LMFDB - Embedded newform 256.8.b.d.129.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,8,Mod(129,256)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("256.129");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	256.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$79.9705665239$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 32)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			129.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.129
Dual form			256.8.b.d.129.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+58.0000i q^{5} +2187.00 q^{9} -8898.00i q^{13} -40094.0 q^{17} +74761.0 q^{25} +233230. i q^{29} -563974. i q^{37} -9530.00 q^{41} +126846. i q^{45} -823543. q^{49} +798602. i q^{53} -3.50533e6i q^{61} +516084. q^{65} -3.91742e6 q^{73} +4.78297e6 q^{81} -2.32545e6i q^{85} -9.24617e6 q^{89} -1.75674e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4374 q^{9} - 80188 q^{17} + 149522 q^{25} - 19060 q^{41} - 1647086 q^{49} + 1032168 q^{65} - 7834836 q^{73} + 9565938 q^{81} - 18492340 q^{89} - 35134812 q^{97}+O(q^{100}) $ 2 * q + 4374 * q^9 - 80188 * q^17 + 149522 * q^25 - 19060 * q^41 - 1647086 * q^49 + 1032168 * q^65 - 7834836 * q^73 + 9565938 * q^81 - 18492340 * q^89 - 35134812 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$-1$	$1$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	0	0
$2$	0	0
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	58.0000i	0.207507i	0.994603	+	0.103754i	$0.0330853\pi$
$5$	58.0000i	0.207507i	−0.994603	+	0.103754i	$0.966915\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	2187.00	1.00000
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	− 8898.00i	− 1.12329i	−0.827379	−	0.561643i	$-0.810169\pi$
$13$	− 8898.00i	− 1.12329i	0.827379	−	0.561643i	$-0.189831\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−40094.0	−1.97928	−0.989642	−	0.143557i	$-0.954146\pi$
$17$	−40094.0	−1.97928	−0.989642	+	0.143557i	$0.954146\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	74761.0	0.956941
$26$	0	0
$27$	0	0
$28$	0	0
$29$	233230.i	1.77579i	0.460047	+	0.887895i	$0.347833\pi$
$29$	233230.i	1.77579i	−0.460047	+	0.887895i	$0.652167\pi$
$30$	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
$35$	0	0
$36$	0	0
$37$	− 563974.i	− 1.83043i	−0.402966	−	0.915215i	$-0.632021\pi$
$37$	− 563974.i	− 1.83043i	0.402966	−	0.915215i	$-0.367979\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9530.00	−0.0215948	−0.0107974	−	0.999942i	$-0.503437\pi$
$41$	−9530.00	−0.0215948	−0.0107974	+	0.999942i	$0.503437\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	126846.i	0.207507i
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−823543.	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	798602.i	0.736826i	0.929662	+	0.368413i	$0.120099\pi$
$53$	798602.i	0.736826i	−0.929662	+	0.368413i	$0.879901\pi$
$54$	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
$61$	− 3.50533e6i	− 1.97731i	−0.150208	−	0.988654i	$-0.547994\pi$
$61$	− 3.50533e6i	− 1.97731i	0.150208	−	0.988654i	$-0.452006\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	516084.	0.233090
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−3.91742e6	−1.17861	−0.589305	−	0.807911i	$-0.700598\pi$
$73$	−3.91742e6	−1.17861	−0.589305	+	0.807911i	$0.700598\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	4.78297e6	1.00000
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	− 2.32545e6i	− 0.410716i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.24617e6	−1.39026	−0.695131	−	0.718883i	$-0.744654\pi$
$89$	−9.24617e6	−1.39026	−0.695131	+	0.718883i	$0.744654\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.75674e7	−1.95437	−0.977185	−	0.212392i	$-0.931875\pi$
$97$	−1.75674e7	−1.95437	−0.977185	+	0.212392i	$0.931875\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.33042e7i	− 1.28488i	−0.766334	−	0.642442i	$-0.777921\pi$
$101$	− 1.33042e7i	− 1.28488i	0.766334	−	0.642442i	$-0.222079\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	− 2.41162e7i	− 1.78368i	−0.452352	−	0.891839i	$-0.649415\pi$
$109$	− 2.41162e7i	− 1.78368i	0.452352	−	0.891839i	$-0.350585\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.91247e7	−1.89883	−0.949417	−	0.314017i	$-0.898325\pi$
$113$	−2.91247e7	−1.89883	−0.949417	+	0.314017i	$0.898325\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.94599e7i	− 1.12329i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.94872e7	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.86739e6i	0.406079i
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−4.60312e7	−1.52943	−0.764716	−	0.644367i	$-0.777121\pi$
$137$	−4.60312e7	−1.52943	−0.764716	+	0.644367i	$0.777121\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.35273e7	−0.368489
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 7.31577e7i	− 1.81179i	−0.423502	−	0.905895i	$-0.639199\pi$
$149$	− 7.31577e7i	− 1.81179i	0.423502	−	0.905895i	$-0.360801\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−8.76856e7	−1.97928
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 9.09715e7i	− 1.87610i	−0.346497	−	0.938051i	$-0.612629\pi$
$157$	− 9.09715e7i	− 1.87610i	0.346497	−	0.938051i	$-0.387371\pi$
$158$	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
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−1.64259e7	−0.261773
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 6.55688e7i	− 0.962800i	−0.876501	−	0.481400i	$-0.840129\pi$
$173$	− 6.55688e7i	− 0.962800i	0.876501	−	0.481400i	$-0.159871\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	1.45860e8i	1.82836i	0.405309	+	0.914180i	$0.367164\pi$
$181$	1.45860e8i	1.82836i	−0.405309	+	0.914180i	$0.632836\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.27105e7	0.379827
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−1.05243e8	−1.05377	−0.526883	−	0.849938i	$-0.676639\pi$
$193$	−1.05243e8	−1.05377	−0.526883	+	0.849938i	$0.676639\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.02733e8i	− 0.957363i	−0.877989	−	0.478681i	$-0.841115\pi$
$197$	− 1.02733e8i	− 0.957363i	0.877989	−	0.478681i	$-0.158885\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 552740.i	− 0.00448108i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.56756e8i	2.22330i
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	1.63502e8	0.956941
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 2.17910e8i	− 1.19909i	−0.800339	−	0.599547i	$-0.795347\pi$
$229$	− 2.17910e8i	− 1.19909i	0.800339	−	0.599547i	$-0.204653\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.90325e8	1.50362	0.751811	−	0.659379i	$-0.229181\pi$
$233$	2.90325e8	1.50362	0.751811	+	0.659379i	$0.229181\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	1.08961e8	0.501433	0.250717	−	0.968061i	$-0.419334\pi$
$241$	1.08961e8	0.501433	0.250717	+	0.968061i	$0.419334\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 4.77655e7i	− 0.207507i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.30304e8	−0.846323	−0.423161	−	0.906054i	$-0.639080\pi$
$257$	−2.30304e8	−0.846323	−0.423161	+	0.906054i	$0.639080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	5.10074e8i	1.77579i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	−4.63189e7	−0.152897
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.80336e7i	0.244427i	0.992504	+	0.122213i	$0.0389992\pi$
$269$	7.80336e7i	0.244427i	−0.992504	+	0.122213i	$0.961001\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.35122e8i	1.51277i	0.654126	+	0.756386i	$0.273037\pi$
$277$	5.35122e8i	1.51277i	−0.654126	+	0.756386i	$0.726963\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.34409e8	1.70568	0.852839	−	0.522174i	$-0.174879\pi$
$281$	6.34409e8	1.70568	0.852839	+	0.522174i	$0.174879\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.19719e9	2.91757
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.87629e8i	1.36479i	0.730983	+	0.682396i	$0.239062\pi$
$293$	5.87629e8i	1.36479i	−0.730983	+	0.682396i	$0.760938\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.03309e8	0.410306
$306$	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		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−5.25457e8	−0.968573	−0.484286	−	0.874910i	$-0.660921\pi$
$313$	−5.25457e8	−0.968573	−0.484286	+	0.874910i	$0.660921\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.13287e9i	1.99744i	0.0505827	+	0.998720i	$0.483892\pi$
$317$	1.13287e9i	1.99744i	−0.0505827	+	0.998720i	$0.516108\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 6.65223e8i	− 1.07492i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	− 1.23341e9i	− 1.83043i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	6.05044e8	0.861157	0.430578	−	0.902553i	$-0.358310\pi$
$337$	6.05044e8	0.861157	0.430578	+	0.902553i	$0.358310\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	1.50468e9i	1.89477i	0.320097	+	0.947385i	$0.396285\pi$
$349$	1.50468e9i	1.89477i	−0.320097	+	0.947385i	$0.603715\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.64963e9	−1.99607	−0.998035	−	0.0626535i	$-0.980044\pi$
$353$	−1.64963e9	−1.99607	−0.998035	+	0.0626535i	$0.980044\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	8.93872e8	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2.27210e8i	− 0.244570i
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−2.08421e7	−0.0215948
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.03979e9i	1.03744i	0.854944	+	0.518720i	$0.173591\pi$
$373$	1.03979e9i	1.03744i	−0.854944	+	0.518720i	$0.826409\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.07528e9	1.99472
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 1.94162e9i	− 1.67240i	−0.548422	−	0.836202i	$-0.684771\pi$
$389$	− 1.94162e9i	− 1.67240i	0.548422	−	0.836202i	$-0.315229\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.34628e9i	− 1.07986i	−0.841709	−	0.539932i	$-0.818450\pi$
$397$	− 1.34628e9i	− 1.07986i	0.841709	−	0.539932i	$-0.181550\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.84817e8	−0.685248	−0.342624	−	0.939473i	$-0.611316\pi$
$401$	−8.84817e8	−0.685248	−0.342624	+	0.939473i	$0.611316\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.77412e8i	0.207507i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−2.38968e9	−1.72706	−0.863531	−	0.504296i	$-0.831752\pi$
$409$	−2.38968e9	−1.72706	−0.863531	+	0.504296i	$0.831752\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	− 1.58512e9i	− 1.03532i	−0.855586	−	0.517661i	$-0.826803\pi$
$421$	− 1.58512e9i	− 1.03532i	0.855586	−	0.517661i	$-0.173197\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.99747e9	−1.89406
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	−1.34596e9	−0.796757	−0.398378	−	0.917221i	$-0.630427\pi$
$433$	−1.34596e9	−0.796757	−0.398378	+	0.917221i	$0.630427\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	−1.80109e9	−1.00000
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	− 5.36278e8i	− 0.288489i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.71111e9	1.41346	0.706732	−	0.707481i	$-0.250168\pi$
$449$	2.71111e9	1.41346	0.706732	+	0.707481i	$0.250168\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.02241e9	−0.501095	−0.250548	−	0.968104i	$-0.580611\pi$
$457$	−1.02241e9	−0.501095	−0.250548	+	0.968104i	$0.580611\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 4.07657e9i	− 1.93795i	−0.247164	−	0.968974i	$-0.579499\pi$
$461$	− 4.07657e9i	− 1.93795i	0.247164	−	0.968974i	$-0.420501\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	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
$477$	1.74654e9i	0.736826i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−5.01824e9	−2.05610
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.01891e9i	− 0.405545i
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	− 9.35112e9i	− 3.51479i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	7.71643e8	0.266623
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.95022e9i	1.99996i	0.00646007	+	0.999979i	$0.497944\pi$
$509$	5.95022e9i	1.99996i	−0.00646007	+	0.999979i	$0.502056\pi$
$510$	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
$521$	2.38452e9	0.738701	0.369350	−	0.929290i	$-0.379580\pi$
$521$	2.38452e9	0.738701	0.369350	+	0.929290i	$0.379580\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−3.40483e9	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8.47979e7i	0.0242572i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	7.36235e9i	1.99906i	0.0306534	+	0.999530i	$0.490241\pi$
$541$	7.36235e9i	1.99906i	−0.0306534	+	0.999530i	$0.509759\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1.39874e9	0.370126
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	− 7.66616e9i	− 1.97731i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 2.15369e9i	− 0.528067i	−0.964513	−	0.264034i	$-0.914947\pi$
$557$	− 2.15369e9i	− 0.528067i	0.964513	−	0.264034i	$-0.0850530\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	− 1.68923e9i	− 0.394022i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6.84617e9	−1.55796	−0.778978	−	0.627052i	$-0.784262\pi$
$569$	−6.84617e9	−1.55796	−0.778978	+	0.627052i	$0.784262\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.65224e9	−0.574775	−0.287387	−	0.957814i	$-0.592787\pi$
$577$	−2.65224e9	−0.574775	−0.287387	+	0.957814i	$0.592787\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	1.12868e9	0.233090
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	7.08713e9	1.39566	0.697829	−	0.716265i	$-0.254150\pi$
$593$	7.08713e9	1.39566	0.697829	+	0.716265i	$0.254150\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	1.05496e10	1.98233	0.991165	−	0.132637i	$-0.0423443\pi$
$601$	1.05496e10	1.98233	0.991165	+	0.132637i	$0.0423443\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.13026e9i	0.207507i
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	9.50665e9i	1.66692i	0.552577	+	0.833462i	$0.313645\pi$
$613$	9.50665e9i	1.66692i	−0.552577	+	0.833462i	$0.686355\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.16779e9	0.371551	0.185776	−	0.982592i	$-0.440520\pi$
$617$	2.16779e9	0.371551	0.185776	+	0.982592i	$0.440520\pi$
$618$	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
$625$	5.32639e9	0.872676
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.26120e10i	3.62294i
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	7.32789e9i	1.12329i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.92310e9	0.888272	0.444136	−	0.895959i	$-0.353511\pi$
$641$	5.92310e9	0.888272	0.444136	+	0.895959i	$0.353511\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 7.96894e9i	− 1.11997i	−0.828504	−	0.559983i	$-0.810808\pi$
$653$	− 7.96894e9i	− 1.11997i	0.828504	−	0.559983i	$-0.189192\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−8.56739e9	−1.17861
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	− 1.15742e9i	− 0.155879i	−0.996958	−	0.0779394i	$-0.975166\pi$
$661$	− 1.15742e9i	− 0.155879i	0.996958	−	0.0779394i	$-0.0248341\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	1.54182e10	1.94976	0.974879	−	0.222734i	$-0.0714982\pi$
$673$	1.54182e10	1.94976	0.974879	+	0.222734i	$0.0714982\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 4.29286e9i	− 0.531724i	−0.964011	−	0.265862i	$-0.914343\pi$
$677$	− 4.29286e9i	− 0.531724i	0.964011	−	0.265862i	$-0.0856566\pi$
$678$	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
$685$	− 2.66981e9i	− 0.317368i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.10596e9	0.827667
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.82096e8	0.0427423
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.77141e10i	1.94226i	0.238556	+	0.971129i	$0.423326\pi$
$701$	1.77141e10i	1.94226i	−0.238556	+	0.971129i	$0.576674\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 1.64383e10i	− 1.73219i	−0.499877	−	0.866096i	$-0.666621\pi$
$709$	− 1.64383e10i	− 1.73219i	0.499877	−	0.866096i	$-0.333379\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.74365e10i	1.69933i
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	1.04604e10	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.85321e10i	1.73804i	0.494773	+	0.869022i	$0.335251\pi$
$733$	1.85321e10i	1.73804i	−0.494773	+	0.869022i	$0.664749\pi$
$734$	0	0
$735$	0	0
$736$	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
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	4.24315e9	0.375959
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.72709e10i	− 1.44703i	−0.690307	−	0.723516i	$-0.742525\pi$
$757$	− 1.72709e10i	− 1.44703i	0.690307	−	0.723516i	$-0.257475\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.99861e10	1.64392	0.821961	−	0.569543i	$-0.192880\pi$
$761$	1.99861e10	1.64392	0.821961	+	0.569543i	$0.192880\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	− 5.08576e9i	− 0.410716i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−2.52206e10	−1.99992	−0.999960	−	0.00895271i	$-0.997150\pi$
$769$	−2.52206e10	−1.99992	−0.999960	+	0.00895271i	$0.997150\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.55364e10i	1.98853i	0.106964	+	0.994263i	$0.465887\pi$
$773$	2.55364e10i	1.98853i	−0.106964	+	0.994263i	$0.534113\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	5.27635e9	0.389305
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−3.11904e10	−2.22108
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 9.53016e9i	− 0.666801i	−0.942785	−	0.333400i	$-0.891804\pi$
$797$	− 9.53016e9i	− 0.666801i	0.942785	−	0.333400i	$-0.108196\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−2.02214e10	−1.39026
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.84568e10	1.88959	0.944793	−	0.327668i	$-0.106263\pi$
$809$	2.84568e10	1.88959	0.944793	+	0.327668i	$0.106263\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.88014e10i	− 1.81641i	−0.418530	−	0.908203i	$-0.637454\pi$
$821$	− 2.88014e10i	− 1.81641i	0.418530	−	0.908203i	$-0.362546\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	− 2.59446e10i	− 1.58163i	−0.612052	−	0.790817i	$-0.709656\pi$
$829$	− 2.59446e10i	− 1.58163i	0.612052	−	0.790817i	$-0.290344\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.30191e10	1.97928
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−3.71464e10	−2.15343
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 9.52701e8i	− 0.0543198i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	2.32103e9i	0.128044i	0.997948	+	0.0640220i	$0.0203928\pi$
$853$	2.32103e9i	0.128044i	−0.997948	+	0.0640220i	$0.979607\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.11516e10	−1.14792	−0.573960	−	0.818884i	$-0.694593\pi$
$857$	−2.11516e10	−1.14792	−0.573960	+	0.818884i	$0.694593\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	3.80299e9	0.199788
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−3.84199e10	−1.95437
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 5.68052e9i	− 0.284374i	−0.989840	−	0.142187i	$-0.954587\pi$
$877$	− 5.68052e9i	− 0.284374i	0.989840	−	0.142187i	$-0.0454134\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.69850e10	−1.32956	−0.664778	−	0.747041i	$-0.731474\pi$
$881$	−2.69850e10	−1.32956	−0.664778	+	0.747041i	$0.731474\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	− 3.20191e10i	− 1.45839i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−8.45989e9	−0.379398
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	− 2.90963e10i	− 1.28488i
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 4.21633e10i	− 1.75161i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.43415e10	−0.586866	−0.293433	−	0.955980i	$-0.594798\pi$
$929$	−1.43415e10	−0.586866	−0.293433	+	0.955980i	$0.594798\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	5.03482e10	1.99938	0.999690	−	0.0249138i	$-0.00793112\pi$
$937$	5.03482e10	1.99938	0.999690	+	0.0249138i	$0.00793112\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.49808e10i	1.36857i	0.729217	+	0.684283i	$0.239885\pi$
$941$	3.49808e10i	1.36857i	−0.729217	+	0.684283i	$0.760115\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	3.48572e10i	1.32392i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.27653e9	0.197480	0.0987400	−	0.995113i	$-0.468519\pi$
$953$	5.27653e9	0.197480	0.0987400	+	0.995113i	$0.468519\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−2.75126e10	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 6.10411e9i	− 0.218664i
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.63185e10	−1.24594	−0.622970	−	0.782246i	$-0.714074\pi$
$977$	−3.63185e10	−1.24594	−0.622970	+	0.782246i	$0.714074\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 5.27422e10i	− 1.78368i
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	5.95849e9	0.198660
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.44203e10i	− 0.460832i	−0.973092	−	0.230416i	$-0.925991\pi$
$997$	− 1.44203e10i	− 0.460832i	0.973092	−	0.230416i	$-0.0740087\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.8.b.d.129.2		2
4.3	odd	2	CM	256.8.b.d.129.2		2
8.3	odd	2	inner	256.8.b.d.129.1		2
8.5	even	2	inner	256.8.b.d.129.1		2
16.3	odd	4		64.8.a.d.1.1		1
16.5	even	4		32.8.a.a.1.1	✓	1
16.11	odd	4		32.8.a.a.1.1	✓	1
16.13	even	4		64.8.a.d.1.1		1
48.5	odd	4		288.8.a.b.1.1		1
48.11	even	4		288.8.a.b.1.1		1
48.29	odd	4		576.8.a.n.1.1		1
48.35	even	4		576.8.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.8.a.a.1.1	✓	1	16.5	even	4
32.8.a.a.1.1	✓	1	16.11	odd	4
64.8.a.d.1.1		1	16.3	odd	4
64.8.a.d.1.1		1	16.13	even	4
256.8.b.d.129.1		2	8.3	odd	2	inner
256.8.b.d.129.1		2	8.5	even	2	inner
256.8.b.d.129.2		2	1.1	even	1	trivial
256.8.b.d.129.2		2	4.3	odd	2	CM
288.8.a.b.1.1		1	48.5	odd	4
288.8.a.b.1.1		1	48.11	even	4
576.8.a.n.1.1		1	48.29	odd	4
576.8.a.n.1.1		1	48.35	even	4

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	256.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+58.0000i q^{5} +2187.00 q^{9} -8898.00i q^{13} -40094.0 q^{17} +74761.0 q^{25} +233230. i q^{29} -563974. i q^{37} -9530.00 q^{41} +126846. i q^{45} -823543. q^{49} +798602. i q^{53} -3.50533e6i q^{61} +516084. q^{65} -3.91742e6 q^{73} +4.78297e6 q^{81} -2.32545e6i q^{85} -9.24617e6 q^{89} -1.75674e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4374 q^{9} - 80188 q^{17} + 149522 q^{25} - 19060 q^{41} - 1647086 q^{49} + 1032168 q^{65} - 7834836 q^{73} + 9565938 q^{81} - 18492340 q^{89} - 35134812 q^{97}+O(q^{100}) \) 2 * q + 4374 * q^9 - 80188 * q^17 + 149522 * q^25 - 19060 * q^41 - 1647086 * q^49 + 1032168 * q^65 - 7834836 * q^73 + 9565938 * q^81 - 18492340 * q^89 - 35134812 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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