LMFDB - Embedded newform 1152.4.d.l.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1152,4,Mod(577,1152)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1152, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1152.577");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1152.d (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:	$67.9702003266$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			577.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.577
Dual form			1152.4.d.l.577.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 10.3923i	− 0.929516i	−0.885438	−	0.464758i	$-0.846141\pi$
$5$	− 10.3923i	− 0.929516i	0.885438	−	0.464758i	$-0.153859\pi$
$6$	0	0
$7$	−14.6969	−0.793560	−0.396780	−	0.917914i	$-0.629872\pi$
$7$	−14.6969	−0.793560	−0.396780	+	0.917914i	$0.629872\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.65685i	0.155055i	0.996990	+	0.0775275i	$0.0247026\pi$
$11$	5.65685i	0.155055i	−0.996990	+	0.0775275i	$0.975297\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	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
$27$	0	0
$28$	0	0
$29$	− 218.238i	− 1.39744i	−0.715394	−	0.698722i	$-0.753753\pi$
$29$	− 218.238i	− 1.39744i	0.715394	−	0.698722i	$-0.246247\pi$
$30$	0	0
$31$	−338.030	−1.95845	−0.979224	−	0.202780i	$-0.935002\pi$
$31$	−338.030	−1.95845	−0.979224	+	0.202780i	$0.935002\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	152.735i	0.737627i
$36$	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		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	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
$49$	−127.000	−0.370262
$50$	0	0
$51$	0	0
$52$	0	0
$53$	509.223i	1.31976i	0.751372	+	0.659879i	$0.229392\pi$
$53$	509.223i	1.31976i	−0.751372	+	0.659879i	$0.770608\pi$
$54$	0	0
$55$	58.7878	0.144126
$56$	0	0
$57$	0	0
$58$	0	0
$59$	554.372i	1.22327i	0.791139	+	0.611636i	$0.209488\pi$
$59$	554.372i	1.22327i	−0.791139	+	0.611636i	$0.790512\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	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
$73$	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
$75$	0	0
$76$	0	0
$77$	− 83.1384i	− 0.123046i
$78$	0	0
$79$	308.636	0.439547	0.219774	−	0.975551i	$-0.429468\pi$
$79$	308.636	0.439547	0.219774	+	0.975551i	$0.429468\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1227.54i	1.62337i	0.584095	+	0.811685i	$0.301449\pi$
$83$	1227.54i	1.62337i	−0.584095	+	0.811685i	$0.698551\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	1008.05i	0.993120i	0.868003	+	0.496560i	$0.165404\pi$
$101$	1008.05i	0.993120i	−0.868003	+	0.496560i	$0.834596\pi$
$102$	0	0
$103$	1366.82	1.30754	0.653768	−	0.756695i	$-0.273187\pi$
$103$	1366.82	1.30754	0.653768	+	0.756695i	$0.273187\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2183.55i	1.97282i	0.164314	+	0.986408i	$0.447459\pi$
$107$	2183.55i	1.97282i	−0.164314	+	0.986408i	$0.552541\pi$
$108$	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		−	1.00000i	$-0.5\pi$
$113$	0	0			1.00000i	$0.5\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1299.00	0.975958
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1475.71i	− 1.05593i
$126$	0	0
$127$	1748.94	1.22199	0.610996	−	0.791634i	$-0.290769\pi$
$127$	1748.94	1.22199	0.610996	+	0.791634i	$0.290769\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2692.66i	1.79587i	0.440128	+	0.897935i	$0.354933\pi$
$131$	2692.66i	1.79587i	−0.440128	+	0.897935i	$0.645067\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	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
$145$	−2268.00	−1.29895
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3419.07i	− 1.87987i	−0.341350	−	0.939936i	$-0.610884\pi$
$149$	− 3419.07i	− 1.87987i	0.341350	−	0.939936i	$-0.389116\pi$
$150$	0	0
$151$	−3600.75	−1.94056	−0.970281	−	0.241981i	$-0.922203\pi$
$151$	−3600.75	−1.94056	−0.970281	+	0.241981i	$0.922203\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3512.91i	1.82041i
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	2197.00	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3793.19i	1.66700i	0.552519	+	0.833500i	$0.313666\pi$
$173$	3793.19i	1.66700i	−0.552519	+	0.833500i	$0.686334\pi$
$174$	0	0
$175$	−249.848	−0.107924
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4627.31i	1.93219i	0.258196	+	0.966093i	$0.416872\pi$
$179$	4627.31i	1.93219i	−0.258196	+	0.966093i	$0.583128\pi$
$180$	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
$189$	0	0
$190$	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	$-0.655856\pi$
$193$	−2522.00	−0.940609	−0.470304	+	0.882504i	$0.655856\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 72.7461i	− 0.0263094i	−0.999913	−	0.0131547i	$-0.995813\pi$
$197$	− 72.7461i	− 0.0263094i	0.999913	−	0.0131547i	$-0.00418739\pi$
$198$	0	0
$199$	73.4847	0.0261768	0.0130884	−	0.999914i	$-0.495834\pi$
$199$	73.4847	0.0261768	0.0130884	+	0.999914i	$0.495834\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3207.44i	1.10896i
$204$	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
$217$	4968.00	1.55415
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−6657.71	−1.99925	−0.999627	−	0.0273265i	$-0.991301\pi$
$223$	−6657.71	−1.99925	−0.999627	+	0.0273265i	$0.991301\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3365.83i	0.984132i	0.870558	+	0.492066i	$0.163758\pi$
$227$	3365.83i	0.984132i	−0.870558	+	0.492066i	$0.836242\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	1319.82i	0.344165i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4078.59i	1.02565i	0.858493	+	0.512826i	$0.171401\pi$
$251$	4078.59i	1.02565i	−0.858493	+	0.512826i	$0.828599\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	0	0
$260$	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
$265$	5292.00	1.22674
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7264.22i	1.64650i	0.567682	+	0.823248i	$0.307840\pi$
$269$	7264.22i	1.64650i	−0.567682	+	0.823248i	$0.692160\pi$
$270$	0	0
$271$	5217.41	1.16950	0.584751	−	0.811213i	$-0.301192\pi$
$271$	5217.41	1.16950	0.584751	+	0.811213i	$0.301192\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	96.1665i	0.0210875i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	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
$289$	−4913.00	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2899.45i	− 0.578116i	−0.957312	−	0.289058i	$-0.906658\pi$
$293$	− 2899.45i	− 0.578116i	0.957312	−	0.289058i	$-0.0933420\pi$
$294$	0	0
$295$	5761.20	1.13705
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	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
$315$	0	0
$316$	0	0
$317$	− 11275.7i	− 1.99780i	−0.0468563	−	0.998902i	$-0.514920\pi$
$317$	− 11275.7i	− 1.99780i	0.0468563	−	0.998902i	$-0.485080\pi$
$318$	0	0
$319$	1234.54	0.216681
$320$	0	0
$321$	0	0
$322$	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
$337$	−11594.0	−1.87408	−0.937041	−	0.349220i	$-0.886447\pi$
$337$	−11594.0	−1.87408	−0.937041	+	0.349220i	$0.886447\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1912.18i	− 0.303667i
$342$	0	0
$343$	6907.56	1.08739
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6816.51i	− 1.05455i	−0.849694	−	0.527276i	$-0.823213\pi$
$347$	− 6816.51i	− 1.05455i	0.849694	−	0.527276i	$-0.176787\pi$
$348$	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		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	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
$363$	0	0
$364$	0	0
$365$	− 3346.32i	− 0.479875i
$366$	0	0
$367$	−5276.20	−0.750451	−0.375225	−	0.926934i	$-0.622435\pi$
$367$	−5276.20	−0.750451	−0.375225	+	0.926934i	$0.622435\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 7484.02i	− 1.04731i
$372$	0	0
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	−864.000	−0.114373
$386$	0	0
$387$	0	0
$388$	0	0
$389$	14040.0i	1.82997i	0.403493	+	0.914983i	$0.367796\pi$
$389$	14040.0i	1.82997i	−0.403493	+	0.914983i	$0.632204\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 3207.44i	− 0.408566i
$396$	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		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	11270.0	1.36251	0.681254	−	0.732047i	$-0.261435\pi$
$409$	11270.0	1.36251	0.681254	+	0.732047i	$0.261435\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 8147.57i	− 0.970740i
$414$	0	0
$415$	12756.9	1.50895
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 12315.0i	− 1.43586i	−0.696115	−	0.717930i	$-0.745090\pi$
$419$	− 12315.0i	− 1.43586i	0.696115	−	0.717930i	$-0.254910\pi$
$420$	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	$-0.827627\pi$
$433$	−15442.0	−1.71385	−0.856923	+	0.515445i	$0.827627\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−17562.8	−1.90940	−0.954702	−	0.297562i	$-0.903826\pi$
$439$	−17562.8	−1.90940	−0.954702	+	0.297562i	$0.903826\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 14962.4i	− 1.60470i	−0.596851	−	0.802352i	$-0.703582\pi$
$443$	− 14962.4i	− 1.60470i	0.596851	−	0.802352i	$-0.296418\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0		−	1.00000i	$-0.5\pi$
$449$	0	0			1.00000i	$0.5\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2774.00	0.283944	0.141972	−	0.989871i	$-0.454656\pi$
$457$	2774.00	0.283944	0.141972	+	0.989871i	$0.454656\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2629.25i	0.265632i	0.991141	+	0.132816i	$0.0424020\pi$
$461$	2629.25i	0.265632i	−0.991141	+	0.132816i	$0.957598\pi$
$462$	0	0
$463$	−7304.38	−0.733182	−0.366591	−	0.930382i	$-0.619475\pi$
$463$	−7304.38	−0.733182	−0.366591	+	0.930382i	$0.619475\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6612.86i	− 0.655261i	−0.944806	−	0.327630i	$-0.893750\pi$
$467$	− 6612.86i	− 0.655261i	0.944806	−	0.327630i	$-0.106250\pi$
$468$	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
$477$	0	0
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5965.18i	0.558485i
$486$	0	0
$487$	−21295.9	−1.98154	−0.990768	−	0.135571i	$-0.956713\pi$
$487$	−21295.9	−1.98154	−0.990768	+	0.135571i	$0.956713\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 21744.9i	− 1.99865i	−0.0367748	−	0.999324i	$-0.511708\pi$
$491$	− 21744.9i	− 1.99865i	0.0367748	−	0.999324i	$-0.488292\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	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
$505$	10476.0	0.923121
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22561.7i	1.96469i	0.187067	+	0.982347i	$0.440102\pi$
$509$	22561.7i	1.96469i	−0.187067	+	0.982347i	$0.559898\pi$
$510$	0	0
$511$	−4732.41	−0.409686
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 14204.4i	− 1.21538i
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0		−	1.00000i	$-0.5\pi$
$521$	0	0			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	22692.1	1.83376
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 718.420i	− 0.0574111i
$540$	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	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−4536.00	−0.348807
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 16076.9i	− 1.22298i	−0.791252	−	0.611490i	$-0.790570\pi$
$557$	− 16076.9i	− 1.22298i	0.791252	−	0.611490i	$-0.209430\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 25144.7i	− 1.88228i	−0.338017	−	0.941140i	$-0.609756\pi$
$563$	− 25144.7i	− 1.88228i	0.338017	−	0.941140i	$-0.390244\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	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
$577$	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
$579$	0	0
$580$	0	0
$581$	− 18041.0i	− 1.28824i
$582$	0	0
$583$	−2880.60	−0.204635
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 12988.1i	− 0.913250i	−0.889659	−	0.456625i	$-0.849058\pi$
$587$	− 12988.1i	− 0.913250i	0.889659	−	0.456625i	$-0.150942\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	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	$-0.461037\pi$
$601$	3598.00	0.244202	0.122101	+	0.992518i	$0.461037\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 13499.6i	− 0.907169i
$606$	0	0
$607$	−5423.17	−0.362635	−0.181318	−	0.983425i	$-0.558036\pi$
$607$	−5423.17	−0.362635	−0.181318	+	0.983425i	$0.558036\pi$
$608$	0	0
$609$	0	0
$610$	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		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	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
$625$	−13211.0	−0.845504
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	9156.19	0.577658	0.288829	−	0.957381i	$-0.406734\pi$
$631$	9156.19	0.577658	0.288829	+	0.957381i	$0.406734\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 18175.5i	− 1.13586i
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0		−	1.00000i	$-0.5\pi$
$641$	0	0			1.00000i	$0.5\pi$
$642$	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
$649$	−3136.00	−0.189675
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28734.7i	1.72202i	0.508591	+	0.861008i	$0.330166\pi$
$653$	28734.7i	1.72202i	−0.508591	+	0.861008i	$0.669834\pi$
$654$	0	0
$655$	27983.0	1.66929
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 3213.09i	− 0.189931i	−0.995481	−	0.0949654i	$-0.969726\pi$
$659$	− 3213.09i	− 0.189931i	0.995481	−	0.0949654i	$-0.0302740\pi$
$660$	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
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	29342.0	1.68061	0.840305	−	0.542113i	$-0.182376\pi$
$673$	29342.0	1.68061	0.840305	+	0.542113i	$0.182376\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 34762.3i	− 1.97344i	−0.162416	−	0.986722i	$-0.551929\pi$
$677$	− 34762.3i	− 1.97344i	0.162416	−	0.986722i	$-0.448071\pi$
$678$	0	0
$679$	8436.04	0.476798
$680$	0	0
$681$	0	0
$682$	0	0
$683$	11409.9i	0.639219i	0.947549	+	0.319610i	$0.103552\pi$
$683$	11409.9i	0.639219i	−0.947549	+	0.319610i	$0.896448\pi$
$684$	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
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	24806.4i	1.33656i	0.743911	+	0.668278i	$0.232968\pi$
$701$	24806.4i	1.33656i	−0.743911	+	0.668278i	$0.767032\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 14815.3i	− 0.788100i
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	0	0
$711$	0	0
$712$	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
$721$	−20088.0	−1.03761
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 3710.05i	− 0.190052i
$726$	0	0
$727$	38961.6	1.98763	0.993814	−	0.111060i	$-0.0354245\pi$
$727$	38961.6	1.98763	0.993814	+	0.111060i	$0.0354245\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	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
$745$	−35532.0	−1.74737
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 32091.4i	− 1.56555i
$750$	0	0
$751$	−35081.6	−1.70459	−0.852294	−	0.523063i	$-0.824789\pi$
$751$	−35081.6	−1.70459	−0.852294	+	0.523063i	$0.824789\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	37420.1i	1.80378i
$756$	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		−	1.00000i	$-0.5\pi$
$761$	0	0			1.00000i	$0.5\pi$
$762$	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	$-0.0340682\pi$
$769$	42406.0	1.98856	0.994278	+	0.106824i	$0.0340682\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19797.3i	0.921165i	0.887617	+	0.460583i	$0.152360\pi$
$773$	19797.3i	0.921165i	−0.887617	+	0.460583i	$0.847640\pi$
$774$	0	0
$775$	−5746.50	−0.266349
$776$	0	0
$777$	0	0
$778$	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
$787$	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
$795$	0	0
$796$	0	0
$797$	36217.2i	1.60963i	0.593523	+	0.804817i	$0.297737\pi$
$797$	36217.2i	1.60963i	−0.593523	+	0.804817i	$0.702263\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1821.51i	0.0800493i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0		−	1.00000i	$-0.5\pi$
$809$	0	0			1.00000i	$0.5\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	46484.8i	1.97604i	0.154321	+	0.988021i	$0.450681\pi$
$821$	46484.8i	1.97604i	−0.154321	+	0.988021i	$0.549319\pi$
$822$	0	0
$823$	−44340.7	−1.87803	−0.939015	−	0.343877i	$-0.888260\pi$
$823$	−44340.7	−1.87803	−0.939015	+	0.343877i	$0.888260\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	40672.8i	1.71019i	0.518467	+	0.855097i	$0.326503\pi$
$827$	40672.8i	1.71019i	−0.518467	+	0.855097i	$0.673497\pi$
$828$	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
$837$	0	0
$838$	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
$845$	− 22831.9i	− 0.929516i
$846$	0	0
$847$	−19091.3	−0.774481
$848$	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		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	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
$865$	39420.0	1.54950
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1745.91i	0.0681540i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	21688.4i	0.837944i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	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
$889$	−25704.0	−0.969724
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	48088.4	1.79600
$896$	0	0
$897$	0	0
$898$	0	0
$899$	73771.0i	2.73682i
$900$	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
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	−6944.00	−0.251712
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 39573.9i	− 1.42513i
$918$	0	0
$919$	−54217.0	−1.94609	−0.973044	−	0.230622i	$-0.925924\pi$
$919$	−54217.0	−1.94609	−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
$927$	0	0
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	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	$-0.688865\pi$
$937$	−32074.0	−1.11826	−0.559131	+	0.829079i	$0.688865\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 52242.1i	− 1.80982i	−0.425599	−	0.904912i	$-0.639937\pi$
$941$	− 52242.1i	− 1.80982i	0.425599	−	0.904912i	$-0.360063\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20308.1i	0.696858i	0.937335	+	0.348429i	$0.113285\pi$
$947$	20308.1i	0.696858i	−0.937335	+	0.348429i	$0.886715\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	84473.0	2.83552
$962$	0	0
$963$	0	0
$964$	0	0
$965$	26209.4i	0.874311i
$966$	0	0
$967$	−14094.4	−0.468712	−0.234356	−	0.972151i	$-0.575298\pi$
$967$	−14094.4	−0.468712	−0.234356	+	0.972151i	$0.575298\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	59674.2i	1.97223i	0.166066	+	0.986115i	$0.446894\pi$
$971$	59674.2i	1.97223i	−0.166066	+	0.986115i	$0.553106\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	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
$991$	−58126.4	−1.86321	−0.931607	−	0.363466i	$-0.881593\pi$
$991$	−58126.4	−1.86321	−0.931607	+	0.363466i	$0.881593\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 763.675i	− 0.0243318i
$996$	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	1152.4.d.l.577.1	✓	4
3.2	odd	2	inner	1152.4.d.l.577.3	yes	4
4.3	odd	2	inner	1152.4.d.l.577.2	yes	4
8.3	odd	2	inner	1152.4.d.l.577.4	yes	4
8.5	even	2	inner	1152.4.d.l.577.3	yes	4
12.11	even	2	inner	1152.4.d.l.577.4	yes	4
16.3	odd	4		2304.4.a.ca.1.1		4
16.5	even	4		2304.4.a.ca.1.4		4
16.11	odd	4		2304.4.a.ca.1.3		4
16.13	even	4		2304.4.a.ca.1.2		4
24.5	odd	2	CM	1152.4.d.l.577.1	✓	4
24.11	even	2	inner	1152.4.d.l.577.2	yes	4
48.5	odd	4		2304.4.a.ca.1.2		4
48.11	even	4		2304.4.a.ca.1.1		4
48.29	odd	4		2304.4.a.ca.1.4		4
48.35	even	4		2304.4.a.ca.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.d.l.577.1	✓	4	1.1	even	1	trivial
1152.4.d.l.577.1	✓	4	24.5	odd	2	CM
1152.4.d.l.577.2	yes	4	4.3	odd	2	inner
1152.4.d.l.577.2	yes	4	24.11	even	2	inner
1152.4.d.l.577.3	yes	4	3.2	odd	2	inner
1152.4.d.l.577.3	yes	4	8.5	even	2	inner
1152.4.d.l.577.4	yes	4	8.3	odd	2	inner
1152.4.d.l.577.4	yes	4	12.11	even	2	inner
2304.4.a.ca.1.1		4	16.3	odd	4
2304.4.a.ca.1.1		4	48.11	even	4
2304.4.a.ca.1.2		4	16.13	even	4
2304.4.a.ca.1.2		4	48.5	odd	4
2304.4.a.ca.1.3		4	16.11	odd	4
2304.4.a.ca.1.3		4	48.35	even	4
2304.4.a.ca.1.4		4	16.5	even	4
2304.4.a.ca.1.4		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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1152.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-10.3923i q^{5} -14.6969 q^{7} +5.65685i q^{11} +17.0000 q^{25} -218.238i q^{29} -338.030 q^{31} +152.735i q^{35} -127.000 q^{49} +509.223i q^{53} +58.7878 q^{55} +554.372i q^{59} +322.000 q^{73} -83.1384i q^{77} +308.636 q^{79} +1227.54i q^{83} -574.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 68 q^{25} - 508 q^{49} + 1288 q^{73} - 2296 q^{97}+O(q^{100}) \) 4 * q + 68 * q^25 - 508 * q^49 + 1288 * q^73 - 2296 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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