LMFDB - Embedded newform 256.7.c.d.255.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,7,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, 7, names="a")

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

chi := DirichletCharacter("256.255");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 7 $
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:	$58.8938454067$
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:	$\Z[a_1, \ldots, a_{11}]$
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			256.7.c.d.255.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.0000i q^{3} -1387.00 q^{9} +O(q^{10})$	$q-46.0000i q^{3} -1387.00 q^{9} -2338.00i q^{11} -1726.00 q^{17} +2482.00i q^{19} -15625.0 q^{25} +30268.0i q^{27} -107548. q^{33} -134642. q^{41} -74914.0i q^{43} +117649. q^{49} +79396.0i q^{51} +114172. q^{57} +304958. i q^{59} +596626. i q^{67} +593134. q^{73} +718750. i q^{75} +381205. q^{81} -678926. i q^{83} +357262. q^{89} +1.82275e6 q^{97} +3.24281e6i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2774 q^{9}+O(q^{10}) $ 2 * q - 2774 * q^9	$ 2 q - 2774 q^{9} - 3452 q^{17} - 31250 q^{25} - 215096 q^{33} - 269284 q^{41} + 235298 q^{49} + 228344 q^{57} + 1186268 q^{73} + 762410 q^{81} + 714524 q^{89} + 3645508 q^{97}+O(q^{100}) $ 2 * q - 2774 * q^9 - 3452 * q^17 - 31250 * q^25 - 215096 * q^33 - 269284 * q^41 + 235298 * q^49 + 228344 * q^57 + 1186268 * q^73 + 762410 * q^81 + 714524 * q^89 + 3645508 * 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
$3$	− 46.0000i	− 1.70370i	−0.523783	−	0.851852i	$-0.675480\pi$
$3$	− 46.0000i	− 1.70370i	0.523783	−	0.851852i	$-0.324520\pi$
$4$	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
$9$	−1387.00	−1.90261
$10$	0	0
$11$	− 2338.00i	− 1.75657i	−0.478134	−	0.878287i	$-0.658687\pi$
$11$	− 2338.00i	− 1.75657i	0.478134	−	0.878287i	$-0.341313\pi$
$12$	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
$17$	−1726.00	−0.351313	−0.175656	−	0.984452i	$-0.556205\pi$
$17$	−1726.00	−0.351313	−0.175656	+	0.984452i	$0.556205\pi$
$18$	0	0
$19$	2482.00i	0.361860i	0.983496	+	0.180930i	$0.0579108\pi$
$19$	2482.00i	0.361860i	−0.983496	+	0.180930i	$0.942089\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	−15625.0	−1.00000
$26$	0	0
$27$	30268.0i	1.53777i
$28$	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
$33$	−107548.	−2.99268
$34$	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
$41$	−134642.	−1.95357	−0.976785	−	0.214222i	$-0.931278\pi$
$41$	−134642.	−1.95357	−0.976785	+	0.214222i	$0.931278\pi$
$42$	0	0
$43$	− 74914.0i	− 0.942232i	−0.882071	−	0.471116i	$-0.843851\pi$
$43$	− 74914.0i	− 0.942232i	0.882071	−	0.471116i	$-0.156149\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	117649.	1.00000
$50$	0	0
$51$	79396.0i	0.598533i
$52$	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
$57$	114172.	0.616503
$58$	0	0
$59$	304958.i	1.48485i	0.669927	+	0.742427i	$0.266326\pi$
$59$	304958.i	1.48485i	−0.669927	+	0.742427i	$0.733674\pi$
$60$	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
$67$	596626.i	1.98371i	0.127380	+	0.991854i	$0.459343\pi$
$67$	596626.i	1.98371i	−0.127380	+	0.991854i	$0.540657\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	593134.	1.52470	0.762350	−	0.647165i	$-0.224046\pi$
$73$	593134.	1.52470	0.762350	+	0.647165i	$0.224046\pi$
$74$	0	0
$75$	718750.i	1.70370i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	381205.	0.717304
$82$	0	0
$83$	− 678926.i	− 1.18738i	−0.804695	−	0.593688i	$-0.797671\pi$
$83$	− 678926.i	− 1.18738i	0.804695	−	0.593688i	$-0.202329\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	357262.	0.506777	0.253388	−	0.967365i	$-0.418455\pi$
$89$	357262.	0.506777	0.253388	+	0.967365i	$0.418455\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.82275e6	1.99716	0.998580	−	0.0532728i	$-0.0169653\pi$
$97$	1.82275e6	1.99716	0.998580	+	0.0532728i	$0.0169653\pi$
$98$	0	0
$99$	3.24281e6i	3.34207i
$100$	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
$107$	474014.i	0.386937i	0.981107	+	0.193468i	$0.0619737\pi$
$107$	474014.i	0.386937i	−0.981107	+	0.193468i	$0.938026\pi$
$108$	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
$113$	−2.81289e6	−1.94948	−0.974738	−	0.223350i	$-0.928301\pi$
$113$	−2.81289e6	−1.94948	−0.974738	+	0.223350i	$0.928301\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3.69468e6	−2.08555
$122$	0	0
$123$	6.19353e6i	3.32830i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	−3.44604e6	−1.60528
$130$	0	0
$131$	2.95362e6i	1.31383i	0.753963	+	0.656917i	$0.228140\pi$
$131$	2.95362e6i	1.31383i	−0.753963	+	0.656917i	$0.771860\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	80206.0	0.0311921	0.0155961	−	0.999878i	$-0.495035\pi$
$137$	80206.0	0.0311921	0.0155961	+	0.999878i	$0.495035\pi$
$138$	0	0
$139$	− 3.19856e6i	− 1.19100i	−0.803357	−	0.595498i	$-0.796955\pi$
$139$	− 3.19856e6i	− 1.19100i	0.803357	−	0.595498i	$-0.203045\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 5.41185e6i	− 1.70370i
$148$	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
$153$	2.39396e6	0.668410
$154$	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
$163$	7.72059e6i	1.78274i	0.453277	+	0.891370i	$0.350255\pi$
$163$	7.72059e6i	1.78274i	−0.453277	+	0.891370i	$0.649745\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−4.82681e6	−1.00000
$170$	0	0
$171$	− 3.44253e6i	− 0.688478i
$172$	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
$177$	1.40281e7	2.52975
$178$	0	0
$179$	− 3.22888e6i	− 0.562979i	−0.959564	−	0.281490i	$-0.909171\pi$
$179$	− 3.22888e6i	− 0.562979i	0.959564	−	0.281490i	$-0.0908285\pi$
$180$	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
$187$	4.03539e6i	0.617107i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	−1.00100e7	−1.39240	−0.696198	−	0.717850i	$-0.745126\pi$
$193$	−1.00100e7	−1.39240	−0.696198	+	0.717850i	$0.745126\pi$
$194$	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
$201$	2.74448e7	3.37965
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.80292e6	0.635634
$210$	0	0
$211$	− 1.86421e7i	− 1.98448i	−0.124340	−	0.992240i	$-0.539681\pi$
$211$	− 1.86421e7i	− 1.98448i	0.124340	−	0.992240i	$-0.460319\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 2.72842e7i	− 2.59764i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	2.16719e7	1.90261
$226$	0	0
$227$	− 1.97707e7i	− 1.69023i	−0.534586	−	0.845114i	$-0.679533\pi$
$227$	− 1.97707e7i	− 1.69023i	0.534586	−	0.845114i	$-0.320467\pi$
$228$	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
$233$	−1.10622e7	−0.874530	−0.437265	−	0.899333i	$-0.644053\pi$
$233$	−1.10622e7	−0.874530	−0.437265	+	0.899333i	$0.644053\pi$
$234$	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
$241$	−2.65018e7	−1.89332	−0.946659	−	0.322237i	$-0.895565\pi$
$241$	−2.65018e7	−1.89332	−0.946659	+	0.322237i	$0.895565\pi$
$242$	0	0
$243$	4.52994e6i	0.315699i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−3.12306e7	−2.02294
$250$	0	0
$251$	− 1.31193e7i	− 0.829640i	−0.909904	−	0.414820i	$-0.863845\pi$
$251$	− 1.31193e7i	− 0.829640i	0.909904	−	0.414820i	$-0.136155\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.89723e7	−1.11769	−0.558844	−	0.829273i	$-0.688755\pi$
$257$	−1.89723e7	−1.11769	−0.558844	+	0.829273i	$0.688755\pi$
$258$	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
$267$	− 1.64341e7i	− 0.863398i
$268$	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
$275$	3.65312e7i	1.75657i
$276$	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
$281$	−4.28969e7	−1.93333	−0.966667	−	0.256038i	$-0.917583\pi$
$281$	−4.28969e7	−1.93333	−0.966667	+	0.256038i	$0.917583\pi$
$282$	0	0
$283$	1.91505e7i	0.844931i	0.906379	+	0.422466i	$0.138835\pi$
$283$	1.91505e7i	0.844931i	−0.906379	+	0.422466i	$0.861165\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−2.11585e7	−0.876579
$290$	0	0
$291$	− 8.38467e7i	− 3.40257i
$292$	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
$297$	7.07666e7	2.70121
$298$	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
$307$	− 5.97121e6i	− 0.206370i	−0.994662	−	0.103185i	$-0.967097\pi$
$307$	− 5.97121e6i	− 0.206370i	0.994662	−	0.103185i	$-0.0329034\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−9.06351e6	−0.295572	−0.147786	−	0.989019i	$-0.547215\pi$
$313$	−9.06351e6	−0.295572	−0.147786	+	0.989019i	$0.547215\pi$
$314$	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
$321$	2.18046e7	0.659225
$322$	0	0
$323$	− 4.28393e6i	− 0.127126i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 4.59882e6i	− 0.126813i	−0.997988	−	0.0634063i	$-0.979804\pi$
$331$	− 4.59882e6i	− 0.126813i	0.997988	−	0.0634063i	$-0.0201964\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5.36426e7	1.40159	0.700794	−	0.713364i	$-0.252829\pi$
$337$	5.36426e7	1.40159	0.700794	+	0.713364i	$0.252829\pi$
$338$	0	0
$339$	1.29393e8i	3.32133i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 4.72029e7i	− 1.12975i	−0.825178	−	0.564873i	$-0.808925\pi$
$347$	− 4.72029e7i	− 1.12975i	0.825178	−	0.564873i	$-0.191075\pi$
$348$	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
$353$	−6.52211e7	−1.48274	−0.741368	−	0.671099i	$-0.765823\pi$
$353$	−6.52211e7	−1.48274	−0.741368	+	0.671099i	$0.765823\pi$
$354$	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
$361$	4.08856e7	0.869057
$362$	0	0
$363$	1.69955e8i	3.55316i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	1.86748e8	3.71687
$370$	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
$379$	2.72157e7i	0.499921i	0.968256	+	0.249961i	$0.0804177\pi$
$379$	2.72157e7i	0.499921i	−0.968256	+	0.249961i	$0.919582\pi$
$380$	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
$387$	1.03906e8i	1.79270i
$388$	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
$393$	1.35866e8	2.23838
$394$	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
$401$	−7.99344e7	−1.23965	−0.619827	−	0.784738i	$-0.712797\pi$
$401$	−7.99344e7	−1.23965	−0.619827	+	0.784738i	$0.712797\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−1.30356e8	−1.90529	−0.952644	−	0.304088i	$-0.901648\pi$
$409$	−1.30356e8	−1.90529	−0.952644	+	0.304088i	$0.901648\pi$
$410$	0	0
$411$	− 3.68948e6i	− 0.0531422i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.47134e8	−2.02910
$418$	0	0
$419$	− 1.34918e8i	− 1.83412i	−0.398744	−	0.917062i	$-0.630554\pi$
$419$	− 1.34918e8i	− 1.83412i	0.398744	−	0.917062i	$-0.369446\pi$
$420$	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
$425$	2.69688e7	0.351313
$426$	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
$433$	−1.32005e8	−1.62603	−0.813014	−	0.582244i	$-0.802175\pi$
$433$	−1.32005e8	−1.62603	−0.813014	+	0.582244i	$0.802175\pi$
$434$	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
$441$	−1.63179e8	−1.90261
$442$	0	0
$443$	1.59916e8i	1.83942i	0.392595	+	0.919711i	$0.371577\pi$
$443$	1.59916e8i	1.83942i	−0.392595	+	0.919711i	$0.628423\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.25702e8	1.38869	0.694344	−	0.719643i	$-0.255694\pi$
$449$	1.25702e8	1.38869	0.694344	+	0.719643i	$0.255694\pi$
$450$	0	0
$451$	3.14793e8i	3.43159i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1.35637e8	−1.42112	−0.710558	−	0.703639i	$-0.751557\pi$
$457$	−1.35637e8	−1.42112	−0.710558	+	0.703639i	$0.751557\pi$
$458$	0	0
$459$	− 5.22426e7i	− 0.540240i
$460$	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
$467$	− 2.22058e7i	− 0.218030i	−0.994040	−	0.109015i	$-0.965230\pi$
$467$	− 2.22058e7i	− 0.218030i	0.994040	−	0.109015i	$-0.0347696\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.75149e8	−1.65510
$474$	0	0
$475$	− 3.87812e7i	− 0.361860i
$476$	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
$489$	3.55147e8	3.03726
$490$	0	0
$491$	− 8.73643e7i	− 0.738056i	−0.929418	−	0.369028i	$-0.879691\pi$
$491$	− 8.73643e7i	− 0.738056i	0.929418	−	0.369028i	$-0.120309\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 8.32468e7i	− 0.669986i	−0.942221	−	0.334993i	$-0.891266\pi$
$499$	− 8.32468e7i	− 0.669986i	0.942221	−	0.334993i	$-0.108734\pi$
$500$	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
$507$	2.22033e8i	1.70370i
$508$	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
$513$	−7.51252e7	−0.556459
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.98899e8	1.40644	0.703218	−	0.710974i	$-0.251746\pi$
$521$	1.98899e8	1.40644	0.703218	+	0.710974i	$0.251746\pi$
$522$	0	0
$523$	− 2.63549e8i	− 1.84228i	−0.389229	−	0.921141i	$-0.627259\pi$
$523$	− 2.63549e8i	− 1.84228i	0.389229	−	0.921141i	$-0.372741\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	1.48036e8	1.00000
$530$	0	0
$531$	− 4.22977e8i	− 2.82509i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−1.48528e8	−0.959150
$538$	0	0
$539$	− 2.75063e8i	− 1.75657i
$540$	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
$547$	− 1.50088e8i	− 0.917030i	−0.888687	−	0.458515i	$-0.848382\pi$
$547$	− 1.50088e8i	− 0.917030i	0.888687	−	0.458515i	$-0.151618\pi$
$548$	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
$561$	1.85628e8	1.05137
$562$	0	0
$563$	− 2.03362e8i	− 1.13958i	−0.821791	−	0.569789i	$-0.807025\pi$
$563$	− 2.03362e8i	− 1.13958i	0.821791	−	0.569789i	$-0.192975\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.62709e8	1.96889	0.984445	−	0.175695i	$-0.0562174\pi$
$569$	3.62709e8	1.96889	0.984445	+	0.175695i	$0.0562174\pi$
$570$	0	0
$571$	3.58715e8i	1.92682i	0.268035	+	0.963409i	$0.413626\pi$
$571$	3.58715e8i	1.92682i	−0.268035	+	0.963409i	$0.586374\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1.99757e6	−0.0103986	−0.00519929	−	0.999986i	$-0.501655\pi$
$577$	−1.99757e6	−0.0103986	−0.00519929	+	0.999986i	$0.501655\pi$
$578$	0	0
$579$	4.60461e8i	2.37223i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 2.97402e8i	− 1.47038i	−0.677862	−	0.735189i	$-0.737094\pi$
$587$	− 2.97402e8i	− 1.47038i	0.677862	−	0.735189i	$-0.262906\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.68860e8	1.28933	0.644663	−	0.764467i	$-0.276998\pi$
$593$	2.68860e8	1.28933	0.644663	+	0.764467i	$0.276998\pi$
$594$	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
$601$	2.26861e8	1.04505	0.522525	−	0.852624i	$-0.324990\pi$
$601$	2.26861e8	1.04505	0.522525	+	0.852624i	$0.324990\pi$
$602$	0	0
$603$	− 8.27520e8i	− 3.77422i
$604$	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
$617$	−3.44191e8	−1.46536	−0.732679	−	0.680575i	$-0.761730\pi$
$617$	−3.44191e8	−1.46536	−0.732679	+	0.680575i	$0.761730\pi$
$618$	0	0
$619$	4.68505e8i	1.97534i	0.156543	+	0.987671i	$0.449965\pi$
$619$	4.68505e8i	1.97534i	−0.156543	+	0.987671i	$0.550035\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2.44141e8	1.00000
$626$	0	0
$627$	− 2.66934e8i	− 1.08293i
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	−8.57535e8	−3.38096
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4.82154e8	−1.83068	−0.915338	−	0.402687i	$-0.868076\pi$
$641$	−4.82154e8	−1.83068	−0.915338	+	0.402687i	$0.868076\pi$
$642$	0	0
$643$	− 2.83407e8i	− 1.06605i	−0.846099	−	0.533025i	$-0.821055\pi$
$643$	− 2.83407e8i	− 1.06605i	0.846099	−	0.533025i	$-0.178945\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	7.12992e8	2.60826
$650$	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
$657$	−8.22677e8	−2.90090
$658$	0	0
$659$	− 3.12918e8i	− 1.09339i	−0.837332	−	0.546694i	$-0.815886\pi$
$659$	− 3.12918e8i	− 1.09339i	0.837332	−	0.546694i	$-0.184114\pi$
$660$	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
$673$	−2.41386e8	−0.791895	−0.395947	−	0.918273i	$-0.629584\pi$
$673$	−2.41386e8	−0.791895	−0.395947	+	0.918273i	$0.629584\pi$
$674$	0	0
$675$	− 4.72938e8i	− 1.53777i
$676$	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
$681$	−9.09454e8	−2.87965
$682$	0	0
$683$	− 4.93943e8i	− 1.55030i	−0.631779	−	0.775148i	$-0.717675\pi$
$683$	− 4.93943e8i	− 1.55030i	0.631779	−	0.775148i	$-0.282325\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	6.55966e8i	1.98814i	0.108734	+	0.994071i	$0.465320\pi$
$691$	6.55966e8i	1.98814i	−0.108734	+	0.994071i	$0.534680\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.32392e8	0.686314
$698$	0	0
$699$	5.08862e8i	1.48994i
$700$	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
$723$	1.21908e9i	3.22565i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	4.86276e8	1.25516
$730$	0	0
$731$	1.29302e8i	0.331018i
$732$	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
$737$	1.39491e9	3.48453
$738$	0	0
$739$	− 4.94167e8i	− 1.22445i	−0.790685	−	0.612224i	$-0.790275\pi$
$739$	− 4.94167e8i	− 1.22445i	0.790685	−	0.612224i	$-0.209725\pi$
$740$	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
$747$	9.41670e8i	2.25911i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	−6.03488e8	−1.41346
$754$	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
$761$	−2.86987e8	−0.651191	−0.325595	−	0.945509i	$-0.605565\pi$
$761$	−2.86987e8	−0.651191	−0.325595	+	0.945509i	$0.605565\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	6.98978e8	1.53704	0.768519	−	0.639827i	$-0.220994\pi$
$769$	6.98978e8	1.53704	0.768519	+	0.639827i	$0.220994\pi$
$770$	0	0
$771$	8.72725e8i	1.90421i
$772$	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
$779$	− 3.34181e8i	− 0.706919i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	9.26584e8i	1.90091i	0.310868	+	0.950453i	$0.399380\pi$
$787$	9.26584e8i	1.90091i	−0.310868	+	0.950453i	$0.600620\pi$
$788$	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
$801$	−4.95522e8	−0.964197
$802$	0	0
$803$	− 1.38675e9i	− 2.67825i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	8.53143e8	1.61130	0.805649	−	0.592393i	$-0.201817\pi$
$809$	8.53143e8	1.61130	0.805649	+	0.592393i	$0.201817\pi$
$810$	0	0
$811$	3.45583e8i	0.647873i	0.946079	+	0.323937i	$0.105006\pi$
$811$	3.45583e8i	0.647873i	−0.946079	+	0.323937i	$0.894994\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.85937e8	0.340956
$818$	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
$825$	1.68044e9	2.99268
$826$	0	0
$827$	− 5.79438e8i	− 1.02445i	−0.858851	−	0.512225i	$-0.828821\pi$
$827$	− 5.79438e8i	− 1.02445i	0.858851	−	0.512225i	$-0.171179\pi$
$828$	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
$833$	−2.03062e8	−0.351313
$834$	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
$841$	−5.94823e8	−1.00000
$842$	0	0
$843$	1.97326e9i	3.29383i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	8.80924e8	1.43951
$850$	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
$857$	9.11769e8	1.44858	0.724290	−	0.689496i	$-0.242168\pi$
$857$	9.11769e8	1.44858	0.724290	+	0.689496i	$0.242168\pi$
$858$	0	0
$859$	8.15878e8i	1.28720i	0.765363	+	0.643599i	$0.222560\pi$
$859$	8.15878e8i	1.28720i	−0.765363	+	0.643599i	$0.777440\pi$
$860$	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
$867$	9.73291e8i	1.49343i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−2.52816e9	−3.79981
$874$	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
$881$	3.74799e8	0.548114	0.274057	−	0.961714i	$-0.411634\pi$
$881$	3.74799e8	0.548114	0.274057	+	0.961714i	$0.411634\pi$
$882$	0	0
$883$	1.34895e9i	1.95935i	0.200584	+	0.979676i	$0.435716\pi$
$883$	1.34895e9i	1.95935i	−0.200584	+	0.979676i	$0.564284\pi$
$884$	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
$891$	− 8.91257e8i	− 1.26000i
$892$	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
$907$	− 8.05321e8i	− 1.07931i	−0.841885	−	0.539656i	$-0.818554\pi$
$907$	− 8.05321e8i	− 1.07931i	0.841885	−	0.539656i	$-0.181446\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−1.58733e9	−2.08571
$914$	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
$921$	−2.74676e8	−0.351594
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.55501e9	−1.93948	−0.969739	−	0.244146i	$-0.921492\pi$
$929$	−1.55501e9	−1.93948	−0.969739	+	0.244146i	$0.921492\pi$
$930$	0	0
$931$	2.92005e8i	0.361860i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.52100e9	−1.84889	−0.924443	−	0.381320i	$-0.875470\pi$
$937$	−1.52100e9	−1.84889	−0.924443	+	0.381320i	$0.875470\pi$
$938$	0	0
$939$	4.16921e8i	0.503567i
$940$	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
$947$	− 1.69218e9i	− 1.99249i	−0.0865891	−	0.996244i	$-0.527597\pi$
$947$	− 1.69218e9i	− 1.99249i	0.0865891	−	0.996244i	$-0.472403\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.84034e8	−0.443701	−0.221851	−	0.975081i	$-0.571210\pi$
$953$	−3.84034e8	−0.443701	−0.221851	+	0.975081i	$0.571210\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.87504e8	1.00000
$962$	0	0
$963$	− 6.57457e8i	− 0.736188i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	−1.97061e8	−0.216585
$970$	0	0
$971$	− 1.83097e9i	− 1.99997i	−0.00535408	−	0.999986i	$-0.501704\pi$
$971$	− 1.83097e9i	− 1.99997i	0.00535408	−	0.999986i	$-0.498296\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.56343e9	−1.67647	−0.838234	−	0.545310i	$-0.816412\pi$
$977$	−1.56343e9	−1.67647	−0.838234	+	0.545310i	$0.816412\pi$
$978$	0	0
$979$	− 8.35279e8i	− 0.890191i
$980$	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
$993$	−2.11546e8	−0.216051
$994$	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	256.7.c.d.255.1		2
4.3	odd	2	inner	256.7.c.d.255.2		2
8.3	odd	2	CM	256.7.c.d.255.1		2
8.5	even	2	inner	256.7.c.d.255.2		2
16.3	odd	4		32.7.d.a.15.1		1
16.5	even	4		32.7.d.a.15.1		1
16.11	odd	4		8.7.d.a.3.1	✓	1
16.13	even	4		8.7.d.a.3.1	✓	1
48.5	odd	4		288.7.b.a.271.1		1
48.11	even	4		72.7.b.a.19.1		1
48.29	odd	4		72.7.b.a.19.1		1
48.35	even	4		288.7.b.a.271.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.7.d.a.3.1	✓	1	16.11	odd	4
8.7.d.a.3.1	✓	1	16.13	even	4
32.7.d.a.15.1		1	16.3	odd	4
32.7.d.a.15.1		1	16.5	even	4
72.7.b.a.19.1		1	48.11	even	4
72.7.b.a.19.1		1	48.29	odd	4
256.7.c.d.255.1		2	1.1	even	1	trivial
256.7.c.d.255.1		2	8.3	odd	2	CM
256.7.c.d.255.2		2	4.3	odd	2	inner
256.7.c.d.255.2		2	8.5	even	2	inner
288.7.b.a.271.1		1	48.5	odd	4
288.7.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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-46.0000i q^{3} -1387.00 q^{9} +O(q^{10})\)	\(q-46.0000i q^{3} -1387.00 q^{9} -2338.00i q^{11} -1726.00 q^{17} +2482.00i q^{19} -15625.0 q^{25} +30268.0i q^{27} -107548. q^{33} -134642. q^{41} -74914.0i q^{43} +117649. q^{49} +79396.0i q^{51} +114172. q^{57} +304958. i q^{59} +596626. i q^{67} +593134. q^{73} +718750. i q^{75} +381205. q^{81} -678926. i q^{83} +357262. q^{89} +1.82275e6 q^{97} +3.24281e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2774 q^{9}+O(q^{10}) \) 2 * q - 2774 * q^9	\( 2 q - 2774 q^{9} - 3452 q^{17} - 31250 q^{25} - 215096 q^{33} - 269284 q^{41} + 235298 q^{49} + 228344 q^{57} + 1186268 q^{73} + 762410 q^{81} + 714524 q^{89} + 3645508 q^{97}+O(q^{100}) \) 2 * q - 2774 * q^9 - 3452 * q^17 - 31250 * q^25 - 215096 * q^33 - 269284 * q^41 + 235298 * q^49 + 228344 * q^57 + 1186268 * q^73 + 762410 * q^81 + 714524 * q^89 + 3645508 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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