LMFDB - Embedded newform 416.7.h.a.207.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [416,7,Mod(207,416)] 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("416.207"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 7, names="a")

Level:	$N$	$=$	$416 = 2^{5} \cdot 13$
Weight:	$k$	$=$	$7$
Character orbit:	$[\chi]$	$=$	416.h (of order $2$ , degree $1$ , not minimal)

Newform invariants

comment:select newform

sage:traces = [1,0,50,0,-218] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$95.7024987859$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			207.1
Character	$\chi$	$=$	416.207

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

f(q)

=

q+50.0000 q^{3} -218.000 q^{5} +614.000 q^{7} +1771.00 q^{9} -2197.00 q^{13} -10900.0 q^{15} +3170.00 q^{17} +30700.0 q^{21} +31899.0 q^{25} +52100.0 q^{27} +27830.0 q^{31} -133852. q^{35} +13894.0 q^{37} -109850. q^{39} +111490. q^{43} -386078. q^{45} -128554. q^{47} +259347. q^{49} +158500. q^{51} +1.08739e6 q^{63} +478946. q^{65} +317990. q^{71} +1.59495e6 q^{75} +1.31394e6 q^{81} -691060. q^{85} -1.34896e6 q^{91} +1.39150e6 q^{93} +O(q^{100})

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/416\mathbb{Z}\right)^\times$ .

$n$	$261$	$287$	$353$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	50.0000	1.85185	0.925926	−	0.377705i	$-0.123287\pi$
$3$	50.0000	1.85185	0.925926	+	0.377705i	$0.123287\pi$
$4$	0	0
$5$	−218.000	−1.74400	−0.872000	−	0.489506i	$-0.837177\pi$
$5$	−218.000	−1.74400	−0.872000	+	0.489506i	$0.837177\pi$
$6$	0	0
$7$	614.000	1.79009	0.895044	−	0.445978i	$-0.147144\pi$
$7$	614.000	1.79009	0.895044	+	0.445978i	$0.147144\pi$
$8$	0	0
$9$	1771.00	2.42936
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	−2197.00	−1.00000
$13$	−2197.00	−1.00000
$14$	0	0
$15$	−10900.0	−3.22963
$16$	0	0
$17$	3170.00	0.645227	0.322613	−	0.946531i	$-0.395439\pi$
$17$	3170.00	0.645227	0.322613	+	0.946531i	$0.395439\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	30700.0	3.31498
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	31899.0	2.04154
$26$	0	0
$27$	52100.0	2.64695
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	27830.0	0.934175	0.467087	−	0.884211i	$-0.345303\pi$
$31$	27830.0	0.934175	0.467087	+	0.884211i	$0.345303\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−133852.	−3.12191
$36$	0	0
$37$	13894.0	0.274298	0.137149	−	0.990550i	$-0.456206\pi$
$37$	13894.0	0.274298	0.137149	+	0.990550i	$0.456206\pi$
$38$	0	0
$39$	−109850.	−1.85185
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	111490.	1.40227	0.701133	−	0.713030i	$-0.252678\pi$
$43$	111490.	1.40227	0.701133	+	0.713030i	$0.252678\pi$
$44$	0	0
$45$	−386078.	−4.23680
$46$	0	0
$47$	−128554.	−1.23820	−0.619102	−	0.785311i	$-0.712503\pi$
$47$	−128554.	−1.23820	−0.619102	+	0.785311i	$0.712503\pi$
$48$	0	0
$49$	259347.	2.20441
$50$	0	0
$51$	158500.	1.19486
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	1.08739e6	4.34876
$64$	0	0
$65$	478946.	1.74400
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	317990.	0.888461	0.444231	−	0.895913i	$-0.353477\pi$
$71$	317990.	0.888461	0.444231	+	0.895913i	$0.353477\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	1.59495e6	3.78062
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	1.31394e6	2.47241
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	−691060.	−1.12528
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	−1.34896e6	−1.79009
$92$	0	0
$93$	1.39150e6	1.72995
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	−6.69260e6	−5.78132
$106$	0	0
$107$	2.44717e6	1.99762	0.998810	−	0.0487741i	$-0.0155314\pi$
$107$	2.44717e6	1.99762	0.998810	+	0.0487741i	$0.0155314\pi$
$108$	0	0
$109$	2.58127e6	1.99321	0.996607	−	0.0823070i	$-0.0262288\pi$
$109$	2.58127e6	1.99321	0.996607	+	0.0823070i	$0.0262288\pi$
$110$	0	0
$111$	694700.	0.507959
$112$	0	0
$113$	419330.	0.290617	0.145308	−	0.989386i	$-0.453583\pi$
$113$	419330.	0.290617	0.145308	+	0.989386i	$0.453583\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3.89089e6	−2.42936
$118$	0	0
$119$	1.94638e6	1.15501
$120$	0	0
$121$	1.77156e6	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−3.54773e6	−1.81644
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	5.57450e6	2.59679
$130$	0	0
$131$	−3.97192e6	−1.76680	−0.883398	−	0.468624i	$-0.844750\pi$
$131$	−3.97192e6	−1.76680	−0.883398	+	0.468624i	$0.844750\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−1.13578e7	−4.61629
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−5.34784e6	−1.99129	−0.995643	−	0.0932421i	$-0.970277\pi$
$139$	−5.34784e6	−1.99129	−0.995643	+	0.0932421i	$0.970277\pi$
$140$	0	0
$141$	−6.42770e6	−2.29297
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.29674e7	4.08225
$148$	0	0
$149$	6.40951e6	1.93761	0.968804	−	0.247827i	$-0.0797165\pi$
$149$	6.40951e6	1.93761	0.968804	+	0.247827i	$0.0797165\pi$
$150$	0	0
$151$	5.65031e6	1.64112	0.820562	−	0.571557i	$-0.193661\pi$
$151$	5.65031e6	1.64112	0.820562	+	0.571557i	$0.193661\pi$
$152$	0	0
$153$	5.61407e6	1.56749
$154$	0	0
$155$	−6.06694e6	−1.62920
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.80527e6	−1.89057	−0.945287	−	0.326241i	$-0.894218\pi$
$167$	−8.80527e6	−1.89057	−0.945287	+	0.326241i	$0.894218\pi$
$168$	0	0
$169$	4.82681e6	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	1.95860e7	3.65453
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.12106e7	1.95465	0.977325	−	0.211746i	$-0.0679149\pi$
$179$	1.12106e7	1.95465	0.977325	+	0.211746i	$0.0679149\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−3.02889e6	−0.478375
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.19894e7	4.73828
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	2.39473e7	3.22963
$196$	0	0
$197$	−4.37695e6	−0.572497	−0.286249	−	0.958155i	$-0.592408\pi$
$197$	−4.37695e6	−0.572497	−0.286249	+	0.958155i	$0.592408\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1.82029e7	−1.93773	−0.968863	−	0.247598i	$-0.920359\pi$
$211$	−1.82029e7	−1.93773	−0.968863	+	0.247598i	$0.920359\pi$
$212$	0	0
$213$	1.58995e7	1.64530
$214$	0	0
$215$	−2.43048e7	−2.44555
$216$	0	0
$217$	1.70876e7	1.67225
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−6.96449e6	−0.645227
$222$	0	0
$223$	3.08293e6	0.278003	0.139002	−	0.990292i	$-0.455611\pi$
$223$	3.08293e6	0.278003	0.139002	+	0.990292i	$0.455611\pi$
$224$	0	0
$225$	5.64931e7	4.95962
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	7.53559e6	0.627496	0.313748	−	0.949506i	$-0.398415\pi$
$229$	7.53559e6	0.627496	0.313748	+	0.949506i	$0.398415\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.04145e7	−0.823325	−0.411662	−	0.911336i	$-0.635052\pi$
$233$	−1.04145e7	−0.823325	−0.411662	+	0.911336i	$0.635052\pi$
$234$	0	0
$235$	2.80248e7	2.15943
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.69138e7	−1.23893	−0.619465	−	0.785024i	$-0.712650\pi$
$239$	−1.69138e7	−1.23893	−0.619465	+	0.785024i	$0.712650\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	2.77162e7	1.93159
$244$	0	0
$245$	−5.65376e7	−3.84450
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1.21784e7	0.770139	0.385070	−	0.922888i	$-0.374177\pi$
$251$	1.21784e7	0.770139	0.385070	+	0.922888i	$0.374177\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3.45530e7	−2.08384
$256$	0	0
$257$	−1.58527e7	−0.933906	−0.466953	−	0.884282i	$-0.654648\pi$
$257$	−1.58527e7	−0.933906	−0.466953	+	0.884282i	$0.654648\pi$
$258$	0	0
$259$	8.53092e6	0.491017
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	1.63301e7	0.820503	0.410252	−	0.911972i	$-0.365441\pi$
$271$	1.63301e7	0.820503	0.410252	+	0.911972i	$0.365441\pi$
$272$	0	0
$273$	−6.74479e7	−3.31498
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	4.92869e7	2.26944
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−3.82458e7	−1.68742	−0.843712	−	0.536796i	$-0.819635\pi$
$283$	−3.82458e7	−1.68742	−0.843712	+	0.536796i	$0.819635\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−1.40887e7	−0.583682
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.91523e7	−1.95407	−0.977037	−	0.213069i	$-0.931654\pi$
$293$	−4.91523e7	−1.95407	−0.977037	+	0.213069i	$0.931654\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	6.84549e7	2.51018
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−6.13286e7	−2.00000	−1.00000		0.000361172i	$-0.999885\pi$
$313$	−6.13286e7	−2.00000	−1.00000		0.000361172i	$0.999885\pi$
$314$	0	0
$315$	−2.37052e8	−7.58423
$316$	0	0
$317$	−4.54692e7	−1.42738	−0.713690	−	0.700461i	$-0.752978\pi$
$317$	−4.54692e7	−1.42738	−0.713690	+	0.700461i	$0.752978\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	1.22358e8	3.69930
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−7.00821e7	−2.04154
$326$	0	0
$327$	1.29064e8	3.69114
$328$	0	0
$329$	−7.89322e7	−2.21649
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	2.46063e7	0.666366
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−3.55087e7	−0.927779	−0.463890	−	0.885893i	$-0.653547\pi$
$337$	−3.55087e7	−0.927779	−0.463890	+	0.885893i	$0.653547\pi$
$338$	0	0
$339$	2.09665e7	0.538179
$340$	0	0
$341$	0	0
$342$	0	0
$343$	8.70026e7	2.15600
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.81636e7	−0.434723	−0.217361	−	0.976091i	$-0.569745\pi$
$347$	−1.81636e7	−0.434723	−0.217361	+	0.976091i	$0.569745\pi$
$348$	0	0
$349$	5.97550e7	1.40572	0.702859	−	0.711329i	$-0.251906\pi$
$349$	5.97550e7	1.40572	0.702859	+	0.711329i	$0.251906\pi$
$350$	0	0
$351$	−1.14464e8	−2.64695
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	−6.93218e7	−1.54948
$356$	0	0
$357$	9.73190e7	2.13891
$358$	0	0
$359$	8.10358e7	1.75143	0.875716	−	0.482826i	$-0.160390\pi$
$359$	8.10358e7	1.75143	0.875716	+	0.482826i	$0.160390\pi$
$360$	0	0
$361$	4.70459e7	1.00000
$362$	0	0
$363$	8.85780e7	1.85185
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	−1.77387e8	−3.36378
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6.77127e6	−0.120524	−0.0602621	−	0.998183i	$-0.519194\pi$
$383$	−6.77127e6	−0.120524	−0.0602621	+	0.998183i	$0.519194\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	1.97449e8	3.40660
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−1.98596e8	−3.27184
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.09696e8	1.75315	0.876577	−	0.481261i	$-0.159821\pi$
$397$	1.09696e8	1.75315	0.876577	+	0.481261i	$0.159821\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−6.11425e7	−0.934175
$404$	0	0
$405$	−2.86439e8	−4.31189
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.67392e8	−3.68757
$418$	0	0
$419$	−1.45495e8	−1.97791	−0.988955	−	0.148219i	$-0.952646\pi$
$419$	−1.45495e8	−1.97791	−0.988955	+	0.148219i	$0.952646\pi$
$420$	0	0
$421$	−1.45559e8	−1.95072	−0.975358	−	0.220627i	$-0.929190\pi$
$421$	−1.45559e8	−1.95072	−0.975358	+	0.220627i	$0.929190\pi$
$422$	0	0
$423$	−2.27669e8	−3.00804
$424$	0	0
$425$	1.01120e8	1.31725
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−1.47692e8	−1.84469	−0.922347	−	0.386363i	$-0.873731\pi$
$431$	−1.47692e8	−1.84469	−0.922347	+	0.386363i	$0.873731\pi$
$432$	0	0
$433$	5.54331e7	0.682819	0.341409	−	0.939915i	$-0.389096\pi$
$433$	5.54331e7	0.682819	0.341409	+	0.939915i	$0.389096\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	4.59304e8	5.35530
$442$	0	0
$443$	−8.60423e7	−0.989693	−0.494847	−	0.868980i	$-0.664776\pi$
$443$	−8.60423e7	−0.989693	−0.494847	+	0.868980i	$0.664776\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.20476e8	3.58816
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	2.82516e8	3.03912
$454$	0	0
$455$	2.94073e8	3.12191
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	1.65157e8	1.70789
$460$	0	0
$461$	1.72671e8	1.76245	0.881226	−	0.472696i	$-0.156719\pi$
$461$	1.72671e8	1.76245	0.881226	+	0.472696i	$0.156719\pi$
$462$	0	0
$463$	−1.05552e8	−1.06347	−0.531733	−	0.846912i	$-0.678459\pi$
$463$	−1.05552e8	−1.06347	−0.531733	+	0.846912i	$0.678459\pi$
$464$	0	0
$465$	−3.03347e8	−3.01704
$466$	0	0
$467$	−2.87454e7	−0.282239	−0.141120	−	0.989993i	$-0.545070\pi$
$467$	−2.87454e7	−0.282239	−0.141120	+	0.989993i	$0.545070\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.01712e7	0.0925476	0.0462738	−	0.998929i	$-0.485265\pi$
$479$	1.01712e7	0.0925476	0.0462738	+	0.998929i	$0.485265\pi$
$480$	0	0
$481$	−3.05251e7	−0.274298
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.97305e8	−1.70825	−0.854126	−	0.520066i	$-0.825907\pi$
$487$	−1.97305e8	−1.70825	−0.854126	+	0.520066i	$0.825907\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.99411e7	0.675345	0.337672	−	0.941264i	$-0.390360\pi$
$491$	7.99411e7	0.675345	0.337672	+	0.941264i	$0.390360\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.95246e8	1.59042
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	−4.40264e8	−3.50106
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.41340e8	1.85185
$508$	0	0
$509$	−2.61109e8	−1.98001	−0.990006	−	0.141024i	$-0.954961\pi$
$509$	−2.61109e8	−1.98001	−0.990006	+	0.141024i	$0.954961\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−2.81764e8	−1.99238	−0.996191	−	0.0872021i	$-0.972207\pi$
$521$	−2.81764e8	−1.99238	−0.996191	+	0.0872021i	$0.972207\pi$
$522$	0	0
$523$	8.89336e7	0.621671	0.310836	−	0.950464i	$-0.399391\pi$
$523$	8.89336e7	0.621671	0.310836	+	0.950464i	$0.399391\pi$
$524$	0	0
$525$	9.79299e8	6.76764
$526$	0	0
$527$	8.82211e7	0.602755
$528$	0	0
$529$	1.48036e8	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−5.33483e8	−3.48385
$536$	0	0
$537$	5.60529e8	3.61972
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3.94438e7	0.249108	0.124554	−	0.992213i	$-0.460250\pi$
$541$	3.94438e7	0.249108	0.124554	+	0.992213i	$0.460250\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.62717e8	−3.47617
$546$	0	0
$547$	−3.05993e8	−1.86960	−0.934801	−	0.355171i	$-0.884423\pi$
$547$	−3.05993e8	−1.86960	−0.934801	+	0.355171i	$0.884423\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.51445e8	−0.885880
$556$	0	0
$557$	1.62805e8	0.942111	0.471055	−	0.882104i	$-0.343873\pi$
$557$	1.62805e8	0.942111	0.471055	+	0.882104i	$0.343873\pi$
$558$	0	0
$559$	−2.44944e8	−1.40227
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3.35941e8	−1.88251	−0.941255	−	0.337697i	$-0.890352\pi$
$563$	−3.35941e8	−1.88251	−0.941255	+	0.337697i	$0.890352\pi$
$564$	0	0
$565$	−9.14139e7	−0.506836
$566$	0	0
$567$	8.06760e8	4.42583
$568$	0	0
$569$	−3.45458e8	−1.87524	−0.937622	−	0.347656i	$-0.886978\pi$
$569$	−3.45458e8	−1.87524	−0.937622	+	0.347656i	$0.886978\pi$
$570$	0	0
$571$	−6.07173e7	−0.326140	−0.163070	−	0.986615i	$-0.552140\pi$
$571$	−6.07173e7	−0.326140	−0.163070	+	0.986615i	$0.552140\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	8.48213e8	4.23680
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−2.18848e8	−1.06018
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	−4.24311e8	−2.01434
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−3.37535e8	−1.55487	−0.777437	−	0.628961i	$-0.783480\pi$
$601$	−3.37535e8	−1.55487	−0.777437	+	0.628961i	$0.783480\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.86200e8	−1.74400
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.82433e8	1.23820
$612$	0	0
$613$	8.30155e7	0.360394	0.180197	−	0.983631i	$-0.442326\pi$
$613$	8.30155e7	0.360394	0.180197	+	0.983631i	$0.442326\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.74984e8	1.12633
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.40440e7	0.176984
$630$	0	0
$631$	−4.93350e8	−1.96366	−0.981832	−	0.189754i	$-0.939231\pi$
$631$	−4.93350e8	−1.96366	−0.981832	+	0.189754i	$0.939231\pi$
$632$	0	0
$633$	−9.10143e8	−3.58838
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−5.69785e8	−2.20441
$638$	0	0
$639$	5.63160e8	2.15839
$640$	0	0
$641$	4.15127e8	1.57618	0.788091	−	0.615558i	$-0.211069\pi$
$641$	4.15127e8	1.57618	0.788091	+	0.615558i	$0.211069\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	−1.21524e9	−4.52880
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	8.54381e8	3.09677
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	8.65878e8	3.08129
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4.36753e8	1.52609	0.763044	−	0.646346i	$-0.223704\pi$
$659$	4.36753e8	1.52609	0.763044	+	0.646346i	$0.223704\pi$
$660$	0	0
$661$	3.83714e8	1.32863	0.664314	−	0.747454i	$-0.268724\pi$
$661$	3.83714e8	1.32863	0.664314	+	0.747454i	$0.268724\pi$
$662$	0	0
$663$	−3.48224e8	−1.19486
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	1.54147e8	0.514821
$670$	0	0
$671$	0	0
$672$	0	0
$673$	6.07111e8	1.99170	0.995848	−	0.0910264i	$-0.0290148\pi$
$673$	6.07111e8	1.99170	0.995848	+	0.0910264i	$0.0290148\pi$
$674$	0	0
$675$	1.66194e9	5.40385
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3.76780e8	1.16203
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.16583e9	3.47280
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−5.20726e8	−1.52468
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	1.40124e9	3.99894
$706$	0	0
$707$	0	0
$708$	0	0
$709$	2.61192e8	0.732859	0.366430	−	0.930446i	$-0.380580\pi$
$709$	2.61192e8	0.732859	0.366430	+	0.930446i	$0.380580\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.45689e8	−2.29431
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	4.27945e8	1.10460
$730$	0	0
$731$	3.53423e8	0.904780
$732$	0	0
$733$	6.63862e8	1.68564	0.842821	−	0.538193i	$-0.180893\pi$
$733$	6.63862e8	1.68564	0.842821	+	0.538193i	$0.180893\pi$
$734$	0	0
$735$	−2.82688e9	−7.11944
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−3.66280e8	−0.892989	−0.446495	−	0.894786i	$-0.647328\pi$
$743$	−3.66280e8	−0.892989	−0.446495	+	0.894786i	$0.647328\pi$
$744$	0	0
$745$	−1.39727e9	−3.37919
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.50256e9	3.57591
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	6.08920e8	1.42618
$754$	0	0
$755$	−1.23177e9	−2.86212
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	1.58490e9	3.56803
$764$	0	0
$765$	−1.22387e9	−2.73369
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	−7.92634e8	−1.72946
$772$	0	0
$773$	7.84772e8	1.69905	0.849523	−	0.527552i	$-0.176890\pi$
$773$	7.84772e8	1.69905	0.849523	+	0.527552i	$0.176890\pi$
$774$	0	0
$775$	8.87749e8	1.90715
$776$	0	0
$777$	4.26546e8	0.909290
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	2.57469e8	0.520229
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	−4.07516e8	−0.798922
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.63710e8	−1.06466	−0.532329	−	0.846538i	$-0.678683\pi$
$809$	−5.63710e8	−1.06466	−0.532329	+	0.846538i	$0.678683\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	8.16504e8	1.51945
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−2.38900e9	−4.34876
$820$	0	0
$821$	−7.52144e8	−1.35916	−0.679581	−	0.733600i	$-0.737838\pi$
$821$	−7.52144e8	−1.35916	−0.679581	+	0.733600i	$0.737838\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8.22130e8	1.42235
$834$	0	0
$835$	1.91955e9	3.29716
$836$	0	0
$837$	1.44994e9	2.47272
$838$	0	0
$839$	7.83672e8	1.32693	0.663466	−	0.748207i	$-0.269085\pi$
$839$	7.83672e8	1.32693	0.663466	+	0.748207i	$0.269085\pi$
$840$	0	0
$841$	5.94823e8	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.05224e9	−1.74400
$846$	0	0
$847$	1.08774e9	1.79009
$848$	0	0
$849$	−1.91229e9	−3.12486
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1.14583e9	1.84617	0.923086	−	0.384593i	$-0.125658\pi$
$853$	1.14583e9	1.84617	0.923086	+	0.384593i	$0.125658\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.10042e8	−0.174831	−0.0874153	−	0.996172i	$-0.527861\pi$
$857$	−1.10042e8	−0.174831	−0.0874153	+	0.996172i	$0.527861\pi$
$858$	0	0
$859$	1.25286e9	1.97661	0.988307	−	0.152475i	$-0.0487243\pi$
$859$	1.25286e9	1.97661	0.988307	+	0.152475i	$0.0487243\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	5.47423e8	0.851708	0.425854	−	0.904792i	$-0.359974\pi$
$863$	5.47423e8	0.851708	0.425854	+	0.904792i	$0.359974\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−7.04433e8	−1.08089
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.17831e9	−3.25158
$876$	0	0
$877$	−2.72943e8	−0.404645	−0.202322	−	0.979319i	$-0.564849\pi$
$877$	−2.72943e8	−0.404645	−0.202322	+	0.979319i	$0.564849\pi$
$878$	0	0
$879$	−2.45762e9	−3.61866
$880$	0	0
$881$	−3.77235e8	−0.551676	−0.275838	−	0.961204i	$-0.588955\pi$
$881$	−3.77235e8	−0.551676	−0.275838	+	0.961204i	$0.588955\pi$
$882$	0	0
$883$	−1.25086e9	−1.81688	−0.908442	−	0.418011i	$-0.862727\pi$
$883$	−1.25086e9	−1.81688	−0.908442	+	0.418011i	$0.862727\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−2.44391e9	−3.40891
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	3.42274e9	4.64848
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.36131e8	0.182446	0.0912232	−	0.995830i	$-0.470922\pi$
$907$	1.36131e8	0.182446	0.0912232	+	0.995830i	$0.470922\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.43876e9	−3.16272
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−6.98624e8	−0.888461
$924$	0	0
$925$	4.43205e8	0.559989
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.61380e9	−1.96169	−0.980844	−	0.194793i	$-0.937597\pi$
$937$	−1.61380e9	−1.96169	−0.980844	+	0.194793i	$0.937597\pi$
$938$	0	0
$939$	−3.06643e9	−3.70370
$940$	0	0
$941$	−1.11296e9	−1.33571	−0.667854	−	0.744292i	$-0.732787\pi$
$941$	−1.11296e9	−1.33571	−0.667854	+	0.744292i	$0.732787\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−6.97369e9	−8.26356
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−2.27346e9	−2.64330
$952$	0	0
$953$	1.68629e9	1.94829	0.974146	−	0.225921i	$-0.0725390\pi$
$953$	1.68629e9	1.94829	0.974146	+	0.225921i	$0.0725390\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.12995e8	−0.127318
$962$	0	0
$963$	4.33394e9	4.85293
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.27382e9	1.40873	0.704365	−	0.709838i	$-0.251232\pi$
$967$	1.27382e9	1.40873	0.704365	+	0.709838i	$0.251232\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.85702e7	0.0858223	0.0429111	−	0.999079i	$-0.486337\pi$
$971$	7.85702e7	0.0858223	0.0429111	+	0.999079i	$0.486337\pi$
$972$	0	0
$973$	−3.28357e9	−3.56458
$974$	0	0
$975$	−3.50411e9	−3.78062
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	4.57143e9	4.84223
$982$	0	0
$983$	−1.53648e9	−1.61758	−0.808792	−	0.588095i	$-0.799878\pi$
$983$	−1.53648e9	−1.61758	−0.808792	+	0.588095i	$0.799878\pi$
$984$	0	0
$985$	9.54176e8	0.998435
$986$	0	0
$987$	−3.94661e9	−4.10462
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	7.23877e8	0.726053

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.7.h.a.207.1		1
4.3	odd	2		104.7.h.b.51.1	yes	1
8.3	odd	2		416.7.h.b.207.1		1
8.5	even	2		104.7.h.a.51.1	✓	1
13.12	even	2		416.7.h.b.207.1		1
52.51	odd	2		104.7.h.a.51.1	✓	1
104.51	odd	2	CM	416.7.h.a.207.1		1
104.77	even	2		104.7.h.b.51.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.7.h.a.51.1	✓	1	8.5	even	2
104.7.h.a.51.1	✓	1	52.51	odd	2
104.7.h.b.51.1	yes	1	4.3	odd	2
104.7.h.b.51.1	yes	1	104.77	even	2
416.7.h.a.207.1		1	1.1	even	1	trivial
416.7.h.a.207.1		1	104.51	odd	2	CM
416.7.h.b.207.1		1	8.3	odd	2
416.7.h.b.207.1		1	13.12	even	2

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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

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Modular forms

Varieties

Fields

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

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	416.7.h.a.207.1

Level	$416$
Weight	$7$
Character	416.207
Self dual	yes
Analytic conductor	$95.702$
Analytic rank	$0$
Dimension	$1$
CM discriminant	-104
Inner twists	$2$