LMFDB - Embedded newform 4950.2.a.w.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4950,2,Mod(1,4950)] 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("4950.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4950, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4950.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,1,0,0,-4,1,0,0,-1,0,0,-4,0,1,-2,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.5259490005$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 990)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4950.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -4.00000 q^{7} +1.00000 q^{8} -1.00000 q^{11} -4.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} +8.00000 q^{19} -1.00000 q^{22} +4.00000 q^{23} -4.00000 q^{28} -8.00000 q^{31} +1.00000 q^{32} -2.00000 q^{34} -4.00000 q^{37} +8.00000 q^{38} +4.00000 q^{41} -8.00000 q^{43} -1.00000 q^{44} +4.00000 q^{46} -12.0000 q^{47} +9.00000 q^{49} +2.00000 q^{53} -4.00000 q^{56} -12.0000 q^{59} +2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +8.00000 q^{67} -2.00000 q^{68} +8.00000 q^{71} -4.00000 q^{73} -4.00000 q^{74} +8.00000 q^{76} +4.00000 q^{77} -4.00000 q^{79} +4.00000 q^{82} -12.0000 q^{83} -8.00000 q^{86} -1.00000 q^{88} -16.0000 q^{89} +4.00000 q^{92} -12.0000 q^{94} -16.0000 q^{97} +9.00000 q^{98} +O(q^{100})$

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−4.00000	−1.06904
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	−4.00000	−0.755929
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	8.00000	1.29777
$39$	0	0
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	4.00000	0.589768
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	0	0
$51$	0	0
$52$	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
$56$	−4.00000	−0.534522
$57$	0	0
$58$	0	0
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	8.00000	0.917663
$77$	4.00000	0.455842
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	4.00000	0.441726
$83$	−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
$85$	0	0
$86$	−8.00000	−0.862662
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	0	0
$92$	4.00000	0.417029
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	0	0
$106$	2.00000	0.194257
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	0	0
$112$	−4.00000	−0.377964
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	−12.0000	−1.10469
$119$	8.00000	0.733359
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.00000	0.181071
$123$	0	0
$124$	−8.00000	−0.718421
$125$	0	0
$126$	0	0
$127$	−20.0000	−1.77471	−0.887357	−	0.461084i	$-0.847461\pi$
$127$	−20.0000	−1.77471	−0.887357	+	0.461084i	$0.847461\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−32.0000	−2.77475
$134$	8.00000	0.691095
$135$	0	0
$136$	−2.00000	−0.171499
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	8.00000	0.671345
$143$	0	0
$144$	0	0
$145$	0	0
$146$	−4.00000	−0.331042
$147$	0	0
$148$	−4.00000	−0.328798
$149$	16.0000	1.31077	0.655386	−	0.755295i	$-0.272506\pi$
$149$	16.0000	1.31077	0.655386	+	0.755295i	$0.272506\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	8.00000	0.648886
$153$	0	0
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	0	0
$161$	−16.0000	−1.26098
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	4.00000	0.312348
$165$	0	0
$166$	−12.0000	−0.931381
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−16.0000	−1.19925
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	0	0
$184$	4.00000	0.294884
$185$	0	0
$186$	0	0
$187$	2.00000	0.146254
$188$	−12.0000	−0.875190
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	9.00000	0.642857
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	4.00000	0.278693
$207$	0	0
$208$	0	0
$209$	−8.00000	−0.553372
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	2.00000	0.137361
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	32.0000	2.17230
$218$	−14.0000	−0.948200
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	20.0000	1.33930	0.669650	−	0.742677i	$-0.266444\pi$
$223$	20.0000	1.33930	0.669650	+	0.742677i	$0.266444\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	−6.00000	−0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	−12.0000	−0.781133
$237$	0	0
$238$	8.00000	0.518563
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−30.0000	−1.93247	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−30.0000	−1.93247	−0.966235	+	0.257663i	$0.917048\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	2.00000	0.128037
$245$	0	0
$246$	0	0
$247$	0	0
$248$	−8.00000	−0.508001
$249$	0	0
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−20.0000	−1.25491
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.0000	1.62184	0.810918	−	0.585160i	$-0.198968\pi$
$257$	26.0000	1.62184	0.810918	+	0.585160i	$0.198968\pi$
$258$	0	0
$259$	16.0000	0.994192
$260$	0	0
$261$	0	0
$262$	−4.00000	−0.247121
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	0	0
$266$	−32.0000	−1.96205
$267$	0	0
$268$	8.00000	0.488678
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	18.0000	1.08742
$275$	0	0
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	28.0000	1.67034	0.835170	−	0.549992i	$-0.185369\pi$
$281$	28.0000	1.67034	0.835170	+	0.549992i	$0.185369\pi$
$282$	0	0
$283$	−32.0000	−1.90220	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−32.0000	−1.90220	−0.951101	+	0.308879i	$0.900046\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	0	0
$287$	−16.0000	−0.944450
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	0	0
$292$	−4.00000	−0.234082
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	0	0
$295$	0	0
$296$	−4.00000	−0.232495
$297$	0	0
$298$	16.0000	0.926855
$299$	0	0
$300$	0	0
$301$	32.0000	1.84445
$302$	−12.0000	−0.690522
$303$	0	0
$304$	8.00000	0.458831
$305$	0	0
$306$	0	0
$307$	−8.00000	−0.456584	−0.228292	−	0.973593i	$-0.573314\pi$
$307$	−8.00000	−0.456584	−0.228292	+	0.973593i	$0.573314\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	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
$313$	−24.0000	−1.35656	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	−24.0000	−1.35656	−0.678280	+	0.734803i	$0.737274\pi$
$314$	−4.00000	−0.225733
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	−16.0000	−0.891645
$323$	−16.0000	−0.890264
$324$	0	0
$325$	0	0
$326$	−8.00000	−0.443079
$327$	0	0
$328$	4.00000	0.220863
$329$	48.0000	2.64633
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−12.0000	−0.658586
$333$	0	0
$334$	8.00000	0.437741
$335$	0	0
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	−13.0000	−0.707107
$339$	0	0
$340$	0	0
$341$	8.00000	0.433224
$342$	0	0
$343$	−8.00000	−0.431959
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−2.00000	−0.107521
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	0	0
$355$	0	0
$356$	−16.0000	−0.847998
$357$	0	0
$358$	−12.0000	−0.634220
$359$	−16.0000	−0.844448	−0.422224	−	0.906492i	$-0.638750\pi$
$359$	−16.0000	−0.844448	−0.422224	+	0.906492i	$0.638750\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	10.0000	0.525588
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	−8.00000	−0.414224	−0.207112	−	0.978317i	$-0.566407\pi$
$373$	−8.00000	−0.414224	−0.207112	+	0.978317i	$0.566407\pi$
$374$	2.00000	0.103418
$375$	0	0
$376$	−12.0000	−0.618853
$377$	0	0
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	24.0000	1.22795
$383$	−20.0000	−1.02195	−0.510976	−	0.859595i	$-0.670716\pi$
$383$	−20.0000	−1.02195	−0.510976	+	0.859595i	$0.670716\pi$
$384$	0	0
$385$	0	0
$386$	−4.00000	−0.203595
$387$	0	0
$388$	−16.0000	−0.812277
$389$	−4.00000	−0.202808	−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000	−0.202808	−0.101404	+	0.994845i	$0.532333\pi$
$390$	0	0
$391$	−8.00000	−0.404577
$392$	9.00000	0.454569
$393$	0	0
$394$	6.00000	0.302276
$395$	0	0
$396$	0	0
$397$	20.0000	1.00377	0.501886	−	0.864934i	$-0.332640\pi$
$397$	20.0000	1.00377	0.501886	+	0.864934i	$0.332640\pi$
$398$	8.00000	0.401004
$399$	0	0
$400$	0	0
$401$	32.0000	1.59800	0.799002	−	0.601329i	$-0.205362\pi$
$401$	32.0000	1.59800	0.799002	+	0.601329i	$0.205362\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.00000	0.198273
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	0	0
$412$	4.00000	0.197066
$413$	48.0000	2.36193
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	−8.00000	−0.391293
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	−8.00000	−0.389434
$423$	0	0
$424$	2.00000	0.0971286
$425$	0	0
$426$	0	0
$427$	−8.00000	−0.387147
$428$	−12.0000	−0.580042
$429$	0	0
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	16.0000	0.768911	0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000	0.768911	0.384455	+	0.923144i	$0.374389\pi$
$434$	32.0000	1.53605
$435$	0	0
$436$	−14.0000	−0.670478
$437$	32.0000	1.53077
$438$	0	0
$439$	−20.0000	−0.954548	−0.477274	−	0.878755i	$-0.658375\pi$
$439$	−20.0000	−0.954548	−0.477274	+	0.878755i	$0.658375\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	0	0
$445$	0	0
$446$	20.0000	0.947027
$447$	0	0
$448$	−4.00000	−0.188982
$449$	−8.00000	−0.377543	−0.188772	−	0.982021i	$-0.560451\pi$
$449$	−8.00000	−0.377543	−0.188772	+	0.982021i	$0.560451\pi$
$450$	0	0
$451$	−4.00000	−0.188353
$452$	−6.00000	−0.282216
$453$	0	0
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	12.0000	0.561336	0.280668	−	0.959805i	$-0.409444\pi$
$457$	12.0000	0.561336	0.280668	+	0.959805i	$0.409444\pi$
$458$	6.00000	0.280362
$459$	0	0
$460$	0	0
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	0	0
$463$	−36.0000	−1.67306	−0.836531	−	0.547920i	$-0.815420\pi$
$463$	−36.0000	−1.67306	−0.836531	+	0.547920i	$0.815420\pi$
$464$	0	0
$465$	0	0
$466$	6.00000	0.277945
$467$	−32.0000	−1.48078	−0.740392	−	0.672176i	$-0.765360\pi$
$467$	−32.0000	−1.48078	−0.740392	+	0.672176i	$0.765360\pi$
$468$	0	0
$469$	−32.0000	−1.47762
$470$	0	0
$471$	0	0
$472$	−12.0000	−0.552345
$473$	8.00000	0.367840
$474$	0	0
$475$	0	0
$476$	8.00000	0.366679
$477$	0	0
$478$	24.0000	1.09773
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	0	0
$482$	−30.0000	−1.36646
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	0	0
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	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
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−8.00000	−0.359211
$497$	−32.0000	−1.43540
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	−4.00000	−0.178529
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	0	0
$506$	−4.00000	−0.177822
$507$	0	0
$508$	−20.0000	−0.887357
$509$	36.0000	1.59567	0.797836	−	0.602875i	$-0.205978\pi$
$509$	36.0000	1.59567	0.797836	+	0.602875i	$0.205978\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	1.00000	0.0441942
$513$	0	0
$514$	26.0000	1.14681
$515$	0	0
$516$	0	0
$517$	12.0000	0.527759
$518$	16.0000	0.703000
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	32.0000	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	8.00000	0.348817
$527$	16.0000	0.696971
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	−32.0000	−1.38738
$533$	0	0
$534$	0	0
$535$	0	0
$536$	8.00000	0.345547
$537$	0	0
$538$	12.0000	0.517357
$539$	−9.00000	−0.387657
$540$	0	0
$541$	18.0000	0.773880	0.386940	−	0.922105i	$-0.373532\pi$
$541$	18.0000	0.773880	0.386940	+	0.922105i	$0.373532\pi$
$542$	20.0000	0.859074
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	18.0000	0.768922
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	16.0000	0.680389
$554$	8.00000	0.339887
$555$	0	0
$556$	0	0
$557$	−14.0000	−0.593199	−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000	−0.593199	−0.296600	+	0.955002i	$0.595853\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	28.0000	1.18111
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	0	0
$566$	−32.0000	−1.34506
$567$	0	0
$568$	8.00000	0.335673
$569$	36.0000	1.50920	0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000	1.50920	0.754599	+	0.656186i	$0.227831\pi$
$570$	0	0
$571$	8.00000	0.334790	0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000	0.334790	0.167395	+	0.985890i	$0.446465\pi$
$572$	0	0
$573$	0	0
$574$	−16.0000	−0.667827
$575$	0	0
$576$	0	0
$577$	−8.00000	−0.333044	−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000	−0.333044	−0.166522	+	0.986038i	$0.553254\pi$
$578$	−13.0000	−0.540729
$579$	0	0
$580$	0	0
$581$	48.0000	1.99138
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	−4.00000	−0.165521
$585$	0	0
$586$	22.0000	0.908812
$587$	−24.0000	−0.990586	−0.495293	−	0.868726i	$-0.664939\pi$
$587$	−24.0000	−0.990586	−0.495293	+	0.868726i	$0.664939\pi$
$588$	0	0
$589$	−64.0000	−2.63707
$590$	0	0
$591$	0	0
$592$	−4.00000	−0.164399
$593$	34.0000	1.39621	0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000	1.39621	0.698106	+	0.715994i	$0.254026\pi$
$594$	0	0
$595$	0	0
$596$	16.0000	0.655386
$597$	0	0
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	−26.0000	−1.06056	−0.530281	−	0.847822i	$-0.677914\pi$
$601$	−26.0000	−1.06056	−0.530281	+	0.847822i	$0.677914\pi$
$602$	32.0000	1.30422
$603$	0	0
$604$	−12.0000	−0.488273
$605$	0	0
$606$	0	0
$607$	36.0000	1.46119	0.730597	−	0.682808i	$-0.239242\pi$
$607$	36.0000	1.46119	0.730597	+	0.682808i	$0.239242\pi$
$608$	8.00000	0.324443
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−32.0000	−1.29247	−0.646234	−	0.763139i	$-0.723657\pi$
$613$	−32.0000	−1.29247	−0.646234	+	0.763139i	$0.723657\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	4.00000	0.161165
$617$	−2.00000	−0.0805170	−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000	−0.0805170	−0.0402585	+	0.999189i	$0.512818\pi$
$618$	0	0
$619$	12.0000	0.482321	0.241160	−	0.970485i	$-0.422472\pi$
$619$	12.0000	0.482321	0.241160	+	0.970485i	$0.422472\pi$
$620$	0	0
$621$	0	0
$622$	−8.00000	−0.320771
$623$	64.0000	2.56411
$624$	0	0
$625$	0	0
$626$	−24.0000	−0.959233
$627$	0	0
$628$	−4.00000	−0.159617
$629$	8.00000	0.318981
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	−22.0000	−0.873732
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8.00000	−0.315981	−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000	−0.315981	−0.157991	+	0.987441i	$0.550502\pi$
$642$	0	0
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	−16.0000	−0.629512
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−38.0000	−1.48705	−0.743527	−	0.668705i	$-0.766849\pi$
$653$	−38.0000	−1.48705	−0.743527	+	0.668705i	$0.766849\pi$
$654$	0	0
$655$	0	0
$656$	4.00000	0.156174
$657$	0	0
$658$	48.0000	1.87123
$659$	−20.0000	−0.779089	−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000	−0.779089	−0.389545	+	0.921008i	$0.627368\pi$
$660$	0	0
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	−12.0000	−0.465690
$665$	0	0
$666$	0	0
$667$	0	0
$668$	8.00000	0.309529
$669$	0	0
$670$	0	0
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	28.0000	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$673$	28.0000	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	−13.0000	−0.500000
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	64.0000	2.45609
$680$	0	0
$681$	0	0
$682$	8.00000	0.306336
$683$	−40.0000	−1.53056	−0.765279	−	0.643699i	$-0.777399\pi$
$683$	−40.0000	−1.53056	−0.765279	+	0.643699i	$0.777399\pi$
$684$	0	0
$685$	0	0
$686$	−8.00000	−0.305441
$687$	0	0
$688$	−8.00000	−0.304997
$689$	0	0
$690$	0	0
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−2.00000	−0.0760286
$693$	0	0
$694$	−4.00000	−0.151838
$695$	0	0
$696$	0	0
$697$	−8.00000	−0.303022
$698$	2.00000	0.0757011
$699$	0	0
$700$	0	0
$701$	−8.00000	−0.302156	−0.151078	−	0.988522i	$-0.548274\pi$
$701$	−8.00000	−0.302156	−0.151078	+	0.988522i	$0.548274\pi$
$702$	0	0
$703$	−32.0000	−1.20690
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−26.0000	−0.978523
$707$	0	0
$708$	0	0
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	0	0
$711$	0	0
$712$	−16.0000	−0.599625
$713$	−32.0000	−1.19841
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	−16.0000	−0.597115
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−16.0000	−0.595871
$722$	45.0000	1.67473
$723$	0	0
$724$	10.0000	0.371647
$725$	0	0
$726$	0	0
$727$	−36.0000	−1.33517	−0.667583	−	0.744535i	$-0.732671\pi$
$727$	−36.0000	−1.33517	−0.667583	+	0.744535i	$0.732671\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	48.0000	1.77292	0.886460	−	0.462805i	$-0.153157\pi$
$733$	48.0000	1.77292	0.886460	+	0.462805i	$0.153157\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	4.00000	0.147442
$737$	−8.00000	−0.294684
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	0	0
$742$	−8.00000	−0.293689
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	0	0
$746$	−8.00000	−0.292901
$747$	0	0
$748$	2.00000	0.0731272
$749$	48.0000	1.75388
$750$	0	0
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−20.0000	−0.726912	−0.363456	−	0.931611i	$-0.618403\pi$
$757$	−20.0000	−0.726912	−0.363456	+	0.931611i	$0.618403\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	0	0
$761$	20.0000	0.724999	0.362500	−	0.931984i	$-0.381923\pi$
$761$	20.0000	0.724999	0.362500	+	0.931984i	$0.381923\pi$
$762$	0	0
$763$	56.0000	2.02734
$764$	24.0000	0.868290
$765$	0	0
$766$	−20.0000	−0.722629
$767$	0	0
$768$	0	0
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	0	0
$772$	−4.00000	−0.143963
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	0	0
$775$	0	0
$776$	−16.0000	−0.574367
$777$	0	0
$778$	−4.00000	−0.143407
$779$	32.0000	1.14652
$780$	0	0
$781$	−8.00000	−0.286263
$782$	−8.00000	−0.286079
$783$	0	0
$784$	9.00000	0.321429
$785$	0	0
$786$	0	0
$787$	48.0000	1.71102	0.855508	−	0.517790i	$-0.173245\pi$
$787$	48.0000	1.71102	0.855508	+	0.517790i	$0.173245\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	0	0
$791$	24.0000	0.853342
$792$	0	0
$793$	0	0
$794$	20.0000	0.709773
$795$	0	0
$796$	8.00000	0.283552
$797$	−10.0000	−0.354218	−0.177109	−	0.984191i	$-0.556675\pi$
$797$	−10.0000	−0.354218	−0.177109	+	0.984191i	$0.556675\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	0	0
$801$	0	0
$802$	32.0000	1.12996
$803$	4.00000	0.141157
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	28.0000	0.984428	0.492214	−	0.870474i	$-0.336188\pi$
$809$	28.0000	0.984428	0.492214	+	0.870474i	$0.336188\pi$
$810$	0	0
$811$	8.00000	0.280918	0.140459	−	0.990086i	$-0.455142\pi$
$811$	8.00000	0.280918	0.140459	+	0.990086i	$0.455142\pi$
$812$	0	0
$813$	0	0
$814$	4.00000	0.140200
$815$	0	0
$816$	0	0
$817$	−64.0000	−2.23908
$818$	22.0000	0.769212
$819$	0	0
$820$	0	0
$821$	32.0000	1.11681	0.558404	−	0.829569i	$-0.311414\pi$
$821$	32.0000	1.11681	0.558404	+	0.829569i	$0.311414\pi$
$822$	0	0
$823$	28.0000	0.976019	0.488009	−	0.872838i	$-0.337723\pi$
$823$	28.0000	0.976019	0.488009	+	0.872838i	$0.337723\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	48.0000	1.67013
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−18.0000	−0.623663
$834$	0	0
$835$	0	0
$836$	−8.00000	−0.276686
$837$	0	0
$838$	−36.0000	−1.24360
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	6.00000	0.206774
$843$	0	0
$844$	−8.00000	−0.275371
$845$	0	0
$846$	0	0
$847$	−4.00000	−0.137442
$848$	2.00000	0.0686803
$849$	0	0
$850$	0	0
$851$	−16.0000	−0.548473
$852$	0	0
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−12.0000	−0.410152
$857$	22.0000	0.751506	0.375753	−	0.926720i	$-0.377384\pi$
$857$	22.0000	0.751506	0.375753	+	0.926720i	$0.377384\pi$
$858$	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
$861$	0	0
$862$	16.0000	0.544962
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	0	0
$866$	16.0000	0.543702
$867$	0	0
$868$	32.0000	1.08615
$869$	4.00000	0.135691
$870$	0	0
$871$	0	0
$872$	−14.0000	−0.474100
$873$	0	0
$874$	32.0000	1.08242
$875$	0	0
$876$	0	0
$877$	40.0000	1.35070	0.675352	−	0.737496i	$-0.263992\pi$
$877$	40.0000	1.35070	0.675352	+	0.737496i	$0.263992\pi$
$878$	−20.0000	−0.674967
$879$	0	0
$880$	0	0
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	0	0
$883$	40.0000	1.34611	0.673054	−	0.739594i	$-0.264982\pi$
$883$	40.0000	1.34611	0.673054	+	0.739594i	$0.264982\pi$
$884$	0	0
$885$	0	0
$886$	16.0000	0.537531
$887$	32.0000	1.07445	0.537227	−	0.843437i	$-0.319472\pi$
$887$	32.0000	1.07445	0.537227	+	0.843437i	$0.319472\pi$
$888$	0	0
$889$	80.0000	2.68311
$890$	0	0
$891$	0	0
$892$	20.0000	0.669650
$893$	−96.0000	−3.21252
$894$	0	0
$895$	0	0
$896$	−4.00000	−0.133631
$897$	0	0
$898$	−8.00000	−0.266963
$899$	0	0
$900$	0	0
$901$	−4.00000	−0.133259
$902$	−4.00000	−0.133185
$903$	0	0
$904$	−6.00000	−0.199557
$905$	0	0
$906$	0	0
$907$	−32.0000	−1.06254	−0.531271	−	0.847202i	$-0.678286\pi$
$907$	−32.0000	−1.06254	−0.531271	+	0.847202i	$0.678286\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	0	0
$911$	−48.0000	−1.59031	−0.795155	−	0.606406i	$-0.792611\pi$
$911$	−48.0000	−1.59031	−0.795155	+	0.606406i	$0.792611\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	12.0000	0.396925
$915$	0	0
$916$	6.00000	0.198246
$917$	16.0000	0.528367
$918$	0	0
$919$	4.00000	0.131948	0.0659739	−	0.997821i	$-0.478985\pi$
$919$	4.00000	0.131948	0.0659739	+	0.997821i	$0.478985\pi$
$920$	0	0
$921$	0	0
$922$	32.0000	1.05386
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−36.0000	−1.18303
$927$	0	0
$928$	0	0
$929$	−16.0000	−0.524943	−0.262471	−	0.964940i	$-0.584538\pi$
$929$	−16.0000	−0.524943	−0.262471	+	0.964940i	$0.584538\pi$
$930$	0	0
$931$	72.0000	2.35970
$932$	6.00000	0.196537
$933$	0	0
$934$	−32.0000	−1.04707
$935$	0	0
$936$	0	0
$937$	28.0000	0.914720	0.457360	−	0.889282i	$-0.348795\pi$
$937$	28.0000	0.914720	0.457360	+	0.889282i	$0.348795\pi$
$938$	−32.0000	−1.04484
$939$	0	0
$940$	0	0
$941$	−56.0000	−1.82555	−0.912774	−	0.408465i	$-0.866064\pi$
$941$	−56.0000	−1.82555	−0.912774	+	0.408465i	$0.866064\pi$
$942$	0	0
$943$	16.0000	0.521032
$944$	−12.0000	−0.390567
$945$	0	0
$946$	8.00000	0.260102
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	8.00000	0.259281
$953$	26.0000	0.842223	0.421111	−	0.907009i	$-0.361640\pi$
$953$	26.0000	0.842223	0.421111	+	0.907009i	$0.361640\pi$
$954$	0	0
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	24.0000	0.775405
$959$	−72.0000	−2.32500
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	0	0
$964$	−30.0000	−0.966235
$965$	0	0
$966$	0	0
$967$	28.0000	0.900419	0.450210	−	0.892923i	$-0.351349\pi$
$967$	28.0000	0.900419	0.450210	+	0.892923i	$0.351349\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	0	0
$971$	−4.00000	−0.128366	−0.0641831	−	0.997938i	$-0.520444\pi$
$971$	−4.00000	−0.128366	−0.0641831	+	0.997938i	$0.520444\pi$
$972$	0	0
$973$	0	0
$974$	12.0000	0.384505
$975$	0	0
$976$	2.00000	0.0640184
$977$	22.0000	0.703842	0.351921	−	0.936030i	$-0.385529\pi$
$977$	22.0000	0.703842	0.351921	+	0.936030i	$0.385529\pi$
$978$	0	0
$979$	16.0000	0.511362
$980$	0	0
$981$	0	0
$982$	−4.00000	−0.127645
$983$	4.00000	0.127580	0.0637901	−	0.997963i	$-0.479681\pi$
$983$	4.00000	0.127580	0.0637901	+	0.997963i	$0.479681\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	−8.00000	−0.254000
$993$	0	0
$994$	−32.0000	−1.01498
$995$	0	0
$996$	0	0
$997$	−8.00000	−0.253363	−0.126681	−	0.991943i	$-0.540433\pi$
$997$	−8.00000	−0.253363	−0.126681	+	0.991943i	$0.540433\pi$
$998$	−20.0000	−0.633089
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4950.2.a.w.1.1		1
3.2	odd	2		4950.2.a.b.1.1		1
5.2	odd	4		990.2.c.e.199.2	yes	2
5.3	odd	4		990.2.c.e.199.1	yes	2
5.4	even	2		4950.2.a.s.1.1		1
15.2	even	4		990.2.c.a.199.1	✓	2
15.8	even	4		990.2.c.a.199.2	yes	2
15.14	odd	2		4950.2.a.bt.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
990.2.c.a.199.1	✓	2	15.2	even	4
990.2.c.a.199.2	yes	2	15.8	even	4
990.2.c.e.199.1	yes	2	5.3	odd	4
990.2.c.e.199.2	yes	2	5.2	odd	4
4950.2.a.b.1.1		1	3.2	odd	2
4950.2.a.s.1.1		1	5.4	even	2
4950.2.a.w.1.1		1	1.1	even	1	trivial
4950.2.a.bt.1.1		1	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4950.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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