LMFDB - Embedded newform 768.2.c.j.767.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,2,Mod(767,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("768.767");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	768.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:	$6.13251087523$
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} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			767.1
Root			$0.517638i$ of defining polynomial
Character	$\chi$	$=$	768.767
Dual form			768.2.c.j.767.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} -2.82843i q^{5} -4.89898i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} -2.82843i q^{5} -4.89898i q^{7} +3.00000 q^{9} -3.46410 q^{11} +4.89898i q^{15} +8.48528i q^{21} -3.00000 q^{25} -5.19615 q^{27} +2.82843i q^{29} +4.89898i q^{31} +6.00000 q^{33} -13.8564 q^{35} -8.48528i q^{45} -17.0000 q^{49} -14.1421i q^{53} +9.79796i q^{55} -10.3923 q^{59} -14.6969i q^{63} -14.0000 q^{73} +5.19615 q^{75} +16.9706i q^{77} -14.6969i q^{79} +9.00000 q^{81} +17.3205 q^{83} -4.89898i q^{87} -8.48528i q^{93} +2.00000 q^{97} -10.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−1.73205	−1.00000
$3$	−1.73205	−1.00000
$4$	0	0
$5$	− 2.82843i	− 1.26491i	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	− 2.82843i	− 1.26491i	0.774597	−	0.632456i	$-0.217953\pi$
$6$	0	0
$7$	− 4.89898i	− 1.85164i	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	− 4.89898i	− 1.85164i	0.377964	−	0.925820i	$-0.376624\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−3.46410	−1.04447	−0.522233	−	0.852803i	$-0.674901\pi$
$11$	−3.46410	−1.04447	−0.522233	+	0.852803i	$0.674901\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	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	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	8.48528i	1.85164i
$22$	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
$27$	−5.19615	−1.00000
$28$	0	0
$29$	2.82843i	0.525226i	0.964901	+	0.262613i	$0.0845842\pi$
$29$	2.82843i	0.525226i	−0.964901	+	0.262613i	$0.915416\pi$
$30$	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
$35$	−13.8564	−2.34216
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	− 8.48528i	− 1.26491i
$46$	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
$53$	− 14.1421i	− 1.94257i	−0.237915	−	0.971286i	$-0.576464\pi$
$53$	− 14.1421i	− 1.94257i	0.237915	−	0.971286i	$-0.423536\pi$
$54$	0	0
$55$	9.79796i	1.32116i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.3923	−1.35296	−0.676481	−	0.736460i	$-0.736496\pi$
$59$	−10.3923	−1.35296	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	− 14.6969i	− 1.85164i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	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	$-0.805631\pi$
$73$	−14.0000	−1.63858	−0.819288	+	0.573382i	$0.805631\pi$
$74$	0	0
$75$	5.19615	0.600000
$76$	0	0
$77$	16.9706i	1.93398i
$78$	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
$83$	17.3205	1.90117	0.950586	−	0.310460i	$-0.100483\pi$
$83$	17.3205	1.90117	0.950586	+	0.310460i	$0.100483\pi$
$84$	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
$93$	− 8.48528i	− 0.879883i
$94$	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
$99$	−10.3923	−1.04447
$100$	0	0
$101$	19.7990i	1.97007i	0.172345	+	0.985037i	$0.444865\pi$
$101$	19.7990i	1.97007i	−0.172345	+	0.985037i	$0.555135\pi$
$102$	0	0
$103$	14.6969i	1.44813i	0.689730	+	0.724066i	$0.257729\pi$
$103$	14.6969i	1.44813i	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	24.0000	2.34216
$106$	0	0
$107$	17.3205	1.67444	0.837218	−	0.546869i	$-0.184180\pi$
$107$	17.3205	1.67444	0.837218	+	0.546869i	$0.184180\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	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
$125$	− 5.65685i	− 0.505964i
$126$	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
$131$	−3.46410	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$131$	−3.46410	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$132$	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	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	8.00000	0.664364
$146$	0	0
$147$	29.4449	2.42857
$148$	0	0
$149$	− 2.82843i	− 0.231714i	−0.993266	−	0.115857i	$-0.963039\pi$
$149$	− 2.82843i	− 0.231714i	0.993266	−	0.115857i	$-0.0369614\pi$
$150$	0	0
$151$	− 24.4949i	− 1.99337i	−0.0813788	−	0.996683i	$-0.525932\pi$
$151$	− 24.4949i	− 1.99337i	0.0813788	−	0.996683i	$-0.474068\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	13.8564	1.11297
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	24.4949i	1.94257i
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	− 16.9706i	− 1.32116i
$166$	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
$173$	− 19.7990i	− 1.50529i	−0.658427	−	0.752645i	$-0.728778\pi$
$173$	− 19.7990i	− 1.50529i	0.658427	−	0.752645i	$-0.271222\pi$
$174$	0	0
$175$	14.6969i	1.11098i
$176$	0	0
$177$	18.0000	1.35296
$178$	0	0
$179$	24.2487	1.81243	0.906217	−	0.422813i	$-0.138957\pi$
$179$	24.2487	1.81243	0.906217	+	0.422813i	$0.138957\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	25.4558i	1.85164i
$190$	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
$197$	− 14.1421i	− 1.00759i	−0.863825	−	0.503793i	$-0.831938\pi$
$197$	− 14.1421i	− 1.00759i	0.863825	−	0.503793i	$-0.168062\pi$
$198$	0	0
$199$	− 24.4949i	− 1.73640i	−0.496217	−	0.868199i	$-0.665278\pi$
$199$	− 24.4949i	− 1.73640i	0.496217	−	0.868199i	$-0.334722\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	13.8564	0.972529
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	24.0000	1.62923
$218$	0	0
$219$	24.2487	1.63858
$220$	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
$227$	−10.3923	−0.689761	−0.344881	−	0.938647i	$-0.612081\pi$
$227$	−10.3923	−0.689761	−0.344881	+	0.938647i	$0.612081\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	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
$237$	25.4558i	1.65353i
$238$	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
$243$	−15.5885	−1.00000
$244$	0	0
$245$	48.0833i	3.07193i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−30.0000	−1.90117
$250$	0	0
$251$	−31.1769	−1.96787	−0.983935	−	0.178529i	$-0.942866\pi$
$251$	−31.1769	−1.96787	−0.983935	+	0.178529i	$0.942866\pi$
$252$	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
$261$	8.48528i	0.525226i
$262$	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
$269$	− 31.1127i	− 1.89697i	−0.316815	−	0.948487i	$-0.602613\pi$
$269$	− 31.1127i	− 1.89697i	0.316815	−	0.948487i	$-0.397387\pi$
$270$	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
$275$	10.3923	0.626680
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	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	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	−3.46410	−0.203069
$292$	0	0
$293$	− 14.1421i	− 0.826192i	−0.910687	−	0.413096i	$-0.864447\pi$
$293$	− 14.1421i	− 0.826192i	0.910687	−	0.413096i	$-0.135553\pi$
$294$	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	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	− 25.4558i	− 1.44813i
$310$	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	$-0.0893093\pi$
$313$	34.0000	1.92179	0.960897	+	0.276907i	$0.0893093\pi$
$314$	0	0
$315$	−41.5692	−2.34216
$316$	0	0
$317$	− 31.1127i	− 1.74746i	−0.486408	−	0.873732i	$-0.661693\pi$
$317$	− 31.1127i	− 1.74746i	0.486408	−	0.873732i	$-0.338307\pi$
$318$	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	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	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
$341$	− 16.9706i	− 0.919007i
$342$	0	0
$343$	48.9898i	2.64520i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.2487	1.30174	0.650870	−	0.759190i	$-0.274404\pi$
$347$	24.2487	1.30174	0.650870	+	0.759190i	$0.274404\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	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
$363$	−1.73205	−0.0909091
$364$	0	0
$365$	39.5980i	2.07265i
$366$	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
$371$	−69.2820	−3.59694
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	9.79796i	0.505964i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	− 8.48528i	− 0.434714i
$382$	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
$389$	31.1127i	1.57748i	0.614729	+	0.788738i	$0.289265\pi$
$389$	31.1127i	1.57748i	−0.614729	+	0.788738i	$0.710735\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	6.00000	0.302660
$394$	0	0
$395$	−41.5692	−2.09157
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 25.4558i	− 1.26491i
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	50.9117i	2.50520i
$414$	0	0
$415$	− 48.9898i	− 2.40481i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−10.3923	−0.507697	−0.253849	−	0.967244i	$-0.581697\pi$
$419$	−10.3923	−0.507697	−0.253849	+	0.967244i	$0.581697\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	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
$435$	−13.8564	−0.664364
$436$	0	0
$437$	0	0
$438$	0	0
$439$	− 24.4949i	− 1.16908i	−0.811366	−	0.584539i	$-0.801275\pi$
$439$	− 24.4949i	− 1.16908i	0.811366	−	0.584539i	$-0.198725\pi$
$440$	0	0
$441$	−51.0000	−2.42857
$442$	0	0
$443$	−31.1769	−1.48126	−0.740630	−	0.671913i	$-0.765473\pi$
$443$	−31.1769	−1.48126	−0.740630	+	0.671913i	$0.765473\pi$
$444$	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
$453$	42.4264i	1.99337i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	38.0000	1.77757	0.888783	−	0.458329i	$-0.151552\pi$
$457$	38.0000	1.77757	0.888783	+	0.458329i	$0.151552\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 19.7990i	− 0.922131i	−0.887366	−	0.461065i	$-0.847467\pi$
$461$	− 19.7990i	− 0.922131i	0.887366	−	0.461065i	$-0.152533\pi$
$462$	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
$467$	17.3205	0.801498	0.400749	−	0.916188i	$-0.368750\pi$
$467$	17.3205	0.801498	0.400749	+	0.916188i	$0.368750\pi$
$468$	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
$477$	− 42.4264i	− 1.94257i
$478$	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
$485$	− 5.65685i	− 0.256865i
$486$	0	0
$487$	− 44.0908i	− 1.99795i	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	− 44.0908i	− 1.99795i	0.0453143	−	0.998973i	$-0.485571\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−38.1051	−1.71966	−0.859830	−	0.510581i	$-0.829431\pi$
$491$	−38.1051	−1.71966	−0.859830	+	0.510581i	$0.829431\pi$
$492$	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	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	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
$507$	22.5167	1.00000
$508$	0	0
$509$	2.82843i	0.125368i	0.998033	+	0.0626839i	$0.0199660\pi$
$509$	2.82843i	0.125368i	−0.998033	+	0.0626839i	$0.980034\pi$
$510$	0	0
$511$	68.5857i	3.03405i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	41.5692	1.83176
$516$	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	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	− 25.4558i	− 1.11098i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−31.1769	−1.35296
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 48.9898i	− 2.11801i
$536$	0	0
$537$	−42.0000	−1.81243
$538$	0	0
$539$	58.8897	2.53656
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	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
$557$	14.1421i	0.599222i	0.954062	+	0.299611i	$0.0968568\pi$
$557$	14.1421i	0.599222i	−0.954062	+	0.299611i	$0.903143\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−38.1051	−1.60594	−0.802970	−	0.596020i	$-0.796748\pi$
$563$	−38.1051	−1.60594	−0.802970	+	0.596020i	$0.796748\pi$
$564$	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	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	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
$579$	45.0333	1.87152
$580$	0	0
$581$	− 84.8528i	− 3.52029i
$582$	0	0
$583$	48.9898i	2.02895i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	17.3205	0.714894	0.357447	−	0.933933i	$-0.383647\pi$
$587$	17.3205	0.714894	0.357447	+	0.933933i	$0.383647\pi$
$588$	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
$597$	42.4264i	1.73640i
$598$	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	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2.82843i	− 0.114992i
$606$	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		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	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	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	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	$-0.968911\pi$
$631$	− 4.89898i	− 0.195025i	0.995234	−	0.0975126i	$-0.0310886\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	13.8564	0.549875
$636$	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	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	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
$651$	−41.5692	−1.62923
$652$	0	0
$653$	48.0833i	1.88164i	0.338902	+	0.940822i	$0.389945\pi$
$653$	48.0833i	1.88164i	−0.338902	+	0.940822i	$0.610055\pi$
$654$	0	0
$655$	9.79796i	0.382838i
$656$	0	0
$657$	−42.0000	−1.63858
$658$	0	0
$659$	24.2487	0.944596	0.472298	−	0.881439i	$-0.343425\pi$
$659$	24.2487	0.944596	0.472298	+	0.881439i	$0.343425\pi$
$660$	0	0
$661$	0	0		−	1.00000i	$-0.5\pi$
$661$	0	0			1.00000i	$0.5\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	25.4558i	0.984180i
$670$	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
$675$	15.5885	0.600000
$676$	0	0
$677$	− 2.82843i	− 0.108705i	−0.998522	−	0.0543526i	$-0.982690\pi$
$677$	− 2.82843i	− 0.108705i	0.998522	−	0.0543526i	$-0.0173095\pi$
$678$	0	0
$679$	− 9.79796i	− 0.376011i
$680$	0	0
$681$	18.0000	0.689761
$682$	0	0
$683$	51.9615	1.98825	0.994126	−	0.108227i	$-0.0345173\pi$
$683$	51.9615	1.98825	0.994126	+	0.108227i	$0.0345173\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	50.9117i	1.93398i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	36.7696i	1.38877i	0.719605	+	0.694383i	$0.244323\pi$
$701$	36.7696i	1.38877i	−0.719605	+	0.694383i	$0.755677\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	96.9948	3.64787
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	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
$723$	17.3205	0.644157
$724$	0	0
$725$	− 8.48528i	− 0.315135i
$726$	0	0
$727$	53.8888i	1.99862i	0.0370879	+	0.999312i	$0.488192\pi$
$727$	53.8888i	1.99862i	−0.0370879	+	0.999312i	$0.511808\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0		−	1.00000i	$-0.5\pi$
$733$	0	0			1.00000i	$0.5\pi$
$734$	0	0
$735$	− 83.2827i	− 3.07193i
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	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
$747$	51.9615	1.90117
$748$	0	0
$749$	− 84.8528i	− 3.10045i
$750$	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
$755$	−69.2820	−2.52143
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	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
$773$	19.7990i	0.712120i	0.934463	+	0.356060i	$0.115880\pi$
$773$	19.7990i	0.712120i	−0.934463	+	0.356060i	$0.884120\pi$
$774$	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	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	69.2820	2.45718
$796$	0	0
$797$	− 53.7401i	− 1.90357i	−0.306762	−	0.951786i	$-0.599246\pi$
$797$	− 53.7401i	− 1.90357i	0.306762	−	0.951786i	$-0.400754\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	48.4974	1.71144
$804$	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	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	− 42.4264i	− 1.48796i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 48.0833i	− 1.67812i	−0.544041	−	0.839059i	$-0.683106\pi$
$821$	− 48.0833i	− 1.67812i	0.544041	−	0.839059i	$-0.316894\pi$
$822$	0	0
$823$	34.2929i	1.19537i	0.801730	+	0.597687i	$0.203913\pi$
$823$	34.2929i	1.19537i	−0.801730	+	0.597687i	$0.796087\pi$
$824$	0	0
$825$	−18.0000	−0.626680
$826$	0	0
$827$	−10.3923	−0.361376	−0.180688	−	0.983540i	$-0.557832\pi$
$827$	−10.3923	−0.361376	−0.180688	+	0.983540i	$0.557832\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 25.4558i	− 0.879883i
$838$	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
$845$	36.7696i	1.26491i
$846$	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		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	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	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	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
$867$	−29.4449	−1.00000
$868$	0	0
$869$	50.9117i	1.72706i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	−27.7128	−0.936864
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	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	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	− 50.9117i	− 1.71138i
$886$	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
$891$	−31.1769	−1.04447
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 68.5857i	− 2.29257i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−13.8564	−0.462137
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	59.3970i	1.97007i
$910$	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
$917$	16.9706i	0.560417i
$918$	0	0
$919$	34.2929i	1.13122i	0.824674	+	0.565608i	$0.191359\pi$
$919$	34.2929i	1.13122i	−0.824674	+	0.565608i	$0.808641\pi$
$920$	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	$-0.896288\pi$
$937$	−58.0000	−1.89478	−0.947389	+	0.320085i	$0.896288\pi$
$938$	0	0
$939$	−58.8897	−1.92179
$940$	0	0
$941$	48.0833i	1.56747i	0.621096	+	0.783735i	$0.286688\pi$
$941$	48.0833i	1.56747i	−0.621096	+	0.783735i	$0.713312\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	72.0000	2.34216
$946$	0	0
$947$	24.2487	0.787977	0.393989	−	0.919115i	$-0.371095\pi$
$947$	24.2487	0.787977	0.393989	+	0.919115i	$0.371095\pi$
$948$	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
$957$	16.9706i	0.548580i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	7.00000	0.225806
$962$	0	0
$963$	51.9615	1.67444
$964$	0	0
$965$	73.5391i	2.36731i
$966$	0	0
$967$	− 4.89898i	− 0.157541i	−0.996893	−	0.0787703i	$-0.974901\pi$
$967$	− 4.89898i	− 0.157541i	0.996893	−	0.0787703i	$-0.0250994\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−3.46410	−0.111168	−0.0555842	−	0.998454i	$-0.517702\pi$
$971$	−3.46410	−0.111168	−0.0555842	+	0.998454i	$0.517702\pi$
$972$	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
$995$	−69.2820	−2.19639
$996$	0	0
$997$	0	0		−	1.00000i	$-0.5\pi$
$997$	0	0			1.00000i	$0.5\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.2.f.a.191.1	✓	4	16.13	even	4
384.2.f.a.191.1	✓	4	48.5	odd	4
384.2.f.a.191.2	yes	4	16.11	odd	4
384.2.f.a.191.2	yes	4	48.35	even	4
384.2.f.a.191.3	yes	4	16.3	odd	4
384.2.f.a.191.3	yes	4	48.11	even	4
384.2.f.a.191.4	yes	4	16.5	even	4
384.2.f.a.191.4	yes	4	48.29	odd	4
768.2.c.j.767.1		4	1.1	even	1	trivial
768.2.c.j.767.1		4	24.5	odd	2	CM
768.2.c.j.767.2		4	8.3	odd	2	inner
768.2.c.j.767.2		4	12.11	even	2	inner
768.2.c.j.767.3		4	4.3	odd	2	inner
768.2.c.j.767.3		4	24.11	even	2	inner
768.2.c.j.767.4		4	3.2	odd	2	inner
768.2.c.j.767.4		4	8.5	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:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} -2.82843i q^{5} -4.89898i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} -2.82843i q^{5} -4.89898i q^{7} +3.00000 q^{9} -3.46410 q^{11} +4.89898i q^{15} +8.48528i q^{21} -3.00000 q^{25} -5.19615 q^{27} +2.82843i q^{29} +4.89898i q^{31} +6.00000 q^{33} -13.8564 q^{35} -8.48528i q^{45} -17.0000 q^{49} -14.1421i q^{53} +9.79796i q^{55} -10.3923 q^{59} -14.6969i q^{63} -14.0000 q^{73} +5.19615 q^{75} +16.9706i q^{77} -14.6969i q^{79} +9.00000 q^{81} +17.3205 q^{83} -4.89898i q^{87} -8.48528i q^{93} +2.00000 q^{97} -10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} - 12 q^{25} + 24 q^{33} - 68 q^{49} - 56 q^{73} + 36 q^{81} + 8 q^{97}+O(q^{100}) \) 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

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