LMFDB - Embedded newform 5328.2.e.f.2591.13

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 = [24,0,0,0,0,0,0,0,0,0,0,0,0] 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			2591.13
Character	$\chi$	$=$	5328.2591
Dual form			5328.2.e.f.2591.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.617709i q^{5} -2.19022i q^{7} -0.928117 q^{11} -0.951287 q^{13} +0.493268i q^{17} +2.39074i q^{19} -5.17950 q^{23} +4.61844 q^{25} +4.22724i q^{29} +2.47929i q^{31} +1.35292 q^{35} -1.00000 q^{37} -7.06368i q^{41} -9.01128i q^{43} +0.0135125 q^{47} +2.20293 q^{49} -7.85933i q^{53} -0.573306i q^{55} -10.4192 q^{59} -5.10465 q^{61} -0.587619i q^{65} -11.1628i q^{67} +0.138737 q^{71} -6.72224 q^{73} +2.03278i q^{77} +10.7014i q^{79} -14.6492 q^{83} -0.304696 q^{85} +13.8407i q^{89} +2.08353i q^{91} -1.47678 q^{95} +9.48506 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 32 q^{25} - 24 q^{37} - 80 q^{49} - 48 q^{61} - 48 q^{73} - 40 q^{85} + 24 q^{97}+O(q^{100}) $ 24 * q - 32 * q^25 - 24 * q^37 - 80 * q^49 - 48 * q^61 - 48 * q^73 - 40 * q^85 + 24 * 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
$5$	0.617709i	0.276248i	0.990415	+	0.138124i	$0.0441072\pi$
$5$	0.617709i	0.276248i	−0.990415	+	0.138124i	$0.955893\pi$
$6$	0	0
$7$	− 2.19022i	− 0.827826i	−0.910316	−	0.413913i	$-0.864162\pi$
$7$	− 2.19022i	− 0.827826i	0.910316	−	0.413913i	$-0.135838\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.928117	−0.279838	−0.139919	−	0.990163i	$-0.544684\pi$
$11$	−0.928117	−0.279838	−0.139919	+	0.990163i	$0.544684\pi$
$12$	0	0
$13$	−0.951287	−0.263840	−0.131920	−	0.991260i	$-0.542114\pi$
$13$	−0.951287	−0.263840	−0.131920	+	0.991260i	$0.542114\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.493268i	0.119635i	0.998209	+	0.0598175i	$0.0190519\pi$
$17$	0.493268i	0.119635i	−0.998209	+	0.0598175i	$0.980948\pi$
$18$	0	0
$19$	2.39074i	0.548473i	0.961662	+	0.274237i	$0.0884251\pi$
$19$	2.39074i	0.548473i	−0.961662	+	0.274237i	$0.911575\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.17950	−1.08000	−0.540000	−	0.841665i	$-0.681576\pi$
$23$	−5.17950	−1.08000	−0.540000	+	0.841665i	$0.681576\pi$
$24$	0	0
$25$	4.61844	0.923687
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.22724i	0.784980i	0.919756	+	0.392490i	$0.128386\pi$
$29$	4.22724i	0.784980i	−0.919756	+	0.392490i	$0.871614\pi$
$30$	0	0
$31$	2.47929i	0.445293i	0.974899	+	0.222646i	$0.0714695\pi$
$31$	2.47929i	0.445293i	−0.974899	+	0.222646i	$0.928530\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.35292	0.228685
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 7.06368i	− 1.10316i	−0.834121	−	0.551581i	$-0.814025\pi$
$41$	− 7.06368i	− 1.10316i	0.834121	−	0.551581i	$-0.185975\pi$
$42$	0	0
$43$	− 9.01128i	− 1.37421i	−0.726560	−	0.687103i	$-0.758882\pi$
$43$	− 9.01128i	− 1.37421i	0.726560	−	0.687103i	$-0.241118\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.0135125	0.00197099	0.000985497	−	1.00000i	$-0.499686\pi$
$47$	0.0135125	0.00197099	0.000985497		1.00000i	$0.499686\pi$
$48$	0	0
$49$	2.20293	0.314704
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 7.85933i	− 1.07956i	−0.841806	−	0.539781i	$-0.818507\pi$
$53$	− 7.85933i	− 1.07956i	0.841806	−	0.539781i	$-0.181493\pi$
$54$	0	0
$55$	− 0.573306i	− 0.0773046i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.4192	−1.35647	−0.678235	−	0.734845i	$-0.737255\pi$
$59$	−10.4192	−1.35647	−0.678235	+	0.734845i	$0.737255\pi$
$60$	0	0
$61$	−5.10465	−0.653584	−0.326792	−	0.945096i	$-0.605968\pi$
$61$	−5.10465	−0.653584	−0.326792	+	0.945096i	$0.605968\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 0.587619i	− 0.0728851i
$66$	0	0
$67$	− 11.1628i	− 1.36376i	−0.731466	−	0.681878i	$-0.761164\pi$
$67$	− 11.1628i	− 1.36376i	0.731466	−	0.681878i	$-0.238836\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.138737	0.0164650	0.00823250	−	0.999966i	$-0.497379\pi$
$71$	0.138737	0.0164650	0.00823250	+	0.999966i	$0.497379\pi$
$72$	0	0
$73$	−6.72224	−0.786779	−0.393390	−	0.919372i	$-0.628698\pi$
$73$	−6.72224	−0.786779	−0.393390	+	0.919372i	$0.628698\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.03278i	0.231657i
$78$	0	0
$79$	10.7014i	1.20401i	0.798494	+	0.602003i	$0.205631\pi$
$79$	10.7014i	1.20401i	−0.798494	+	0.602003i	$0.794369\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.6492	−1.60796	−0.803981	−	0.594655i	$-0.797289\pi$
$83$	−14.6492	−1.60796	−0.803981	+	0.594655i	$0.797289\pi$
$84$	0	0
$85$	−0.304696	−0.0330489
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.8407i	1.46711i	0.679628	+	0.733556i	$0.262141\pi$
$89$	13.8407i	1.46711i	−0.679628	+	0.733556i	$0.737859\pi$
$90$	0	0
$91$	2.08353i	0.218413i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.47678	−0.151515
$96$	0	0
$97$	9.48506	0.963062	0.481531	−	0.876429i	$-0.340081\pi$
$97$	9.48506	0.963062	0.481531	+	0.876429i	$0.340081\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.8378i	1.77492i	0.460880	+	0.887462i	$0.347534\pi$
$101$	17.8378i	1.77492i	−0.460880	+	0.887462i	$0.652466\pi$
$102$	0	0
$103$	− 7.30200i	− 0.719488i	−0.933051	−	0.359744i	$-0.882864\pi$
$103$	− 7.30200i	− 0.719488i	0.933051	−	0.359744i	$-0.117136\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.09598	−0.685994	−0.342997	−	0.939337i	$-0.611442\pi$
$107$	−7.09598	−0.685994	−0.342997	+	0.939337i	$0.611442\pi$
$108$	0	0
$109$	−12.2856	−1.17675	−0.588373	−	0.808590i	$-0.700231\pi$
$109$	−12.2856	−1.17675	−0.588373	+	0.808590i	$0.700231\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0.865395i	0.0814095i	0.999171	+	0.0407048i	$0.0129603\pi$
$113$	0.865395i	0.0814095i	−0.999171	+	0.0407048i	$0.987040\pi$
$114$	0	0
$115$	− 3.19943i	− 0.298348i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.08037	0.0990370
$120$	0	0
$121$	−10.1386	−0.921691
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.94139i	0.531414i
$126$	0	0
$127$	− 0.685276i	− 0.0608084i	−0.999538	−	0.0304042i	$-0.990321\pi$
$127$	− 0.685276i	− 0.0608084i	0.999538	−	0.0304042i	$-0.00967945\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.496062	0.0433411	0.0216706	−	0.999765i	$-0.493102\pi$
$131$	0.496062	0.0433411	0.0216706	+	0.999765i	$0.493102\pi$
$132$	0	0
$133$	5.23625	0.454040
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.97949i	− 0.510863i	−0.966827	−	0.255431i	$-0.917783\pi$
$137$	− 5.97949i	− 0.510863i	0.966827	−	0.255431i	$-0.0822175\pi$
$138$	0	0
$139$	− 3.61827i	− 0.306898i	−0.988157	−	0.153449i	$-0.950962\pi$
$139$	− 3.61827i	− 0.306898i	0.988157	−	0.153449i	$-0.0490381\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.882905	0.0738323
$144$	0	0
$145$	−2.61121	−0.216849
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.9982i	1.14678i	0.819283	+	0.573389i	$0.194372\pi$
$149$	13.9982i	1.14678i	−0.819283	+	0.573389i	$0.805628\pi$
$150$	0	0
$151$	− 20.6128i	− 1.67745i	−0.544558	−	0.838723i	$-0.683303\pi$
$151$	− 20.6128i	− 1.67745i	0.544558	−	0.838723i	$-0.316697\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.53148	−0.123011
$156$	0	0
$157$	7.43200	0.593138	0.296569	−	0.955011i	$-0.404157\pi$
$157$	7.43200	0.593138	0.296569	+	0.955011i	$0.404157\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	11.3443i	0.894053i
$162$	0	0
$163$	9.31945i	0.729956i	0.931016	+	0.364978i	$0.118923\pi$
$163$	9.31945i	0.729956i	−0.931016	+	0.364978i	$0.881077\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.06779	−0.392157	−0.196079	−	0.980588i	$-0.562821\pi$
$167$	−5.06779	−0.392157	−0.196079	+	0.980588i	$0.562821\pi$
$168$	0	0
$169$	−12.0951	−0.930389
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.53127i	− 0.116420i	−0.998304	−	0.0582101i	$-0.981461\pi$
$173$	− 1.53127i	− 0.116420i	0.998304	−	0.0582101i	$-0.0185393\pi$
$174$	0	0
$175$	− 10.1154i	− 0.764652i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.8900	−1.56139	−0.780695	−	0.624913i	$-0.785134\pi$
$179$	−20.8900	−1.56139	−0.780695	+	0.624913i	$0.785134\pi$
$180$	0	0
$181$	−20.3812	−1.51492	−0.757462	−	0.652879i	$-0.773561\pi$
$181$	−20.3812	−1.51492	−0.757462	+	0.652879i	$0.773561\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 0.617709i	− 0.0454149i
$186$	0	0
$187$	− 0.457810i	− 0.0334784i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.41530	0.319480	0.159740	−	0.987159i	$-0.448934\pi$
$191$	4.41530	0.319480	0.159740	+	0.987159i	$0.448934\pi$
$192$	0	0
$193$	−9.58738	−0.690114	−0.345057	−	0.938582i	$-0.612140\pi$
$193$	−9.58738	−0.690114	−0.345057	+	0.938582i	$0.612140\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.05322i	− 0.146286i	−0.997321	−	0.0731430i	$-0.976697\pi$
$197$	− 2.05322i	− 0.146286i	0.997321	−	0.0731430i	$-0.0233030\pi$
$198$	0	0
$199$	− 2.71751i	− 0.192639i	−0.995350	−	0.0963197i	$-0.969293\pi$
$199$	− 2.71751i	− 0.192639i	0.995350	−	0.0963197i	$-0.0307071\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	9.25861	0.649827
$204$	0	0
$205$	4.36330	0.304746
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 2.21888i	− 0.153483i
$210$	0	0
$211$	− 10.6745i	− 0.734862i	−0.930051	−	0.367431i	$-0.880237\pi$
$211$	− 10.6745i	− 0.734862i	0.930051	−	0.367431i	$-0.119763\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.56635	0.379622
$216$	0	0
$217$	5.43019	0.368625
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 0.469239i	− 0.0315644i
$222$	0	0
$223$	14.8269i	0.992885i	0.868070	+	0.496443i	$0.165361\pi$
$223$	14.8269i	0.992885i	−0.868070	+	0.496443i	$0.834639\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.98340	0.264387	0.132194	−	0.991224i	$-0.457798\pi$
$227$	3.98340	0.264387	0.132194	+	0.991224i	$0.457798\pi$
$228$	0	0
$229$	−15.0344	−0.993504	−0.496752	−	0.867893i	$-0.665474\pi$
$229$	−15.0344	−0.993504	−0.496752	+	0.867893i	$0.665474\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 8.64444i	− 0.566316i	−0.959073	−	0.283158i	$-0.908618\pi$
$233$	− 8.64444i	− 0.566316i	0.959073	−	0.283158i	$-0.0913821\pi$
$234$	0	0
$235$	0.00834676i	0 0.000544483i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5.04326	−0.326221	−0.163111	−	0.986608i	$-0.552153\pi$
$239$	−5.04326	−0.326221	−0.163111	+	0.986608i	$0.552153\pi$
$240$	0	0
$241$	6.70104	0.431652	0.215826	−	0.976432i	$-0.430756\pi$
$241$	6.70104	0.431652	0.215826	+	0.976432i	$0.430756\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.36077i	0.0869362i
$246$	0	0
$247$	− 2.27428i	− 0.144709i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.0813	−0.951922	−0.475961	−	0.879466i	$-0.657900\pi$
$251$	−15.0813	−0.951922	−0.475961	+	0.879466i	$0.657900\pi$
$252$	0	0
$253$	4.80718	0.302225
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.67923i	0.541396i	0.962664	+	0.270698i	$0.0872544\pi$
$257$	8.67923i	0.541396i	−0.962664	+	0.270698i	$0.912746\pi$
$258$	0	0
$259$	2.19022i	0.136094i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.8591	−1.71787	−0.858933	−	0.512089i	$-0.828872\pi$
$263$	−27.8591	−1.71787	−0.858933	+	0.512089i	$0.828872\pi$
$264$	0	0
$265$	4.85478	0.298227
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.04120i	0.307367i	0.988120	+	0.153684i	$0.0491137\pi$
$269$	5.04120i	0.307367i	−0.988120	+	0.153684i	$0.950886\pi$
$270$	0	0
$271$	− 14.8728i	− 0.903457i	−0.892156	−	0.451728i	$-0.850808\pi$
$271$	− 14.8728i	− 0.903457i	0.892156	−	0.451728i	$-0.149192\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.28645	−0.258483
$276$	0	0
$277$	−12.6300	−0.758862	−0.379431	−	0.925220i	$-0.623880\pi$
$277$	−12.6300	−0.758862	−0.379431	+	0.925220i	$0.623880\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.4269i	0.979944i	0.871738	+	0.489972i	$0.162993\pi$
$281$	16.4269i	0.979944i	−0.871738	+	0.489972i	$0.837007\pi$
$282$	0	0
$283$	− 9.73779i	− 0.578852i	−0.957200	−	0.289426i	$-0.906536\pi$
$283$	− 9.73779i	− 0.578852i	0.957200	−	0.289426i	$-0.0934643\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.4710	−0.913226
$288$	0	0
$289$	16.7567	0.985687
$290$	0	0
$291$	0	0
$292$	0	0
$293$	12.1430i	0.709400i	0.934980	+	0.354700i	$0.115417\pi$
$293$	12.1430i	0.709400i	−0.934980	+	0.354700i	$0.884583\pi$
$294$	0	0
$295$	− 6.43606i	− 0.374722i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.92719	0.284947
$300$	0	0
$301$	−19.7367	−1.13760
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 3.15319i	− 0.180551i
$306$	0	0
$307$	− 9.31509i	− 0.531640i	−0.964023	−	0.265820i	$-0.914357\pi$
$307$	− 9.31509i	− 0.531640i	0.964023	−	0.265820i	$-0.0856427\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.50573	0.0853819	0.0426909	−	0.999088i	$-0.486407\pi$
$311$	1.50573	0.0853819	0.0426909	+	0.999088i	$0.486407\pi$
$312$	0	0
$313$	12.9055	0.729459	0.364730	−	0.931113i	$-0.381161\pi$
$313$	12.9055	0.729459	0.364730	+	0.931113i	$0.381161\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.4389i	1.37263i	0.727306	+	0.686313i	$0.240772\pi$
$317$	24.4389i	1.37263i	−0.727306	+	0.686313i	$0.759228\pi$
$318$	0	0
$319$	− 3.92338i	− 0.219667i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.17927	−0.0656166
$324$	0	0
$325$	−4.39346	−0.243705
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 0.0295953i	− 0.00163164i
$330$	0	0
$331$	− 10.1317i	− 0.556887i	−0.960453	−	0.278443i	$-0.910182\pi$
$331$	− 10.1317i	− 0.556887i	0.960453	−	0.278443i	$-0.0898184\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.89538	0.376735
$336$	0	0
$337$	−5.20293	−0.283421	−0.141711	−	0.989908i	$-0.545260\pi$
$337$	−5.20293	−0.283421	−0.141711	+	0.989908i	$0.545260\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2.30107i	− 0.124610i
$342$	0	0
$343$	− 20.1565i	− 1.08835i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.54179	−0.458547	−0.229274	−	0.973362i	$-0.573635\pi$
$347$	−8.54179	−0.458547	−0.229274	+	0.973362i	$0.573635\pi$
$348$	0	0
$349$	−1.67931	−0.0898914	−0.0449457	−	0.998989i	$-0.514311\pi$
$349$	−1.67931	−0.0898914	−0.0449457	+	0.998989i	$0.514311\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 26.1781i	− 1.39332i	−0.717401	−	0.696660i	$-0.754669\pi$
$353$	− 26.1781i	− 1.39332i	0.717401	−	0.696660i	$-0.245331\pi$
$354$	0	0
$355$	0.0856988i	0.00454842i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−27.8712	−1.47099	−0.735494	−	0.677531i	$-0.763050\pi$
$359$	−27.8712	−1.47099	−0.735494	+	0.677531i	$0.763050\pi$
$360$	0	0
$361$	13.2844	0.699177
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 4.15239i	− 0.217346i
$366$	0	0
$367$	15.2478i	0.795928i	0.917401	+	0.397964i	$0.130283\pi$
$367$	15.2478i	0.795928i	−0.917401	+	0.397964i	$0.869717\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−17.2137	−0.893689
$372$	0	0
$373$	8.34334	0.432002	0.216001	−	0.976393i	$-0.430699\pi$
$373$	8.34334	0.432002	0.216001	+	0.976393i	$0.430699\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.02132i	− 0.207109i
$378$	0	0
$379$	− 31.0493i	− 1.59490i	−0.603387	−	0.797448i	$-0.706183\pi$
$379$	− 31.0493i	− 1.59490i	0.603387	−	0.797448i	$-0.293817\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0.864805	0.0441895	0.0220947	−	0.999756i	$-0.492966\pi$
$383$	0.864805	0.0441895	0.0220947	+	0.999756i	$0.492966\pi$
$384$	0	0
$385$	−1.25567	−0.0639948
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 24.0697i	− 1.22038i	−0.792254	−	0.610192i	$-0.791092\pi$
$389$	− 24.0697i	− 1.22038i	0.792254	−	0.610192i	$-0.208908\pi$
$390$	0	0
$391$	− 2.55488i	− 0.129206i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.61038	−0.332604
$396$	0	0
$397$	−16.4323	−0.824716	−0.412358	−	0.911022i	$-0.635295\pi$
$397$	−16.4323	−0.824716	−0.412358	+	0.911022i	$0.635295\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 12.6766i	− 0.633041i	−0.948586	−	0.316520i	$-0.897485\pi$
$401$	− 12.6766i	− 0.633041i	0.948586	−	0.316520i	$-0.102515\pi$
$402$	0	0
$403$	− 2.35851i	− 0.117486i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.928117	0.0460050
$408$	0	0
$409$	16.7657	0.829010	0.414505	−	0.910047i	$-0.363955\pi$
$409$	16.7657	0.829010	0.414505	+	0.910047i	$0.363955\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	22.8205i	1.12292i
$414$	0	0
$415$	− 9.04896i	− 0.444196i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	24.5357	1.19865	0.599323	−	0.800507i	$-0.295437\pi$
$419$	24.5357	1.19865	0.599323	+	0.800507i	$0.295437\pi$
$420$	0	0
$421$	23.6837	1.15427	0.577136	−	0.816648i	$-0.304170\pi$
$421$	23.6837	1.15427	0.577136	+	0.816648i	$0.304170\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.27812i	0.110505i
$426$	0	0
$427$	11.1803i	0.541054i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.6189	−0.848671	−0.424336	−	0.905505i	$-0.639492\pi$
$431$	−17.6189	−0.848671	−0.424336	+	0.905505i	$0.639492\pi$
$432$	0	0
$433$	−37.3003	−1.79254	−0.896269	−	0.443511i	$-0.853733\pi$
$433$	−37.3003	−1.79254	−0.896269	+	0.443511i	$0.853733\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 12.3828i	− 0.592351i
$438$	0	0
$439$	18.5277i	0.884277i	0.896947	+	0.442139i	$0.145780\pi$
$439$	18.5277i	0.884277i	−0.896947	+	0.442139i	$0.854220\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.8332	−0.657237	−0.328618	−	0.944463i	$-0.606583\pi$
$443$	−13.8332	−0.657237	−0.328618	+	0.944463i	$0.606583\pi$
$444$	0	0
$445$	−8.54953	−0.405287
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.96758i	0.0928559i	0.998922	+	0.0464279i	$0.0147838\pi$
$449$	1.96758i	0.0928559i	−0.998922	+	0.0464279i	$0.985216\pi$
$450$	0	0
$451$	6.55592i	0.308706i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.28702	−0.0603362
$456$	0	0
$457$	27.4413	1.28365	0.641825	−	0.766851i	$-0.278178\pi$
$457$	27.4413	1.28365	0.641825	+	0.766851i	$0.278178\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	27.6821i	1.28928i	0.764485	+	0.644641i	$0.222993\pi$
$461$	27.6821i	1.28928i	−0.764485	+	0.644641i	$0.777007\pi$
$462$	0	0
$463$	− 15.0508i	− 0.699472i	−0.936848	−	0.349736i	$-0.886271\pi$
$463$	− 15.0508i	− 0.699472i	0.936848	−	0.349736i	$-0.113729\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.3869	1.40614	0.703068	−	0.711122i	$-0.251813\pi$
$467$	30.3869	1.40614	0.703068	+	0.711122i	$0.251813\pi$
$468$	0	0
$469$	−24.4491	−1.12895
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.36352i	0.384555i
$474$	0	0
$475$	11.0415i	0.506618i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−0.664004	−0.0303391	−0.0151695	−	0.999885i	$-0.504829\pi$
$479$	−0.664004	−0.0303391	−0.0151695	+	0.999885i	$0.504829\pi$
$480$	0	0
$481$	0.951287	0.0433750
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.85901i	0.266044i
$486$	0	0
$487$	20.8164i	0.943280i	0.881791	+	0.471640i	$0.156338\pi$
$487$	20.8164i	0.943280i	−0.881791	+	0.471640i	$0.843662\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.31850	−0.104633	−0.0523163	−	0.998631i	$-0.516660\pi$
$491$	−2.31850	−0.104633	−0.0523163	+	0.998631i	$0.516660\pi$
$492$	0	0
$493$	−2.08516	−0.0939110
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 0.303864i	− 0.0136302i
$498$	0	0
$499$	10.6938i	0.478719i	0.970931	+	0.239360i	$0.0769375\pi$
$499$	10.6938i	0.478719i	−0.970931	+	0.239360i	$0.923063\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.33386	0.0594741	0.0297370	−	0.999558i	$-0.490533\pi$
$503$	1.33386	0.0594741	0.0297370	+	0.999558i	$0.490533\pi$
$504$	0	0
$505$	−11.0186	−0.490319
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 8.59739i	− 0.381073i	−0.981680	−	0.190536i	$-0.938977\pi$
$509$	− 8.59739i	− 0.381073i	0.981680	−	0.190536i	$-0.0610227\pi$
$510$	0	0
$511$	14.7232i	0.651316i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.51051	0.198757
$516$	0	0
$517$	−0.0125411	−0.000551558 0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 23.9002i	− 1.04709i	−0.851999	−	0.523544i	$-0.824610\pi$
$521$	− 23.9002i	− 1.04709i	0.851999	−	0.523544i	$-0.175390\pi$
$522$	0	0
$523$	− 3.55714i	− 0.155543i	−0.996971	−	0.0777714i	$-0.975220\pi$
$523$	− 3.55714i	− 0.155543i	0.996971	−	0.0777714i	$-0.0247804\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.22295	−0.0532726
$528$	0	0
$529$	3.82725	0.166402
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.71959i	0.291058i
$534$	0	0
$535$	− 4.38325i	− 0.189504i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.04457	−0.0880660
$540$	0	0
$541$	−12.0717	−0.519003	−0.259502	−	0.965743i	$-0.583558\pi$
$541$	−12.0717	−0.519003	−0.259502	+	0.965743i	$0.583558\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 7.58892i	− 0.325073i
$546$	0	0
$547$	11.4387i	0.489084i	0.969639	+	0.244542i	$0.0786377\pi$
$547$	11.4387i	0.489084i	−0.969639	+	0.244542i	$0.921362\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10.1062	−0.430540
$552$	0	0
$553$	23.4385	0.996708
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.7645i	1.13405i	0.823702	+	0.567023i	$0.191905\pi$
$557$	26.7645i	1.13405i	−0.823702	+	0.567023i	$0.808095\pi$
$558$	0	0
$559$	8.57231i	0.362570i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.90246	−0.333049	−0.166525	−	0.986037i	$-0.553254\pi$
$563$	−7.90246	−0.333049	−0.166525	+	0.986037i	$0.553254\pi$
$564$	0	0
$565$	−0.534562	−0.0224892
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 26.4231i	− 1.10772i	−0.832611	−	0.553858i	$-0.813155\pi$
$569$	− 26.4231i	− 1.10772i	0.832611	−	0.553858i	$-0.186845\pi$
$570$	0	0
$571$	− 7.60950i	− 0.318447i	−0.987243	−	0.159224i	$-0.949101\pi$
$571$	− 7.60950i	− 0.318447i	0.987243	−	0.159224i	$-0.0508991\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−23.9212	−0.997583
$576$	0	0
$577$	−28.6004	−1.19065	−0.595324	−	0.803485i	$-0.702976\pi$
$577$	−28.6004	−1.19065	−0.595324	+	0.803485i	$0.702976\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	32.0851i	1.33111i
$582$	0	0
$583$	7.29437i	0.302102i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−37.9300	−1.56554	−0.782770	−	0.622311i	$-0.786194\pi$
$587$	−37.9300	−1.56554	−0.782770	+	0.622311i	$0.786194\pi$
$588$	0	0
$589$	−5.92732	−0.244231
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 32.9885i	− 1.35468i	−0.735672	−	0.677338i	$-0.763133\pi$
$593$	− 32.9885i	− 1.35468i	0.735672	−	0.677338i	$-0.236867\pi$
$594$	0	0
$595$	0.667352i	0.0273588i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0.333772	0.0136375	0.00681877	−	0.999977i	$-0.497830\pi$
$599$	0.333772	0.0136375	0.00681877	+	0.999977i	$0.497830\pi$
$600$	0	0
$601$	10.8840	0.443968	0.221984	−	0.975050i	$-0.428747\pi$
$601$	10.8840	0.443968	0.221984	+	0.975050i	$0.428747\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 6.26270i	− 0.254615i
$606$	0	0
$607$	− 23.5072i	− 0.954127i	−0.878869	−	0.477063i	$-0.841701\pi$
$607$	− 23.5072i	− 0.954127i	0.878869	−	0.477063i	$-0.158299\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.0128542	−0.000520026 0
$612$	0	0
$613$	36.3165	1.46681	0.733405	−	0.679792i	$-0.237930\pi$
$613$	36.3165	1.46681	0.733405	+	0.679792i	$0.237930\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 20.8556i	− 0.839615i	−0.907613	−	0.419808i	$-0.862098\pi$
$617$	− 20.8556i	− 0.839615i	0.907613	−	0.419808i	$-0.137902\pi$
$618$	0	0
$619$	8.68092i	0.348916i	0.984665	+	0.174458i	$0.0558173\pi$
$619$	8.68092i	0.348916i	−0.984665	+	0.174458i	$0.944183\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	30.3142	1.21451
$624$	0	0
$625$	19.4221	0.776885
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 0.493268i	− 0.0196679i
$630$	0	0
$631$	24.2557i	0.965603i	0.875730	+	0.482802i	$0.160381\pi$
$631$	24.2557i	0.965603i	−0.875730	+	0.482802i	$0.839619\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.423301	0.0167982
$636$	0	0
$637$	−2.09562	−0.0830313
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.4106i	0.490190i	0.969499	+	0.245095i	$0.0788191\pi$
$641$	12.4106i	0.490190i	−0.969499	+	0.245095i	$0.921181\pi$
$642$	0	0
$643$	− 18.5803i	− 0.732736i	−0.930470	−	0.366368i	$-0.880601\pi$
$643$	− 18.5803i	− 0.732736i	0.930470	−	0.366368i	$-0.119399\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.8420	1.68429	0.842146	−	0.539249i	$-0.181292\pi$
$647$	42.8420	1.68429	0.842146	+	0.539249i	$0.181292\pi$
$648$	0	0
$649$	9.67028	0.379592
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.4053i	0.876789i	0.898783	+	0.438395i	$0.144453\pi$
$653$	22.4053i	0.876789i	−0.898783	+	0.438395i	$0.855547\pi$
$654$	0	0
$655$	0.306422i	0.0119729i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14.2469	0.554980	0.277490	−	0.960728i	$-0.410497\pi$
$659$	14.2469	0.554980	0.277490	+	0.960728i	$0.410497\pi$
$660$	0	0
$661$	3.69032	0.143537	0.0717684	−	0.997421i	$-0.477136\pi$
$661$	3.69032	0.143537	0.0717684	+	0.997421i	$0.477136\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.23448i	0.125428i
$666$	0	0
$667$	− 21.8950i	− 0.847779i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.73772	0.182897
$672$	0	0
$673$	23.1734	0.893269	0.446635	−	0.894716i	$-0.352622\pi$
$673$	23.1734	0.893269	0.446635	+	0.894716i	$0.352622\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4.72382i	0.181551i	0.995871	+	0.0907755i	$0.0289346\pi$
$677$	4.72382i	0.181551i	−0.995871	+	0.0907755i	$0.971065\pi$
$678$	0	0
$679$	− 20.7744i	− 0.797248i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.2237	0.926894	0.463447	−	0.886125i	$-0.346612\pi$
$683$	24.2237	0.926894	0.463447	+	0.886125i	$0.346612\pi$
$684$	0	0
$685$	3.69359	0.141125
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.47647i	0.284831i
$690$	0	0
$691$	7.00816i	0.266603i	0.991076	+	0.133302i	$0.0425579\pi$
$691$	7.00816i	0.266603i	−0.991076	+	0.133302i	$0.957442\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2.23504	0.0847799
$696$	0	0
$697$	3.48428	0.131977
$698$	0	0
$699$	0	0
$700$	0	0
$701$	16.6100i	0.627350i	0.949530	+	0.313675i	$0.101560\pi$
$701$	16.6100i	0.627350i	−0.949530	+	0.313675i	$0.898440\pi$
$702$	0	0
$703$	− 2.39074i	− 0.0901684i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	39.0687	1.46933
$708$	0	0
$709$	−28.9216	−1.08617	−0.543087	−	0.839676i	$-0.682745\pi$
$709$	−28.9216	−1.08617	−0.543087	+	0.839676i	$0.682745\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 12.8415i	− 0.480917i
$714$	0	0
$715$	0.545379i	0.0203960i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	34.5397	1.28811	0.644056	−	0.764978i	$-0.277250\pi$
$719$	34.5397	1.28811	0.644056	+	0.764978i	$0.277250\pi$
$720$	0	0
$721$	−15.9930	−0.595611
$722$	0	0
$723$	0	0
$724$	0	0
$725$	19.5233i	0.725076i
$726$	0	0
$727$	0.281173i	0.0104281i	0.999986	+	0.00521407i	$0.00165970\pi$
$727$	0.281173i	0.0104281i	−0.999986	+	0.00521407i	$0.998340\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.44497	0.164403
$732$	0	0
$733$	−11.5874	−0.427990	−0.213995	−	0.976835i	$-0.568648\pi$
$733$	−11.5874	−0.427990	−0.213995	+	0.976835i	$0.568648\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.3604i	0.381630i
$738$	0	0
$739$	31.2107i	1.14811i	0.818818	+	0.574053i	$0.194630\pi$
$739$	31.2107i	1.14811i	−0.818818	+	0.574053i	$0.805370\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−49.6171	−1.82027	−0.910137	−	0.414306i	$-0.864024\pi$
$743$	−49.6171	−1.82027	−0.910137	+	0.414306i	$0.864024\pi$
$744$	0	0
$745$	−8.64682	−0.316795
$746$	0	0
$747$	0	0
$748$	0	0
$749$	15.5418i	0.567884i
$750$	0	0
$751$	− 2.40788i	− 0.0878646i	−0.999035	−	0.0439323i	$-0.986011\pi$
$751$	− 2.40788i	− 0.0878646i	0.999035	−	0.0439323i	$-0.0139886\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	12.7327	0.463391
$756$	0	0
$757$	10.0559	0.365489	0.182745	−	0.983160i	$-0.441502\pi$
$757$	10.0559	0.365489	0.182745	+	0.983160i	$0.441502\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.69736i	0.170279i	0.996369	+	0.0851396i	$0.0271336\pi$
$761$	4.69736i	0.170279i	−0.996369	+	0.0851396i	$0.972866\pi$
$762$	0	0
$763$	26.9082i	0.974141i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.91169	0.357890
$768$	0	0
$769$	−29.5175	−1.06443	−0.532215	−	0.846610i	$-0.678640\pi$
$769$	−29.5175	−1.06443	−0.532215	+	0.846610i	$0.678640\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 35.0139i	− 1.25936i	−0.776853	−	0.629682i	$-0.783185\pi$
$773$	− 35.0139i	− 1.25936i	0.776853	−	0.629682i	$-0.216815\pi$
$774$	0	0
$775$	11.4504i	0.411311i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	16.8874	0.605054
$780$	0	0
$781$	−0.128764	−0.00460753
$782$	0	0
$783$	0	0
$784$	0	0
$785$	4.59081i	0.163853i
$786$	0	0
$787$	30.2196i	1.07721i	0.842558	+	0.538605i	$0.181049\pi$
$787$	30.2196i	1.07721i	−0.842558	+	0.538605i	$0.818951\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.89541	0.0673929
$792$	0	0
$793$	4.85599	0.172441
$794$	0	0
$795$	0	0
$796$	0	0
$797$	7.37048i	0.261076i	0.991443	+	0.130538i	$0.0416704\pi$
$797$	7.37048i	0.261076i	−0.991443	+	0.130538i	$0.958330\pi$
$798$	0	0
$799$	0.00666526i	0 0.000235800i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.23903	0.220171
$804$	0	0
$805$	−7.00745	−0.246980
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 31.6065i	− 1.11122i	−0.831442	−	0.555612i	$-0.812484\pi$
$809$	− 31.6065i	− 1.11122i	0.831442	−	0.555612i	$-0.187516\pi$
$810$	0	0
$811$	6.97244i	0.244836i	0.992479	+	0.122418i	$0.0390648\pi$
$811$	6.97244i	0.244836i	−0.992479	+	0.122418i	$0.960935\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.75671	−0.201649
$816$	0	0
$817$	21.5436	0.753715
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 11.3856i	− 0.397360i	−0.980064	−	0.198680i	$-0.936335\pi$
$821$	− 11.3856i	− 0.397360i	0.980064	−	0.198680i	$-0.0636654\pi$
$822$	0	0
$823$	− 13.5638i	− 0.472805i	−0.971655	−	0.236402i	$-0.924032\pi$
$823$	− 13.5638i	− 0.472805i	0.971655	−	0.236402i	$-0.0759684\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.2898	0.914187	0.457093	−	0.889419i	$-0.348890\pi$
$827$	26.2898	0.914187	0.457093	+	0.889419i	$0.348890\pi$
$828$	0	0
$829$	2.83065	0.0983125	0.0491562	−	0.998791i	$-0.484347\pi$
$829$	2.83065	0.0983125	0.0491562	+	0.998791i	$0.484347\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.08663i	0.0376496i
$834$	0	0
$835$	− 3.13042i	− 0.108333i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.47816	0.292699	0.146349	−	0.989233i	$-0.453248\pi$
$839$	8.47816	0.292699	0.146349	+	0.989233i	$0.453248\pi$
$840$	0	0
$841$	11.1304	0.383807
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 7.47122i	− 0.257018i
$846$	0	0
$847$	22.2058i	0.763000i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.17950	0.177551
$852$	0	0
$853$	−0.941831	−0.0322477	−0.0161238	−	0.999870i	$-0.505133\pi$
$853$	−0.941831	−0.0322477	−0.0161238	+	0.999870i	$0.505133\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 22.7274i	− 0.776354i	−0.921585	−	0.388177i	$-0.873105\pi$
$857$	− 22.7274i	− 0.776354i	0.921585	−	0.388177i	$-0.126895\pi$
$858$	0	0
$859$	44.8717i	1.53100i	0.643435	+	0.765501i	$0.277509\pi$
$859$	44.8717i	1.53100i	−0.643435	+	0.765501i	$0.722491\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−31.4814	−1.07164	−0.535819	−	0.844333i	$-0.679997\pi$
$863$	−31.4814	−1.07164	−0.535819	+	0.844333i	$0.679997\pi$
$864$	0	0
$865$	0.945878	0.0321608
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 9.93219i	− 0.336926i
$870$	0	0
$871$	10.6190i	0.359813i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	13.0130	0.439919
$876$	0	0
$877$	0.516099	0.0174274	0.00871370	−	0.999962i	$-0.497226\pi$
$877$	0.516099	0.0174274	0.00871370	+	0.999962i	$0.497226\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	38.8512i	1.30893i	0.756092	+	0.654465i	$0.227106\pi$
$881$	38.8512i	1.30893i	−0.756092	+	0.654465i	$0.772894\pi$
$882$	0	0
$883$	− 27.0921i	− 0.911722i	−0.890051	−	0.455861i	$-0.849331\pi$
$883$	− 27.0921i	− 0.911722i	0.890051	−	0.455861i	$-0.150669\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.0649	1.41240	0.706200	−	0.708012i	$-0.250408\pi$
$887$	42.0649	1.41240	0.706200	+	0.708012i	$0.250408\pi$
$888$	0	0
$889$	−1.50091	−0.0503388
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0.0323047i	0.00108104i
$894$	0	0
$895$	− 12.9039i	− 0.431330i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.4805	−0.349546
$900$	0	0
$901$	3.87675	0.129153
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 12.5897i	− 0.418495i
$906$	0	0
$907$	55.9874i	1.85903i	0.368783	+	0.929516i	$0.379775\pi$
$907$	55.9874i	1.85903i	−0.368783	+	0.929516i	$0.620225\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	32.1509	1.06521	0.532604	−	0.846364i	$-0.321213\pi$
$911$	32.1509	1.06521	0.532604	+	0.846364i	$0.321213\pi$
$912$	0	0
$913$	13.5962	0.449969
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1.08649i	− 0.0358789i
$918$	0	0
$919$	− 42.9429i	− 1.41656i	−0.705933	−	0.708278i	$-0.749472\pi$
$919$	− 42.9429i	− 1.41656i	0.705933	−	0.708278i	$-0.250528\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.131978	−0.00434412
$924$	0	0
$925$	−4.61844	−0.151853
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.40658i	0.144575i	0.997384	+	0.0722876i	$0.0230299\pi$
$929$	4.40658i	0.144575i	−0.997384	+	0.0722876i	$0.976970\pi$
$930$	0	0
$931$	5.26662i	0.172607i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.282793	0.00924833
$936$	0	0
$937$	12.2331	0.399638	0.199819	−	0.979833i	$-0.435965\pi$
$937$	12.2331	0.399638	0.199819	+	0.979833i	$0.435965\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7.57422i	0.246913i	0.992350	+	0.123456i	$0.0393979\pi$
$941$	7.57422i	0.246913i	−0.992350	+	0.123456i	$0.960602\pi$
$942$	0	0
$943$	36.5863i	1.19142i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.4734	1.21772	0.608860	−	0.793278i	$-0.291627\pi$
$947$	37.4734	1.21772	0.608860	+	0.793278i	$0.291627\pi$
$948$	0	0
$949$	6.39478	0.207583
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 39.0563i	− 1.26516i	−0.774496	−	0.632579i	$-0.781996\pi$
$953$	− 39.0563i	− 1.26516i	0.774496	−	0.632579i	$-0.218004\pi$
$954$	0	0
$955$	2.72737i	0.0882556i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13.0964	−0.422905
$960$	0	0
$961$	24.8531	0.801714
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 5.92221i	− 0.190643i
$966$	0	0
$967$	4.01357i	0.129068i	0.997916	+	0.0645339i	$0.0205561\pi$
$967$	4.01357i	0.129068i	−0.997916	+	0.0645339i	$0.979444\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42.9072	−1.37696	−0.688478	−	0.725257i	$-0.741721\pi$
$971$	−42.9072	−1.37696	−0.688478	+	0.725257i	$0.741721\pi$
$972$	0	0
$973$	−7.92482	−0.254058
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 44.6971i	− 1.42999i	−0.699131	−	0.714994i	$-0.746430\pi$
$977$	− 44.6971i	− 1.42999i	0.699131	−	0.714994i	$-0.253570\pi$
$978$	0	0
$979$	− 12.8458i	− 0.410554i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−22.5277	−0.718521	−0.359261	−	0.933237i	$-0.616971\pi$
$983$	−22.5277	−0.718521	−0.359261	+	0.933237i	$0.616971\pi$
$984$	0	0
$985$	1.26829	0.0404112
$986$	0	0
$987$	0	0
$988$	0	0
$989$	46.6739i	1.48414i
$990$	0	0
$991$	− 14.8043i	− 0.470274i	−0.971962	−	0.235137i	$-0.924446\pi$
$991$	− 14.8043i	− 0.470274i	0.971962	−	0.235137i	$-0.0755539\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.67863	0.0532162
$996$	0	0
$997$	−30.6958	−0.972145	−0.486072	−	0.873919i	$-0.661571\pi$
$997$	−30.6958	−0.972145	−0.486072	+	0.873919i	$0.661571\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5328.2.e.f.2591.13	yes	24
3.2	odd	2	inner	5328.2.e.f.2591.11	✓	24
4.3	odd	2	inner	5328.2.e.f.2591.14	yes	24
12.11	even	2	inner	5328.2.e.f.2591.12	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5328.2.e.f.2591.11	✓	24	3.2	odd	2	inner
5328.2.e.f.2591.12	yes	24	12.11	even	2	inner
5328.2.e.f.2591.13	yes	24	1.1	even	1	trivial
5328.2.e.f.2591.14	yes	24	4.3	odd	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)

\(f(q)\)	\(=\)	\(q+0.617709i q^{5} -2.19022i q^{7} -0.928117 q^{11} -0.951287 q^{13} +0.493268i q^{17} +2.39074i q^{19} -5.17950 q^{23} +4.61844 q^{25} +4.22724i q^{29} +2.47929i q^{31} +1.35292 q^{35} -1.00000 q^{37} -7.06368i q^{41} -9.01128i q^{43} +0.0135125 q^{47} +2.20293 q^{49} -7.85933i q^{53} -0.573306i q^{55} -10.4192 q^{59} -5.10465 q^{61} -0.587619i q^{65} -11.1628i q^{67} +0.138737 q^{71} -6.72224 q^{73} +2.03278i q^{77} +10.7014i q^{79} -14.6492 q^{83} -0.304696 q^{85} +13.8407i q^{89} +2.08353i q^{91} -1.47678 q^{95} +9.48506 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 32 q^{25} - 24 q^{37} - 80 q^{49} - 48 q^{61} - 48 q^{73} - 40 q^{85} + 24 q^{97}+O(q^{100}) \) 24 * q - 32 * q^25 - 24 * q^37 - 80 * q^49 - 48 * q^61 - 48 * q^73 - 40 * q^85 + 24 * q^97

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