.85411 q^{59} +5.74368 q^{61} +1.36110i q^{65} +10.1661i q^{67} +2.03339 q^{71} +0.419197 q^{73} -24.3941i q^{77} +0.532846i q^{79} -3.34689 q^{83} +5.43533 q^{85} -4.38205i q^{89} +6.78602i q^{91} -1.68996 q^{95} -3.65391 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 16 q^{13} + 24 q^{37} - 16 q^{49} + 48 q^{61} + 48 q^{73} + 56 q^{85} - 8 q^{97}+O(q^{100}) $ 24 * q + 16 * q^13 + 24 * q^37 - 16 * q^49 + 48 * q^61 + 48 * q^73 + 56 * q^85 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5328\mathbb{Z}\right)^\times$.

$n$	$1297$	$1333$	$1999$	$2369$
$\chi(n)$	$1$	$1$	$-1$	$-1$

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
$5$	− 0.827995i	− 0.370291i	−0.982711	−	0.185145i	$-0.940724\pi$
$5$	− 0.827995i	− 0.370291i	0.982711	−	0.185145i	$-0.0592756\pi$
$6$	0	0
$7$	− 4.12813i	− 1.56029i	−0.625602	−	0.780143i	$-0.715146\pi$
$7$	− 4.12813i	− 1.56029i	0.625602	−	0.780143i	$-0.284854\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.90923	1.78170	0.890851	−	0.454297i	$-0.150109\pi$
$11$	5.90923	1.78170	0.890851	+	0.454297i	$0.150109\pi$
$12$	0	0
$13$	−1.64385	−0.455922	−0.227961	−	0.973670i	$-0.573206\pi$
$13$	−1.64385	−0.455922	−0.227961	+	0.973670i	$0.573206\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.56445i	1.59211i	0.605223	+	0.796056i	$0.293084\pi$
$17$	6.56445i	1.59211i	−0.605223	+	0.796056i	$0.706916\pi$
$18$	0	0
$19$	− 2.04102i	− 0.468243i	−0.972207	−	0.234122i	$-0.924779\pi$
$19$	− 2.04102i	− 0.468243i	0.972207	−	0.234122i	$-0.0752214\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.81174	1.42035	0.710173	−	0.704027i	$-0.248617\pi$
$23$	6.81174	1.42035	0.710173	+	0.704027i	$0.248617\pi$
$24$	0	0
$25$	4.31442	0.862885
$26$	0	0
$27$	0	0
$28$	0	0
$29$	9.84801i	1.82873i	0.404891	+	0.914365i	$0.367309\pi$
$29$	9.84801i	1.82873i	−0.404891	+	0.914365i	$0.632691\pi$
$30$	0	0
$31$	10.7545i	1.93157i	0.259337	+	0.965787i	$0.416496\pi$
$31$	10.7545i	1.93157i	−0.259337	+	0.965787i	$0.583504\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.41807	−0.577759
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.00378i	0.469111i	0.972103	+	0.234556i	$0.0753635\pi$
$41$	3.00378i	0.469111i	−0.972103	+	0.234556i	$0.924636\pi$
$42$	0	0
$43$	11.5653i	1.76369i	0.471536	+	0.881847i	$0.343700\pi$
$43$	11.5653i	1.76369i	−0.471536	+	0.881847i	$0.656300\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.55822	−0.664884	−0.332442	−	0.943124i	$-0.607873\pi$
$47$	−4.55822	−0.664884	−0.332442	+	0.943124i	$0.607873\pi$
$48$	0	0
$49$	−10.0414	−1.43449
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 3.86226i	− 0.530522i	−0.964177	−	0.265261i	$-0.914542\pi$
$53$	− 3.86226i	− 0.530522i	0.964177	−	0.265261i	$-0.0854581\pi$
$54$	0	0
$55$	− 4.89282i	− 0.659747i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.85411	0.501763	0.250881	−	0.968018i	$-0.419280\pi$
$59$	3.85411	0.501763	0.250881	+	0.968018i	$0.419280\pi$
$60$	0	0
$61$	5.74368	0.735403	0.367701	−	0.929944i	$-0.380145\pi$
$61$	5.74368	0.735403	0.367701	+	0.929944i	$0.380145\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.36110i	0.168824i
$66$	0	0
$67$	10.1661i	1.24199i	0.783816	+	0.620993i	$0.213271\pi$
$67$	10.1661i	1.24199i	−0.783816	+	0.620993i	$0.786729\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.03339	0.241319	0.120660	−	0.992694i	$-0.461499\pi$
$71$	2.03339	0.241319	0.120660	+	0.992694i	$0.461499\pi$
$72$	0	0
$73$	0.419197	0.0490633	0.0245316	−	0.999699i	$-0.492191\pi$
$73$	0.419197	0.0490633	0.0245316	+	0.999699i	$0.492191\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 24.3941i	− 2.77996i
$78$	0	0
$79$	0.532846i	0.0599498i	0.999551	+	0.0299749i	$0.00954274\pi$
$79$	0.532846i	0.0599498i	−0.999551	+	0.0299749i	$0.990457\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.34689	−0.367369	−0.183684	−	0.982985i	$-0.558802\pi$
$83$	−3.34689	−0.367369	−0.183684	+	0.982985i	$0.558802\pi$
$84$	0	0
$85$	5.43533	0.589544
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 4.38205i	− 0.464496i	−0.972657	−	0.232248i	$-0.925392\pi$
$89$	− 4.38205i	− 0.464496i	0.972657	−	0.232248i	$-0.0746081\pi$
$90$	0	0
$91$	6.78602i	0.711368i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.68996	−0.173386
$96$	0	0
$97$	−3.65391	−0.370998	−0.185499	−	0.982644i	$-0.559390\pi$
$97$	−3.65391	−0.370998	−0.185499	+	0.982644i	$0.559390\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 14.5620i	− 1.44898i	−0.689286	−	0.724489i	$-0.742076\pi$
$101$	− 14.5620i	− 1.44898i	0.689286	−	0.724489i	$-0.257924\pi$
$102$	0	0
$103$	12.8417i	1.26533i	0.774427	+	0.632663i	$0.218038\pi$
$103$	12.8417i	1.26533i	−0.774427	+	0.632663i	$0.781962\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.51926	0.920261	0.460131	−	0.887851i	$-0.347803\pi$
$107$	9.51926	0.920261	0.460131	+	0.887851i	$0.347803\pi$
$108$	0	0
$109$	−3.84351	−0.368141	−0.184071	−	0.982913i	$-0.558927\pi$
$109$	−3.84351	−0.368141	−0.184071	+	0.982913i	$0.558927\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2.39788i	− 0.225573i	−0.993619	−	0.112787i	$-0.964022\pi$
$113$	− 2.39788i	− 0.225573i	0.993619	−	0.112787i	$-0.0359777\pi$
$114$	0	0
$115$	− 5.64009i	− 0.525941i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	27.0989	2.48415
$120$	0	0
$121$	23.9190	2.17446
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 7.71230i	− 0.689809i
$126$	0	0
$127$	− 8.91572i	− 0.791142i	−0.918435	−	0.395571i	$-0.870547\pi$
$127$	− 8.91572i	− 0.791142i	0.918435	−	0.395571i	$-0.129453\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.0337	0.964018	0.482009	−	0.876166i	$-0.339907\pi$
$131$	11.0337	0.964018	0.482009	+	0.876166i	$0.339907\pi$
$132$	0	0
$133$	−8.42561	−0.730593
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.2842i	1.47669i	0.674422	+	0.738347i	$0.264393\pi$
$137$	17.2842i	1.47669i	−0.674422	+	0.738347i	$0.735607\pi$
$138$	0	0
$139$	16.8099i	1.42580i	0.701265	+	0.712900i	$0.252619\pi$
$139$	16.8099i	1.42580i	−0.701265	+	0.712900i	$0.747381\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−9.71389	−0.812317
$144$	0	0
$145$	8.15411	0.677162
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 4.89413i	− 0.400943i	−0.979700	−	0.200471i	$-0.935753\pi$
$149$	− 4.89413i	− 0.400943i	0.979700	−	0.200471i	$-0.0642474\pi$
$150$	0	0
$151$	− 19.7755i	− 1.60931i	−0.593745	−	0.804653i	$-0.702351\pi$
$151$	− 19.7755i	− 1.60931i	0.593745	−	0.804653i	$-0.297649\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.90471	0.715244
$156$	0	0
$157$	−11.8079	−0.942370	−0.471185	−	0.882035i	$-0.656174\pi$
$157$	−11.8079	−0.942370	−0.471185	+	0.882035i	$0.656174\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 28.1197i	− 2.21614i
$162$	0	0
$163$	9.34325i	0.731820i	0.930650	+	0.365910i	$0.119242\pi$
$163$	9.34325i	0.731820i	−0.930650	+	0.365910i	$0.880758\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.30062	−0.642322	−0.321161	−	0.947025i	$-0.604073\pi$
$167$	−8.30062	−0.642322	−0.321161	+	0.947025i	$0.604073\pi$
$168$	0	0
$169$	−10.2978	−0.792135
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.5516i	0.802225i	0.916029	+	0.401112i	$0.131376\pi$
$173$	10.5516i	0.802225i	−0.916029	+	0.401112i	$0.868624\pi$
$174$	0	0
$175$	− 17.8105i	− 1.34635i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.21126	−0.613738	−0.306869	−	0.951752i	$-0.599281\pi$
$179$	−8.21126	−0.613738	−0.306869	+	0.951752i	$0.599281\pi$
$180$	0	0
$181$	22.1727	1.64809	0.824043	−	0.566527i	$-0.191713\pi$
$181$	22.1727	1.64809	0.824043	+	0.566527i	$0.191713\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 0.827995i	− 0.0608754i
$186$	0	0
$187$	38.7909i	2.83667i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.8143	−1.50607	−0.753034	−	0.657982i	$-0.771410\pi$
$191$	−20.8143	−1.50607	−0.753034	+	0.657982i	$0.771410\pi$
$192$	0	0
$193$	−12.5618	−0.904215	−0.452108	−	0.891963i	$-0.649328\pi$
$193$	−12.5618	−0.904215	−0.452108	+	0.891963i	$0.649328\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.7425i	− 0.979116i	−0.871971	−	0.489558i	$-0.837158\pi$
$197$	− 13.7425i	− 0.979116i	0.871971	−	0.489558i	$-0.162842\pi$
$198$	0	0
$199$	− 19.5410i	− 1.38522i	−0.721310	−	0.692612i	$-0.756460\pi$
$199$	− 19.5410i	− 1.38522i	0.721310	−	0.692612i	$-0.243540\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	40.6538	2.85334
$204$	0	0
$205$	2.48711	0.173708
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 12.0609i	− 0.834269i
$210$	0	0
$211$	− 18.0836i	− 1.24493i	−0.782649	−	0.622464i	$-0.786132\pi$
$211$	− 18.0836i	− 1.24493i	0.782649	−	0.622464i	$-0.213868\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.57603	0.653080
$216$	0	0
$217$	44.3961	3.01381
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 10.7910i	− 0.725879i
$222$	0	0
$223$	6.64384i	0.444904i	0.974944	+	0.222452i	$0.0714061\pi$
$223$	6.64384i	0.444904i	−0.974944	+	0.222452i	$0.928594\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.06466	−0.468898	−0.234449	−	0.972128i	$-0.575329\pi$
$227$	−7.06466	−0.468898	−0.234449	+	0.972128i	$0.575329\pi$
$228$	0	0
$229$	−14.0974	−0.931581	−0.465790	−	0.884895i	$-0.654230\pi$
$229$	−14.0974	−0.931581	−0.465790	+	0.884895i	$0.654230\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 19.1196i	− 1.25256i	−0.779597	−	0.626282i	$-0.784576\pi$
$233$	− 19.1196i	− 1.25256i	0.779597	−	0.626282i	$-0.215424\pi$
$234$	0	0
$235$	3.77418i	0.246201i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.886993	0.0573748	0.0286874	−	0.999588i	$-0.490867\pi$
$239$	0.886993	0.0573748	0.0286874	+	0.999588i	$0.490867\pi$
$240$	0	0
$241$	−4.95612	−0.319251	−0.159626	−	0.987178i	$-0.551029\pi$
$241$	−4.95612	−0.319251	−0.159626	+	0.987178i	$0.551029\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	8.31426i	0.531179i
$246$	0	0
$247$	3.35514i	0.213482i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−0.175153	−0.0110556	−0.00552778	−	0.999985i	$-0.501760\pi$
$251$	−0.175153	−0.0110556	−0.00552778	+	0.999985i	$0.501760\pi$
$252$	0	0
$253$	40.2522	2.53063
$254$	0	0
$255$	0	0
$256$	0	0
$257$	27.5610i	1.71921i	0.510959	+	0.859605i	$0.329291\pi$
$257$	27.5610i	1.71921i	−0.510959	+	0.859605i	$0.670709\pi$
$258$	0	0
$259$	− 4.12813i	− 0.256509i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.2510	1.00208	0.501039	−	0.865424i	$-0.332951\pi$
$263$	16.2510	1.00208	0.501039	+	0.865424i	$0.332951\pi$
$264$	0	0
$265$	−3.19793	−0.196447
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 19.6499i	− 1.19807i	−0.800722	−	0.599036i	$-0.795550\pi$
$269$	− 19.6499i	− 1.19807i	0.800722	−	0.599036i	$-0.204450\pi$
$270$	0	0
$271$	9.40618i	0.571385i	0.958321	+	0.285692i	$0.0922236\pi$
$271$	9.40618i	0.571385i	−0.958321	+	0.285692i	$0.907776\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	25.4949	1.53740
$276$	0	0
$277$	−0.463418	−0.0278441	−0.0139220	−	0.999903i	$-0.504432\pi$
$277$	−0.463418	−0.0278441	−0.0139220	+	0.999903i	$0.504432\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.42818i	0.502783i	0.967886	+	0.251391i	$0.0808881\pi$
$281$	8.42818i	0.502783i	−0.967886	+	0.251391i	$0.919112\pi$
$282$	0	0
$283$	− 19.0241i	− 1.13087i	−0.824794	−	0.565433i	$-0.808709\pi$
$283$	− 19.0241i	− 1.13087i	0.824794	−	0.565433i	$-0.191291\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.4000	0.731948
$288$	0	0
$289$	−26.0920	−1.53482
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 10.1296i	− 0.591779i	−0.955222	−	0.295889i	$-0.904384\pi$
$293$	− 10.1296i	− 0.591779i	0.955222	−	0.295889i	$-0.0956160\pi$
$294$	0	0
$295$	− 3.19119i	− 0.185798i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−11.1975	−0.647567
$300$	0	0
$301$	47.7431	2.75187
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 4.75574i	− 0.272313i
$306$	0	0
$307$	− 0.806986i	− 0.0460572i	−0.999735	−	0.0230286i	$-0.992669\pi$
$307$	− 0.806986i	− 0.0460572i	0.999735	−	0.0230286i	$-0.00733087\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.06313	0.400514	0.200257	−	0.979743i	$-0.435822\pi$
$311$	7.06313	0.400514	0.200257	+	0.979743i	$0.435822\pi$
$312$	0	0
$313$	−31.9085	−1.80358	−0.901788	−	0.432179i	$-0.857745\pi$
$313$	−31.9085	−1.80358	−0.901788	+	0.432179i	$0.857745\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.4427i	0.586523i	0.956032	+	0.293261i	$0.0947406\pi$
$317$	10.4427i	0.586523i	−0.956032	+	0.293261i	$0.905259\pi$
$318$	0	0
$319$	58.1942i	3.25825i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13.3982	0.745496
$324$	0	0
$325$	−7.09227	−0.393408
$326$	0	0
$327$	0	0
$328$	0	0
$329$	18.8169i	1.03741i
$330$	0	0
$331$	9.49008i	0.521622i	0.965390	+	0.260811i	$0.0839900\pi$
$331$	9.49008i	0.521622i	−0.965390	+	0.260811i	$0.916010\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.41748	0.459896
$336$	0	0
$337$	33.1042	1.80330	0.901650	−	0.432466i	$-0.142357\pi$
$337$	33.1042	1.80330	0.901650	+	0.432466i	$0.142357\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	63.5511i	3.44149i
$342$	0	0
$343$	12.5554i	0.677930i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.8451	0.689564	0.344782	−	0.938683i	$-0.387953\pi$
$347$	12.8451	0.689564	0.344782	+	0.938683i	$0.387953\pi$
$348$	0	0
$349$	−1.28109	−0.0685752	−0.0342876	−	0.999412i	$-0.510916\pi$
$349$	−1.28109	−0.0685752	−0.0342876	+	0.999412i	$0.510916\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 3.46649i	− 0.184503i	−0.995736	−	0.0922514i	$-0.970594\pi$
$353$	− 3.46649i	− 0.184503i	0.995736	−	0.0922514i	$-0.0294064\pi$
$354$	0	0
$355$	− 1.68364i	− 0.0893583i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.1684	0.800558	0.400279	−	0.916393i	$-0.368913\pi$
$359$	15.1684	0.800558	0.400279	+	0.916393i	$0.368913\pi$
$360$	0	0
$361$	14.8342	0.780748
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 0.347093i	− 0.0181677i
$366$	0	0
$367$	− 7.59145i	− 0.396270i	−0.980175	−	0.198135i	$-0.936511\pi$
$367$	− 7.59145i	− 0.396270i	0.980175	−	0.198135i	$-0.0634885\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−15.9439	−0.827765
$372$	0	0
$373$	−2.89739	−0.150021	−0.0750105	−	0.997183i	$-0.523899\pi$
$373$	−2.89739	−0.150021	−0.0750105	+	0.997183i	$0.523899\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 16.1887i	− 0.833758i
$378$	0	0
$379$	− 0.117094i	− 0.00601471i	−0.999995	−	0.00300736i	$-0.999043\pi$
$379$	− 0.117094i	− 0.00601471i	0.999995	−	0.00300736i	$-0.000957273\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.3177	−0.578307	−0.289154	−	0.957283i	$-0.593374\pi$
$383$	−11.3177	−0.578307	−0.289154	+	0.957283i	$0.593374\pi$
$384$	0	0
$385$	−20.1982	−1.02939
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 29.2798i	− 1.48454i	−0.670099	−	0.742272i	$-0.733748\pi$
$389$	− 29.2798i	− 1.48454i	0.670099	−	0.742272i	$-0.266252\pi$
$390$	0	0
$391$	44.7153i	2.26135i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0.441194	0.0221989
$396$	0	0
$397$	−28.0503	−1.40781	−0.703903	−	0.710296i	$-0.748561\pi$
$397$	−28.0503	−1.40781	−0.703903	+	0.710296i	$0.748561\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 10.0208i	− 0.500415i	−0.968192	−	0.250207i	$-0.919501\pi$
$401$	− 10.0208i	− 0.500415i	0.968192	−	0.250207i	$-0.0804988\pi$
$402$	0	0
$403$	− 17.6789i	− 0.880647i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5.90923	0.292910
$408$	0	0
$409$	−28.9804	−1.43299	−0.716493	−	0.697594i	$-0.754254\pi$
$409$	−28.9804	−1.43299	−0.716493	+	0.697594i	$0.754254\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 15.9103i	− 0.782893i
$414$	0	0
$415$	2.77121i	0.136033i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	35.4762	1.73312	0.866562	−	0.499069i	$-0.166325\pi$
$419$	35.4762	1.73312	0.866562	+	0.499069i	$0.166325\pi$
$420$	0	0
$421$	26.6996	1.30126	0.650629	−	0.759395i	$-0.274505\pi$
$421$	26.6996	1.30126	0.650629	+	0.759395i	$0.274505\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	28.3218i	1.37381i
$426$	0	0
$427$	− 23.7106i	− 1.14744i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5.31642	0.256083	0.128041	−	0.991769i	$-0.459131\pi$
$431$	5.31642	0.256083	0.128041	+	0.991769i	$0.459131\pi$
$432$	0	0
$433$	−32.3998	−1.55703	−0.778517	−	0.627623i	$-0.784028\pi$
$433$	−32.3998	−1.55703	−0.778517	+	0.627623i	$0.784028\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 13.9029i	− 0.665067i
$438$	0	0
$439$	29.1579i	1.39163i	0.718221	+	0.695815i	$0.244957\pi$
$439$	29.1579i	1.39163i	−0.718221	+	0.695815i	$0.755043\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	27.5734	1.31005	0.655027	−	0.755606i	$-0.272657\pi$
$443$	27.5734	1.31005	0.655027	+	0.755606i	$0.272657\pi$
$444$	0	0
$445$	−3.62832	−0.171999
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11.7395i	0.554023i	0.960867	+	0.277012i	$0.0893441\pi$
$449$	11.7395i	0.554023i	−0.960867	+	0.277012i	$0.910656\pi$
$450$	0	0
$451$	17.7500i	0.835816i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.61879	0.263413
$456$	0	0
$457$	−42.2259	−1.97524	−0.987622	−	0.156853i	$-0.949865\pi$
$457$	−42.2259	−1.97524	−0.987622	+	0.156853i	$0.949865\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 30.1906i	− 1.40612i	−0.711132	−	0.703058i	$-0.751817\pi$
$461$	− 30.1906i	− 1.40612i	0.711132	−	0.703058i	$-0.248183\pi$
$462$	0	0
$463$	24.8349i	1.15418i	0.816682	+	0.577088i	$0.195811\pi$
$463$	24.8349i	1.15418i	−0.816682	+	0.577088i	$0.804189\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.9297	−1.06106	−0.530530	−	0.847666i	$-0.678007\pi$
$467$	−22.9297	−1.06106	−0.530530	+	0.847666i	$0.678007\pi$
$468$	0	0
$469$	41.9670	1.93785
$470$	0	0
$471$	0	0
$472$	0	0
$473$	68.3422i	3.14238i
$474$	0	0
$475$	− 8.80585i	− 0.404040i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	23.5450	1.07580	0.537900	−	0.843009i	$-0.319218\pi$
$479$	23.5450	1.07580	0.537900	+	0.843009i	$0.319218\pi$
$480$	0	0
$481$	−1.64385	−0.0749531
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.02542i	0.137377i
$486$	0	0
$487$	21.3742i	0.968556i	0.874914	+	0.484278i	$0.160918\pi$
$487$	21.3742i	0.968556i	−0.874914	+	0.484278i	$0.839082\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.3900	−1.19097	−0.595483	−	0.803368i	$-0.703039\pi$
$491$	−26.3900	−1.19097	−0.595483	+	0.803368i	$0.703039\pi$
$492$	0	0
$493$	−64.6468	−2.91154
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.39411i	− 0.376527i
$498$	0	0
$499$	− 13.8093i	− 0.618190i	−0.951031	−	0.309095i	$-0.899974\pi$
$499$	− 13.8093i	− 0.618190i	0.951031	−	0.309095i	$-0.100026\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.13714	−0.0952905	−0.0476453	−	0.998864i	$-0.515172\pi$
$503$	−2.13714	−0.0952905	−0.0476453	+	0.998864i	$0.515172\pi$
$504$	0	0
$505$	−12.0573	−0.536543
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 36.1875i	− 1.60398i	−0.597335	−	0.801992i	$-0.703774\pi$
$509$	− 36.1875i	− 1.60398i	0.597335	−	0.801992i	$-0.296226\pi$
$510$	0	0
$511$	− 1.73050i	− 0.0765527i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.6328	0.468538
$516$	0	0
$517$	−26.9356	−1.18463
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 40.3694i	− 1.76861i	−0.466906	−	0.884307i	$-0.654631\pi$
$521$	− 40.3694i	− 1.76861i	0.466906	−	0.884307i	$-0.345369\pi$
$522$	0	0
$523$	− 12.8543i	− 0.562078i	−0.959696	−	0.281039i	$-0.909321\pi$
$523$	− 12.8543i	− 0.562078i	0.959696	−	0.281039i	$-0.0906790\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−70.5977	−3.07528
$528$	0	0
$529$	23.3998	1.01738
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.93776i	− 0.213878i
$534$	0	0
$535$	− 7.88190i	− 0.340764i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−59.3372	−2.55583
$540$	0	0
$541$	13.8111	0.593786	0.296893	−	0.954911i	$-0.404049\pi$
$541$	13.8111	0.593786	0.296893	+	0.954911i	$0.404049\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.18240i	0.136319i
$546$	0	0
$547$	− 24.6873i	− 1.05555i	−0.849383	−	0.527777i	$-0.823026\pi$
$547$	− 24.6873i	− 1.05555i	0.849383	−	0.527777i	$-0.176974\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	20.1000	0.856290
$552$	0	0
$553$	2.19966	0.0935389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 17.5725i	− 0.744572i	−0.928118	−	0.372286i	$-0.878574\pi$
$557$	− 17.5725i	− 0.744572i	0.928118	−	0.372286i	$-0.121426\pi$
$558$	0	0
$559$	− 19.0116i	− 0.804107i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18.2356	0.768537	0.384269	−	0.923221i	$-0.374454\pi$
$563$	18.2356	0.768537	0.384269	+	0.923221i	$0.374454\pi$
$564$	0	0
$565$	−1.98543	−0.0835277
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 12.1120i	− 0.507762i	−0.967235	−	0.253881i	$-0.918293\pi$
$569$	− 12.1120i	− 0.507762i	0.967235	−	0.253881i	$-0.0817072\pi$
$570$	0	0
$571$	17.5981i	0.736457i	0.929735	+	0.368229i	$0.120036\pi$
$571$	17.5981i	0.736457i	−0.929735	+	0.368229i	$0.879964\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	29.3887	1.22559
$576$	0	0
$577$	−1.55766	−0.0648463	−0.0324231	−	0.999474i	$-0.510322\pi$
$577$	−1.55766	−0.0648463	−0.0324231	+	0.999474i	$0.510322\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13.8164i	0.573200i
$582$	0	0
$583$	− 22.8230i	− 0.945231i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.1792	−0.543962	−0.271981	−	0.962303i	$-0.587679\pi$
$587$	−13.1792	−0.543962	−0.271981	+	0.962303i	$0.587679\pi$
$588$	0	0
$589$	21.9503	0.904446
$590$	0	0
$591$	0	0
$592$	0	0
$593$	25.9023i	1.06368i	0.846845	+	0.531840i	$0.178499\pi$
$593$	25.9023i	1.06368i	−0.846845	+	0.531840i	$0.821501\pi$
$594$	0	0
$595$	− 22.4377i	− 0.919858i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−24.3946	−0.996737	−0.498368	−	0.866965i	$-0.666067\pi$
$599$	−24.3946	−0.996737	−0.498368	+	0.866965i	$0.666067\pi$
$600$	0	0
$601$	36.2586	1.47902	0.739509	−	0.673147i	$-0.235058\pi$
$601$	36.2586	1.47902	0.739509	+	0.673147i	$0.235058\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 19.8049i	− 0.805182i
$606$	0	0
$607$	22.8527i	0.927563i	0.885950	+	0.463781i	$0.153508\pi$
$607$	22.8527i	0.927563i	−0.885950	+	0.463781i	$0.846492\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.49303	0.303135
$612$	0	0
$613$	11.2231	0.453296	0.226648	−	0.973977i	$-0.427223\pi$
$613$	11.2231	0.453296	0.226648	+	0.973977i	$0.427223\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 12.7900i	− 0.514908i	−0.966291	−	0.257454i	$-0.917116\pi$
$617$	− 12.7900i	− 0.514908i	0.966291	−	0.257454i	$-0.0828836\pi$
$618$	0	0
$619$	− 10.7660i	− 0.432722i	−0.976313	−	0.216361i	$-0.930581\pi$
$619$	− 10.7660i	− 0.432722i	0.976313	−	0.216361i	$-0.0694188\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−18.0897	−0.724747
$624$	0	0
$625$	15.1864	0.607455
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.56445i	0.261742i
$630$	0	0
$631$	− 13.6486i	− 0.543344i	−0.962390	−	0.271672i	$-0.912423\pi$
$631$	− 13.6486i	− 0.543344i	0.962390	−	0.271672i	$-0.0875765\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−7.38217	−0.292953
$636$	0	0
$637$	16.5066	0.654016
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 5.96867i	− 0.235748i	−0.993029	−	0.117874i	$-0.962392\pi$
$641$	− 5.96867i	− 0.235748i	0.993029	−	0.117874i	$-0.0376079\pi$
$642$	0	0
$643$	− 18.5413i	− 0.731198i	−0.930773	−	0.365599i	$-0.880864\pi$
$643$	− 18.5413i	− 0.731198i	0.930773	−	0.365599i	$-0.119136\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.8794	0.624283	0.312142	−	0.950036i	$-0.398954\pi$
$647$	15.8794	0.624283	0.312142	+	0.950036i	$0.398954\pi$
$648$	0	0
$649$	22.7749	0.893991
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10.2761i	0.402135i	0.979577	+	0.201067i	$0.0644410\pi$
$653$	10.2761i	0.402135i	−0.979577	+	0.201067i	$0.935559\pi$
$654$	0	0
$655$	− 9.13584i	− 0.356967i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.39542	0.210176	0.105088	−	0.994463i	$-0.466488\pi$
$659$	5.39542	0.210176	0.105088	+	0.994463i	$0.466488\pi$
$660$	0	0
$661$	26.1950	1.01887	0.509434	−	0.860510i	$-0.329855\pi$
$661$	26.1950	1.01887	0.509434	+	0.860510i	$0.329855\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.97636i	0.270532i
$666$	0	0
$667$	67.0821i	2.59743i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	33.9407	1.31027
$672$	0	0
$673$	12.2457	0.472039	0.236019	−	0.971748i	$-0.424157\pi$
$673$	12.2457	0.472039	0.236019	+	0.971748i	$0.424157\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 15.7060i	− 0.603630i	−0.953367	−	0.301815i	$-0.902408\pi$
$677$	− 15.7060i	− 0.603630i	0.953367	−	0.301815i	$-0.0975925\pi$
$678$	0	0
$679$	15.0838i	0.578863i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	33.2828	1.27353	0.636765	−	0.771058i	$-0.280272\pi$
$683$	33.2828	1.27353	0.636765	+	0.771058i	$0.280272\pi$
$684$	0	0
$685$	14.3113	0.546806
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.34897i	0.241876i
$690$	0	0
$691$	− 22.1731i	− 0.843503i	−0.906712	−	0.421751i	$-0.861416\pi$
$691$	− 22.1731i	− 0.843503i	0.906712	−	0.421751i	$-0.138584\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13.9185	0.527961
$696$	0	0
$697$	−19.7181	−0.746878
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.5727i	0.852561i	0.904591	+	0.426280i	$0.140176\pi$
$701$	22.5727i	0.852561i	−0.904591	+	0.426280i	$0.859824\pi$
$702$	0	0
$703$	− 2.04102i	− 0.0769787i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−60.1140	−2.26082
$708$	0	0
$709$	3.73100	0.140121	0.0700604	−	0.997543i	$-0.477681\pi$
$709$	3.73100	0.140121	0.0700604	+	0.997543i	$0.477681\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	73.2572i	2.74350i
$714$	0	0
$715$	8.04306i	0.300793i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−41.1923	−1.53621	−0.768106	−	0.640323i	$-0.778801\pi$
$719$	−41.1923	−1.53621	−0.768106	+	0.640323i	$0.778801\pi$
$720$	0	0
$721$	53.0120	1.97427
$722$	0	0
$723$	0	0
$724$	0	0
$725$	42.4885i	1.57798i
$726$	0	0
$727$	− 25.8695i	− 0.959445i	−0.877420	−	0.479723i	$-0.840737\pi$
$727$	− 25.8695i	− 0.959445i	0.877420	−	0.479723i	$-0.159263\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−75.9199	−2.80800
$732$	0	0
$733$	35.4736	1.31025	0.655124	−	0.755522i	$-0.272617\pi$
$733$	35.4736	1.31025	0.655124	+	0.755522i	$0.272617\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	60.0739i	2.21285i
$738$	0	0
$739$	30.8916i	1.13637i	0.822902	+	0.568183i	$0.192353\pi$
$739$	30.8916i	1.13637i	−0.822902	+	0.568183i	$0.807647\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17.1706	−0.629928	−0.314964	−	0.949104i	$-0.601992\pi$
$743$	−17.1706	−0.629928	−0.314964	+	0.949104i	$0.601992\pi$
$744$	0	0
$745$	−4.05232	−0.148465
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 39.2967i	− 1.43587i
$750$	0	0
$751$	− 40.8445i	− 1.49044i	−0.666820	−	0.745219i	$-0.732345\pi$
$751$	− 40.8445i	− 1.49044i	0.666820	−	0.745219i	$-0.267655\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−16.3740	−0.595911
$756$	0	0
$757$	−37.5719	−1.36558	−0.682788	−	0.730617i	$-0.739233\pi$
$757$	−37.5719	−1.36558	−0.682788	+	0.730617i	$0.739233\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.82990i	0.175084i	0.996161	+	0.0875419i	$0.0279012\pi$
$761$	4.82990i	0.175084i	−0.996161	+	0.0875419i	$0.972099\pi$
$762$	0	0
$763$	15.8665i	0.574405i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−6.33559	−0.228765
$768$	0	0
$769$	34.9687	1.26100	0.630502	−	0.776188i	$-0.282849\pi$
$769$	34.9687	1.26100	0.630502	+	0.776188i	$0.282849\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 10.5329i	− 0.378841i	−0.981896	−	0.189420i	$-0.939339\pi$
$773$	− 10.5329i	− 0.378841i	0.981896	−	0.189420i	$-0.0606609\pi$
$774$	0	0
$775$	46.3997i	1.66673i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.13079	0.219658
$780$	0	0
$781$	12.0158	0.429959
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.77685i	0.348951i
$786$	0	0
$787$	− 15.7275i	− 0.560623i	−0.959909	−	0.280312i	$-0.909562\pi$
$787$	− 15.7275i	− 0.560623i	0.959909	−	0.280312i	$-0.0904378\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.89874	−0.351959
$792$	0	0
$793$	−9.44174	−0.335286
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 46.8183i	− 1.65839i	−0.558959	−	0.829195i	$-0.688799\pi$
$797$	− 46.8183i	− 1.65839i	0.558959	−	0.829195i	$-0.311201\pi$
$798$	0	0
$799$	− 29.9222i	− 1.05857i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	2.47713	0.0874161
$804$	0	0
$805$	−23.2830	−0.820618
$806$	0	0
$807$	0	0
$808$	0	0
$809$	47.3503i	1.66475i	0.554213	+	0.832375i	$0.313019\pi$
$809$	47.3503i	1.66475i	−0.554213	+	0.832375i	$0.686981\pi$
$810$	0	0
$811$	− 37.0982i	− 1.30269i	−0.758780	−	0.651347i	$-0.774204\pi$
$811$	− 37.0982i	− 1.30269i	0.758780	−	0.651347i	$-0.225796\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	7.73617	0.270986
$816$	0	0
$817$	23.6051	0.825838
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 52.7570i	− 1.84123i	−0.390471	−	0.920615i	$-0.627688\pi$
$821$	− 52.7570i	− 1.84123i	0.390471	−	0.920615i	$-0.372312\pi$
$822$	0	0
$823$	− 21.2754i	− 0.741613i	−0.928710	−	0.370806i	$-0.879081\pi$
$823$	− 21.2754i	− 0.741613i	0.928710	−	0.370806i	$-0.120919\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	45.6314	1.58676	0.793380	−	0.608726i	$-0.208319\pi$
$827$	45.6314	1.58676	0.793380	+	0.608726i	$0.208319\pi$
$828$	0	0
$829$	−19.6501	−0.682477	−0.341239	−	0.939977i	$-0.610846\pi$
$829$	−19.6501	−0.682477	−0.341239	+	0.939977i	$0.610846\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 65.9165i	− 2.28387i
$834$	0	0
$835$	6.87288i	0.237846i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.1248	1.07455	0.537273	−	0.843408i	$-0.319455\pi$
$839$	31.1248	1.07455	0.537273	+	0.843408i	$0.319455\pi$
$840$	0	0
$841$	−67.9833	−2.34425
$842$	0	0
$843$	0	0
$844$	0	0
$845$	8.52649i	0.293320i
$846$	0	0
$847$	− 98.7409i	− 3.39278i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.81174	0.233503
$852$	0	0
$853$	−55.2258	−1.89089	−0.945447	−	0.325775i	$-0.894375\pi$
$853$	−55.2258	−1.89089	−0.945447	+	0.325775i	$0.894375\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 33.7374i	− 1.15245i	−0.817292	−	0.576224i	$-0.804526\pi$
$857$	− 33.7374i	− 1.15245i	0.817292	−	0.576224i	$-0.195474\pi$
$858$	0	0
$859$	− 23.1124i	− 0.788586i	−0.918985	−	0.394293i	$-0.870990\pi$
$859$	− 23.1124i	− 0.788586i	0.918985	−	0.394293i	$-0.129010\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−56.5722	−1.92574	−0.962871	−	0.269963i	$-0.912989\pi$
$863$	−56.5722	−1.92574	−0.962871	+	0.269963i	$0.912989\pi$
$864$	0	0
$865$	8.73669	0.297056
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.14871i	0.106813i
$870$	0	0
$871$	− 16.7115i	− 0.566249i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−31.8373	−1.07630
$876$	0	0
$877$	40.5492	1.36925	0.684624	−	0.728896i	$-0.259966\pi$
$877$	40.5492	1.36925	0.684624	+	0.728896i	$0.259966\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	30.9339i	1.04219i	0.853499	+	0.521095i	$0.174476\pi$
$881$	30.9339i	1.04219i	−0.853499	+	0.521095i	$0.825524\pi$
$882$	0	0
$883$	− 42.1028i	− 1.41687i	−0.705775	−	0.708437i	$-0.749401\pi$
$883$	− 42.1028i	− 1.41687i	0.705775	−	0.708437i	$-0.250599\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−31.8465	−1.06930	−0.534650	−	0.845074i	$-0.679557\pi$
$887$	−31.8465	−1.06930	−0.534650	+	0.845074i	$0.679557\pi$
$888$	0	0
$889$	−36.8052	−1.23441
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.30344i	0.311328i
$894$	0	0
$895$	6.79888i	0.227262i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−105.911	−3.53233
$900$	0	0
$901$	25.3536	0.844650
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 18.3589i	− 0.610271i
$906$	0	0
$907$	− 36.5757i	− 1.21448i	−0.794520	−	0.607238i	$-0.792277\pi$
$907$	− 36.5757i	− 1.21448i	0.794520	−	0.607238i	$-0.207723\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	43.2345	1.43242	0.716212	−	0.697883i	$-0.245874\pi$
$911$	43.2345	1.43242	0.716212	+	0.697883i	$0.245874\pi$
$912$	0	0
$913$	−19.7776	−0.654541
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 45.5485i	− 1.50414i
$918$	0	0
$919$	19.2080i	0.633614i	0.948490	+	0.316807i	$0.102611\pi$
$919$	19.2080i	0.633614i	−0.948490	+	0.316807i	$0.897389\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−3.34259	−0.110023
$924$	0	0
$925$	4.31442	0.141857
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 40.9760i	− 1.34438i	−0.740379	−	0.672189i	$-0.765354\pi$
$929$	− 40.9760i	− 1.34438i	0.740379	−	0.672189i	$-0.234646\pi$
$930$	0	0
$931$	20.4948i	0.671691i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	32.1186	1.05039
$936$	0	0
$937$	−52.1956	−1.70516	−0.852578	−	0.522599i	$-0.824962\pi$
$937$	−52.1956	−1.70516	−0.852578	+	0.522599i	$0.824962\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 14.9832i	− 0.488438i	−0.969720	−	0.244219i	$-0.921468\pi$
$941$	− 14.9832i	− 0.488438i	0.969720	−	0.244219i	$-0.0785316\pi$
$942$	0	0
$943$	20.4610i	0.666300i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.3569	1.44141	0.720703	−	0.693244i	$-0.243819\pi$
$947$	44.3569	1.44141	0.720703	+	0.693244i	$0.243819\pi$
$948$	0	0
$949$	−0.689096	−0.0223690
$950$	0	0
$951$	0	0
$952$	0	0
$953$	8.74054i	0.283134i	0.989929	+	0.141567i	$0.0452140\pi$
$953$	8.74054i	0.283134i	−0.989929	+	0.141567i	$0.954786\pi$
$954$	0	0
$955$	17.2341i	0.557683i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	71.3516	2.30406
$960$	0	0
$961$	−84.6603	−2.73098
$962$	0	0
$963$	0	0
$964$	0	0
$965$	10.4011i	0.334823i
$966$	0	0
$967$	44.4982i	1.43096i	0.698631	+	0.715482i	$0.253793\pi$
$967$	44.4982i	1.43096i	−0.698631	+	0.715482i	$0.746207\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.1277	0.998936	0.499468	−	0.866332i	$-0.333529\pi$
$971$	31.1277	0.998936	0.499468	+	0.866332i	$0.333529\pi$
$972$	0	0
$973$	69.3936	2.22466
$974$	0	0
$975$	0	0
$976$	0	0
$977$	29.3393i	0.938646i	0.883026	+	0.469323i	$0.155502\pi$
$977$	29.3393i	0.938646i	−0.883026	+	0.469323i	$0.844498\pi$
$978$	0	0
$979$	− 25.8946i	− 0.827594i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−3.77643	−0.120449	−0.0602247	−	0.998185i	$-0.519182\pi$
$983$	−3.77643	−0.120449	−0.0602247	+	0.998185i	$0.519182\pi$
$984$	0	0
$985$	−11.3788	−0.362557
$986$	0	0
$987$	0	0
$988$	0	0
$989$	78.7799i	2.50506i
$990$	0	0
$991$	− 6.04274i	− 0.191954i	−0.995384	−	0.0959770i	$-0.969402\pi$
$991$	− 6.04274i	− 0.191954i	0.995384	−	0.0959770i	$-0.0305975\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16.1798	−0.512936
$996$	0	0
$997$	−8.19498	−0.259537	−0.129769	−	0.991544i	$-0.541423\pi$
$997$	−8.19498	−0.259537	−0.129769	+	0.991544i	$0.541423\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5328.2.e.g.2591.9	✓	24
3.2	odd	2	inner	5328.2.e.g.2591.15	yes	24
4.3	odd	2	inner	5328.2.e.g.2591.10	yes	24
12.11	even	2	inner	5328.2.e.g.2591.16	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5328.2.e.g.2591.9	✓	24	1.1	even	1	trivial
5328.2.e.g.2591.10	yes	24	4.3	odd	2	inner
5328.2.e.g.2591.15	yes	24	3.2	odd	2	inner
5328.2.e.g.2591.16	yes	24	12.11	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5328.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.827995i q^{5} -4.12813i q^{7} +5.90923 q^{11} -1.64385 q^{13} +6.56445i q^{17} -2.04102i q^{19} +6.81174 q^{23} +4.31442 q^{25} +9.84801i q^{29} +10.7545i q^{31} -3.41807 q^{35} +1.00000 q^{37} +3.00378i q^{41} +11.5653i q^{43} -4.55822 q^{47} -10.0414 q^{49} -3.86226i q^{53} -4.89282i q^{55} +3.85411 q^{59} +5.74368 q^{61} +1.36110i q^{65} +10.1661i q^{67} +2.03339 q^{71} +0.419197 q^{73} -24.3941i q^{77} +0.532846i q^{79} -3.34689 q^{83} +5.43533 q^{85} -4.38205i q^{89} +6.78602i q^{91} -1.68996 q^{95} -3.65391 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 16 q^{13} + 24 q^{37} - 16 q^{49} + 48 q^{61} + 48 q^{73} + 56 q^{85} - 8 q^{97}+O(q^{100}) \) 24 * q + 16 * q^13 + 24 * q^37 - 16 * q^49 + 48 * q^61 + 48 * q^73 + 56 * q^85 - 8 * q^97

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