LMFDB - Embedded newform 256.8.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,8,Mod(1,256)]

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, 0]))

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

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

chi := DirichletCharacter("256.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	256.a (trivial)

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:	yes
Analytic conductor:	$79.9705665239$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 128)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	256.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+36.7696 q^{3} -835.000 q^{9} +O(q^{10})$	$q+36.7696 q^{3} -835.000 q^{9} -511.945 q^{11} +22182.0 q^{17} -2961.36 q^{19} -78125.0 q^{25} -111118. q^{27} -18824.0 q^{33} +236886. q^{41} -1.01914e6 q^{43} -823543. q^{49} +815622. q^{51} -108888. q^{57} -2.98191e6 q^{59} +3.06762e6 q^{67} -4.86561e6 q^{73} -2.87262e6 q^{75} -2.25960e6 q^{81} +9.24221e6 q^{83} -7.07312e6 q^{89} -9.93889e6 q^{97} +427474. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 1670 q^{9}+O(q^{10}) $ 2 * q - 1670 * q^9	$ 2 q - 1670 q^{9} + 44364 q^{17} - 156250 q^{25} - 37648 q^{33} + 473772 q^{41} - 1647086 q^{49} - 217776 q^{57} - 9731228 q^{73} - 4519198 q^{81} - 14146236 q^{89} - 19877780 q^{97}+O(q^{100}) $ 2 * q - 1670 * q^9 + 44364 * q^17 - 156250 * q^25 - 37648 * q^33 + 473772 * q^41 - 1647086 * q^49 - 217776 * q^57 - 9731228 * q^73 - 4519198 * q^81 - 14146236 * q^89 - 19877780 * q^97

Display 100 coefficients 10 coefficients

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
$3$	36.7696	0.786256	0.393128	−	0.919484i	$-0.371393\pi$
$3$	36.7696	0.786256	0.393128	+	0.919484i	$0.371393\pi$
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−835.000	−0.381802
$10$	0	0
$11$	−511.945	−0.115971	−0.0579855	−	0.998317i	$-0.518468\pi$
$11$	−511.945	−0.115971	−0.0579855	+	0.998317i	$0.518468\pi$
$12$	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
$17$	22182.0	1.09504	0.547519	−	0.836793i	$-0.315572\pi$
$17$	22182.0	1.09504	0.547519	+	0.836793i	$0.315572\pi$
$18$	0	0
$19$	−2961.36	−0.0990499	−0.0495250	−	0.998773i	$-0.515771\pi$
$19$	−2961.36	−0.0990499	−0.0495250	+	0.998773i	$0.515771\pi$
$20$	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
$25$	−78125.0	−1.00000
$26$	0	0
$27$	−111118.	−1.08645
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−18824.0	−0.0911828
$34$	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
$41$	236886.	0.536779	0.268390	−	0.963310i	$-0.413509\pi$
$41$	236886.	0.536779	0.268390	+	0.963310i	$0.413509\pi$
$42$	0	0
$43$	−1.01914e6	−1.95477	−0.977383	−	0.211475i	$-0.932173\pi$
$43$	−1.01914e6	−1.95477	−0.977383	+	0.211475i	$0.932173\pi$
$44$	0	0
$45$	0	0
$46$	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
$51$	815622.	0.860981
$52$	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
$57$	−108888.	−0.0778786
$58$	0	0
$59$	−2.98191e6	−1.89022	−0.945111	−	0.326750i	$-0.894047\pi$
$59$	−2.98191e6	−1.89022	−0.945111	+	0.326750i	$0.894047\pi$
$60$	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
$67$	3.06762e6	1.24606	0.623032	−	0.782196i	$-0.285901\pi$
$67$	3.06762e6	1.24606	0.623032	+	0.782196i	$0.285901\pi$
$68$	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
$73$	−4.86561e6	−1.46389	−0.731944	−	0.681365i	$-0.761387\pi$
$73$	−4.86561e6	−1.46389	−0.731944	+	0.681365i	$0.761387\pi$
$74$	0	0
$75$	−2.87262e6	−0.786256
$76$	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
$81$	−2.25960e6	−0.472426
$82$	0	0
$83$	9.24221e6	1.77420	0.887099	−	0.461578i	$-0.152717\pi$
$83$	9.24221e6	1.77420	0.887099	+	0.461578i	$0.152717\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.07312e6	−1.06352	−0.531760	−	0.846895i	$-0.678469\pi$
$89$	−7.07312e6	−1.06352	−0.531760	+	0.846895i	$0.678469\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−9.93889e6	−1.10570	−0.552849	−	0.833281i	$-0.686460\pi$
$97$	−9.93889e6	−1.10570	−0.552849	+	0.833281i	$0.686460\pi$
$98$	0	0
$99$	427474.	0.0442779
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.19039e7	0.939389	0.469695	−	0.882829i	$-0.344364\pi$
$107$	1.19039e7	0.939389	0.469695	+	0.882829i	$0.344364\pi$
$108$	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
$113$	−2.16700e7	−1.41281	−0.706405	−	0.707808i	$-0.749684\pi$
$113$	−2.16700e7	−1.41281	−0.706405	+	0.707808i	$0.749684\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.92251e7	−0.986551
$122$	0	0
$123$	8.71019e6	0.422046
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	−3.74734e7	−1.53695
$130$	0	0
$131$	−5.13948e7	−1.99742	−0.998711	−	0.0507560i	$-0.983837\pi$
$131$	−5.13948e7	−1.99742	−0.998711	+	0.0507560i	$0.983837\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.84166e7	−1.94095	−0.970475	−	0.241201i	$-0.922459\pi$
$137$	−5.84166e7	−1.94095	−0.970475	+	0.241201i	$0.922459\pi$
$138$	0	0
$139$	3.38949e7	1.07049	0.535244	−	0.844697i	$-0.320219\pi$
$139$	3.38949e7	1.07049	0.535244	+	0.844697i	$0.320219\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.02813e7	−0.786256
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	−1.85220e7	−0.418087
$154$	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
$163$	−9.43411e7	−1.70626	−0.853128	−	0.521702i	$-0.825297\pi$
$163$	−9.43411e7	−1.70626	−0.853128	+	0.521702i	$0.825297\pi$
$164$	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
$169$	−6.27485e7	−1.00000
$170$	0	0
$171$	2.47274e6	0.0378174
$172$	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
$177$	−1.09644e8	−1.48620
$178$	0	0
$179$	6.70974e7	0.874420	0.437210	−	0.899359i	$-0.355967\pi$
$179$	6.70974e7	0.874420	0.437210	+	0.899359i	$0.355967\pi$
$180$	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
$187$	−1.13560e7	−0.126993
$188$	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
$193$	−2.25308e7	−0.225593	−0.112797	−	0.993618i	$-0.535981\pi$
$193$	−2.25308e7	−0.225593	−0.112797	+	0.993618i	$0.535981\pi$
$194$	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		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	1.12795e8	0.979725
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.51606e6	0.0114869
$210$	0	0
$211$	2.53627e8	1.85869	0.929346	−	0.369210i	$-0.120372\pi$
$211$	2.53627e8	1.85869	0.929346	+	0.369210i	$0.120372\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.78906e8	−1.15099
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	6.52344e7	0.381802
$226$	0	0
$227$	2.15553e8	1.22310	0.611552	−	0.791204i	$-0.290545\pi$
$227$	2.15553e8	1.22310	0.611552	+	0.791204i	$0.290545\pi$
$228$	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
$233$	−1.01581e8	−0.526099	−0.263050	−	0.964782i	$-0.584728\pi$
$233$	−1.01581e8	−0.526099	−0.263050	+	0.964782i	$0.584728\pi$
$234$	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
$241$	2.67977e8	1.23321	0.616606	−	0.787272i	$-0.288507\pi$
$241$	2.67977e8	1.23321	0.616606	+	0.787272i	$0.288507\pi$
$242$	0	0
$243$	1.59930e8	0.715002
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	3.39832e8	1.39497
$250$	0	0
$251$	−1.17757e8	−0.470034	−0.235017	−	0.971991i	$-0.575515\pi$
$251$	−1.17757e8	−0.470034	−0.235017	+	0.971991i	$0.575515\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.43843e8	1.63103	0.815517	−	0.578733i	$-0.196453\pi$
$257$	4.43843e8	1.63103	0.815517	+	0.578733i	$0.196453\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−2.60075e8	−0.836199
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.99957e7	0.115971
$276$	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
$281$	5.46754e8	1.47001	0.735005	−	0.678062i	$-0.237180\pi$
$281$	5.46754e8	1.47001	0.735005	+	0.678062i	$0.237180\pi$
$282$	0	0
$283$	2.08205e8	0.546058	0.273029	−	0.962006i	$-0.411975\pi$
$283$	2.08205e8	0.546058	0.273029	+	0.962006i	$0.411975\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8.17025e7	0.199110
$290$	0	0
$291$	−3.65449e8	−0.869362
$292$	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
$297$	5.68861e7	0.125997
$298$	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
$307$	−1.00385e9	−1.98010	−0.990048	−	0.140733i	$-0.955054\pi$
$307$	−1.00385e9	−1.98010	−0.990048	+	0.140733i	$0.955054\pi$
$308$	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
$313$	9.84035e8	1.81387	0.906933	−	0.421275i	$-0.138417\pi$
$313$	9.84035e8	1.81387	0.906933	+	0.421275i	$0.138417\pi$
$314$	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
$321$	4.37701e8	0.738600
$322$	0	0
$323$	−6.56890e7	−0.108463
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	9.99518e8	1.51493	0.757465	−	0.652876i	$-0.226438\pi$
$331$	9.99518e8	1.51493	0.757465	+	0.652876i	$0.226438\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5.51176e8	0.784487	0.392244	−	0.919861i	$-0.371699\pi$
$337$	5.51176e8	0.784487	0.392244	+	0.919861i	$0.371699\pi$
$338$	0	0
$339$	−7.96795e8	−1.11083
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.60950e8	0.849211	0.424606	−	0.905378i	$-0.360413\pi$
$347$	6.60950e8	0.849211	0.424606	+	0.905378i	$0.360413\pi$
$348$	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
$353$	3.10363e8	0.375542	0.187771	−	0.982213i	$-0.439874\pi$
$353$	3.10363e8	0.375542	0.187771	+	0.982213i	$0.439874\pi$
$354$	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
$361$	−8.85102e8	−0.990189
$362$	0	0
$363$	−7.06898e8	−0.775681
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	−1.97800e8	−0.204943
$370$	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
$379$	2.11851e9	1.99891	0.999457	−	0.0329408i	$-0.0104873\pi$
$379$	2.11851e9	1.99891	0.999457	+	0.0329408i	$0.0104873\pi$
$380$	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
$387$	8.50983e8	0.746333
$388$	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
$393$	−1.88976e9	−1.57049
$394$	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
$401$	−2.24350e9	−1.73748	−0.868740	−	0.495268i	$-0.835070\pi$
$401$	−2.24350e9	−1.73748	−0.868740	+	0.495268i	$0.835070\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.14006e9	1.54666	0.773330	−	0.634004i	$-0.218590\pi$
$409$	2.14006e9	1.54666	0.773330	+	0.634004i	$0.218590\pi$
$410$	0	0
$411$	−2.14795e9	−1.52608
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.24630e9	0.841678
$418$	0	0
$419$	3.00841e9	1.99797	0.998983	−	0.0450790i	$-0.0143540\pi$
$419$	3.00841e9	1.99797	0.998983	+	0.0450790i	$0.0143540\pi$
$420$	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
$425$	−1.73297e9	−1.09504
$426$	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
$433$	−3.31027e9	−1.95955	−0.979774	−	0.200109i	$-0.935870\pi$
$433$	−3.31027e9	−1.95955	−0.979774	+	0.200109i	$0.935870\pi$
$434$	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
$441$	6.87658e8	0.381802
$442$	0	0
$443$	−1.65968e9	−0.907010	−0.453505	−	0.891254i	$-0.649826\pi$
$443$	−1.65968e9	−0.907010	−0.453505	+	0.891254i	$0.649826\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2.27126e9	−1.18414	−0.592072	−	0.805885i	$-0.701690\pi$
$449$	−2.27126e9	−1.18414	−0.592072	+	0.805885i	$0.701690\pi$
$450$	0	0
$451$	−1.21273e8	−0.0622508
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.04270e9	−1.98137	−0.990684	−	0.136183i	$-0.956517\pi$
$457$	−4.04270e9	−1.98137	−0.990684	+	0.136183i	$0.956517\pi$
$458$	0	0
$459$	−2.46481e9	−1.18970
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.69178e9	1.22301	0.611505	−	0.791241i	$-0.290565\pi$
$467$	2.69178e9	1.22301	0.611505	+	0.791241i	$0.290565\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.21745e8	0.226696
$474$	0	0
$475$	2.31356e8	0.0990499
$476$	0	0
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	−3.46888e9	−1.34155
$490$	0	0
$491$	−4.00291e9	−1.52613	−0.763064	−	0.646323i	$-0.776306\pi$
$491$	−4.00291e9	−1.52613	−0.763064	+	0.646323i	$0.776306\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.40494e9	−1.22676	−0.613378	−	0.789790i	$-0.710190\pi$
$499$	−3.40494e9	−1.22676	−0.613378	+	0.789790i	$0.710190\pi$
$500$	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
$505$	0	0
$506$	0	0
$507$	−2.30723e9	−0.786256
$508$	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
$513$	3.29060e8	0.107613
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−5.14691e9	−1.59446	−0.797232	−	0.603673i	$-0.793703\pi$
$521$	−5.14691e9	−1.59446	−0.797232	+	0.603673i	$0.793703\pi$
$522$	0	0
$523$	5.47033e9	1.67208	0.836040	−	0.548668i	$-0.184865\pi$
$523$	5.47033e9	1.67208	0.836040	+	0.548668i	$0.184865\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−3.40483e9	−1.00000
$530$	0	0
$531$	2.48990e9	0.721690
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.46714e9	0.687518
$538$	0	0
$539$	4.21609e8	0.115971
$540$	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
$547$	−7.32742e9	−1.91424	−0.957118	−	0.289698i	$-0.906445\pi$
$547$	−7.32742e9	−1.91424	−0.957118	+	0.289698i	$0.906445\pi$
$548$	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
$561$	−4.17554e8	−0.0998487
$562$	0	0
$563$	−6.60784e8	−0.156056	−0.0780279	−	0.996951i	$-0.524862\pi$
$563$	−6.60784e8	−0.156056	−0.0780279	+	0.996951i	$0.524862\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	8.34926e9	1.90001	0.950003	−	0.312241i	$-0.101080\pi$
$569$	8.34926e9	1.90001	0.950003	+	0.312241i	$0.101080\pi$
$570$	0	0
$571$	8.70729e9	1.95730	0.978648	−	0.205545i	$-0.0658966\pi$
$571$	8.70729e9	1.95730	0.978648	+	0.205545i	$0.0658966\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	6.48604e9	1.40561	0.702805	−	0.711383i	$-0.251931\pi$
$577$	6.48604e9	1.40561	0.702805	+	0.711383i	$0.251931\pi$
$578$	0	0
$579$	−8.28447e8	−0.177374
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.32738e9	1.29119	0.645596	−	0.763679i	$-0.276609\pi$
$587$	6.32738e9	1.29119	0.645596	+	0.763679i	$0.276609\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4.79445e9	−0.944164	−0.472082	−	0.881555i	$-0.656497\pi$
$593$	−4.79445e9	−0.944164	−0.472082	+	0.881555i	$0.656497\pi$
$594$	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
$601$	8.35888e9	1.57068	0.785339	−	0.619066i	$-0.212489\pi$
$601$	8.35888e9	1.57068	0.785339	+	0.619066i	$0.212489\pi$
$602$	0	0
$603$	−2.56147e9	−0.475749
$604$	0	0
$605$	0	0
$606$	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
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.14929e10	−1.96984	−0.984919	−	0.173018i	$-0.944648\pi$
$617$	−1.14929e10	−1.96984	−0.984919	+	0.173018i	$0.944648\pi$
$618$	0	0
$619$	8.97316e9	1.52065	0.760323	−	0.649545i	$-0.225041\pi$
$619$	8.97316e9	1.52065	0.760323	+	0.649545i	$0.225041\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	6.10352e9	1.00000
$626$	0	0
$627$	5.57447e7	0.00903165
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	9.32576e9	1.46141
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.12547e9	1.06859	0.534294	−	0.845299i	$-0.320577\pi$
$641$	7.12547e9	1.06859	0.534294	+	0.845299i	$0.320577\pi$
$642$	0	0
$643$	−1.24491e10	−1.84671	−0.923355	−	0.383948i	$-0.874564\pi$
$643$	−1.24491e10	−1.84671	−0.923355	+	0.383948i	$0.874564\pi$
$644$	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
$649$	1.52658e9	0.219211
$650$	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
$657$	4.06279e9	0.558914
$658$	0	0
$659$	1.18365e10	1.61111	0.805556	−	0.592520i	$-0.201867\pi$
$659$	1.18365e10	1.61111	0.805556	+	0.592520i	$0.201867\pi$
$660$	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
$673$	1.54576e10	1.95474	0.977369	−	0.211539i	$-0.0678477\pi$
$673$	1.54576e10	1.95474	0.977369	+	0.211539i	$0.0678477\pi$
$674$	0	0
$675$	8.68106e9	1.08645
$676$	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
$681$	7.92579e9	0.961674
$682$	0	0
$683$	−1.61382e10	−1.93813	−0.969066	−	0.246800i	$-0.920621\pi$
$683$	−1.61382e10	−1.93813	−0.969066	+	0.246800i	$0.920621\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−6.69881e9	−0.772369	−0.386184	−	0.922422i	$-0.626207\pi$
$691$	−6.69881e9	−0.772369	−0.386184	+	0.922422i	$0.626207\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.25461e9	0.587794
$698$	0	0
$699$	−3.73510e9	−0.413649
$700$	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		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	9.85339e9	0.969620
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	1.08223e10	1.03460
$730$	0	0
$731$	−2.26066e10	−2.14055
$732$	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
$737$	−1.57046e9	−0.144507
$738$	0	0
$739$	3.66788e8	0.0334318	0.0167159	−	0.999860i	$-0.494679\pi$
$739$	3.66788e8	0.0334318	0.0167159	+	0.999860i	$0.494679\pi$
$740$	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
$745$	0	0
$746$	0	0
$747$	−7.71724e9	−0.677392
$748$	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
$753$	−4.32988e9	−0.369567
$754$	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
$761$	3.03427e9	0.249579	0.124789	−	0.992183i	$-0.460175\pi$
$761$	3.03427e9	0.249579	0.124789	+	0.992183i	$0.460175\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−7.12479e9	−0.564976	−0.282488	−	0.959271i	$-0.591160\pi$
$769$	−7.12479e9	−0.564976	−0.282488	+	0.959271i	$0.591160\pi$
$770$	0	0
$771$	1.63199e10	1.28241
$772$	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
$779$	−7.01505e8	−0.0531680
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−4.21697e9	−0.308382	−0.154191	−	0.988041i	$-0.549277\pi$
$787$	−4.21697e9	−0.308382	−0.154191	+	0.988041i	$0.549277\pi$
$788$	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
$801$	5.90605e9	0.406054
$802$	0	0
$803$	2.49093e9	0.169768
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.03026e10	−1.34813	−0.674064	−	0.738673i	$-0.735453\pi$
$809$	−2.03026e10	−1.34813	−0.674064	+	0.738673i	$0.735453\pi$
$810$	0	0
$811$	−1.18458e10	−0.779812	−0.389906	−	0.920855i	$-0.627492\pi$
$811$	−1.18458e10	−0.779812	−0.389906	+	0.920855i	$0.627492\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	3.01805e9	0.193619
$818$	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		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	1.47062e9	0.0911828
$826$	0	0
$827$	2.04957e10	1.26006	0.630032	−	0.776569i	$-0.283042\pi$
$827$	2.04957e10	1.26006	0.630032	+	0.776569i	$0.283042\pi$
$828$	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
$833$	−1.82678e10	−1.09504
$834$	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
$841$	−1.72499e10	−1.00000
$842$	0	0
$843$	2.01039e10	1.15580
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	7.65561e9	0.429341
$850$	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
$857$	−3.44377e10	−1.86897	−0.934483	−	0.356008i	$-0.884137\pi$
$857$	−3.44377e10	−1.86897	−0.934483	+	0.356008i	$0.884137\pi$
$858$	0	0
$859$	3.15983e10	1.70093	0.850467	−	0.526029i	$-0.176320\pi$
$859$	3.15983e10	1.70093	0.850467	+	0.526029i	$0.176320\pi$
$860$	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
$865$	0	0
$866$	0	0
$867$	3.00416e9	0.156551
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	8.29897e9	0.422157
$874$	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
$881$	−3.38273e10	−1.66668	−0.833338	−	0.552763i	$-0.813573\pi$
$881$	−3.38273e10	−1.66668	−0.833338	+	0.552763i	$0.813573\pi$
$882$	0	0
$883$	−3.97855e10	−1.94474	−0.972371	−	0.233439i	$-0.925002\pi$
$883$	−3.97855e10	−1.94474	−0.972371	+	0.233439i	$0.925002\pi$
$884$	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
$891$	1.15679e9	0.0547877
$892$	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
$907$	1.76352e10	0.784790	0.392395	−	0.919797i	$-0.371647\pi$
$907$	1.76352e10	0.784790	0.392395	+	0.919797i	$0.371647\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−4.73150e9	−0.205756
$914$	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
$921$	−3.69112e10	−1.55686
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4.75217e10	−1.94463	−0.972315	−	0.233673i	$-0.924925\pi$
$929$	−4.75217e10	−1.94463	−0.972315	+	0.233673i	$0.924925\pi$
$930$	0	0
$931$	2.43881e9	0.0990499
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	4.18278e10	1.66102	0.830512	−	0.557000i	$-0.188048\pi$
$937$	4.18278e10	1.66102	0.830512	+	0.557000i	$0.188048\pi$
$938$	0	0
$939$	3.61825e10	1.42616
$940$	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
$947$	−4.76750e10	−1.82417	−0.912087	−	0.409998i	$-0.865530\pi$
$947$	−4.76750e10	−1.82417	−0.912087	+	0.409998i	$0.865530\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.62577e10	−1.73125	−0.865624	−	0.500694i	$-0.833078\pi$
$953$	−4.62577e10	−1.73125	−0.865624	+	0.500694i	$0.833078\pi$
$954$	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
$963$	−9.93974e9	−0.358660
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	−2.41535e9	−0.0852801
$970$	0	0
$971$	−2.82185e10	−0.989161	−0.494581	−	0.869132i	$-0.664678\pi$
$971$	−2.82185e10	−0.989161	−0.494581	+	0.869132i	$0.664678\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.63345e10	−1.24649	−0.623245	−	0.782027i	$-0.714186\pi$
$977$	−3.63345e10	−1.24649	−0.623245	+	0.782027i	$0.714186\pi$
$978$	0	0
$979$	3.62105e9	0.123337
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	3.67518e10	1.19112
$994$	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	256.8.a.g.1.2		2
4.3	odd	2	inner	256.8.a.g.1.1		2
8.3	odd	2	CM	256.8.a.g.1.2		2
8.5	even	2	inner	256.8.a.g.1.1		2
16.3	odd	4		128.8.b.a.65.1	✓	2
16.5	even	4		128.8.b.a.65.1	✓	2
16.11	odd	4		128.8.b.a.65.2	yes	2
16.13	even	4		128.8.b.a.65.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.8.b.a.65.1	✓	2	16.3	odd	4
128.8.b.a.65.1	✓	2	16.5	even	4
128.8.b.a.65.2	yes	2	16.11	odd	4
128.8.b.a.65.2	yes	2	16.13	even	4
256.8.a.g.1.1		2	4.3	odd	2	inner
256.8.a.g.1.1		2	8.5	even	2	inner
256.8.a.g.1.2		2	1.1	even	1	trivial
256.8.a.g.1.2		2	8.3	odd	2	CM

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.a (trivial)

\(f(q)\)	\(=\)	\(q+36.7696 q^{3} -835.000 q^{9} +O(q^{10})\)	\(q+36.7696 q^{3} -835.000 q^{9} -511.945 q^{11} +22182.0 q^{17} -2961.36 q^{19} -78125.0 q^{25} -111118. q^{27} -18824.0 q^{33} +236886. q^{41} -1.01914e6 q^{43} -823543. q^{49} +815622. q^{51} -108888. q^{57} -2.98191e6 q^{59} +3.06762e6 q^{67} -4.86561e6 q^{73} -2.87262e6 q^{75} -2.25960e6 q^{81} +9.24221e6 q^{83} -7.07312e6 q^{89} -9.93889e6 q^{97} +427474. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 1670 q^{9}+O(q^{10}) \) 2 * q - 1670 * q^9	\( 2 q - 1670 q^{9} + 44364 q^{17} - 156250 q^{25} - 37648 q^{33} + 473772 q^{41} - 1647086 q^{49} - 217776 q^{57} - 9731228 q^{73} - 4519198 q^{81} - 14146236 q^{89} - 19877780 q^{97}+O(q^{100}) \) 2 * q - 1670 * q^9 + 44364 * q^17 - 156250 * q^25 - 37648 * q^33 + 473772 * q^41 - 1647086 * q^49 - 217776 * q^57 - 9731228 * q^73 - 4519198 * q^81 - 14146236 * q^89 - 19877780 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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