LMFDB - Embedded newform 192.8.f.b.95.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,8,Mod(95,192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(192, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("192.95");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	192.f (of order $2$, degree $1$, 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:	$59.9779248930$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			95.3
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	192.95
Dual form			192.8.f.b.95.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+46.7654 q^{3} -508.000i q^{7} +2187.00 q^{9} +O(q^{10})$	$q+46.7654 q^{3} -508.000i q^{7} +2187.00 q^{9} -6117.60i q^{13} +16589.6 q^{19} -23756.8i q^{21} -78125.0 q^{25} +102276. q^{27} -178916. i q^{31} -549081. i q^{37} -286092. i q^{39} +124843. q^{43} +565479. q^{49} +775818. q^{57} +266244. i q^{61} -1.11100e6i q^{63} -4.90862e6 q^{67} +6.27481e6 q^{73} -3.65354e6 q^{75} -8.76304e6i q^{79} +4.78297e6 q^{81} -3.10774e6 q^{91} -8.36707e6i q^{93} +1.22452e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8748 q^{9}+O(q^{10}) $ 4 * q + 8748 * q^9	$ 4 q + 8748 q^{9} - 312500 q^{25} + 2261916 q^{49} + 3103272 q^{57} + 25099240 q^{73} + 19131876 q^{81} + 48980792 q^{97}+O(q^{100}) $ 4 * q + 8748 * q^9 - 312500 * q^25 + 2261916 * q^49 + 3103272 * q^57 + 25099240 * q^73 + 19131876 * q^81 + 48980792 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/192\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$133$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	46.7654	1.00000
$3$	46.7654	1.00000
$4$	0	0
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	− 508.000i	− 0.559784i	−0.960031	−	0.279892i	$-0.909701\pi$
$7$	− 508.000i	− 0.559784i	0.960031	−	0.279892i	$-0.0902987\pi$
$8$	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
$13$	− 6117.60i	− 0.772289i	−0.922438	−	0.386144i	$-0.873807\pi$
$13$	− 6117.60i	− 0.772289i	0.922438	−	0.386144i	$-0.126193\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	16589.6	0.554878	0.277439	−	0.960743i	$-0.410514\pi$
$19$	16589.6	0.554878	0.277439	+	0.960743i	$0.410514\pi$
$20$	0	0
$21$	− 23756.8i	− 0.559784i
$22$	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
$27$	102276.	1.00000
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	− 178916.i	− 1.07866i	−0.842096	−	0.539328i	$-0.818678\pi$
$31$	− 178916.i	− 1.07866i	0.842096	−	0.539328i	$-0.181322\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 549081.i	− 1.78209i	−0.453912	−	0.891046i	$-0.649972\pi$
$37$	− 549081.i	− 1.78209i	0.453912	−	0.891046i	$-0.350028\pi$
$38$	0	0
$39$	− 286092.i	− 0.772289i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	124843.	0.239455	0.119727	−	0.992807i	$-0.461798\pi$
$43$	124843.	0.239455	0.119727	+	0.992807i	$0.461798\pi$
$44$	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
$49$	565479.	0.686642
$50$	0	0
$51$	0	0
$52$	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
$57$	775818.	0.554878
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	266244.i	0.150185i	0.997177	+	0.0750923i	$0.0239251\pi$
$61$	266244.i	0.150185i	−0.997177	+	0.0750923i	$0.976075\pi$
$62$	0	0
$63$	− 1.11100e6i	− 0.559784i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.90862e6	−1.99387	−0.996937	−	0.0782078i	$-0.975080\pi$
$67$	−4.90862e6	−1.99387	−0.996937	+	0.0782078i	$0.975080\pi$
$68$	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
$73$	6.27481e6	1.88786	0.943932	−	0.330141i	$-0.107096\pi$
$73$	6.27481e6	1.88786	0.943932	+	0.330141i	$0.107096\pi$
$74$	0	0
$75$	−3.65354e6	−1.00000
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 8.76304e6i	− 1.99968i	−0.0179303	−	0.999839i	$-0.505708\pi$
$79$	− 8.76304e6i	− 1.99968i	0.0179303	−	0.999839i	$-0.494292\pi$
$80$	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
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	−3.10774e6	−0.432315
$92$	0	0
$93$	− 8.36707e6i	− 1.07866i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.22452e7	1.36227	0.681137	−	0.732156i	$-0.261486\pi$
$97$	1.22452e7	1.36227	0.681137	+	0.732156i	$0.261486\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	− 8.02622e6i	− 0.723737i	−0.932229	−	0.361868i	$-0.882139\pi$
$103$	− 8.02622e6i	− 0.723737i	0.932229	−	0.361868i	$-0.117861\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	− 2.11700e7i	− 1.56577i	−0.622167	−	0.782884i	$-0.713748\pi$
$109$	− 2.11700e7i	− 1.56577i	0.622167	−	0.782884i	$-0.286252\pi$
$110$	0	0
$111$	− 2.56780e7i	− 1.78209i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.33792e7i	− 0.772289i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.94872e7	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 4.51256e7i	− 1.95484i	−0.211317	−	0.977418i	$-0.567775\pi$
$127$	− 4.51256e7i	− 1.95484i	0.211317	−	0.977418i	$-0.432225\pi$
$128$	0	0
$129$	5.83832e6	0.239455
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	− 8.42751e6i	− 0.310612i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	−3.08109e7	−0.973088	−0.486544	−	0.873656i	$-0.661743\pi$
$139$	−3.08109e7	−0.973088	−0.486544	+	0.873656i	$0.661743\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2.64448e7	0.686642
$148$	0	0
$149$	0	0		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	7.70389e7i	1.82092i	0.413597	+	0.910460i	$0.364272\pi$
$151$	7.70389e7i	1.82092i	−0.413597	+	0.910460i	$0.635728\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.17601e7i	1.06745i	0.845659	+	0.533723i	$0.179208\pi$
$157$	5.17601e7i	1.06745i	−0.845659	+	0.533723i	$0.820792\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−6.77910e7	−1.22607	−0.613035	−	0.790056i	$-0.710052\pi$
$163$	−6.77910e7	−1.22607	−0.613035	+	0.790056i	$0.710052\pi$
$164$	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
$169$	2.53234e7	0.403570
$170$	0	0
$171$	3.62814e7	0.554878
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	3.96875e7i	0.559784i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	1.54508e8i	1.93677i	0.249468	+	0.968383i	$0.419744\pi$
$181$	1.54508e8i	1.93677i	−0.249468	+	0.968383i	$0.580256\pi$
$182$	0	0
$183$	1.24510e7i	0.150185i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 5.19561e7i	− 0.559784i
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−9.65180e7	−0.966402	−0.483201	−	0.875509i	$-0.660526\pi$
$193$	−9.65180e7	−0.966402	−0.483201	+	0.875509i	$0.660526\pi$
$194$	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
$199$	1.44785e8i	1.30238i	0.758916	+	0.651188i	$0.225729\pi$
$199$	1.44785e8i	1.30238i	−0.758916	+	0.651188i	$0.774271\pi$
$200$	0	0
$201$	−2.29554e8	−1.99387
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	1.60698e8	1.17766	0.588832	−	0.808256i	$-0.299588\pi$
$211$	1.60698e8	1.17766	0.588832	+	0.808256i	$0.299588\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−9.08893e7	−0.603815
$218$	0	0
$219$	2.93444e8	1.88786
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2.62897e8i	1.58752i	0.608233	+	0.793759i	$0.291879\pi$
$223$	2.62897e8i	1.58752i	−0.608233	+	0.793759i	$0.708121\pi$
$224$	0	0
$225$	−1.70859e8	−1.00000
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	3.02698e8i	1.66565i	0.553534	+	0.832827i	$0.313279\pi$
$229$	3.02698e8i	1.66565i	−0.553534	+	0.832827i	$0.686721\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4.09807e8i	− 1.99968i
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−5.75871e7	−0.265012	−0.132506	−	0.991182i	$-0.542302\pi$
$241$	−5.75871e7	−0.265012	−0.132506	+	0.991182i	$0.542302\pi$
$242$	0	0
$243$	2.23677e8	1.00000
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 1.01488e8i	− 0.428526i
$248$	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
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	−2.78933e8	−0.997587
$260$	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
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	4.85556e8i	1.48200i	0.671508	+	0.740998i	$0.265647\pi$
$271$	4.85556e8i	1.48200i	−0.671508	+	0.740998i	$0.734353\pi$
$272$	0	0
$273$	−1.45335e8	−0.432315
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.07448e8i	1.99993i	0.00827024	+	0.999966i	$0.497367\pi$
$277$	7.07448e8i	1.99993i	−0.00827024	+	0.999966i	$0.502633\pi$
$278$	0	0
$279$	− 3.91289e8i	− 1.07866i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	−6.17390e8	−1.61923	−0.809613	−	0.586964i	$-0.800323\pi$
$283$	−6.17390e8	−1.61923	−0.809613	+	0.586964i	$0.800323\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4.10339e8	1.00000
$290$	0	0
$291$	5.72651e8	1.36227
$292$	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
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	− 6.34201e7i	− 0.134043i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−9.98019e8	−1.96859	−0.984293	−	0.176541i	$-0.943509\pi$
$307$	−9.98019e8	−1.96859	−0.984293	+	0.176541i	$0.943509\pi$
$308$	0	0
$309$	− 3.75349e8i	− 0.723737i
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	1.08488e9	1.99976	0.999880	−	0.0155022i	$-0.00493470\pi$
$313$	1.08488e9	1.99976	0.999880	+	0.0155022i	$0.00493470\pi$
$314$	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
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.77938e8i	0.772289i
$326$	0	0
$327$	− 9.90022e8i	− 1.56577i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.29087e8	−0.650351	−0.325175	−	0.945654i	$-0.605423\pi$
$331$	−4.29087e8	−0.650351	−0.325175	+	0.945654i	$0.605423\pi$
$332$	0	0
$333$	− 1.20084e9i	− 1.78209i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.27571e9	1.81571	0.907854	−	0.419286i	$-0.137719\pi$
$337$	1.27571e9	1.81571	0.907854	+	0.419286i	$0.137719\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 7.05623e8i	− 0.944155i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	− 1.42801e9i	− 1.79822i	−0.437728	−	0.899108i	$-0.644217\pi$
$349$	− 1.42801e9i	− 1.79822i	0.437728	−	0.899108i	$-0.355783\pi$
$350$	0	0
$351$	− 6.25683e8i	− 0.772289i
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	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
$361$	−6.18657e8	−0.692110
$362$	0	0
$363$	9.11325e8	1.00000
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 1.26600e9i	− 1.33691i	−0.743754	−	0.668454i	$-0.766956\pi$
$367$	− 1.26600e9i	− 1.33691i	0.743754	−	0.668454i	$-0.233044\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1.90994e9i	1.90563i	0.303548	+	0.952816i	$0.401829\pi$
$373$	1.90994e9i	1.90563i	−0.303548	+	0.952816i	$0.598171\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−2.58413e8	−0.243824	−0.121912	−	0.992541i	$-0.538903\pi$
$379$	−2.58413e8	−0.243824	−0.121912	+	0.992541i	$0.538903\pi$
$380$	0	0
$381$	− 2.11032e9i	− 1.95484i
$382$	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
$387$	2.73031e8	0.239455
$388$	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
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 1.60493e9i	− 1.28733i	−0.765308	−	0.643665i	$-0.777413\pi$
$397$	− 1.60493e9i	− 1.28733i	0.765308	−	0.643665i	$-0.222587\pi$
$398$	0	0
$399$	− 3.94116e8i	− 0.310612i
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	−1.09454e9	−0.833034
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	2.12727e9	1.53741	0.768707	−	0.639601i	$-0.220900\pi$
$409$	2.12727e9	1.53741	0.768707	+	0.639601i	$0.220900\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.44088e9	−0.973088
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	2.09095e9i	1.36570i	0.730558	+	0.682850i	$0.239260\pi$
$421$	2.09095e9i	1.36570i	−0.730558	+	0.682850i	$0.760740\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.35252e8	0.0840709
$428$	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
$433$	1.11566e9	0.660424	0.330212	−	0.943907i	$-0.392880\pi$
$433$	1.11566e9	0.660424	0.330212	+	0.943907i	$0.392880\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	3.25154e9i	1.83427i	0.398576	+	0.917135i	$0.369505\pi$
$439$	3.25154e9i	1.83427i	−0.398576	+	0.917135i	$0.630495\pi$
$440$	0	0
$441$	1.23670e9	0.686642
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.60275e9i	1.82092i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.06709e9	−1.99332	−0.996661	−	0.0816509i	$-0.973981\pi$
$457$	−4.06709e9	−1.99332	−0.996661	+	0.0816509i	$0.973981\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	1.02273e9i	0.478881i	0.970911	+	0.239440i	$0.0769639\pi$
$463$	1.02273e9i	0.478881i	−0.970911	+	0.239440i	$0.923036\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	0	0
$469$	2.49358e9i	1.11614i
$470$	0	0
$471$	2.42058e9i	1.06745i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.29606e9	−0.554878
$476$	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
$481$	−3.35906e9	−1.37629
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	4.34606e9i	1.70508i	0.522664	+	0.852539i	$0.324938\pi$
$487$	4.34606e9i	1.70508i	−0.522664	+	0.852539i	$0.675062\pi$
$488$	0	0
$489$	−3.17027e9	−1.22607
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	4.26639e9	1.53712	0.768562	−	0.639775i	$-0.220972\pi$
$499$	4.26639e9	1.53712	0.768562	+	0.639775i	$0.220972\pi$
$500$	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
$507$	1.18426e9	0.403570
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	− 3.18760e9i	− 1.05680i
$512$	0	0
$513$	1.69671e9	0.554878
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	2.18274e9	0.667185	0.333592	−	0.942717i	$-0.391739\pi$
$523$	2.18274e9	0.667185	0.333592	+	0.942717i	$0.391739\pi$
$524$	0	0
$525$	1.85600e9i	0.559784i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−3.40483e9	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 6.37614e9i	− 1.73128i	−0.500668	−	0.865640i	$-0.666912\pi$
$541$	− 6.37614e9i	− 1.73128i	0.500668	−	0.865640i	$-0.333088\pi$
$542$	0	0
$543$	7.22564e9i	1.93677i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.77362e9	1.24707	0.623537	−	0.781793i	$-0.285695\pi$
$547$	4.77362e9	1.24707	0.623537	+	0.781793i	$0.285695\pi$
$548$	0	0
$549$	5.82275e8i	0.150185i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−4.45163e9	−1.11939
$554$	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
$559$	− 7.63738e8i	− 0.184928i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 2.42975e9i	− 0.559784i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	6.49838e9	1.46076	0.730379	−	0.683042i	$-0.239343\pi$
$571$	6.49838e9	1.46076	0.730379	+	0.683042i	$0.239343\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−4.23933e9	−0.918718	−0.459359	−	0.888251i	$-0.651921\pi$
$577$	−4.23933e9	−0.918718	−0.459359	+	0.888251i	$0.651921\pi$
$578$	0	0
$579$	−4.51370e9	−0.966402
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	− 2.96814e9i	− 0.598523i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.77091e9i	1.30238i
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	7.41654e9	1.39361	0.696804	−	0.717262i	$-0.254605\pi$
$601$	7.41654e9	1.39361	0.696804	+	0.717262i	$0.254605\pi$
$602$	0	0
$603$	−1.07352e10	−1.99387
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 2.36131e9i	− 0.428541i	−0.976774	−	0.214270i	$-0.931263\pi$
$607$	− 2.36131e9i	− 0.428541i	0.976774	−	0.214270i	$-0.0687374\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.67486e9i	0.293675i	0.989161	+	0.146837i	$0.0469094\pi$
$613$	1.67486e9i	0.293675i	−0.989161	+	0.146837i	$0.953091\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	2.08193e9	0.352816	0.176408	−	0.984317i	$-0.443552\pi$
$619$	2.08193e9	0.352816	0.176408	+	0.984317i	$0.443552\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	6.10352e9	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 1.17017e10i	− 1.85415i	−0.374870	−	0.927077i	$-0.622313\pi$
$631$	− 1.17017e10i	− 1.85415i	0.374870	−	0.927077i	$-0.377687\pi$
$632$	0	0
$633$	7.51509e9	1.17766
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 3.45938e9i	− 0.530286i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	−1.34415e10	−1.99393	−0.996966	−	0.0778386i	$-0.975198\pi$
$643$	−1.34415e10	−1.99393	−0.996966	+	0.0778386i	$0.975198\pi$
$644$	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
$651$	−4.25047e9	−0.603815
$652$	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
$657$	1.37230e10	1.88786
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	− 1.32938e10i	− 1.79038i	−0.445688	−	0.895188i	$-0.647041\pi$
$661$	− 1.32938e10i	− 1.79038i	0.445688	−	0.895188i	$-0.352959\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	1.22945e10i	1.58752i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	4.99918e9	0.632187	0.316094	−	0.948728i	$-0.397629\pi$
$673$	4.99918e9	0.632187	0.316094	+	0.948728i	$0.397629\pi$
$674$	0	0
$675$	−7.99030e9	−1.00000
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	− 6.22056e9i	− 0.762580i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	1.41558e10i	1.66565i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.33859e9	0.961434	0.480717	−	0.876876i	$-0.340376\pi$
$691$	8.33859e9	0.961434	0.480717	+	0.876876i	$0.340376\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	− 9.10902e9i	− 0.988845i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 1.87359e10i	− 1.97430i	−0.159811	−	0.987148i	$-0.551088\pi$
$709$	− 1.87359e10i	− 1.97430i	0.159811	−	0.987148i	$-0.448912\pi$
$710$	0	0
$711$	− 1.91648e10i	− 1.99968i
$712$	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
$721$	−4.07732e9	−0.405136
$722$	0	0
$723$	−2.69308e9	−0.265012
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8.01595e9i	− 0.773722i	−0.922138	−	0.386861i	$-0.873559\pi$
$727$	− 8.01595e9i	− 0.773722i	0.922138	−	0.386861i	$-0.126441\pi$
$728$	0	0
$729$	1.04604e10	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	7.98653e9i	0.749021i	0.927223	+	0.374511i	$0.122189\pi$
$733$	7.98653e9i	0.749021i	−0.927223	+	0.374511i	$0.877811\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	1.09057e10	0.994026	0.497013	−	0.867743i	$-0.334430\pi$
$739$	1.09057e10	0.994026	0.497013	+	0.867743i	$0.334430\pi$
$740$	0	0
$741$	− 4.74615e9i	− 0.428526i
$742$	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
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 1.45957e10i	− 1.25744i	−0.777633	−	0.628718i	$-0.783580\pi$
$751$	− 1.45957e10i	− 1.25744i	0.777633	−	0.628718i	$-0.216420\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 2.27817e10i	− 1.90875i	−0.298603	−	0.954377i	$-0.596521\pi$
$757$	− 2.27817e10i	− 1.90875i	0.298603	−	0.954377i	$-0.403479\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	−1.07543e10	−0.876493
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−6.30053e9	−0.499615	−0.249807	−	0.968296i	$-0.580367\pi$
$769$	−6.30053e9	−0.499615	−0.249807	+	0.968296i	$0.580367\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	1.39778e10i	1.07866i
$776$	0	0
$777$	−1.30444e10	−0.997587
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.14622e10	0.838214	0.419107	−	0.907937i	$-0.362343\pi$
$787$	1.14622e10	0.838214	0.419107	+	0.907937i	$0.362343\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.62877e9	0.115986
$794$	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
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	2.91438e10	1.91855	0.959276	−	0.282472i	$-0.0911544\pi$
$811$	2.91438e10	1.91855	0.959276	+	0.282472i	$0.0911544\pi$
$812$	0	0
$813$	2.27072e10i	1.48200i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.07109e9	0.132868
$818$	0	0
$819$	−6.79663e9	−0.432315
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	3.18623e10i	1.99240i	0.0870772	+	0.996202i	$0.472247\pi$
$823$	3.18623e10i	1.99240i	−0.0870772	+	0.996202i	$0.527753\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	3.21908e10i	1.96242i	0.192950	+	0.981209i	$0.438195\pi$
$829$	3.21908e10i	1.96242i	−0.192950	+	0.981209i	$0.561805\pi$
$830$	0	0
$831$	3.30841e10i	1.99993i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.82988e10i	− 1.07866i
$838$	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
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 9.89948e9i	− 0.559784i
$848$	0	0
$849$	−2.88725e10	−1.61923
$850$	0	0
$851$	0	0
$852$	0	0
$853$	2.68069e10i	1.47885i	0.673237	+	0.739427i	$0.264904\pi$
$853$	2.68069e10i	1.47885i	−0.673237	+	0.739427i	$0.735096\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−3.20767e10	−1.72669	−0.863343	−	0.504617i	$-0.831634\pi$
$859$	−3.20767e10	−1.72669	−0.863343	+	0.504617i	$0.831634\pi$
$860$	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
$867$	1.91896e10	1.00000
$868$	0	0
$869$	0	0
$870$	0	0
$871$	3.00290e10i	1.53985i
$872$	0	0
$873$	2.67802e10	1.36227
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.67376e10i	0.837905i	0.908008	+	0.418953i	$0.137603\pi$
$877$	1.67376e10i	0.837905i	−0.908008	+	0.418953i	$0.862397\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	−2.39614e10	−1.17125	−0.585626	−	0.810581i	$-0.699151\pi$
$883$	−2.39614e10	−1.17125	−0.585626	+	0.810581i	$0.699151\pi$
$884$	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
$889$	−2.29238e10	−1.09429
$890$	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
$901$	0	0
$902$	0	0
$903$	− 2.96587e9i	− 0.134043i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.63161e10	0.726092	0.363046	−	0.931771i	$-0.381737\pi$
$907$	1.63161e10	0.726092	0.363046	+	0.931771i	$0.381737\pi$
$908$	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
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	3.77715e10i	1.60532i	0.596440	+	0.802658i	$0.296581\pi$
$919$	3.77715e10i	1.60532i	−0.596440	+	0.802658i	$0.703419\pi$
$920$	0	0
$921$	−4.66727e10	−1.96859
$922$	0	0
$923$	0	0
$924$	0	0
$925$	4.28969e10i	1.78209i
$926$	0	0
$927$	− 1.75533e10i	− 0.723737i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	9.38106e9	0.381003
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.58514e9	0.142370	0.0711848	−	0.997463i	$-0.477322\pi$
$937$	3.58514e9	0.142370	0.0711848	+	0.997463i	$0.477322\pi$
$938$	0	0
$939$	5.07350e10	1.99976
$940$	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
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	− 3.83868e10i	− 1.45798i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−4.49832e9	−0.163500
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4.22432e10i	1.50233i	0.660117	+	0.751163i	$0.270507\pi$
$967$	4.22432e10i	1.50233i	−0.660117	+	0.751163i	$0.729493\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	1.56519e10i	0.544719i
$974$	0	0
$975$	2.23509e10i	0.772289i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 4.62987e10i	− 1.56577i
$982$	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
$991$	− 4.67016e10i	− 1.52431i	−0.647394	−	0.762156i	$-0.724141\pi$
$991$	− 4.67016e10i	− 1.52431i	0.647394	−	0.762156i	$-0.275859\pi$
$992$	0	0
$993$	−2.00664e10	−0.650351
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 2.76522e10i	− 0.883682i	−0.897093	−	0.441841i	$-0.854326\pi$
$997$	− 2.76522e10i	− 0.883682i	0.897093	−	0.441841i	$-0.145674\pi$
$998$	0	0
$999$	− 5.61577e10i	− 1.78209i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.8.f.b.95.3	yes	4
3.2	odd	2	CM	192.8.f.b.95.3	yes	4
4.3	odd	2	inner	192.8.f.b.95.2	yes	4
8.3	odd	2	inner	192.8.f.b.95.4	yes	4
8.5	even	2	inner	192.8.f.b.95.1	✓	4
12.11	even	2	inner	192.8.f.b.95.2	yes	4
24.5	odd	2	inner	192.8.f.b.95.1	✓	4
24.11	even	2	inner	192.8.f.b.95.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.8.f.b.95.1	✓	4	8.5	even	2	inner
192.8.f.b.95.1	✓	4	24.5	odd	2	inner
192.8.f.b.95.2	yes	4	4.3	odd	2	inner
192.8.f.b.95.2	yes	4	12.11	even	2	inner
192.8.f.b.95.3	yes	4	1.1	even	1	trivial
192.8.f.b.95.3	yes	4	3.2	odd	2	CM
192.8.f.b.95.4	yes	4	8.3	odd	2	inner
192.8.f.b.95.4	yes	4	24.11	even	2	inner

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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	192.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+46.7654 q^{3} -508.000i q^{7} +2187.00 q^{9} +O(q^{10})\)	\(q+46.7654 q^{3} -508.000i q^{7} +2187.00 q^{9} -6117.60i q^{13} +16589.6 q^{19} -23756.8i q^{21} -78125.0 q^{25} +102276. q^{27} -178916. i q^{31} -549081. i q^{37} -286092. i q^{39} +124843. q^{43} +565479. q^{49} +775818. q^{57} +266244. i q^{61} -1.11100e6i q^{63} -4.90862e6 q^{67} +6.27481e6 q^{73} -3.65354e6 q^{75} -8.76304e6i q^{79} +4.78297e6 q^{81} -3.10774e6 q^{91} -8.36707e6i q^{93} +1.22452e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8748 q^{9}+O(q^{10}) \) 4 * q + 8748 * q^9	\( 4 q + 8748 q^{9} - 312500 q^{25} + 2261916 q^{49} + 3103272 q^{57} + 25099240 q^{73} + 19131876 q^{81} + 48980792 q^{97}+O(q^{100}) \) 4 * q + 8748 * q^9 - 312500 * q^25 + 2261916 * q^49 + 3103272 * q^57 + 25099240 * q^73 + 19131876 * q^81 + 48980792 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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