LMFDB - Embedded newform 192.4.f.a.95.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [192,4,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, 4, names="a")

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

chi := DirichletCharacter("192.95");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 192 = 2^{6} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
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:	$11.3283667211$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
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			192.4.f.a.95.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615i q^{3} -10.3923 q^{5} +34.0000i q^{7} -27.0000 q^{9} +O(q^{10})$	$q+5.19615i q^{3} -10.3923 q^{5} +34.0000i q^{7} -27.0000 q^{9} -72.7461i q^{11} -54.0000i q^{15} -176.669 q^{21} -17.0000 q^{25} -140.296i q^{27} -218.238 q^{29} -70.0000i q^{31} +378.000 q^{33} -353.338i q^{35} +280.592 q^{45} -813.000 q^{49} -509.223 q^{53} +756.000i q^{55} +717.069i q^{59} -918.000i q^{63} +322.000 q^{73} -88.3346i q^{75} +2473.37 q^{77} +1370.00i q^{79} +729.000 q^{81} +883.346i q^{83} -1134.00i q^{87} +363.731 q^{93} -574.000 q^{97} +1964.15i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 108 q^{9}+O(q^{10}) $ 4 * q - 108 * q^9	$ 4 q - 108 q^{9} - 68 q^{25} + 1512 q^{33} - 3252 q^{49} + 1288 q^{73} + 2916 q^{81} - 2296 q^{97}+O(q^{100}) $ 4 * q - 108 * q^9 - 68 * q^25 + 1512 * q^33 - 3252 * q^49 + 1288 * q^73 + 2916 * q^81 - 2296 * 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
$3$	5.19615i	1.00000i
$3$	5.19615i	1.00000i
$4$	0	0
$5$	−10.3923	−0.929516	−0.464758	−	0.885438i	$-0.653859\pi$
$5$	−10.3923	−0.929516	−0.464758	+	0.885438i	$0.653859\pi$
$6$	0	0
$7$	34.0000i	1.83583i	0.396780	+	0.917914i	$0.370128\pi$
$7$	34.0000i	1.83583i	−0.396780	+	0.917914i	$0.629872\pi$
$8$	0	0
$9$	−27.0000	−1.00000
$10$	0	0
$11$	− 72.7461i	− 1.99398i	−0.0775275	−	0.996990i	$-0.524703\pi$
$11$	− 72.7461i	− 1.99398i	0.0775275	−	0.996990i	$-0.475297\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	− 54.0000i	− 0.929516i
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−176.669	−1.83583
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−17.0000	−0.136000
$26$	0	0
$27$	− 140.296i	− 1.00000i
$28$	0	0
$29$	−218.238	−1.39744	−0.698722	−	0.715394i	$-0.746247\pi$
$29$	−218.238	−1.39744	−0.698722	+	0.715394i	$0.746247\pi$
$30$	0	0
$31$	− 70.0000i	− 0.405560i	−0.979224	−	0.202780i	$-0.935002\pi$
$31$	− 70.0000i	− 0.405560i	0.979224	−	0.202780i	$-0.0649977\pi$
$32$	0	0
$33$	378.000	1.99398
$34$	0	0
$35$	− 353.338i	− 1.70643i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	280.592	0.929516
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−813.000	−2.37026
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−509.223	−1.31976	−0.659879	−	0.751372i	$-0.729392\pi$
$53$	−509.223	−1.31976	−0.659879	+	0.751372i	$0.729392\pi$
$54$	0	0
$55$	756.000i	1.85344i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	717.069i	1.58228i	0.611636	+	0.791139i	$0.290512\pi$
$59$	717.069i	1.58228i	−0.611636	+	0.791139i	$0.709488\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	− 918.000i	− 1.83583i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	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
$73$	322.000	0.516264	0.258132	−	0.966110i	$-0.416893\pi$
$73$	322.000	0.516264	0.258132	+	0.966110i	$0.416893\pi$
$74$	0	0
$75$	− 88.3346i	− 0.136000i
$76$	0	0
$77$	2473.37	3.66060
$78$	0	0
$79$	1370.00i	1.95110i	0.219774	+	0.975551i	$0.429468\pi$
$79$	1370.00i	1.95110i	−0.219774	+	0.975551i	$0.570532\pi$
$80$	0	0
$81$	729.000	1.00000
$82$	0	0
$83$	883.346i	1.16819i	0.811685	+	0.584095i	$0.198551\pi$
$83$	883.346i	1.16819i	−0.811685	+	0.584095i	$0.801449\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1134.00i	− 1.39744i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	363.731	0.405560
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−574.000	−0.600834	−0.300417	−	0.953808i	$-0.597126\pi$
$97$	−574.000	−0.600834	−0.300417	+	0.953808i	$0.597126\pi$
$98$	0	0
$99$	1964.15i	1.99398i
$100$	0	0
$101$	−1008.05	−0.993120	−0.496560	−	0.868003i	$-0.665404\pi$
$101$	−1008.05	−0.993120	−0.496560	+	0.868003i	$0.665404\pi$
$102$	0	0
$103$	− 1582.00i	− 1.51339i	−0.653768	−	0.756695i	$-0.726813\pi$
$103$	− 1582.00i	− 1.51339i	0.653768	−	0.756695i	$-0.273187\pi$
$104$	0	0
$105$	1836.00	1.70643
$106$	0	0
$107$	− 363.731i	− 0.328628i	−0.986408	−	0.164314i	$-0.947459\pi$
$107$	− 363.731i	− 0.328628i	0.986408	−	0.164314i	$-0.0525410\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3961.00	−2.97596
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1475.71	1.05593
$126$	0	0
$127$	2266.00i	1.58327i	0.610996	+	0.791634i	$0.290769\pi$
$127$	2266.00i	1.58327i	−0.610996	+	0.791634i	$0.709231\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1319.82i	− 0.880255i	−0.897935	−	0.440128i	$-0.854933\pi$
$131$	− 1319.82i	− 0.880255i	0.897935	−	0.440128i	$-0.145067\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1458.00i	0.929516i
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2268.00	1.29895
$146$	0	0
$147$	− 4224.47i	− 2.37026i
$148$	0	0
$149$	−3419.07	−1.87987	−0.939936	−	0.341350i	$-0.889116\pi$
$149$	−3419.07	−1.87987	−0.939936	+	0.341350i	$0.889116\pi$
$150$	0	0
$151$	898.000i	0.483962i	0.970281	+	0.241981i	$0.0777971\pi$
$151$	898.000i	0.483962i	−0.970281	+	0.241981i	$0.922203\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	727.461i	0.376975i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	− 2646.00i	− 1.31976i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	−3928.29	−1.85344
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	2197.00	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3793.19	−1.66700	−0.833500	−	0.552519i	$-0.813666\pi$
$173$	−3793.19	−1.66700	−0.833500	+	0.552519i	$0.813666\pi$
$174$	0	0
$175$	− 578.000i	− 0.249673i
$176$	0	0
$177$	−3726.00	−1.58228
$178$	0	0
$179$	− 1236.68i	− 0.516392i	−0.966093	−	0.258196i	$-0.916872\pi$
$179$	− 1236.68i	− 0.516392i	0.966093	−	0.258196i	$-0.0831280\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	4770.07	1.83583
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	2522.00	0.940609	0.470304	−	0.882504i	$-0.344144\pi$
$193$	2522.00	0.940609	0.470304	+	0.882504i	$0.344144\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	72.7461	0.0263094	0.0131547	−	0.999913i	$-0.495813\pi$
$197$	72.7461	0.0263094	0.0131547	+	0.999913i	$0.495813\pi$
$198$	0	0
$199$	− 5614.00i	− 1.99983i	−0.0130884	−	0.999914i	$-0.504166\pi$
$199$	− 5614.00i	− 1.99983i	0.0130884	−	0.999914i	$-0.495834\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 7420.11i	− 2.56546i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	2380.00	0.744539
$218$	0	0
$219$	1673.16i	0.516264i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 182.000i	− 0.0546530i	−0.999627	−	0.0273265i	$-0.991301\pi$
$223$	− 182.000i	− 0.0546530i	0.999627	−	0.0273265i	$-0.00869938\pi$
$224$	0	0
$225$	459.000	0.136000
$226$	0	0
$227$	5954.79i	1.74112i	0.492066	+	0.870558i	$0.336242\pi$
$227$	5954.79i	1.74112i	−0.492066	+	0.870558i	$0.663758\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	12852.0i	3.66060i
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−7118.73	−1.95110
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−6230.00	−1.66518	−0.832592	−	0.553886i	$-0.813144\pi$
$241$	−6230.00	−1.66518	−0.832592	+	0.553886i	$0.813144\pi$
$242$	0	0
$243$	3788.00i	1.00000i
$244$	0	0
$245$	8448.94	2.20320
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−4590.00	−1.16819
$250$	0	0
$251$	6827.74i	1.71699i	0.512826	+	0.858493i	$0.328599\pi$
$251$	6827.74i	1.71699i	−0.512826	+	0.858493i	$0.671401\pi$
$252$	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
$259$	0	0
$260$	0	0
$261$	5892.44	1.39744
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	5292.00	1.22674
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7264.22	1.64650	0.823248	−	0.567682i	$-0.192160\pi$
$269$	7264.22	1.64650	0.823248	+	0.567682i	$0.192160\pi$
$270$	0	0
$271$	− 7238.00i	− 1.62243i	−0.584751	−	0.811213i	$-0.698808\pi$
$271$	− 7238.00i	− 1.62243i	0.584751	−	0.811213i	$-0.301192\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1236.68i	0.271181i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	1890.00i	0.405560i
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	4913.00	1.00000
$290$	0	0
$291$	− 2982.59i	− 0.600834i
$292$	0	0
$293$	2899.45	0.578116	0.289058	−	0.957312i	$-0.406658\pi$
$293$	2899.45	0.578116	0.289058	+	0.957312i	$0.406658\pi$
$294$	0	0
$295$	− 7452.00i	− 1.47075i
$296$	0	0
$297$	−10206.0	−1.99398
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 5238.00i	− 0.993120i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	8220.31	1.51339
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	7378.00	1.33236	0.666181	−	0.745790i	$-0.267928\pi$
$313$	7378.00	1.33236	0.666181	+	0.745790i	$0.267928\pi$
$314$	0	0
$315$	9540.14i	1.70643i
$316$	0	0
$317$	−11275.7	−1.99780	−0.998902	−	0.0468563i	$-0.985080\pi$
$317$	−11275.7	−1.99780	−0.998902	+	0.0468563i	$0.985080\pi$
$318$	0	0
$319$	15876.0i	2.78647i
$320$	0	0
$321$	1890.00	0.328628
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11594.0	1.87408	0.937041	−	0.349220i	$-0.113553\pi$
$337$	11594.0	1.87408	0.937041	+	0.349220i	$0.113553\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5092.23	−0.808679
$342$	0	0
$343$	− 15980.0i	− 2.51557i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10984.7i	1.69939i	0.527276	+	0.849694i	$0.323213\pi$
$347$	10984.7i	1.69939i	−0.527276	+	0.849694i	$0.676787\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	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
$361$	−6859.00	−1.00000
$362$	0	0
$363$	− 20582.0i	− 2.97596i
$364$	0	0
$365$	−3346.32	−0.479875
$366$	0	0
$367$	13034.0i	1.85387i	0.375225	+	0.926934i	$0.377565\pi$
$367$	13034.0i	1.85387i	−0.375225	+	0.926934i	$0.622435\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 17313.6i	− 2.42285i
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	7668.00i	1.05593i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	−11774.5	−1.58327
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−25704.0	−3.40259
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14040.0	1.82997	0.914983	−	0.403493i	$-0.132204\pi$
$389$	14040.0	1.82997	0.914983	+	0.403493i	$0.132204\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	6858.00	0.880255
$394$	0	0
$395$	− 14237.5i	− 1.81358i
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−7575.99	−0.929516
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−11270.0	−1.36251	−0.681254	−	0.732047i	$-0.738565\pi$
$409$	−11270.0	−1.36251	−0.681254	+	0.732047i	$0.738565\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−24380.3	−2.90479
$414$	0	0
$415$	− 9180.00i	− 1.08585i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11940.8i	1.39223i	0.717930	+	0.696115i	$0.245090\pi$
$419$	11940.8i	1.39223i	−0.717930	+	0.696115i	$0.754910\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	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
$433$	15442.0	1.71385	0.856923	−	0.515445i	$-0.172373\pi$
$433$	15442.0	1.71385	0.856923	+	0.515445i	$0.172373\pi$
$434$	0	0
$435$	11784.9i	1.29895i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	5474.00i	0.595125i	0.954702	+	0.297562i	$0.0961736\pi$
$439$	5474.00i	0.595125i	−0.954702	+	0.297562i	$0.903826\pi$
$440$	0	0
$441$	21951.0	2.37026
$442$	0	0
$443$	− 11130.2i	− 1.19370i	−0.802352	−	0.596851i	$-0.796418\pi$
$443$	− 11130.2i	− 1.19370i	0.802352	−	0.596851i	$-0.203582\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 17766.0i	− 1.87987i
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−4666.14	−0.483962
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−2774.00	−0.283944	−0.141972	−	0.989871i	$-0.545344\pi$
$457$	−2774.00	−0.283944	−0.141972	+	0.989871i	$0.545344\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2629.25	−0.265632	−0.132816	−	0.991141i	$-0.542402\pi$
$461$	−2629.25	−0.265632	−0.132816	+	0.991141i	$0.542402\pi$
$462$	0	0
$463$	18538.0i	1.86076i	0.366591	+	0.930382i	$0.380525\pi$
$463$	18538.0i	1.86076i	−0.366591	+	0.930382i	$0.619475\pi$
$464$	0	0
$465$	−3780.00	−0.376975
$466$	0	0
$467$	− 19069.9i	− 1.88961i	−0.327630	−	0.944806i	$-0.606250\pi$
$467$	− 19069.9i	− 1.88961i	0.327630	−	0.944806i	$-0.393750\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	13749.0	1.31976
$478$	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
$485$	5965.18	0.558485
$486$	0	0
$487$	2914.00i	0.271142i	0.990768	+	0.135571i	$0.0432868\pi$
$487$	2914.00i	0.271142i	−0.990768	+	0.135571i	$0.956713\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	800.207i	0.0735496i	0.999324	+	0.0367748i	$0.0117084\pi$
$491$	800.207i	0.0735496i	−0.999324	+	0.0367748i	$0.988292\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	− 20412.0i	− 1.85344i
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	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
$505$	10476.0	0.923121
$506$	0	0
$507$	11415.9i	1.00000i
$508$	0	0
$509$	22561.7	1.96469	0.982347	−	0.187067i	$-0.0598980\pi$
$509$	22561.7	1.96469	0.982347	+	0.187067i	$0.0598980\pi$
$510$	0	0
$511$	10948.0i	0.947771i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16440.6i	1.40672i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	− 19710.0i	− 1.66700i
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	3003.38	0.249673
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	− 19360.9i	− 1.58228i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	3780.00i	0.305465i
$536$	0	0
$537$	6426.00	0.516392
$538$	0	0
$539$	59142.6i	4.72626i
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−46580.0	−3.58189
$554$	0	0
$555$	0	0
$556$	0	0
$557$	16076.9	1.22298	0.611490	−	0.791252i	$-0.290570\pi$
$557$	16076.9	1.22298	0.611490	+	0.791252i	$0.290570\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9030.91i	0.676035i	0.941140	+	0.338017i	$0.109756\pi$
$563$	9030.91i	0.676035i	−0.941140	+	0.338017i	$0.890244\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	24786.0i	1.83583i
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	10906.0	0.786868	0.393434	−	0.919353i	$-0.371287\pi$
$577$	10906.0	0.786868	0.393434	+	0.919353i	$0.371287\pi$
$578$	0	0
$579$	13104.7i	0.940609i
$580$	0	0
$581$	−30033.8	−2.14460
$582$	0	0
$583$	37044.0i	2.63157i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 25305.3i	− 1.77932i	−0.456625	−	0.889659i	$-0.650942\pi$
$587$	− 25305.3i	− 1.77932i	0.456625	−	0.889659i	$-0.349058\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	378.000i	0.0263094i
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29171.2	1.99983
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−3598.00	−0.244202	−0.122101	−	0.992518i	$-0.538963\pi$
$601$	−3598.00	−0.244202	−0.122101	+	0.992518i	$0.538963\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	41163.9	2.76620
$606$	0	0
$607$	− 29414.0i	− 1.96685i	−0.181318	−	0.983425i	$-0.558036\pi$
$607$	− 29414.0i	− 1.96685i	0.181318	−	0.983425i	$-0.441964\pi$
$608$	0	0
$609$	38556.0	2.56546
$610$	0	0
$611$	0	0
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−13211.0	−0.845504
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 30350.0i	− 1.91476i	−0.288829	−	0.957381i	$-0.593266\pi$
$631$	− 30350.0i	− 1.91476i	0.288829	−	0.957381i	$-0.406734\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 23549.0i	− 1.47167i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	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
$649$	52164.0	3.15503
$650$	0	0
$651$	12366.8i	0.744539i
$652$	0	0
$653$	−28734.7	−1.72202	−0.861008	−	0.508591i	$-0.830166\pi$
$653$	−28734.7	−1.72202	−0.861008	+	0.508591i	$0.830166\pi$
$654$	0	0
$655$	13716.0i	0.818211i
$656$	0	0
$657$	−8694.00	−0.516264
$658$	0	0
$659$	33681.5i	1.99096i	0.0949654	+	0.995481i	$0.469726\pi$
$659$	33681.5i	1.99096i	−0.0949654	+	0.995481i	$0.530274\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	945.700	0.0546530
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−29342.0	−1.68061	−0.840305	−	0.542113i	$-0.817624\pi$
$673$	−29342.0	−1.68061	−0.840305	+	0.542113i	$0.817624\pi$
$674$	0	0
$675$	2385.03i	0.136000i
$676$	0	0
$677$	−34762.3	−1.97344	−0.986722	−	0.162416i	$-0.948071\pi$
$677$	−34762.3	−1.97344	−0.986722	+	0.162416i	$0.948071\pi$
$678$	0	0
$679$	− 19516.0i	− 1.10303i
$680$	0	0
$681$	−30942.0	−1.74112
$682$	0	0
$683$	− 33827.0i	− 1.89510i	−0.319610	−	0.947549i	$-0.603552\pi$
$683$	− 33827.0i	− 1.89510i	0.319610	−	0.947549i	$-0.396448\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	−66781.0	−3.66060
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	24806.4	1.33656	0.668278	−	0.743911i	$-0.267032\pi$
$701$	24806.4	1.33656	0.668278	+	0.743911i	$0.267032\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 34273.8i	− 1.82320i
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	− 36990.0i	− 1.95110i
$712$	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
$721$	53788.0	2.77832
$722$	0	0
$723$	− 32372.0i	− 1.66518i
$724$	0	0
$725$	3710.05	0.190052
$726$	0	0
$727$	4354.00i	0.222120i	0.993814	+	0.111060i	$0.0354245\pi$
$727$	4354.00i	0.222120i	−0.993814	+	0.111060i	$0.964575\pi$
$728$	0	0
$729$	−19683.0	−1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	43902.0i	2.20320i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	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
$745$	35532.0	1.74737
$746$	0	0
$747$	− 23850.3i	− 1.16819i
$748$	0	0
$749$	12366.8	0.603304
$750$	0	0
$751$	21530.0i	1.04613i	0.852294	+	0.523063i	$0.175211\pi$
$751$	21530.0i	1.04613i	−0.852294	+	0.523063i	$0.824789\pi$
$752$	0	0
$753$	−35478.0	−1.71699
$754$	0	0
$755$	− 9332.29i	− 0.449850i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−42406.0	−1.98856	−0.994278	−	0.106824i	$-0.965932\pi$
$769$	−42406.0	−1.98856	−0.994278	+	0.106824i	$0.965932\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−19797.3	−0.921165	−0.460583	−	0.887617i	$-0.652360\pi$
$773$	−19797.3	−0.921165	−0.460583	+	0.887617i	$0.652360\pi$
$774$	0	0
$775$	1190.00i	0.0551562i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	30618.0i	1.39744i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	27498.0i	1.22674i
$796$	0	0
$797$	−36217.2	−1.60963	−0.804817	−	0.593523i	$-0.797737\pi$
$797$	−36217.2	−1.60963	−0.804817	+	0.593523i	$0.797737\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 23424.3i	− 1.02942i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	37746.0i	1.64650i
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	37609.8	1.62243
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−46484.8	−1.97604	−0.988021	−	0.154321i	$-0.950681\pi$
$821$	−46484.8	−1.97604	−0.988021	+	0.154321i	$0.950681\pi$
$822$	0	0
$823$	− 16238.0i	− 0.687753i	−0.939015	−	0.343877i	$-0.888260\pi$
$823$	− 16238.0i	− 0.687753i	0.939015	−	0.343877i	$-0.111740\pi$
$824$	0	0
$825$	−6426.00	−0.271181
$826$	0	0
$827$	24660.9i	1.03693i	0.855097	+	0.518467i	$0.173497\pi$
$827$	24660.9i	1.03693i	−0.855097	+	0.518467i	$0.826503\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−9820.73	−0.405560
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	23239.0	0.952848
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−22831.9	−0.929516
$846$	0	0
$847$	− 134674.i	− 5.46335i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	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
$865$	39420.0	1.54950
$866$	0	0
$867$	25528.7i	1.00000i
$868$	0	0
$869$	99662.2	3.89046
$870$	0	0
$871$	0	0
$872$	0	0
$873$	15498.0	0.600834
$874$	0	0
$875$	50174.0i	1.93851i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	15066.0i	0.578116i
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	38721.7	1.47075
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−77044.0	−2.90661
$890$	0	0
$891$	− 53031.9i	− 1.99398i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	12852.0i	0.479994i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15276.7i	0.566748i
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	27217.4	0.993120
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	64260.0	2.32935
$914$	0	0
$915$	0	0
$916$	0	0
$917$	44874.0	1.61600
$918$	0	0
$919$	12850.0i	0.461243i	0.973044	+	0.230622i	$0.0740759\pi$
$919$	12850.0i	0.461243i	−0.973044	+	0.230622i	$0.925924\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	42714.0i	1.51339i
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	32074.0	1.11826	0.559131	−	0.829079i	$-0.311135\pi$
$937$	32074.0	1.11826	0.559131	+	0.829079i	$0.311135\pi$
$938$	0	0
$939$	38337.2i	1.33236i
$940$	0	0
$941$	52242.1	1.80982	0.904912	−	0.425599i	$-0.139937\pi$
$941$	52242.1	1.80982	0.904912	+	0.425599i	$0.139937\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−49572.0	−1.70643
$946$	0	0
$947$	54632.3i	1.87467i	0.348429	+	0.937335i	$0.386715\pi$
$947$	54632.3i	1.87467i	−0.348429	+	0.937335i	$0.613285\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 58590.0i	− 1.99780i
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−82494.1	−2.78647
$958$	0	0
$959$	0	0
$960$	0	0
$961$	24891.0	0.835521
$962$	0	0
$963$	9820.73i	0.328628i
$964$	0	0
$965$	−26209.4	−0.874311
$966$	0	0
$967$	58466.0i	1.94430i	0.234356	+	0.972151i	$0.424702\pi$
$967$	58466.0i	1.94430i	−0.234356	+	0.972151i	$0.575298\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 10049.4i	− 0.332131i	−0.986115	−	0.166066i	$-0.946894\pi$
$971$	− 10049.4i	− 0.332131i	0.986115	−	0.166066i	$-0.0531063\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0		−	1.00000i	$-0.5\pi$
$983$	0	0			1.00000i	$0.5\pi$
$984$	0	0
$985$	−756.000	−0.0244550
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 22678.0i	− 0.726933i	−0.931607	−	0.363466i	$-0.881593\pi$
$991$	− 22678.0i	− 0.726933i	0.931607	−	0.363466i	$-0.118407\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	58342.4i	1.85887i
$996$	0	0
$997$	0	0	1.00000			$0$
$997$	0	0	−1.00000			$\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	192.4.f.a.95.3	yes	4
3.2	odd	2	inner	192.4.f.a.95.2	yes	4
4.3	odd	2	inner	192.4.f.a.95.1	✓	4
8.3	odd	2	inner	192.4.f.a.95.4	yes	4
8.5	even	2	inner	192.4.f.a.95.2	yes	4
12.11	even	2	inner	192.4.f.a.95.4	yes	4
16.3	odd	4		768.4.c.r.767.4		4
16.5	even	4		768.4.c.r.767.3		4
16.11	odd	4		768.4.c.r.767.1		4
16.13	even	4		768.4.c.r.767.2		4
24.5	odd	2	CM	192.4.f.a.95.3	yes	4
24.11	even	2	inner	192.4.f.a.95.1	✓	4
48.5	odd	4		768.4.c.r.767.2		4
48.11	even	4		768.4.c.r.767.4		4
48.29	odd	4		768.4.c.r.767.3		4
48.35	even	4		768.4.c.r.767.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
192.4.f.a.95.1	✓	4	4.3	odd	2	inner
192.4.f.a.95.1	✓	4	24.11	even	2	inner
192.4.f.a.95.2	yes	4	3.2	odd	2	inner
192.4.f.a.95.2	yes	4	8.5	even	2	inner
192.4.f.a.95.3	yes	4	1.1	even	1	trivial
192.4.f.a.95.3	yes	4	24.5	odd	2	CM
192.4.f.a.95.4	yes	4	8.3	odd	2	inner
192.4.f.a.95.4	yes	4	12.11	even	2	inner
768.4.c.r.767.1		4	16.11	odd	4
768.4.c.r.767.1		4	48.35	even	4
768.4.c.r.767.2		4	16.13	even	4
768.4.c.r.767.2		4	48.5	odd	4
768.4.c.r.767.3		4	16.5	even	4
768.4.c.r.767.3		4	48.29	odd	4
768.4.c.r.767.4		4	16.3	odd	4
768.4.c.r.767.4		4	48.11	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:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	192.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.19615i q^{3} -10.3923 q^{5} +34.0000i q^{7} -27.0000 q^{9} +O(q^{10})\)	\(q+5.19615i q^{3} -10.3923 q^{5} +34.0000i q^{7} -27.0000 q^{9} -72.7461i q^{11} -54.0000i q^{15} -176.669 q^{21} -17.0000 q^{25} -140.296i q^{27} -218.238 q^{29} -70.0000i q^{31} +378.000 q^{33} -353.338i q^{35} +280.592 q^{45} -813.000 q^{49} -509.223 q^{53} +756.000i q^{55} +717.069i q^{59} -918.000i q^{63} +322.000 q^{73} -88.3346i q^{75} +2473.37 q^{77} +1370.00i q^{79} +729.000 q^{81} +883.346i q^{83} -1134.00i q^{87} +363.731 q^{93} -574.000 q^{97} +1964.15i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 108 q^{9}+O(q^{10}) \) 4 * q - 108 * q^9	\( 4 q - 108 q^{9} - 68 q^{25} + 1512 q^{33} - 3252 q^{49} + 1288 q^{73} + 2916 q^{81} - 2296 q^{97}+O(q^{100}) \) 4 * q - 108 * q^9 - 68 * q^25 + 1512 * q^33 - 3252 * q^49 + 1288 * q^73 + 2916 * q^81 - 2296 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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