LMFDB - Embedded newform 256.9.c.f.255.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,9,Mod(255,256)]

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([1, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	256.c (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:	$104.288924176$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			255.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.9.c.f.255.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-34.0000i q^{3} +5405.00 q^{9} +O(q^{10})$	$q-34.0000i q^{3} +5405.00 q^{9} -27166.0i q^{11} +162434. q^{17} +72286.0i q^{19} -390625. q^{25} -406844. i q^{27} -923644. q^{33} +4.09901e6 q^{41} +5.42640e6i q^{43} +5.76480e6 q^{49} -5.52276e6i q^{51} +2.45772e6 q^{57} -2.41781e7i q^{59} +1.39443e7i q^{67} -3.35676e7 q^{73} +1.32812e7i q^{75} +2.16295e7 q^{81} -3.02100e7i q^{83} +9.55198e7 q^{89} -7.74182e7 q^{97} -1.46832e8i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10810 q^{9}+O(q^{10}) $ 2 * q + 10810 * q^9	$ 2 q + 10810 q^{9} + 324868 q^{17} - 781250 q^{25} - 1847288 q^{33} + 8198012 q^{41} + 11529602 q^{49} + 4915448 q^{57} - 67135108 q^{73} + 43259018 q^{81} + 191039612 q^{89} - 154836476 q^{97}+O(q^{100}) $ 2 * q + 10810 * q^9 + 324868 * q^17 - 781250 * q^25 - 1847288 * q^33 + 8198012 * q^41 + 11529602 * q^49 + 4915448 * q^57 - 67135108 * q^73 + 43259018 * q^81 + 191039612 * q^89 - 154836476 * q^97

Display 100 coefficients 10 coefficients

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
$3$	− 34.0000i	− 0.419753i	−0.977728	−	0.209877i	$-0.932694\pi$
$3$	− 34.0000i	− 0.419753i	0.977728	−	0.209877i	$-0.0673062\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	5405.00	0.823807
$10$	0	0
$11$	− 27166.0i	− 1.85547i	−0.373234	−	0.927737i	$-0.621751\pi$
$11$	− 27166.0i	− 1.85547i	0.373234	−	0.927737i	$-0.378249\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	162434.	1.94483	0.972414	−	0.233261i	$-0.0749397\pi$
$17$	162434.	1.94483	0.972414	+	0.233261i	$0.0749397\pi$
$18$	0	0
$19$	72286.0i	0.554677i	0.960772	+	0.277338i	$0.0894523\pi$
$19$	72286.0i	0.554677i	−0.960772	+	0.277338i	$0.910548\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−390625.	−1.00000
$26$	0	0
$27$	− 406844.i	− 0.765549i
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	−923644.	−0.778841
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.09901e6	1.45058	0.725292	−	0.688441i	$-0.241705\pi$
$41$	4.09901e6	1.45058	0.725292	+	0.688441i	$0.241705\pi$
$42$	0	0
$43$	5.42640e6i	1.58722i	0.608424	+	0.793612i	$0.291802\pi$
$43$	5.42640e6i	1.58722i	−0.608424	+	0.793612i	$0.708198\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	5.76480e6	1.00000
$50$	0	0
$51$	− 5.52276e6i	− 0.816348i
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.45772e6	0.232827
$58$	0	0
$59$	− 2.41781e7i	− 1.99533i	−0.0683312	−	0.997663i	$-0.521767\pi$
$59$	− 2.41781e7i	− 1.99533i	0.0683312	−	0.997663i	$-0.478233\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.39443e7i	0.691986i	0.938237	+	0.345993i	$0.112458\pi$
$67$	1.39443e7i	0.691986i	−0.938237	+	0.345993i	$0.887542\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−3.35676e7	−1.18203	−0.591015	−	0.806661i	$-0.701272\pi$
$73$	−3.35676e7	−1.18203	−0.591015	+	0.806661i	$0.701272\pi$
$74$	0	0
$75$	1.32812e7i	0.419753i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	2.16295e7	0.502466
$82$	0	0
$83$	− 3.02100e7i	− 0.636558i	−0.947997	−	0.318279i	$-0.896895\pi$
$83$	− 3.02100e7i	− 0.636558i	0.947997	−	0.318279i	$-0.103105\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.55198e7	1.52242	0.761208	−	0.648508i	$-0.224607\pi$
$89$	9.55198e7	1.52242	0.761208	+	0.648508i	$0.224607\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.74182e7	−0.874493	−0.437247	−	0.899342i	$-0.644046\pi$
$97$	−7.74182e7	−0.874493	−0.437247	+	0.899342i	$0.644046\pi$
$98$	0	0
$99$	− 1.46832e8i	− 1.52855i
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.84965e8i	− 1.41109i	−0.708664	−	0.705546i	$-0.750702\pi$
$107$	− 1.84965e8i	− 1.41109i	0.708664	−	0.705546i	$-0.249298\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.22024e7	−0.442831	−0.221415	−	0.975180i	$-0.571068\pi$
$113$	−7.22024e7	−0.442831	−0.221415	+	0.975180i	$0.571068\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−5.23633e8	−2.44279
$122$	0	0
$123$	− 1.39366e8i	− 0.608887i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	1.84498e8	0.666242
$130$	0	0
$131$	− 3.39909e8i	− 1.15419i	−0.816677	−	0.577095i	$-0.804186\pi$
$131$	− 3.39909e8i	− 1.15419i	0.816677	−	0.577095i	$-0.195814\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.39511e8	0.963767	0.481884	−	0.876235i	$-0.339953\pi$
$137$	3.39511e8	0.963767	0.481884	+	0.876235i	$0.339953\pi$
$138$	0	0
$139$	− 7.32208e8i	− 1.96144i	−0.195418	−	0.980720i	$-0.562606\pi$
$139$	− 7.32208e8i	− 1.96144i	0.195418	−	0.980720i	$-0.437394\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 1.96003e8i	− 0.419753i
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	8.77956e8	1.60216
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 1.14307e9i	− 1.61929i	−0.586921	−	0.809644i	$-0.699660\pi$
$163$	− 1.14307e9i	− 1.61929i	0.586921	−	0.809644i	$-0.300340\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−8.15731e8	−1.00000
$170$	0	0
$171$	3.90706e8i	0.456947i
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8.22055e8	−0.837544
$178$	0	0
$179$	− 1.90643e9i	− 1.85699i	−0.371349	−	0.928493i	$-0.621105\pi$
$179$	− 1.90643e9i	− 1.85699i	0.371349	−	0.928493i	$-0.378895\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 4.41268e9i	− 3.60858i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	1.43626e9	1.03515	0.517574	−	0.855638i	$-0.326835\pi$
$193$	1.43626e9	1.03515	0.517574	+	0.855638i	$0.326835\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	4.74106e8	0.290463
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.96372e9	1.02919
$210$	0	0
$211$	2.52282e9i	1.27279i	0.771363	+	0.636395i	$0.219575\pi$
$211$	2.52282e9i	1.27279i	−0.771363	+	0.636395i	$0.780425\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.14130e9i	0.496160i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	−2.11133e9	−0.823807
$226$	0	0
$227$	− 3.87828e9i	− 1.46062i	−0.683118	−	0.730308i	$-0.739377\pi$
$227$	− 3.87828e9i	− 1.46062i	0.683118	−	0.730308i	$-0.260623\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.69938e8	−0.159447	−0.0797236	−	0.996817i	$-0.525404\pi$
$233$	−4.69938e8	−0.159447	−0.0797236	+	0.996817i	$0.525404\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−5.80472e8	−0.172073	−0.0860366	−	0.996292i	$-0.527420\pi$
$241$	−5.80472e8	−0.172073	−0.0860366	+	0.996292i	$0.527420\pi$
$242$	0	0
$243$	− 3.40471e9i	− 0.976460i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.02714e9	−0.267197
$250$	0	0
$251$	3.70800e8i	0.0934210i	0.998908	+	0.0467105i	$0.0148738\pi$
$251$	3.70800e8i	0.0934210i	−0.998908	+	0.0467105i	$0.985126\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−8.43940e9	−1.93455	−0.967273	−	0.253738i	$-0.918340\pi$
$257$	−8.43940e9	−1.93455	−0.967273	+	0.253738i	$0.918340\pi$
$258$	0	0
$259$	0	0
$260$	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
$267$	− 3.24767e9i	− 0.639039i
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.06117e10i	1.85547i
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.21245e9	0.354852	0.177426	−	0.984134i	$-0.443223\pi$
$281$	2.21245e9	0.354852	0.177426	+	0.984134i	$0.443223\pi$
$282$	0	0
$283$	1.07196e10i	1.67122i	0.549321	+	0.835611i	$0.314886\pi$
$283$	1.07196e10i	1.67122i	−0.549321	+	0.835611i	$0.685114\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.94090e10	2.78236
$290$	0	0
$291$	2.63222e9i	0.367071i
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.10523e10	−1.42046
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.68482e9i	0.752551i	0.926508	+	0.376276i	$0.122795\pi$
$307$	6.68482e9i	0.752551i	−0.926508	+	0.376276i	$0.877205\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	1.26772e10	1.32083	0.660415	−	0.750901i	$-0.270380\pi$
$313$	1.26772e10	1.32083	0.660415	+	0.750901i	$0.270380\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−6.28882e9	−0.592310
$322$	0	0
$323$	1.17417e10i	1.07875i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.39214e10i	1.99285i	0.0844976	+	0.996424i	$0.473071\pi$
$331$	2.39214e10i	1.99285i	−0.0844976	+	0.996424i	$0.526929\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−2.57940e10	−1.99986	−0.999930	−	0.0118516i	$-0.996227\pi$
$337$	−2.57940e10	−1.99986	−0.999930	+	0.0118516i	$0.996227\pi$
$338$	0	0
$339$	2.45488e9i	0.185880i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.92343e9i	0.546507i	0.961942	+	0.273253i	$0.0880997\pi$
$347$	7.92343e9i	0.546507i	−0.961942	+	0.273253i	$0.911900\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.37121e10	0.883093	0.441546	−	0.897238i	$-0.354430\pi$
$353$	1.37121e10	0.883093	0.441546	+	0.897238i	$0.354430\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	1.17583e10	0.692334
$362$	0	0
$363$	1.78035e10i	1.02537i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	2.21551e10	1.19500
$370$	0	0
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 7.66145e9i	− 0.371325i	−0.982614	−	0.185663i	$-0.940557\pi$
$379$	− 7.66145e9i	− 0.371325i	0.982614	−	0.185663i	$-0.0594431\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	2.93297e10i	1.30757i
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−1.15569e10	−0.484474
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.85927e10	0.719061	0.359530	−	0.933133i	$-0.382937\pi$
$401$	1.85927e10	0.719061	0.359530	+	0.933133i	$0.382937\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	6.23419e9	0.222785	0.111393	−	0.993776i	$-0.464469\pi$
$409$	6.23419e9	0.222785	0.111393	+	0.993776i	$0.464469\pi$
$410$	0	0
$411$	− 1.15434e10i	− 0.404544i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.48951e10	−0.823321
$418$	0	0
$419$	5.40872e10i	1.75484i	0.479720	+	0.877422i	$0.340738\pi$
$419$	5.40872e10i	1.75484i	−0.479720	+	0.877422i	$0.659262\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.34508e10	−1.94483
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−6.86319e10	−1.95243	−0.976213	−	0.216813i	$-0.930434\pi$
$433$	−6.86319e10	−1.95243	−0.976213	+	0.216813i	$0.930434\pi$
$434$	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
$441$	3.11587e10	0.823807
$442$	0	0
$443$	6.61486e10i	1.71754i	0.512364	+	0.858768i	$0.328770\pi$
$443$	6.61486e10i	1.71754i	−0.512364	+	0.858768i	$0.671230\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.89524e10	0.958405	0.479202	−	0.877704i	$-0.340926\pi$
$449$	3.89524e10	0.958405	0.479202	+	0.877704i	$0.340926\pi$
$450$	0	0
$451$	− 1.11354e11i	− 2.69152i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.31242e10	−0.988681	−0.494340	−	0.869268i	$-0.664590\pi$
$457$	−4.31242e10	−0.988681	−0.494340	+	0.869268i	$0.664590\pi$
$458$	0	0
$459$	− 6.60853e10i	− 1.48886i
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 9.41185e10i	− 1.97883i	−0.145128	−	0.989413i	$-0.546359\pi$
$467$	− 9.41185e10i	− 1.97883i	0.145128	−	0.989413i	$-0.453641\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.47414e11	2.94505
$474$	0	0
$475$	− 2.82367e10i	− 0.554677i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	−3.88645e10	−0.679701
$490$	0	0
$491$	− 9.95056e10i	− 1.71207i	−0.516918	−	0.856035i	$-0.672921\pi$
$491$	− 9.95056e10i	− 1.71207i	0.516918	−	0.856035i	$-0.327079\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	1.02919e11i	1.65995i	0.557801	+	0.829975i	$0.311645\pi$
$499$	1.02919e11i	1.65995i	−0.557801	+	0.829975i	$0.688355\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.77348e10i	0.419753i
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	2.94091e10	0.424632
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.27450e10	−0.987308	−0.493654	−	0.869658i	$-0.664339\pi$
$521$	−7.27450e10	−0.987308	−0.493654	+	0.869658i	$0.664339\pi$
$522$	0	0
$523$	1.41570e9i	0.0189219i	0.999955	+	0.00946097i	$0.00301157\pi$
$523$	1.41570e9i	0.0189219i	−0.999955	+	0.00946097i	$0.996988\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	7.83110e10	1.00000
$530$	0	0
$531$	− 1.30683e11i	− 1.64376i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−6.48186e10	−0.779476
$538$	0	0
$539$	− 1.56607e11i	− 1.85547i
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 1.99228e10i	− 0.222537i	−0.993790	−	0.111268i	$-0.964509\pi$
$547$	− 1.99228e10i	− 0.222537i	0.993790	−	0.111268i	$-0.0354913\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−1.50031e11	−1.51471
$562$	0	0
$563$	− 1.38789e11i	− 1.38141i	−0.723136	−	0.690705i	$-0.757300\pi$
$563$	− 1.38789e11i	− 1.38141i	0.723136	−	0.690705i	$-0.242700\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.44287e11	1.37651	0.688255	−	0.725468i	$-0.258377\pi$
$569$	1.44287e11	1.37651	0.688255	+	0.725468i	$0.258377\pi$
$570$	0	0
$571$	− 1.64593e11i	− 1.54834i	−0.632978	−	0.774170i	$-0.718168\pi$
$571$	− 1.64593e11i	− 1.54834i	0.632978	−	0.774170i	$-0.281832\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	2.21678e11	1.99995	0.999976	−	0.00693236i	$-0.00220666\pi$
$577$	2.21678e11	1.99995	0.999976	+	0.00693236i	$0.00220666\pi$
$578$	0	0
$579$	− 4.88327e10i	− 0.434507i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.29666e11i	1.09213i	0.837742	+	0.546066i	$0.183875\pi$
$587$	1.29666e11i	1.09213i	−0.837742	+	0.546066i	$0.816125\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−2.45734e11	−1.98722	−0.993612	−	0.112847i	$-0.964003\pi$
$593$	−2.45734e11	−1.98722	−0.993612	+	0.112847i	$0.964003\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	2.48165e11	1.90214	0.951069	−	0.308978i	$-0.0999870\pi$
$601$	2.48165e11	1.90214	0.951069	+	0.308978i	$0.0999870\pi$
$602$	0	0
$603$	7.53689e10i	0.570063i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.32420e11	−0.913722	−0.456861	−	0.889538i	$-0.651026\pi$
$617$	−1.32420e11	−0.913722	−0.456861	+	0.889538i	$0.651026\pi$
$618$	0	0
$619$	− 9.06943e10i	− 0.617756i	−0.951102	−	0.308878i	$-0.900046\pi$
$619$	− 9.06943e10i	− 0.617756i	0.951102	−	0.308878i	$-0.0999536\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.52588e11	1.00000
$626$	0	0
$627$	− 6.67665e10i	− 0.432005i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	8.57759e10	0.534258
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9.79076e9	0.0579942	0.0289971	−	0.999579i	$-0.490769\pi$
$641$	9.79076e9	0.0579942	0.0289971	+	0.999579i	$0.490769\pi$
$642$	0	0
$643$	− 7.65969e10i	− 0.448092i	−0.974579	−	0.224046i	$-0.928073\pi$
$643$	− 7.65969e10i	− 0.448092i	0.974579	−	0.224046i	$-0.0719266\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	−6.56822e11	−3.70228
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0		−	1.00000i	$-0.5\pi$
$653$	0	0			1.00000i	$0.5\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.81433e11	−0.973764
$658$	0	0
$659$	3.62926e11i	1.92432i	0.272493	+	0.962158i	$0.412152\pi$
$659$	3.62926e11i	1.92432i	−0.272493	+	0.962158i	$0.587848\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	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
$673$	7.87721e9	0.0383983	0.0191992	−	0.999816i	$-0.493888\pi$
$673$	7.87721e9	0.0383983	0.0191992	+	0.999816i	$0.493888\pi$
$674$	0	0
$675$	1.58923e11i	0.765549i
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−1.31862e11	−0.613098
$682$	0	0
$683$	1.64741e11i	0.757040i	0.925593	+	0.378520i	$0.123567\pi$
$683$	1.64741e11i	0.757040i	−0.925593	+	0.378520i	$0.876433\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	2.82749e11i	1.24019i	0.784526	+	0.620095i	$0.212906\pi$
$691$	2.82749e11i	1.24019i	−0.784526	+	0.620095i	$0.787094\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	6.65818e11	2.82114
$698$	0	0
$699$	1.59779e10i	0.0669285i
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	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	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	1.97361e10i	0.0722283i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	2.61512e10	0.0925936
$730$	0	0
$731$	8.81432e11i	3.08688i
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.78810e11	1.28396
$738$	0	0
$739$	− 3.80751e11i	− 1.27662i	−0.769778	−	0.638312i	$-0.779633\pi$
$739$	− 3.80751e11i	− 1.27662i	0.769778	−	0.638312i	$-0.220367\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.63285e11i	− 0.524401i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	1.26072e10	0.0392138
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.45467e10	0.162641	0.0813204	−	0.996688i	$-0.474086\pi$
$761$	5.45467e10	0.162641	0.0813204	+	0.996688i	$0.474086\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−6.94260e11	−1.98526	−0.992628	−	0.121201i	$-0.961325\pi$
$769$	−6.94260e11	−1.98526	−0.992628	+	0.121201i	$0.961325\pi$
$770$	0	0
$771$	2.86940e11i	0.812032i
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.96301e11i	0.804605i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	6.21875e11i	1.62108i	0.585684	+	0.810540i	$0.300826\pi$
$787$	6.21875e11i	1.62108i	−0.585684	+	0.810540i	$0.699174\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	5.16285e11	1.25418
$802$	0	0
$803$	9.11896e11i	2.19323i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−5.68377e11	−1.32691	−0.663456	−	0.748215i	$-0.730911\pi$
$809$	−5.68377e11	−1.32691	−0.663456	+	0.748215i	$0.730911\pi$
$810$	0	0
$811$	7.82841e11i	1.80963i	0.425804	+	0.904815i	$0.359991\pi$
$811$	7.82841e11i	1.80963i	−0.425804	+	0.904815i	$0.640009\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−3.92253e11	−0.880396
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	3.60798e11	0.778841
$826$	0	0
$827$	− 8.84989e11i	− 1.89198i	−0.324200	−	0.945988i	$-0.605095\pi$
$827$	− 8.84989e11i	− 1.89198i	0.324200	−	0.945988i	$-0.394905\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	9.36400e11	1.94483
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−5.00246e11	−1.00000
$842$	0	0
$843$	− 7.52232e10i	− 0.148950i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3.64468e11	0.701501
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.07825e12	1.99892	0.999462	−	0.0328011i	$-0.0104428\pi$
$857$	1.07825e12	1.99892	0.999462	+	0.0328011i	$0.0104428\pi$
$858$	0	0
$859$	4.32719e11i	0.794755i	0.917655	+	0.397377i	$0.130080\pi$
$859$	4.32719e11i	0.794755i	−0.917655	+	0.397377i	$0.869920\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 6.59908e11i	− 1.16790i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−4.18446e11	−0.720414
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−9.39089e11	−1.55885	−0.779423	−	0.626498i	$-0.784488\pi$
$881$	−9.39089e11	−1.55885	−0.779423	+	0.626498i	$0.784488\pi$
$882$	0	0
$883$	− 1.17202e12i	− 1.92793i	−0.266030	−	0.963965i	$-0.585712\pi$
$883$	− 1.17202e12i	− 1.92793i	0.266030	−	0.963965i	$-0.414288\pi$
$884$	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
$891$	− 5.87587e11i	− 0.932313i
$892$	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
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.17048e11i	0.468486i	0.972178	+	0.234243i	$0.0752611\pi$
$907$	3.17048e11i	0.468486i	−0.972178	+	0.234243i	$0.924739\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−8.20684e11	−1.18112
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	2.27284e11	0.315886
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.12157e12	−1.50579	−0.752895	−	0.658140i	$-0.771343\pi$
$929$	−1.12157e12	−1.50579	−0.752895	+	0.658140i	$0.771343\pi$
$930$	0	0
$931$	4.16714e11i	0.554677i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.03224e9	0.00393373	0.00196687	−	0.999998i	$-0.499374\pi$
$937$	3.03224e9	0.00393373	0.00196687	+	0.999998i	$0.499374\pi$
$938$	0	0
$939$	− 4.31026e11i	− 0.554422i
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.59570e11i	1.19310i	0.802576	+	0.596550i	$0.203462\pi$
$947$	9.59570e11i	1.19310i	−0.802576	+	0.596550i	$0.796538\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1.57684e12	−1.91169	−0.955843	−	0.293879i	$-0.905054\pi$
$953$	−1.57684e12	−1.91169	−0.955843	+	0.293879i	$0.905054\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.52891e11	1.00000
$962$	0	0
$963$	− 9.99738e11i	− 1.16247i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	3.99218e11	0.452809
$970$	0	0
$971$	− 8.99918e11i	− 1.01234i	−0.862434	−	0.506170i	$-0.831061\pi$
$971$	− 8.99918e11i	− 1.01234i	0.862434	−	0.506170i	$-0.168939\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.30947e12	1.43720	0.718598	−	0.695425i	$-0.244784\pi$
$977$	1.30947e12	1.43720	0.718598	+	0.695425i	$0.244784\pi$
$978$	0	0
$979$	− 2.59489e12i	− 2.82480i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	8.13327e11	0.836504
$994$	0	0
$995$	0	0
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.9.c.f.255.1		2
4.3	odd	2	inner	256.9.c.f.255.2		2
8.3	odd	2	CM	256.9.c.f.255.1		2
8.5	even	2	inner	256.9.c.f.255.2		2
16.3	odd	4		32.9.d.a.15.1		1
16.5	even	4		32.9.d.a.15.1		1
16.11	odd	4		8.9.d.a.3.1	✓	1
16.13	even	4		8.9.d.a.3.1	✓	1
48.5	odd	4		288.9.b.a.271.1		1
48.11	even	4		72.9.b.a.19.1		1
48.29	odd	4		72.9.b.a.19.1		1
48.35	even	4		288.9.b.a.271.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.9.d.a.3.1	✓	1	16.11	odd	4
8.9.d.a.3.1	✓	1	16.13	even	4
32.9.d.a.15.1		1	16.3	odd	4
32.9.d.a.15.1		1	16.5	even	4
72.9.b.a.19.1		1	48.11	even	4
72.9.b.a.19.1		1	48.29	odd	4
256.9.c.f.255.1		2	1.1	even	1	trivial
256.9.c.f.255.1		2	8.3	odd	2	CM
256.9.c.f.255.2		2	4.3	odd	2	inner
256.9.c.f.255.2		2	8.5	even	2	inner
288.9.b.a.271.1		1	48.5	odd	4
288.9.b.a.271.1		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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-34.0000i q^{3} +5405.00 q^{9} +O(q^{10})\)	\(q-34.0000i q^{3} +5405.00 q^{9} -27166.0i q^{11} +162434. q^{17} +72286.0i q^{19} -390625. q^{25} -406844. i q^{27} -923644. q^{33} +4.09901e6 q^{41} +5.42640e6i q^{43} +5.76480e6 q^{49} -5.52276e6i q^{51} +2.45772e6 q^{57} -2.41781e7i q^{59} +1.39443e7i q^{67} -3.35676e7 q^{73} +1.32812e7i q^{75} +2.16295e7 q^{81} -3.02100e7i q^{83} +9.55198e7 q^{89} -7.74182e7 q^{97} -1.46832e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10810 q^{9}+O(q^{10}) \) 2 * q + 10810 * q^9	\( 2 q + 10810 q^{9} + 324868 q^{17} - 781250 q^{25} - 1847288 q^{33} + 8198012 q^{41} + 11529602 q^{49} + 4915448 q^{57} - 67135108 q^{73} + 43259018 q^{81} + 191039612 q^{89} - 154836476 q^{97}+O(q^{100}) \) 2 * q + 10810 * q^9 + 324868 * q^17 - 781250 * q^25 - 1847288 * q^33 + 8198012 * q^41 + 11529602 * q^49 + 4915448 * q^57 - 67135108 * q^73 + 43259018 * q^81 + 191039612 * q^89 - 154836476 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more