LMFDB - Embedded newform 192.8.c.a.191.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,8,Mod(191,192)]

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, 0, 1]))

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

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

chi := DirichletCharacter("192.191");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	192.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:	$59.9779248930$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			191.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	192.191
Dual form			192.8.c.a.191.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-46.7654i q^{3} -1742.44i q^{7} -2187.00 q^{9} -14614.0 q^{13} +16589.6i q^{19} -81486.0 q^{21} +78125.0 q^{25} +102276. i q^{27} -279356. i q^{31} -279710. q^{37} +683429. i q^{39} -124843. i q^{43} -2.21256e6 q^{49} +775818. q^{57} +3.53555e6 q^{61} +3.81072e6i q^{63} +4.90862e6i q^{67} +6.27481e6 q^{73} -3.65354e6i q^{75} +157149. i q^{79} +4.78297e6 q^{81} +2.54641e7i q^{91} -1.30642e7 q^{93} -1.22452e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4374 q^{9} - 29228 q^{13} - 162972 q^{21} + 156250 q^{25} - 559420 q^{37} - 4425130 q^{49} + 1551636 q^{57} + 7071092 q^{61} + 12549620 q^{73} + 9565938 q^{81} - 26128332 q^{93} - 24490396 q^{97}+O(q^{100}) $ 2 * q - 4374 * q^9 - 29228 * q^13 - 162972 * q^21 + 156250 * q^25 - 559420 * q^37 - 4425130 * q^49 + 1551636 * q^57 + 7071092 * q^61 + 12549620 * q^73 + 9565938 * q^81 - 26128332 * q^93 - 24490396 * q^97

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
$3$	− 46.7654i	− 1.00000i
$3$	− 46.7654i	− 1.00000i
$4$	0	0
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	− 1742.44i	− 1.92006i	−0.279892	−	0.960031i	$-0.590299\pi$
$7$	− 1742.44i	− 1.92006i	0.279892	−	0.960031i	$-0.409701\pi$
$8$	0	0
$9$	−2187.00	−1.00000
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−14614.0	−1.84488	−0.922438	−	0.386144i	$-0.873807\pi$
$13$	−14614.0	−1.84488	−0.922438	+	0.386144i	$0.873807\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	16589.6i	0.554878i	0.960743	+	0.277439i	$0.0894857\pi$
$19$	16589.6i	0.554878i	−0.960743	+	0.277439i	$0.910514\pi$
$20$	0	0
$21$	−81486.0	−1.92006
$22$	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.i	1.00000i
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	− 279356.i	− 1.68419i	−0.539328	−	0.842096i	$-0.681322\pi$
$31$	− 279356.i	− 1.68419i	0.539328	−	0.842096i	$-0.318678\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−279710.	−0.907825	−0.453912	−	0.891046i	$-0.649972\pi$
$37$	−279710.	−0.907825	−0.453912	+	0.891046i	$0.649972\pi$
$38$	0	0
$39$	683429.i	1.84488i
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	− 124843.i	− 0.239455i	−0.992807	−	0.119727i	$-0.961798\pi$
$43$	− 124843.i	− 0.239455i	0.992807	−	0.119727i	$-0.0382021\pi$
$44$	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
$49$	−2.21256e6	−2.68664
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	775818.	0.554878
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	3.53555e6	1.99435	0.997177	−	0.0750923i	$-0.0239251\pi$
$61$	3.53555e6	1.99435	0.997177	+	0.0750923i	$0.0239251\pi$
$62$	0	0
$63$	3.81072e6i	1.92006i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.90862e6i	1.99387i	0.0782078	+	0.996937i	$0.475080\pi$
$67$	4.90862e6i	1.99387i	−0.0782078	+	0.996937i	$0.524920\pi$
$68$	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
$75$	− 3.65354e6i	− 1.00000i
$76$	0	0
$77$	0	0
$78$	0	0
$79$	157149.i	0.0358605i	0.999839	+	0.0179303i	$0.00570769\pi$
$79$	157149.i	0.0358605i	−0.999839	+	0.0179303i	$0.994292\pi$
$80$	0	0
$81$	4.78297e6	1.00000
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	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
$91$	2.54641e7i	3.54228i
$92$	0	0
$93$	−1.30642e7	−1.68419
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1.22452e7	−1.36227	−0.681137	−	0.732156i	$-0.738514\pi$
$97$	−1.22452e7	−1.36227	−0.681137	+	0.732156i	$0.738514\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	2.06768e7i	1.86446i	0.361868	+	0.932229i	$0.382139\pi$
$103$	2.06768e7i	1.86446i	−0.361868	+	0.932229i	$0.617861\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−1.68240e7	−1.24433	−0.622167	−	0.782884i	$-0.713748\pi$
$109$	−1.68240e7	−1.24433	−0.622167	+	0.782884i	$0.713748\pi$
$110$	0	0
$111$	1.30807e7i	0.907825i
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.19608e7	1.84488
$118$	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
$127$	9.75612e6i	0.422634i	0.977418	+	0.211317i	$0.0677752\pi$
$127$	9.75612e6i	0.422634i	−0.977418	+	0.211317i	$0.932225\pi$
$128$	0	0
$129$	−5.83832e6	−0.239455
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	2.89064e7	1.06540
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	− 3.08109e7i	− 0.973088i	−0.873656	−	0.486544i	$-0.838257\pi$
$139$	− 3.08109e7i	− 0.973088i	0.873656	−	0.486544i	$-0.161743\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.03471e8i	2.68664i
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	3.49967e7i	0.827194i	0.910460	+	0.413597i	$0.135728\pi$
$151$	3.49967e7i	0.827194i	−0.910460	+	0.413597i	$0.864272\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−8.20114e7	−1.69132	−0.845659	−	0.533723i	$-0.820792\pi$
$157$	−8.20114e7	−1.69132	−0.845659	+	0.533723i	$0.820792\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 6.77910e7i	− 1.22607i	−0.790056	−	0.613035i	$-0.789948\pi$
$163$	− 6.77910e7i	− 1.22607i	0.790056	−	0.613035i	$-0.210052\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1.50820e8	2.40357
$170$	0	0
$171$	− 3.62814e7i	− 0.554878i
$172$	0	0
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	− 1.36128e8i	− 1.92006i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	−3.98034e7	−0.498936	−0.249468	−	0.968383i	$-0.580256\pi$
$181$	−3.98034e7	−0.498936	−0.249468	+	0.968383i	$0.580256\pi$
$182$	0	0
$183$	− 1.65341e8i	− 1.99435i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.78210e8	1.92006
$190$	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	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	1.68737e8i	1.51783i	0.651188	+	0.758916i	$0.274271\pi$
$199$	1.68737e8i	1.51783i	−0.651188	+	0.758916i	$0.725729\pi$
$200$	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
$211$	− 1.60698e8i	− 1.17766i	−0.808256	−	0.588832i	$-0.799588\pi$
$211$	− 1.60698e8i	− 1.17766i	0.808256	−	0.588832i	$-0.200412\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−4.86761e8	−3.23375
$218$	0	0
$219$	− 2.93444e8i	− 1.88786i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 2.01450e8i	− 1.21647i	−0.793759	−	0.608233i	$-0.791879\pi$
$223$	− 2.01450e8i	− 1.21647i	0.793759	−	0.608233i	$-0.208121\pi$
$224$	0	0
$225$	−1.70859e8	−1.00000
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	−2.01186e8	−1.10707	−0.553534	−	0.832827i	$-0.686721\pi$
$229$	−2.01186e8	−1.10707	−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
$237$	7.34913e6	0.0358605
$238$	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
$243$	− 2.23677e8i	− 1.00000i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 2.42440e8i	− 1.02368i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	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
$259$	4.87379e8i	1.74308i
$260$	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	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	− 4.40021e8i	− 1.34302i	−0.740998	−	0.671508i	$-0.765647\pi$
$271$	− 4.40021e8i	− 1.34302i	0.740998	−	0.671508i	$-0.234353\pi$
$272$	0	0
$273$	1.19084e9	3.54228
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.85096e6	0.0165405	0.00827024	−	0.999966i	$-0.497367\pi$
$277$	5.85096e6	0.0165405	0.00827024	+	0.999966i	$0.497367\pi$
$278$	0	0
$279$	6.10951e8i	1.68419i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	− 6.17390e8i	− 1.61923i	−0.586964	−	0.809613i	$-0.699677\pi$
$283$	− 6.17390e8i	− 1.61923i	0.586964	−	0.809613i	$-0.300323\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4.10339e8	1.00000
$290$	0	0
$291$	5.72651e8i	1.36227i
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−2.17531e8	−0.459769
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	9.98019e8i	1.96859i	0.176541	+	0.984293i	$0.443509\pi$
$307$	9.98019e8i	1.96859i	−0.176541	+	0.984293i	$0.556491\pi$
$308$	0	0
$309$	9.66957e8	1.86446
$310$	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	$-0.995065\pi$
$313$	−1.08488e9	−1.99976	−0.999880	+	0.0155022i	$0.995065\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.14172e9	−1.84488
$326$	0	0
$327$	7.86782e8i	1.24433i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 4.29087e8i	− 0.650351i	−0.945654	−	0.325175i	$-0.894577\pi$
$331$	− 4.29087e8i	− 0.650351i	0.945654	−	0.325175i	$-0.105423\pi$
$332$	0	0
$333$	6.11726e8	0.907825
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.27571e9	−1.81571	−0.907854	−	0.419286i	$-0.862281\pi$
$337$	−1.27571e9	−1.81571	−0.907854	+	0.419286i	$0.862281\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	2.42029e9i	3.23846i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	6.95221e8	0.875455	0.437728	−	0.899108i	$-0.355783\pi$
$349$	6.95221e8	0.875455	0.437728	+	0.899108i	$0.355783\pi$
$350$	0	0
$351$	− 1.49466e9i	− 1.84488i
$352$	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
$363$	9.11325e8i	1.00000i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.40861e9i	1.48751i	0.668454	+	0.743754i	$0.266956\pi$
$367$	1.40861e9i	1.48751i	−0.668454	+	0.743754i	$0.733044\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−6.08469e8	−0.607096	−0.303548	−	0.952816i	$-0.598171\pi$
$373$	−6.08469e8	−0.607096	−0.303548	+	0.952816i	$0.598171\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	2.58413e8i	0.243824i	0.992541	+	0.121912i	$0.0389026\pi$
$379$	2.58413e8i	0.243824i	−0.992541	+	0.121912i	$0.961097\pi$
$380$	0	0
$381$	4.56249e8	0.422634
$382$	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
$387$	2.73031e8i	0.239455i
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.90824e9	−1.53062	−0.765308	−	0.643665i	$-0.777413\pi$
$397$	−1.90824e9	−1.53062	−0.765308	+	0.643665i	$0.777413\pi$
$398$	0	0
$399$	− 1.35182e9i	− 1.06540i
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	4.08250e9i	3.10713i
$404$	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		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−2.23703e9	−1.46112	−0.730558	−	0.682850i	$-0.760740\pi$
$421$	−2.23703e9	−1.46112	−0.730558	+	0.682850i	$0.760740\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 6.16049e9i	− 3.82928i
$428$	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	$-0.607120\pi$
$433$	−1.11566e9	−0.660424	−0.330212	+	0.943907i	$0.607120\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 1.41308e9i	− 0.797151i	−0.917135	−	0.398576i	$-0.869505\pi$
$439$	− 1.41308e9i	− 0.797151i	0.917135	−	0.398576i	$-0.130495\pi$
$440$	0	0
$441$	4.83888e9	2.68664
$442$	0	0
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	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
$453$	1.63663e9	0.827194
$454$	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	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	− 4.14708e9i	− 1.94182i	−0.239440	−	0.970911i	$-0.576964\pi$
$463$	− 4.14708e9i	− 1.94182i	0.239440	−	0.970911i	$-0.423036\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	8.55299e9	3.82836
$470$	0	0
$471$	3.83529e9i	1.69132i
$472$	0	0
$473$	0	0
$474$	0	0
$475$	1.29606e9i	0.554878i
$476$	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
$481$	4.08768e9	1.67482
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	2.66443e9i	1.04533i	0.852539	+	0.522664i	$0.175062\pi$
$487$	2.66443e9i	1.04533i	−0.852539	+	0.522664i	$0.824938\pi$
$488$	0	0
$489$	−3.17027e9	−1.22607
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 4.26639e9i	− 1.53712i	−0.639775	−	0.768562i	$-0.720972\pi$
$499$	− 4.26639e9i	− 1.53712i	0.639775	−	0.768562i	$-0.279028\pi$
$500$	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
$507$	− 7.05318e9i	− 2.40357i
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	− 1.09335e10i	− 3.62482i
$512$	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
$523$	− 2.18274e9i	− 0.667185i	−0.942717	−	0.333592i	$-0.891739\pi$
$523$	− 2.18274e9i	− 0.667185i	0.942717	−	0.333592i	$-0.108261\pi$
$524$	0	0
$525$	−6.36609e9	−1.92006
$526$	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
$541$	−3.68782e9	−1.00134	−0.500668	−	0.865640i	$-0.666912\pi$
$541$	−3.68782e9	−1.00134	−0.500668	+	0.865640i	$0.666912\pi$
$542$	0	0
$543$	1.86142e9i	0.498936i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	4.77362e9i	1.24707i	0.781793	+	0.623537i	$0.214305\pi$
$547$	4.77362e9i	1.24707i	−0.781793	+	0.623537i	$0.785695\pi$
$548$	0	0
$549$	−7.73224e9	−1.99435
$550$	0	0
$551$	0	0
$552$	0	0
$553$	2.73823e8	0.0688545
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	1.82445e9i	0.441765i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 8.33405e9i	− 1.92006i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	6.49838e9i	1.46076i	0.683042	+	0.730379i	$0.260657\pi$
$571$	6.49838e9i	1.46076i	−0.683042	+	0.730379i	$0.739343\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.23933e9	0.918718	0.459359	−	0.888251i	$-0.348079\pi$
$577$	4.23933e9	0.918718	0.459359	+	0.888251i	$0.348079\pi$
$578$	0	0
$579$	4.51370e9i	0.966402i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	4.63439e9	0.934521
$590$	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
$597$	7.89103e9	1.51783
$598$	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
$603$	− 1.07352e10i	− 1.99387i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.07643e10i	1.95355i	0.214270	+	0.976774i	$0.431263\pi$
$607$	1.07643e10i	1.95355i	−0.214270	+	0.976774i	$0.568737\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.12826e10	−1.97832	−0.989161	−	0.146837i	$-0.953091\pi$
$613$	−1.12826e10	−1.97832	−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
$619$	2.08193e9i	0.352816i	0.984317	+	0.176408i	$0.0564478\pi$
$619$	2.08193e9i	0.352816i	−0.984317	+	0.176408i	$0.943552\pi$
$620$	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
$631$	4.73166e9i	0.749740i	0.927077	+	0.374870i	$0.122313\pi$
$631$	4.73166e9i	0.749740i	−0.927077	+	0.374870i	$0.877687\pi$
$632$	0	0
$633$	−7.51509e9	−1.17766
$634$	0	0
$635$	0	0
$636$	0	0
$637$	3.23344e10	4.95652
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	− 1.34415e10i	− 1.99393i	−0.0778386	−	0.996966i	$-0.524802\pi$
$643$	− 1.34415e10i	− 1.99393i	0.0778386	−	0.996966i	$-0.475198\pi$
$644$	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
$651$	2.27636e10i	3.23375i
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−1.37230e10	−1.88786
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−6.61860e9	−0.891376	−0.445688	−	0.895188i	$-0.647041\pi$
$661$	−6.61860e9	−0.891376	−0.445688	+	0.895188i	$0.647041\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−9.42087e9	−1.21647
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−4.99918e9	−0.632187	−0.316094	−	0.948728i	$-0.602371\pi$
$673$	−4.99918e9	−0.632187	−0.316094	+	0.948728i	$0.602371\pi$
$674$	0	0
$675$	7.99030e9i	1.00000i
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	2.13366e10i	2.61565i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	9.40855e9i	1.10707i
$688$	0	0
$689$	0	0
$690$	0	0
$691$	8.33859e9i	0.961434i	0.876876	+	0.480717i	$0.159624\pi$
$691$	8.33859e9i	0.961434i	−0.876876	+	0.480717i	$0.840376\pi$
$692$	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	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	− 4.64027e9i	− 0.503732i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	3.03318e9	0.319622	0.159811	−	0.987148i	$-0.448912\pi$
$709$	3.03318e9	0.319622	0.159811	+	0.987148i	$0.448912\pi$
$710$	0	0
$711$	− 3.43685e8i	− 0.0358605i
$712$	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
$721$	3.60281e10	3.57988
$722$	0	0
$723$	2.69308e9i	0.265012i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.91072e10i	− 1.84428i	−0.386861	−	0.922138i	$-0.626441\pi$
$727$	− 1.91072e10i	− 1.84428i	0.386861	−	0.922138i	$-0.373559\pi$
$728$	0	0
$729$	−1.04604e10	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−1.97732e10	−1.85445	−0.927223	−	0.374511i	$-0.877811\pi$
$733$	−1.97732e10	−1.85445	−0.927223	+	0.374511i	$0.877811\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 1.09057e10i	− 0.994026i	−0.867743	−	0.497013i	$-0.834430\pi$
$739$	− 1.09057e10i	− 0.994026i	0.867743	−	0.497013i	$-0.165570\pi$
$740$	0	0
$741$	−1.13378e10	−1.02368
$742$	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
$751$	1.80528e10i	1.55527i	0.628718	+	0.777633i	$0.283580\pi$
$751$	1.80528e10i	1.55527i	−0.628718	+	0.777633i	$0.716420\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.12787e9	0.597206	0.298603	−	0.954377i	$-0.403479\pi$
$757$	7.12787e9	0.597206	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
$763$	2.93149e10i	2.38920i
$764$	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	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	− 2.18247e10i	− 1.68419i
$776$	0	0
$777$	2.27924e10	1.74308
$778$	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
$787$	1.14622e10i	0.838214i	0.907937	+	0.419107i	$0.137657\pi$
$787$	1.14622e10i	0.838214i	−0.907937	+	0.419107i	$0.862343\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−5.16685e10	−3.67934
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	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
$811$	− 2.91438e10i	− 1.91855i	−0.282472	−	0.959276i	$-0.591154\pi$
$811$	− 2.91438e10i	− 1.91855i	0.282472	−	0.959276i	$-0.408846\pi$
$812$	0	0
$813$	−2.05777e10	−1.34302
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.07109e9	0.132868
$818$	0	0
$819$	− 5.56899e10i	− 3.54228i
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	− 2.78506e9i	− 0.174154i	−0.996202	−	0.0870772i	$-0.972247\pi$
$823$	− 2.78506e9i	− 0.174154i	0.996202	−	0.0870772i	$-0.0277527\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	6.33017e9	0.385900	0.192950	−	0.981209i	$-0.438195\pi$
$829$	6.33017e9	0.385900	0.192950	+	0.981209i	$0.438195\pi$
$830$	0	0
$831$	− 2.73622e8i	− 0.0165405i
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.85713e10	1.68419
$838$	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
$847$	3.39553e10i	1.92006i
$848$	0	0
$849$	−2.88725e10	−1.61923
$850$	0	0
$851$	0	0
$852$	0	0
$853$	2.44073e10	1.34647	0.673237	−	0.739427i	$-0.264904\pi$
$853$	2.44073e10	1.34647	0.673237	+	0.739427i	$0.264904\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	3.20767e10i	1.72669i	0.504617	+	0.863343i	$0.331634\pi$
$859$	3.20767e10i	1.72669i	−0.504617	+	0.863343i	$0.668366\pi$
$860$	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
$867$	− 1.91896e10i	− 1.00000i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	− 7.17346e10i	− 3.67845i
$872$	0	0
$873$	2.67802e10	1.36227
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.62759e10	1.81602	0.908008	−	0.418953i	$-0.137603\pi$
$877$	3.62759e10	1.81602	0.908008	+	0.418953i	$0.137603\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	− 2.39614e10i	− 1.17125i	−0.810581	−	0.585626i	$-0.800849\pi$
$883$	− 2.39614e10i	− 1.17125i	0.810581	−	0.585626i	$-0.199151\pi$
$884$	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
$889$	1.69995e10	0.811483
$890$	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
$903$	1.01729e10i	0.459769i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.63161e10i	− 0.726092i	−0.931771	−	0.363046i	$-0.881737\pi$
$907$	− 1.63161e10i	− 0.726092i	0.931771	−	0.363046i	$-0.118263\pi$
$908$	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
$919$	− 2.80673e10i	− 1.19288i	−0.802658	−	0.596440i	$-0.796581\pi$
$919$	− 2.80673e10i	− 1.19288i	0.802658	−	0.596440i	$-0.203419\pi$
$920$	0	0
$921$	4.66727e10	1.96859
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−2.18523e10	−0.907825
$926$	0	0
$927$	− 4.52201e10i	− 1.86446i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	− 3.67055e10i	− 1.49076i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.58514e9	−0.142370	−0.0711848	−	0.997463i	$-0.522678\pi$
$937$	−3.58514e9	−0.142370	−0.0711848	+	0.997463i	$0.522678\pi$
$938$	0	0
$939$	5.07350e10i	1.99976i
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	−9.17001e10	−3.48288
$950$	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
$961$	−5.05269e10	−1.83650
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	3.71231e10i	1.32023i	0.751163	+	0.660117i	$0.229493\pi$
$967$	3.71231e10i	1.32023i	−0.751163	+	0.660117i	$0.770507\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−5.36862e10	−1.86839
$974$	0	0
$975$	5.33929e10i	1.84488i
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	3.67941e10	1.24433
$982$	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
$991$	− 3.96695e10i	− 1.29479i	−0.762156	−	0.647394i	$-0.775859\pi$
$991$	− 3.96695e10i	− 1.29479i	0.762156	−	0.647394i	$-0.224141\pi$
$992$	0	0
$993$	−2.00664e10	−0.650351
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−5.61437e10	−1.79419	−0.897093	−	0.441841i	$-0.854326\pi$
$997$	−5.61437e10	−1.79419	−0.897093	+	0.441841i	$0.854326\pi$
$998$	0	0
$999$	− 2.86076e10i	− 0.907825i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.8.c.a.191.1		2
3.2	odd	2	CM	192.8.c.a.191.1		2
4.3	odd	2	inner	192.8.c.a.191.2		2
8.3	odd	2		48.8.c.a.47.1	✓	2
8.5	even	2		48.8.c.a.47.2	yes	2
12.11	even	2	inner	192.8.c.a.191.2		2
24.5	odd	2		48.8.c.a.47.2	yes	2
24.11	even	2		48.8.c.a.47.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.8.c.a.47.1	✓	2	8.3	odd	2
48.8.c.a.47.1	✓	2	24.11	even	2
48.8.c.a.47.2	yes	2	8.5	even	2
48.8.c.a.47.2	yes	2	24.5	odd	2
192.8.c.a.191.1		2	1.1	even	1	trivial
192.8.c.a.191.1		2	3.2	odd	2	CM
192.8.c.a.191.2		2	4.3	odd	2	inner
192.8.c.a.191.2		2	12.11	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-46.7654i q^{3} -1742.44i q^{7} -2187.00 q^{9} -14614.0 q^{13} +16589.6i q^{19} -81486.0 q^{21} +78125.0 q^{25} +102276. i q^{27} -279356. i q^{31} -279710. q^{37} +683429. i q^{39} -124843. i q^{43} -2.21256e6 q^{49} +775818. q^{57} +3.53555e6 q^{61} +3.81072e6i q^{63} +4.90862e6i q^{67} +6.27481e6 q^{73} -3.65354e6i q^{75} +157149. i q^{79} +4.78297e6 q^{81} +2.54641e7i q^{91} -1.30642e7 q^{93} -1.22452e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4374 q^{9} - 29228 q^{13} - 162972 q^{21} + 156250 q^{25} - 559420 q^{37} - 4425130 q^{49} + 1551636 q^{57} + 7071092 q^{61} + 12549620 q^{73} + 9565938 q^{81} - 26128332 q^{93} - 24490396 q^{97}+O(q^{100}) \) 2 * q - 4374 * q^9 - 29228 * q^13 - 162972 * q^21 + 156250 * q^25 - 559420 * q^37 - 4425130 * q^49 + 1551636 * q^57 + 7071092 * q^61 + 12549620 * q^73 + 9565938 * q^81 - 26128332 * q^93 - 24490396 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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