LMFDB - Embedded newform 384.2.f.a.191.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [384,2,Mod(191,384)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("384.191"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	384.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-12,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$3.06625543762$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			191.4
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	384.191
Dual form			384.2.f.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)

$f(q)$	$=$	$q+1.73205i q^{3} +2.82843 q^{5} +4.89898i q^{7} -3.00000 q^{9} -3.46410i q^{11} +4.89898i q^{15} -8.48528 q^{21} +3.00000 q^{25} -5.19615i q^{27} +2.82843 q^{29} +4.89898i q^{31} +6.00000 q^{33} +13.8564i q^{35} -8.48528 q^{45} -17.0000 q^{49} +14.1421 q^{53} -9.79796i q^{55} -10.3923i q^{59} -14.6969i q^{63} +14.0000 q^{73} +5.19615i q^{75} +16.9706 q^{77} -14.6969i q^{79} +9.00000 q^{81} -17.3205i q^{83} +4.89898i q^{87} -8.48528 q^{93} +2.00000 q^{97} +10.3923i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{9} + 12 q^{25} + 24 q^{33} - 68 q^{49} + 56 q^{73} + 36 q^{81} + 8 q^{97}+O(q^{100}) $ 4 * q - 12 * q^9 + 12 * q^25 + 24 * q^33 - 68 * q^49 + 56 * q^73 + 36 * q^81 + 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	1.73205i	1.00000i
$3$	1.73205i	1.00000i
$4$	0	0
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	4.89898i	1.85164i	0.377964	+	0.925820i	$0.376624\pi$
$7$	4.89898i	1.85164i	−0.377964	+	0.925820i	$0.623376\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	− 3.46410i	− 1.04447i	−0.852803	−	0.522233i	$-0.825099\pi$
$11$	− 3.46410i	− 1.04447i	0.852803	−	0.522233i	$-0.174901\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	4.89898i	1.26491i
$16$	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
$21$	−8.48528	−1.85164
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	− 5.19615i	− 1.00000i
$28$	0	0
$29$	2.82843	0.525226	0.262613	−	0.964901i	$-0.415416\pi$
$29$	2.82843	0.525226	0.262613	+	0.964901i	$0.415416\pi$
$30$	0	0
$31$	4.89898i	0.879883i	0.898027	+	0.439941i	$0.145001\pi$
$31$	4.89898i	0.879883i	−0.898027	+	0.439941i	$0.854999\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	13.8564i	2.34216i
$36$	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
$45$	−8.48528	−1.26491
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−17.0000	−2.42857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	14.1421	1.94257	0.971286	−	0.237915i	$-0.0764641\pi$
$53$	14.1421	1.94257	0.971286	+	0.237915i	$0.0764641\pi$
$54$	0	0
$55$	− 9.79796i	− 1.32116i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 10.3923i	− 1.35296i	−0.736460	−	0.676481i	$-0.763504\pi$
$59$	− 10.3923i	− 1.35296i	0.736460	−	0.676481i	$-0.236496\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	− 14.6969i	− 1.85164i
$64$	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
$73$	14.0000	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$73$	14.0000	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$74$	0	0
$75$	5.19615i	0.600000i
$76$	0	0
$77$	16.9706	1.93398
$78$	0	0
$79$	− 14.6969i	− 1.65353i	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	− 14.6969i	− 1.65353i	0.562544	−	0.826767i	$-0.309823\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 17.3205i	− 1.90117i	−0.310460	−	0.950586i	$-0.600483\pi$
$83$	− 17.3205i	− 1.90117i	0.310460	−	0.950586i	$-0.399517\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.89898i	0.525226i
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.48528	−0.879883
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	10.3923i	1.04447i
$100$	0	0
$101$	−19.7990	−1.97007	−0.985037	−	0.172345i	$-0.944865\pi$
$101$	−19.7990	−1.97007	−0.985037	+	0.172345i	$0.944865\pi$
$102$	0	0
$103$	− 14.6969i	− 1.44813i	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	− 14.6969i	− 1.44813i	0.689730	−	0.724066i	$-0.257729\pi$
$104$	0	0
$105$	−24.0000	−2.34216
$106$	0	0
$107$	17.3205i	1.67444i	0.546869	+	0.837218i	$0.315820\pi$
$107$	17.3205i	1.67444i	−0.546869	+	0.837218i	$0.684180\pi$
$108$	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
$121$	−1.00000	−0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	4.89898i	0.434714i	0.976092	+	0.217357i	$0.0697436\pi$
$127$	4.89898i	0.434714i	−0.976092	+	0.217357i	$0.930256\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.46410i	0.302660i	0.988483	+	0.151330i	$0.0483556\pi$
$131$	3.46410i	0.302660i	−0.988483	+	0.151330i	$0.951644\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 14.6969i	− 1.26491i
$136$	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
$145$	8.00000	0.664364
$146$	0	0
$147$	− 29.4449i	− 2.42857i
$148$	0	0
$149$	2.82843	0.231714	0.115857	−	0.993266i	$-0.463039\pi$
$149$	2.82843	0.231714	0.115857	+	0.993266i	$0.463039\pi$
$150$	0	0
$151$	24.4949i	1.99337i	0.0813788	+	0.996683i	$0.474068\pi$
$151$	24.4949i	1.99337i	−0.0813788	+	0.996683i	$0.525932\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	13.8564i	1.11297i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	24.4949i	1.94257i
$160$	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
$165$	16.9706	1.32116
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−19.7990	−1.50529	−0.752645	−	0.658427i	$-0.771222\pi$
$173$	−19.7990	−1.50529	−0.752645	+	0.658427i	$0.771222\pi$
$174$	0	0
$175$	14.6969i	1.11098i
$176$	0	0
$177$	18.0000	1.35296
$178$	0	0
$179$	− 24.2487i	− 1.81243i	−0.422813	−	0.906217i	$-0.638957\pi$
$179$	− 24.2487i	− 1.81243i	0.422813	−	0.906217i	$-0.361043\pi$
$180$	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
$189$	25.4558	1.85164
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−26.0000	−1.87152	−0.935760	−	0.352636i	$-0.885285\pi$
$193$	−26.0000	−1.87152	−0.935760	+	0.352636i	$0.885285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.1421	1.00759	0.503793	−	0.863825i	$-0.331938\pi$
$197$	14.1421	1.00759	0.503793	+	0.863825i	$0.331938\pi$
$198$	0	0
$199$	24.4949i	1.73640i	0.496217	+	0.868199i	$0.334722\pi$
$199$	24.4949i	1.73640i	−0.496217	+	0.868199i	$0.665278\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	13.8564i	0.972529i
$204$	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
$217$	−24.0000	−1.62923
$218$	0	0
$219$	24.2487i	1.63858i
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 14.6969i	− 0.984180i	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	− 14.6969i	− 0.984180i	0.870544	−	0.492090i	$-0.163767\pi$
$224$	0	0
$225$	−9.00000	−0.600000
$226$	0	0
$227$	10.3923i	0.689761i	0.938647	+	0.344881i	$0.112081\pi$
$227$	10.3923i	0.689761i	−0.938647	+	0.344881i	$0.887919\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	29.3939i	1.93398i
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	25.4558	1.65353
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	15.5885i	1.00000i
$244$	0	0
$245$	−48.0833	−3.07193
$246$	0	0
$247$	0	0
$248$	0	0
$249$	30.0000	1.90117
$250$	0	0
$251$	− 31.1769i	− 1.96787i	−0.178529	−	0.983935i	$-0.557134\pi$
$251$	− 31.1769i	− 1.96787i	0.178529	−	0.983935i	$-0.442866\pi$
$252$	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
$261$	−8.48528	−0.525226
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	40.0000	2.45718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−31.1127	−1.89697	−0.948487	−	0.316815i	$-0.897387\pi$
$269$	−31.1127	−1.89697	−0.948487	+	0.316815i	$0.897387\pi$
$270$	0	0
$271$	24.4949i	1.48796i	0.668202	+	0.743980i	$0.267064\pi$
$271$	24.4949i	1.48796i	−0.668202	+	0.743980i	$0.732936\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 10.3923i	− 0.626680i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	− 14.6969i	− 0.879883i
$280$	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
$289$	17.0000	1.00000
$290$	0	0
$291$	3.46410i	0.203069i
$292$	0	0
$293$	14.1421	0.826192	0.413096	−	0.910687i	$-0.364447\pi$
$293$	14.1421	0.826192	0.413096	+	0.910687i	$0.364447\pi$
$294$	0	0
$295$	− 29.3939i	− 1.71138i
$296$	0	0
$297$	−18.0000	−1.04447
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	− 34.2929i	− 1.97007i
$304$	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
$309$	25.4558	1.44813
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−34.0000	−1.92179	−0.960897	−	0.276907i	$-0.910691\pi$
$313$	−34.0000	−1.92179	−0.960897	+	0.276907i	$0.910691\pi$
$314$	0	0
$315$	− 41.5692i	− 2.34216i
$316$	0	0
$317$	−31.1127	−1.74746	−0.873732	−	0.486408i	$-0.838307\pi$
$317$	−31.1127	−1.74746	−0.873732	+	0.486408i	$0.838307\pi$
$318$	0	0
$319$	− 9.79796i	− 0.548580i
$320$	0	0
$321$	−30.0000	−1.67444
$322$	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
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	16.9706	0.919007
$342$	0	0
$343$	− 48.9898i	− 2.64520i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.2487i	1.30174i	0.759190	+	0.650870i	$0.225596\pi$
$347$	24.2487i	1.30174i	−0.759190	+	0.650870i	$0.774404\pi$
$348$	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
$361$	−19.0000	−1.00000
$362$	0	0
$363$	− 1.73205i	− 0.0909091i
$364$	0	0
$365$	39.5980	2.07265
$366$	0	0
$367$	4.89898i	0.255725i	0.991792	+	0.127862i	$0.0408116\pi$
$367$	4.89898i	0.255725i	−0.991792	+	0.127862i	$0.959188\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	69.2820i	3.59694i
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	− 9.79796i	− 0.505964i
$376$	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
$381$	−8.48528	−0.434714
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	48.0000	2.44631
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−31.1127	−1.57748	−0.788738	−	0.614729i	$-0.789265\pi$
$389$	−31.1127	−1.57748	−0.788738	+	0.614729i	$0.789265\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−6.00000	−0.302660
$394$	0	0
$395$	− 41.5692i	− 2.09157i
$396$	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
$405$	25.4558	1.26491
$406$	0	0
$407$	0	0
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	50.9117	2.50520
$414$	0	0
$415$	− 48.9898i	− 2.40481i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.3923i	0.507697i	0.967244	+	0.253849i	$0.0816965\pi$
$419$	10.3923i	0.507697i	−0.967244	+	0.253849i	$0.918303\pi$
$420$	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
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	13.8564i	0.664364i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	24.4949i	1.16908i	0.811366	+	0.584539i	$0.198725\pi$
$439$	24.4949i	1.16908i	−0.811366	+	0.584539i	$0.801275\pi$
$440$	0	0
$441$	51.0000	2.42857
$442$	0	0
$443$	− 31.1769i	− 1.48126i	−0.671913	−	0.740630i	$-0.734527\pi$
$443$	− 31.1769i	− 1.48126i	0.671913	−	0.740630i	$-0.265473\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.89898i	0.231714i
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−42.4264	−1.99337
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−38.0000	−1.77757	−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000	−1.77757	−0.888783	+	0.458329i	$0.848448\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−19.7990	−0.922131	−0.461065	−	0.887366i	$-0.652533\pi$
$461$	−19.7990	−0.922131	−0.461065	+	0.887366i	$0.652533\pi$
$462$	0	0
$463$	− 34.2929i	− 1.59372i	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	− 34.2929i	− 1.59372i	0.604161	−	0.796862i	$-0.293508\pi$
$464$	0	0
$465$	−24.0000	−1.11297
$466$	0	0
$467$	− 17.3205i	− 0.801498i	−0.916188	−	0.400749i	$-0.868750\pi$
$467$	− 17.3205i	− 0.801498i	0.916188	−	0.400749i	$-0.131250\pi$
$468$	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
$477$	−42.4264	−1.94257
$478$	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
$485$	5.65685	0.256865
$486$	0	0
$487$	44.0908i	1.99795i	0.0453143	+	0.998973i	$0.485571\pi$
$487$	44.0908i	1.99795i	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 38.1051i	− 1.71966i	−0.510581	−	0.859830i	$-0.670569\pi$
$491$	− 38.1051i	− 1.71966i	0.510581	−	0.859830i	$-0.329431\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	29.3939i	1.32116i
$496$	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
$505$	−56.0000	−2.49197
$506$	0	0
$507$	22.5167i	1.00000i
$508$	0	0
$509$	2.82843	0.125368	0.0626839	−	0.998033i	$-0.480034\pi$
$509$	2.82843	0.125368	0.0626839	+	0.998033i	$0.480034\pi$
$510$	0	0
$511$	68.5857i	3.03405i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 41.5692i	− 1.83176i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	− 34.2929i	− 1.50529i
$520$	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
$525$	−25.4558	−1.11098
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	31.1769i	1.35296i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	48.9898i	2.11801i
$536$	0	0
$537$	42.0000	1.81243
$538$	0	0
$539$	58.8897i	2.53656i
$540$	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
$553$	72.0000	3.06175
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.1421	0.599222	0.299611	−	0.954062i	$-0.403143\pi$
$557$	14.1421	0.599222	0.299611	+	0.954062i	$0.403143\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	38.1051i	1.60594i	0.596020	+	0.802970i	$0.296748\pi$
$563$	38.1051i	1.60594i	−0.596020	+	0.802970i	$0.703252\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	44.0908i	1.85164i
$568$	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
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	0	0
$579$	− 45.0333i	− 1.87152i
$580$	0	0
$581$	84.8528	3.52029
$582$	0	0
$583$	− 48.9898i	− 2.02895i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.3205i	0.714894i	0.933933	+	0.357447i	$0.116353\pi$
$587$	17.3205i	0.714894i	−0.933933	+	0.357447i	$0.883647\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	24.4949i	1.00759i
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−42.4264	−1.73640
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.82843	−0.114992
$606$	0	0
$607$	44.0908i	1.78959i	0.446476	+	0.894795i	$0.352679\pi$
$607$	44.0908i	1.78959i	−0.446476	+	0.894795i	$0.647321\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	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
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	4.89898i	0.195025i	0.995234	+	0.0975126i	$0.0310886\pi$
$631$	4.89898i	0.195025i	−0.995234	+	0.0975126i	$0.968911\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	13.8564i	0.549875i
$636$	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
$649$	−36.0000	−1.41312
$650$	0	0
$651$	− 41.5692i	− 1.62923i
$652$	0	0
$653$	48.0833	1.88164	0.940822	−	0.338902i	$-0.110055\pi$
$653$	48.0833	1.88164	0.940822	+	0.338902i	$0.110055\pi$
$654$	0	0
$655$	9.79796i	0.382838i
$656$	0	0
$657$	−42.0000	−1.63858
$658$	0	0
$659$	− 24.2487i	− 0.944596i	−0.881439	−	0.472298i	$-0.843425\pi$
$659$	− 24.2487i	− 0.944596i	0.881439	−	0.472298i	$-0.156575\pi$
$660$	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
$669$	25.4558	0.984180
$670$	0	0
$671$	0	0
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	0	0
$675$	− 15.5885i	− 0.600000i
$676$	0	0
$677$	2.82843	0.108705	0.0543526	−	0.998522i	$-0.482690\pi$
$677$	2.82843	0.108705	0.0543526	+	0.998522i	$0.482690\pi$
$678$	0	0
$679$	9.79796i	0.376011i
$680$	0	0
$681$	−18.0000	−0.689761
$682$	0	0
$683$	51.9615i	1.98825i	0.108227	+	0.994126i	$0.465483\pi$
$683$	51.9615i	1.98825i	−0.108227	+	0.994126i	$0.534517\pi$
$684$	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
$693$	−50.9117	−1.93398
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	36.7696	1.38877	0.694383	−	0.719605i	$-0.255677\pi$
$701$	36.7696	1.38877	0.694383	+	0.719605i	$0.255677\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 96.9948i	− 3.64787i
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	44.0908i	1.65353i
$712$	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
$721$	72.0000	2.68142
$722$	0	0
$723$	− 17.3205i	− 0.644157i
$724$	0	0
$725$	8.48528	0.315135
$726$	0	0
$727$	− 53.8888i	− 1.99862i	−0.0370879	−	0.999312i	$-0.511808\pi$
$727$	− 53.8888i	− 1.99862i	0.0370879	−	0.999312i	$-0.488192\pi$
$728$	0	0
$729$	−27.0000	−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
$735$	− 83.2827i	− 3.07193i
$736$	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
$745$	8.00000	0.293097
$746$	0	0
$747$	51.9615i	1.90117i
$748$	0	0
$749$	−84.8528	−3.10045
$750$	0	0
$751$	− 53.8888i	− 1.96643i	−0.182453	−	0.983215i	$-0.558404\pi$
$751$	− 53.8888i	− 1.96643i	0.182453	−	0.983215i	$-0.441596\pi$
$752$	0	0
$753$	54.0000	1.96787
$754$	0	0
$755$	69.2820i	2.52143i
$756$	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
$769$	−26.0000	−0.937584	−0.468792	−	0.883309i	$-0.655311\pi$
$769$	−26.0000	−0.937584	−0.468792	+	0.883309i	$0.655311\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−19.7990	−0.712120	−0.356060	−	0.934463i	$-0.615880\pi$
$773$	−19.7990	−0.712120	−0.356060	+	0.934463i	$0.615880\pi$
$774$	0	0
$775$	14.6969i	0.527930i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	− 14.6969i	− 0.525226i
$784$	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
$795$	69.2820i	2.45718i
$796$	0	0
$797$	−53.7401	−1.90357	−0.951786	−	0.306762i	$-0.900754\pi$
$797$	−53.7401	−1.90357	−0.951786	+	0.306762i	$0.900754\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 48.4974i	− 1.71144i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 53.8888i	− 1.89697i
$808$	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
$813$	−42.4264	−1.48796
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.0833	1.67812	0.839059	−	0.544041i	$-0.183106\pi$
$821$	48.0833	1.67812	0.839059	+	0.544041i	$0.183106\pi$
$822$	0	0
$823$	− 34.2929i	− 1.19537i	−0.801730	−	0.597687i	$-0.796087\pi$
$823$	− 34.2929i	− 1.19537i	0.801730	−	0.597687i	$-0.203913\pi$
$824$	0	0
$825$	18.0000	0.626680
$826$	0	0
$827$	− 10.3923i	− 0.361376i	−0.983540	−	0.180688i	$-0.942168\pi$
$827$	− 10.3923i	− 0.361376i	0.983540	−	0.180688i	$-0.0578324\pi$
$828$	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
$837$	25.4558	0.879883
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−21.0000	−0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	36.7696	1.26491
$846$	0	0
$847$	− 4.89898i	− 0.168331i
$848$	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
$865$	−56.0000	−1.90406
$866$	0	0
$867$	29.4449i	1.00000i
$868$	0	0
$869$	−50.9117	−1.72706
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	− 27.7128i	− 0.936864i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	24.4949i	0.826192i
$880$	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
$885$	50.9117	1.71138
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	0	0
$891$	− 31.1769i	− 1.04447i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 68.5857i	− 2.29257i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.8564i	0.462137i
$900$	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
$909$	59.3970	1.97007
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−60.0000	−1.98571
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−16.9706	−0.560417
$918$	0	0
$919$	− 34.2929i	− 1.13122i	−0.824674	−	0.565608i	$-0.808641\pi$
$919$	− 34.2929i	− 1.13122i	0.824674	−	0.565608i	$-0.191359\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	44.0908i	1.44813i
$928$	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
$937$	58.0000	1.89478	0.947389	−	0.320085i	$-0.103712\pi$
$937$	58.0000	1.89478	0.947389	+	0.320085i	$0.103712\pi$
$938$	0	0
$939$	− 58.8897i	− 1.92179i
$940$	0	0
$941$	48.0833	1.56747	0.783735	−	0.621096i	$-0.213312\pi$
$941$	48.0833	1.56747	0.783735	+	0.621096i	$0.213312\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	72.0000	2.34216
$946$	0	0
$947$	− 24.2487i	− 0.787977i	−0.919115	−	0.393989i	$-0.871095\pi$
$947$	− 24.2487i	− 0.787977i	0.919115	−	0.393989i	$-0.128905\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	− 53.8888i	− 1.74746i
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	16.9706	0.548580
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.00000	0.225806
$962$	0	0
$963$	− 51.9615i	− 1.67444i
$964$	0	0
$965$	−73.5391	−2.36731
$966$	0	0
$967$	4.89898i	0.157541i	0.996893	+	0.0787703i	$0.0250994\pi$
$967$	4.89898i	0.157541i	−0.996893	+	0.0787703i	$0.974901\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 3.46410i	− 0.111168i	−0.998454	−	0.0555842i	$-0.982298\pi$
$971$	− 3.46410i	− 0.111168i	0.998454	−	0.0555842i	$-0.0177021\pi$
$972$	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
$985$	40.0000	1.27451
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	24.4949i	0.778106i	0.921215	+	0.389053i	$0.127198\pi$
$991$	24.4949i	0.778106i	−0.921215	+	0.389053i	$0.872802\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	69.2820i	2.19639i
$996$	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	384.2.f.a.191.4	yes	4
3.2	odd	2	inner	384.2.f.a.191.1	✓	4
4.3	odd	2	inner	384.2.f.a.191.2	yes	4
8.3	odd	2	inner	384.2.f.a.191.3	yes	4
8.5	even	2	inner	384.2.f.a.191.1	✓	4
12.11	even	2	inner	384.2.f.a.191.3	yes	4
16.3	odd	4		768.2.c.j.767.3		4
16.5	even	4		768.2.c.j.767.4		4
16.11	odd	4		768.2.c.j.767.2		4
16.13	even	4		768.2.c.j.767.1		4
24.5	odd	2	CM	384.2.f.a.191.4	yes	4
24.11	even	2	inner	384.2.f.a.191.2	yes	4
48.5	odd	4		768.2.c.j.767.1		4
48.11	even	4		768.2.c.j.767.3		4
48.29	odd	4		768.2.c.j.767.4		4
48.35	even	4		768.2.c.j.767.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.f.a.191.1	✓	4	3.2	odd	2	inner
384.2.f.a.191.1	✓	4	8.5	even	2	inner
384.2.f.a.191.2	yes	4	4.3	odd	2	inner
384.2.f.a.191.2	yes	4	24.11	even	2	inner
384.2.f.a.191.3	yes	4	8.3	odd	2	inner
384.2.f.a.191.3	yes	4	12.11	even	2	inner
384.2.f.a.191.4	yes	4	1.1	even	1	trivial
384.2.f.a.191.4	yes	4	24.5	odd	2	CM
768.2.c.j.767.1		4	16.13	even	4
768.2.c.j.767.1		4	48.5	odd	4
768.2.c.j.767.2		4	16.11	odd	4
768.2.c.j.767.2		4	48.35	even	4
768.2.c.j.767.3		4	16.3	odd	4
768.2.c.j.767.3		4	48.11	even	4
768.2.c.j.767.4		4	16.5	even	4
768.2.c.j.767.4		4	48.29	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	384.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +2.82843 q^{5} +4.89898i q^{7} -3.00000 q^{9} -3.46410i q^{11} +4.89898i q^{15} -8.48528 q^{21} +3.00000 q^{25} -5.19615i q^{27} +2.82843 q^{29} +4.89898i q^{31} +6.00000 q^{33} +13.8564i q^{35} -8.48528 q^{45} -17.0000 q^{49} +14.1421 q^{53} -9.79796i q^{55} -10.3923i q^{59} -14.6969i q^{63} +14.0000 q^{73} +5.19615i q^{75} +16.9706 q^{77} -14.6969i q^{79} +9.00000 q^{81} -17.3205i q^{83} +4.89898i q^{87} -8.48528 q^{93} +2.00000 q^{97} +10.3923i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{9} + 12 q^{25} + 24 q^{33} - 68 q^{49} + 56 q^{73} + 36 q^{81} + 8 q^{97}+O(q^{100}) \) 4 * q - 12 * q^9 + 12 * q^25 + 24 * q^33 - 68 * q^49 + 56 * q^73 + 36 * q^81 + 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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