LMFDB - Embedded newform 5808.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5808,2,Mod(1,5808)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5808, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 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("5808.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$5808 = 2^{4} \cdot 3 \cdot 11^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5808.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-1,0,-2,0,-2,0,1,0,0,0,-4] 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:	yes
Analytic conductor:	$46.3771134940$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 2904)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5808.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

f(q)

=

q-1.00000 q^{3} -2.00000 q^{5} -2.00000 q^{7} +1.00000 q^{9} -4.00000 q^{13} +2.00000 q^{15} +2.00000 q^{17} +2.00000 q^{19} +2.00000 q^{21} -1.00000 q^{25} -1.00000 q^{27} +10.0000 q^{29} +4.00000 q^{35} +10.0000 q^{37} +4.00000 q^{39} +6.00000 q^{41} -10.0000 q^{43} -2.00000 q^{45} -4.00000 q^{47} -3.00000 q^{49} -2.00000 q^{51} +2.00000 q^{53} -2.00000 q^{57} +8.00000 q^{59} +8.00000 q^{61} -2.00000 q^{63} +8.00000 q^{65} +4.00000 q^{67} -12.0000 q^{71} -4.00000 q^{73} +1.00000 q^{75} +14.0000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -4.00000 q^{85} -10.0000 q^{87} +2.00000 q^{89} +8.00000 q^{91} -4.00000 q^{95} -14.0000 q^{97} +O(q^{100})

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	10.0000	1.85695	0.928477	−	0.371391i	$-0.121119\pi$
$29$	10.0000	1.85695	0.928477	+	0.371391i	$0.121119\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−2.00000	−0.280056
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.00000	−0.264906
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	8.00000	0.992278
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	0	0
$87$	−10.0000	−1.07211
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	8.00000	0.838628
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	16.0000	1.57653	0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000	1.57653	0.788263	+	0.615338i	$0.210980\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	−20.0000	−1.93347	−0.966736	−	0.255774i	$-0.917670\pi$
$107$	−20.0000	−1.93347	−0.966736	+	0.255774i	$0.917670\pi$
$108$	0	0
$109$	12.0000	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−4.00000	−0.369800
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−6.00000	−0.541002
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	0	0
$129$	10.0000	0.880451
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	2.00000	0.172133
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	10.0000	0.848189	0.424094	−	0.905618i	$-0.360592\pi$
$139$	10.0000	0.848189	0.424094	+	0.905618i	$0.360592\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−20.0000	−1.66091
$146$	0	0
$147$	3.00000	0.247436
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−2.00000	−0.162758	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	−2.00000	−0.162758	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	0	0
$157$	18.0000	1.43656	0.718278	−	0.695756i	$-0.244931\pi$
$157$	18.0000	1.43656	0.718278	+	0.695756i	$0.244931\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.00000	0.309529	0.154765	−	0.987951i	$-0.450538\pi$
$167$	4.00000	0.309529	0.154765	+	0.987951i	$0.450538\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	−8.00000	−0.601317
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−8.00000	−0.591377
$184$	0	0
$185$	−20.0000	−1.47043
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	0	0
$193$	16.0000	1.15171	0.575853	−	0.817554i	$-0.304670\pi$
$193$	16.0000	1.15171	0.575853	+	0.817554i	$0.304670\pi$
$194$	0	0
$195$	−8.00000	−0.572892
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	−20.0000	−1.40372
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	10.0000	0.688428	0.344214	−	0.938891i	$-0.388145\pi$
$211$	10.0000	0.688428	0.344214	+	0.938891i	$0.388145\pi$
$212$	0	0
$213$	12.0000	0.822226
$214$	0	0
$215$	20.0000	1.36399
$216$	0	0
$217$	0	0
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	−8.00000	−0.538138
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	−14.0000	−0.909398
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−4.00000	−0.257663	−0.128831	−	0.991667i	$-0.541123\pi$
$241$	−4.00000	−0.257663	−0.128831	+	0.991667i	$0.541123\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	12.0000	0.760469
$250$	0	0
$251$	−28.0000	−1.76734	−0.883672	−	0.468106i	$-0.844936\pi$
$251$	−28.0000	−1.76734	−0.883672	+	0.468106i	$0.844936\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	0	0
$259$	−20.0000	−1.24274
$260$	0	0
$261$	10.0000	0.618984
$262$	0	0
$263$	4.00000	0.246651	0.123325	−	0.992366i	$-0.460644\pi$
$263$	4.00000	0.246651	0.123325	+	0.992366i	$0.460644\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	−2.00000	−0.122398
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	0	0
$273$	−8.00000	−0.484182
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	30.0000	1.78331	0.891657	−	0.452711i	$-0.149543\pi$
$283$	30.0000	1.78331	0.891657	+	0.452711i	$0.149543\pi$
$284$	0	0
$285$	4.00000	0.236940
$286$	0	0
$287$	−12.0000	−0.708338
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	14.0000	0.820695
$292$	0	0
$293$	−2.00000	−0.116841	−0.0584206	−	0.998292i	$-0.518606\pi$
$293$	−2.00000	−0.116841	−0.0584206	+	0.998292i	$0.518606\pi$
$294$	0	0
$295$	−16.0000	−0.931556
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	20.0000	1.15278
$302$	0	0
$303$	−2.00000	−0.114897
$304$	0	0
$305$	−16.0000	−0.916157
$306$	0	0
$307$	18.0000	1.02731	0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000	1.02731	0.513657	+	0.857996i	$0.328290\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	0	0
$315$	4.00000	0.225374
$316$	0	0
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	20.0000	1.11629
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	4.00000	0.221880
$326$	0	0
$327$	−12.0000	−0.663602
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	0	0
$333$	10.0000	0.547997
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	4.00000	0.217894	0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000	0.217894	0.108947	+	0.994048i	$0.465252\pi$
$338$	0	0
$339$	10.0000	0.543125
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.0000	1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.0000	1.50312	0.751559	−	0.659665i	$-0.229302\pi$
$347$	28.0000	1.50312	0.751559	+	0.659665i	$0.229302\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	0	0
$351$	4.00000	0.213504
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	24.0000	1.27379
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	8.00000	0.418739
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	6.00000	0.312348
$370$	0	0
$371$	−4.00000	−0.207670
$372$	0	0
$373$	−32.0000	−1.65690	−0.828449	−	0.560065i	$-0.810776\pi$
$373$	−32.0000	−1.65690	−0.828449	+	0.560065i	$0.810776\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	0	0
$377$	−40.0000	−2.06010
$378$	0	0
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	18.0000	0.922168
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−10.0000	−0.508329
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	8.00000	0.403547
$394$	0	0
$395$	−28.0000	−1.40883
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	4.00000	0.200250
$400$	0	0
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	0	0
$408$	0	0
$409$	28.0000	1.38451	0.692255	−	0.721653i	$-0.256617\pi$
$409$	28.0000	1.38451	0.692255	+	0.721653i	$0.256617\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	0	0
$413$	−16.0000	−0.787309
$414$	0	0
$415$	24.0000	1.17811
$416$	0	0
$417$	−10.0000	−0.489702
$418$	0	0
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	−4.00000	−0.194487
$424$	0	0
$425$	−2.00000	−0.0970143
$426$	0	0
$427$	−16.0000	−0.774294
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	20.0000	0.958927
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−6.00000	−0.286364	−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000	−0.286364	−0.143182	+	0.989696i	$0.545733\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−32.0000	−1.52037	−0.760183	−	0.649709i	$-0.774891\pi$
$443$	−32.0000	−1.52037	−0.760183	+	0.649709i	$0.774891\pi$
$444$	0	0
$445$	−4.00000	−0.189618
$446$	0	0
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	2.00000	0.0939682
$454$	0	0
$455$	−16.0000	−0.750092
$456$	0	0
$457$	36.0000	1.68401	0.842004	−	0.539471i	$-0.181376\pi$
$457$	36.0000	1.68401	0.842004	+	0.539471i	$0.181376\pi$
$458$	0	0
$459$	−2.00000	−0.0933520
$460$	0	0
$461$	−34.0000	−1.58354	−0.791769	−	0.610821i	$-0.790840\pi$
$461$	−34.0000	−1.58354	−0.791769	+	0.610821i	$0.790840\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−2.00000	−0.0917663
$476$	0	0
$477$	2.00000	0.0915737
$478$	0	0
$479$	−4.00000	−0.182765	−0.0913823	−	0.995816i	$-0.529129\pi$
$479$	−4.00000	−0.182765	−0.0913823	+	0.995816i	$0.529129\pi$
$480$	0	0
$481$	−40.0000	−1.82384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28.0000	1.27141
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	−4.00000	−0.180517	−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000	−0.180517	−0.0902587	+	0.995918i	$0.528769\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.0000	1.07655
$498$	0	0
$499$	−12.0000	−0.537194	−0.268597	−	0.963253i	$-0.586560\pi$
$499$	−12.0000	−0.537194	−0.268597	+	0.963253i	$0.586560\pi$
$500$	0	0
$501$	−4.00000	−0.178707
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	−4.00000	−0.177998
$506$	0	0
$507$	−3.00000	−0.133235
$508$	0	0
$509$	10.0000	0.443242	0.221621	−	0.975133i	$-0.428865\pi$
$509$	10.0000	0.443242	0.221621	+	0.975133i	$0.428865\pi$
$510$	0	0
$511$	8.00000	0.353899
$512$	0	0
$513$	−2.00000	−0.0883022
$514$	0	0
$515$	−32.0000	−1.41009
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−18.0000	−0.790112
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	−26.0000	−1.13690	−0.568450	−	0.822718i	$-0.692457\pi$
$523$	−26.0000	−1.13690	−0.568450	+	0.822718i	$0.692457\pi$
$524$	0	0
$525$	−2.00000	−0.0872872
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	8.00000	0.347170
$532$	0	0
$533$	−24.0000	−1.03956
$534$	0	0
$535$	40.0000	1.72935
$536$	0	0
$537$	4.00000	0.172613
$538$	0	0
$539$	0	0
$540$	0	0
$541$	28.0000	1.20381	0.601907	−	0.798566i	$-0.294408\pi$
$541$	28.0000	1.20381	0.601907	+	0.798566i	$0.294408\pi$
$542$	0	0
$543$	2.00000	0.0858282
$544$	0	0
$545$	−24.0000	−1.02805
$546$	0	0
$547$	−2.00000	−0.0855138	−0.0427569	−	0.999086i	$-0.513614\pi$
$547$	−2.00000	−0.0855138	−0.0427569	+	0.999086i	$0.513614\pi$
$548$	0	0
$549$	8.00000	0.341432
$550$	0	0
$551$	20.0000	0.852029
$552$	0	0
$553$	−28.0000	−1.19068
$554$	0	0
$555$	20.0000	0.848953
$556$	0	0
$557$	−10.0000	−0.423714	−0.211857	−	0.977301i	$-0.567951\pi$
$557$	−10.0000	−0.423714	−0.211857	+	0.977301i	$0.567951\pi$
$558$	0	0
$559$	40.0000	1.69182
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	0	0
$565$	20.0000	0.841406
$566$	0	0
$567$	−2.00000	−0.0839921
$568$	0	0
$569$	18.0000	0.754599	0.377300	−	0.926091i	$-0.376853\pi$
$569$	18.0000	0.754599	0.377300	+	0.926091i	$0.376853\pi$
$570$	0	0
$571$	38.0000	1.59025	0.795125	−	0.606445i	$-0.207405\pi$
$571$	38.0000	1.59025	0.795125	+	0.606445i	$0.207405\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	0	0
$575$	0	0
$576$	0	0
$577$	14.0000	0.582828	0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000	0.582828	0.291414	+	0.956597i	$0.405874\pi$
$578$	0	0
$579$	−16.0000	−0.664937
$580$	0	0
$581$	24.0000	0.995688
$582$	0	0
$583$	0	0
$584$	0	0
$585$	8.00000	0.330759
$586$	0	0
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−10.0000	−0.411345
$592$	0	0
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	8.00000	0.327968
$596$	0	0
$597$	8.00000	0.327418
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−26.0000	−1.05531	−0.527654	−	0.849460i	$-0.676928\pi$
$607$	−26.0000	−1.05531	−0.527654	+	0.849460i	$0.676928\pi$
$608$	0	0
$609$	20.0000	0.810441
$610$	0	0
$611$	16.0000	0.647291
$612$	0	0
$613$	−40.0000	−1.61558	−0.807792	−	0.589467i	$-0.799338\pi$
$613$	−40.0000	−1.61558	−0.807792	+	0.589467i	$0.799338\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.00000	−0.160257
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.0000	0.797452
$630$	0	0
$631$	32.0000	1.27390	0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000	1.27390	0.636950	+	0.770905i	$0.280196\pi$
$632$	0	0
$633$	−10.0000	−0.397464
$634$	0	0
$635$	36.0000	1.42862
$636$	0	0
$637$	12.0000	0.475457
$638$	0	0
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−22.0000	−0.868948	−0.434474	−	0.900684i	$-0.643066\pi$
$641$	−22.0000	−0.868948	−0.434474	+	0.900684i	$0.643066\pi$
$642$	0	0
$643$	36.0000	1.41970	0.709851	−	0.704352i	$-0.248762\pi$
$643$	36.0000	1.41970	0.709851	+	0.704352i	$0.248762\pi$
$644$	0	0
$645$	−20.0000	−0.787499
$646$	0	0
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	40.0000	1.55818	0.779089	−	0.626913i	$-0.215682\pi$
$659$	40.0000	1.55818	0.779089	+	0.626913i	$0.215682\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	0	0
$663$	8.00000	0.310694
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	0	0
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−32.0000	−1.23351	−0.616755	−	0.787155i	$-0.711553\pi$
$673$	−32.0000	−1.23351	−0.616755	+	0.787155i	$0.711553\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	28.0000	1.07454
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	36.0000	1.37549
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	−8.00000	−0.304776
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.0000	−0.758643
$696$	0	0
$697$	12.0000	0.454532
$698$	0	0
$699$	10.0000	0.378235
$700$	0	0
$701$	14.0000	0.528773	0.264386	−	0.964417i	$-0.414831\pi$
$701$	14.0000	0.528773	0.264386	+	0.964417i	$0.414831\pi$
$702$	0	0
$703$	20.0000	0.754314
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	−4.00000	−0.150435
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.0000	0.448148
$718$	0	0
$719$	−44.0000	−1.64092	−0.820462	−	0.571702i	$-0.806283\pi$
$719$	−44.0000	−1.64092	−0.820462	+	0.571702i	$0.806283\pi$
$720$	0	0
$721$	−32.0000	−1.19174
$722$	0	0
$723$	4.00000	0.148762
$724$	0	0
$725$	−10.0000	−0.371391
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−20.0000	−0.739727
$732$	0	0
$733$	−12.0000	−0.443230	−0.221615	−	0.975134i	$-0.571133\pi$
$733$	−12.0000	−0.443230	−0.221615	+	0.975134i	$0.571133\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	0	0
$738$	0	0
$739$	6.00000	0.220714	0.110357	−	0.993892i	$-0.464801\pi$
$739$	6.00000	0.220714	0.110357	+	0.993892i	$0.464801\pi$
$740$	0	0
$741$	8.00000	0.293887
$742$	0	0
$743$	−16.0000	−0.586983	−0.293492	−	0.955962i	$-0.594817\pi$
$743$	−16.0000	−0.586983	−0.293492	+	0.955962i	$0.594817\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	0	0
$747$	−12.0000	−0.439057
$748$	0	0
$749$	40.0000	1.46157
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	0	0
$753$	28.0000	1.02038
$754$	0	0
$755$	4.00000	0.145575
$756$	0	0
$757$	34.0000	1.23575	0.617876	−	0.786276i	$-0.287994\pi$
$757$	34.0000	1.23575	0.617876	+	0.786276i	$0.287994\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	0	0
$765$	−4.00000	−0.144620
$766$	0	0
$767$	−32.0000	−1.15545
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	0	0
$773$	10.0000	0.359675	0.179838	−	0.983696i	$-0.442443\pi$
$773$	10.0000	0.359675	0.179838	+	0.983696i	$0.442443\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20.0000	0.717496
$778$	0	0
$779$	12.0000	0.429945
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−10.0000	−0.357371
$784$	0	0
$785$	−36.0000	−1.28490
$786$	0	0
$787$	38.0000	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$788$	0	0
$789$	−4.00000	−0.142404
$790$	0	0
$791$	20.0000	0.711118
$792$	0	0
$793$	−32.0000	−1.13635
$794$	0	0
$795$	4.00000	0.141865
$796$	0	0
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	−8.00000	−0.283020
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.0000	1.05605
$808$	0	0
$809$	−46.0000	−1.61727	−0.808637	−	0.588308i	$-0.799794\pi$
$809$	−46.0000	−1.61727	−0.808637	+	0.588308i	$0.799794\pi$
$810$	0	0
$811$	−6.00000	−0.210688	−0.105344	−	0.994436i	$-0.533594\pi$
$811$	−6.00000	−0.210688	−0.105344	+	0.994436i	$0.533594\pi$
$812$	0	0
$813$	−2.00000	−0.0701431
$814$	0	0
$815$	40.0000	1.40114
$816$	0	0
$817$	−20.0000	−0.699711
$818$	0	0
$819$	8.00000	0.279543
$820$	0	0
$821$	42.0000	1.46581	0.732905	−	0.680331i	$-0.238164\pi$
$821$	42.0000	1.46581	0.732905	+	0.680331i	$0.238164\pi$
$822$	0	0
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.0000	0.973655	0.486828	−	0.873498i	$-0.338154\pi$
$827$	28.0000	0.973655	0.486828	+	0.873498i	$0.338154\pi$
$828$	0	0
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	0	0
$831$	28.0000	0.971309
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	71.0000	2.44828
$842$	0	0
$843$	18.0000	0.619953
$844$	0	0
$845$	−6.00000	−0.206406
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−30.0000	−1.02960
$850$	0	0
$851$	0	0
$852$	0	0
$853$	8.00000	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$853$	8.00000	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	−38.0000	−1.29806	−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000	−1.29806	−0.649028	+	0.760765i	$0.724824\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	0	0
$863$	−16.0000	−0.544646	−0.272323	−	0.962206i	$-0.587792\pi$
$863$	−16.0000	−0.544646	−0.272323	+	0.962206i	$0.587792\pi$
$864$	0	0
$865$	−36.0000	−1.22404
$866$	0	0
$867$	13.0000	0.441503
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	−14.0000	−0.473828
$874$	0	0
$875$	−24.0000	−0.811348
$876$	0	0
$877$	8.00000	0.270141	0.135070	−	0.990836i	$-0.456874\pi$
$877$	8.00000	0.270141	0.135070	+	0.990836i	$0.456874\pi$
$878$	0	0
$879$	2.00000	0.0674583
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	16.0000	0.537834
$886$	0	0
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	36.0000	1.20740
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	8.00000	0.267411
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	−20.0000	−0.665558
$904$	0	0
$905$	4.00000	0.132964
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	0	0
$909$	2.00000	0.0663358
$910$	0	0
$911$	60.0000	1.98789	0.993944	−	0.109885i	$-0.0350482\pi$
$911$	60.0000	1.98789	0.993944	+	0.109885i	$0.0350482\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	16.0000	0.528944
$916$	0	0
$917$	16.0000	0.528367
$918$	0	0
$919$	−30.0000	−0.989609	−0.494804	−	0.869004i	$-0.664760\pi$
$919$	−30.0000	−0.989609	−0.494804	+	0.869004i	$0.664760\pi$
$920$	0	0
$921$	−18.0000	−0.593120
$922$	0	0
$923$	48.0000	1.57994
$924$	0	0
$925$	−10.0000	−0.328798
$926$	0	0
$927$	16.0000	0.525509
$928$	0	0
$929$	38.0000	1.24674	0.623370	−	0.781927i	$-0.285763\pi$
$929$	38.0000	1.24674	0.623370	+	0.781927i	$0.285763\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−8.00000	−0.261349	−0.130674	−	0.991425i	$-0.541714\pi$
$937$	−8.00000	−0.261349	−0.130674	+	0.991425i	$0.541714\pi$
$938$	0	0
$939$	−14.0000	−0.456873
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	0	0
$949$	16.0000	0.519382
$950$	0	0
$951$	10.0000	0.324272
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	0	0
$955$	24.0000	0.776622
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−20.0000	−0.644491
$964$	0	0
$965$	−32.0000	−1.03012
$966$	0	0
$967$	−2.00000	−0.0643157	−0.0321578	−	0.999483i	$-0.510238\pi$
$967$	−2.00000	−0.0643157	−0.0321578	+	0.999483i	$0.510238\pi$
$968$	0	0
$969$	−4.00000	−0.128499
$970$	0	0
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	0	0
$973$	−20.0000	−0.641171
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−2.00000	−0.0639857	−0.0319928	−	0.999488i	$-0.510185\pi$
$977$	−2.00000	−0.0639857	−0.0319928	+	0.999488i	$0.510185\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	12.0000	0.383131
$982$	0	0
$983$	−40.0000	−1.27580	−0.637901	−	0.770118i	$-0.720197\pi$
$983$	−40.0000	−1.27580	−0.637901	+	0.770118i	$0.720197\pi$
$984$	0	0
$985$	−20.0000	−0.637253
$986$	0	0
$987$	−8.00000	−0.254643
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	0	0
$995$	16.0000	0.507234
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	−10.0000	−0.316386

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5808.2.a.d.1.1		1
4.3	odd	2		2904.2.a.k.1.1	yes	1
11.10	odd	2		5808.2.a.e.1.1		1
12.11	even	2		8712.2.a.v.1.1		1
44.43	even	2		2904.2.a.j.1.1	✓	1
132.131	odd	2		8712.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2904.2.a.j.1.1	✓	1	44.43	even	2
2904.2.a.k.1.1	yes	1	4.3	odd	2
5808.2.a.d.1.1		1	1.1	even	1	trivial
5808.2.a.e.1.1		1	11.10	odd	2
8712.2.a.t.1.1		1	132.131	odd	2
8712.2.a.v.1.1		1	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	5808.2.a.d.1.1

Level	$5808$
Weight	$2$
Character	5808.1
Self dual	yes
Analytic conductor	$46.377$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$