LMFDB - Embedded newform 2304.4.a.ca.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2304,4,Mod(1,2304)] 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("2304.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2304 = 2^{8} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2304.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$135.940400653$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 1152)
Fricke sign:	$-1$
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.2
Root			$-0.517638$ of defining polynomial
Character	$\chi$	$=$	2304.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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
$5$	−10.3923	−0.929516	−0.464758	−	0.885438i	$-0.653859\pi$
$5$	−10.3923	−0.929516	−0.464758	+	0.885438i	$0.653859\pi$
$6$	0	0
$7$	14.6969	0.793560	0.396780	−	0.917914i	$-0.370128\pi$
$7$	14.6969	0.793560	0.396780	+	0.917914i	$0.370128\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.65685	0.155055	0.0775275	−	0.996990i	$-0.475297\pi$
$11$	5.65685	0.155055	0.0775275	+	0.996990i	$0.475297\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	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		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	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
$29$	218.238	1.39744	0.698722	−	0.715394i	$-0.253753\pi$
$29$	218.238	1.39744	0.698722	+	0.715394i	$0.253753\pi$
$30$	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
$35$	−152.735	−0.737627
$36$	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		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	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
$53$	509.223	1.31976	0.659879	−	0.751372i	$-0.270608\pi$
$53$	509.223	1.31976	0.659879	+	0.751372i	$0.270608\pi$
$54$	0	0
$55$	−58.7878	−0.144126
$56$	0	0
$57$	0	0
$58$	0	0
$59$	554.372	1.22327	0.611636	−	0.791139i	$-0.290512\pi$
$59$	554.372	1.22327	0.611636	+	0.791139i	$0.290512\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	0	0
$63$	0	0
$64$	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	$-0.583107\pi$
$73$	−322.000	−0.516264	−0.258132	+	0.966110i	$0.583107\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	83.1384	0.123046
$78$	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
$83$	−1227.54	−1.62337	−0.811685	−	0.584095i	$-0.801449\pi$
$83$	−1227.54	−1.62337	−0.811685	+	0.584095i	$0.801449\pi$
$84$	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
$101$	1008.05	0.993120	0.496560	−	0.868003i	$-0.334596\pi$
$101$	1008.05	0.993120	0.496560	+	0.868003i	$0.334596\pi$
$102$	0	0
$103$	−1366.82	−1.30754	−0.653768	−	0.756695i	$-0.726813\pi$
$103$	−1366.82	−1.30754	−0.653768	+	0.756695i	$0.726813\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2183.55	1.97282	0.986408	−	0.164314i	$-0.0525410\pi$
$107$	2183.55	1.97282	0.986408	+	0.164314i	$0.0525410\pi$
$108$	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		−	1.00000i	$-0.5\pi$
$113$	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
$125$	1475.71	1.05593
$126$	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
$131$	−2692.66	−1.79587	−0.897935	−	0.440128i	$-0.854933\pi$
$131$	−2692.66	−1.79587	−0.897935	+	0.440128i	$0.854933\pi$
$132$	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		−	1.00000i	$-0.5\pi$
$139$	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
$147$	0	0
$148$	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
$151$	3600.75	1.94056	0.970281	−	0.241981i	$-0.0777971\pi$
$151$	3600.75	1.94056	0.970281	+	0.241981i	$0.0777971\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3512.91	1.82041
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	0	0
$159$	0	0
$160$	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
$165$	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
$173$	−3793.19	−1.66700	−0.833500	−	0.552519i	$-0.813666\pi$
$173$	−3793.19	−1.66700	−0.833500	+	0.552519i	$0.813666\pi$
$174$	0	0
$175$	−249.848	−0.107924
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4627.31	−1.93219	−0.966093	−	0.258196i	$-0.916872\pi$
$179$	−4627.31	−1.93219	−0.966093	+	0.258196i	$0.916872\pi$
$180$	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
$189$	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
$197$	−72.7461	−0.0263094	−0.0131547	−	0.999913i	$-0.504187\pi$
$197$	−72.7461	−0.0263094	−0.0131547	+	0.999913i	$0.504187\pi$
$198$	0	0
$199$	−73.4847	−0.0261768	−0.0130884	−	0.999914i	$-0.504166\pi$
$199$	−73.4847	−0.0261768	−0.0130884	+	0.999914i	$0.504166\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3207.44	1.10896
$204$	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
$217$	−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
$227$	−3365.83	−0.984132	−0.492066	−	0.870558i	$-0.663758\pi$
$227$	−3365.83	−0.984132	−0.492066	+	0.870558i	$0.663758\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	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
$245$	1319.82	0.344165
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	4078.59	1.02565	0.512826	−	0.858493i	$-0.328599\pi$
$251$	4078.59	1.02565	0.512826	+	0.858493i	$0.328599\pi$
$252$	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
$269$	−7264.22	−1.64650	−0.823248	−	0.567682i	$-0.807840\pi$
$269$	−7264.22	−1.64650	−0.823248	+	0.567682i	$0.807840\pi$
$270$	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
$275$	−96.1665	−0.0210875
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	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		−	1.00000i	$-0.5\pi$
$283$	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
$291$	0	0
$292$	0	0
$293$	−2899.45	−0.578116	−0.289058	−	0.957312i	$-0.593342\pi$
$293$	−2899.45	−0.578116	−0.289058	+	0.957312i	$0.593342\pi$
$294$	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		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	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	$-0.732072\pi$
$313$	−7378.00	−1.33236	−0.666181	+	0.745790i	$0.732072\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11275.7	1.99780	0.998902	−	0.0468563i	$-0.0149203\pi$
$317$	11275.7	1.99780	0.998902	+	0.0468563i	$0.0149203\pi$
$318$	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		−	1.00000i	$-0.5\pi$
$331$	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	$-0.886447\pi$
$337$	−11594.0	−1.87408	−0.937041	+	0.349220i	$0.886447\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1912.18	−0.303667
$342$	0	0
$343$	−6907.56	−1.08739
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6816.51	−1.05455	−0.527276	−	0.849694i	$-0.676787\pi$
$347$	−6816.51	−1.05455	−0.527276	+	0.849694i	$0.676787\pi$
$348$	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		−	1.00000i	$-0.5\pi$
$353$	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
$365$	3346.32	0.479875
$366$	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
$371$	7484.02	1.04731
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	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
$381$	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
$389$	14040.0	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
$393$	0	0
$394$	0	0
$395$	−3207.44	−0.408566
$396$	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		−	1.00000i	$-0.5\pi$
$401$	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	$-0.738565\pi$
$409$	−11270.0	−1.36251	−0.681254	+	0.732047i	$0.738565\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8147.57	0.970740
$414$	0	0
$415$	12756.9	1.50895
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12315.0	1.43586	0.717930	−	0.696115i	$-0.245090\pi$
$419$	12315.0	1.43586	0.717930	+	0.696115i	$0.245090\pi$
$420$	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
$433$	−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	$-0.0961736\pi$
$439$	17562.8	1.90940	0.954702	+	0.297562i	$0.0961736\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14962.4	−1.60470	−0.802352	−	0.596851i	$-0.796418\pi$
$443$	−14962.4	−1.60470	−0.802352	+	0.596851i	$0.796418\pi$
$444$	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	$-0.545344\pi$
$457$	−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
$463$	−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
$467$	6612.86	0.655261	0.327630	−	0.944806i	$-0.393750\pi$
$467$	6612.86	0.655261	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
$477$	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
$485$	5965.18	0.558485
$486$	0	0
$487$	21295.9	1.98154	0.990768	−	0.135571i	$-0.0432868\pi$
$487$	21295.9	1.98154	0.990768	+	0.135571i	$0.0432868\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−21744.9	−1.99865	−0.999324	−	0.0367748i	$-0.988292\pi$
$491$	−21744.9	−1.99865	−0.999324	+	0.0367748i	$0.988292\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	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
$507$	0	0
$508$	0	0
$509$	−22561.7	−1.96469	−0.982347	−	0.187067i	$-0.940102\pi$
$509$	−22561.7	−1.96469	−0.982347	+	0.187067i	$0.940102\pi$
$510$	0	0
$511$	−4732.41	−0.409686
$512$	0	0
$513$	0	0
$514$	0	0
$515$	14204.4	1.21538
$516$	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		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	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
$539$	−718.420	−0.0574111
$540$	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		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	4536.00	0.348807
$554$	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
$563$	25144.7	1.88228	0.941140	−	0.338017i	$-0.109756\pi$
$563$	25144.7	1.88228	0.941140	+	0.338017i	$0.109756\pi$
$564$	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		−	1.00000i	$-0.5\pi$
$571$	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
$579$	0	0
$580$	0	0
$581$	−18041.0	−1.28824
$582$	0	0
$583$	2880.60	0.204635
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12988.1	−0.913250	−0.456625	−	0.889659i	$-0.650942\pi$
$587$	−12988.1	−0.913250	−0.456625	+	0.889659i	$0.650942\pi$
$588$	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	$-0.538963\pi$
$601$	−3598.00	−0.244202	−0.122101	+	0.992518i	$0.538963\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	13499.6	0.907169
$606$	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		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	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
$631$	−9156.19	−0.577658	−0.288829	−	0.957381i	$-0.593266\pi$
$631$	−9156.19	−0.577658	−0.288829	+	0.957381i	$0.593266\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−18175.5	−1.13586
$636$	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		−	1.00000i	$-0.5\pi$
$643$	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
$649$	3136.00	0.189675
$650$	0	0
$651$	0	0
$652$	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
$655$	27983.0	1.66929
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3213.09	0.189931	0.0949654	−	0.995481i	$-0.469726\pi$
$659$	3213.09	0.189931	0.0949654	+	0.995481i	$0.469726\pi$
$660$	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
$669$	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
$677$	−34762.3	−1.97344	−0.986722	−	0.162416i	$-0.948071\pi$
$677$	−34762.3	−1.97344	−0.986722	+	0.162416i	$0.948071\pi$
$678$	0	0
$679$	−8436.04	−0.476798
$680$	0	0
$681$	0	0
$682$	0	0
$683$	11409.9	0.639219	0.319610	−	0.947549i	$-0.396448\pi$
$683$	11409.9	0.639219	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
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−24806.4	−1.33656	−0.668278	−	0.743911i	$-0.732968\pi$
$701$	−24806.4	−1.33656	−0.668278	+	0.743911i	$0.732968\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14815.3	0.788100
$708$	0	0
$709$	0	0		−	1.00000i	$-0.5\pi$
$709$	0	0			1.00000i	$0.5\pi$
$710$	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
$725$	−3710.05	−0.190052
$726$	0	0
$727$	−38961.6	−1.98763	−0.993814	−	0.111060i	$-0.964575\pi$
$727$	−38961.6	−1.98763	−0.993814	+	0.111060i	$0.964575\pi$
$728$	0	0
$729$	0	0
$730$	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
$735$	0	0
$736$	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
$747$	0	0
$748$	0	0
$749$	32091.4	1.56555
$750$	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
$755$	−37420.1	−1.80378
$756$	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		−	1.00000i	$-0.5\pi$
$761$	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
$773$	19797.3	0.921165	0.460583	−	0.887617i	$-0.347640\pi$
$773$	19797.3	0.921165	0.460583	+	0.887617i	$0.347640\pi$
$774$	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		−	1.00000i	$-0.5\pi$
$787$	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
$795$	0	0
$796$	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
$803$	−1821.51	−0.0800493
$804$	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		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	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
$821$	46484.8	1.97604	0.988021	−	0.154321i	$-0.0493191\pi$
$821$	46484.8	1.97604	0.988021	+	0.154321i	$0.0493191\pi$
$822$	0	0
$823$	44340.7	1.87803	0.939015	−	0.343877i	$-0.111740\pi$
$823$	44340.7	1.87803	0.939015	+	0.343877i	$0.111740\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	40672.8	1.71019	0.855097	−	0.518467i	$-0.173497\pi$
$827$	40672.8	1.71019	0.855097	+	0.518467i	$0.173497\pi$
$828$	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
$837$	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
$845$	22831.9	0.929516
$846$	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		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	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
$867$	0	0
$868$	0	0
$869$	1745.91	0.0681540
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	21688.4	0.837944
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	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		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	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
$899$	−73771.0	−2.73682
$900$	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
$909$	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
$917$	−39573.9	−1.42513
$918$	0	0
$919$	54217.0	1.94609	0.973044	−	0.230622i	$-0.0740759\pi$
$919$	54217.0	1.94609	0.973044	+	0.230622i	$0.0740759\pi$
$920$	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	$-0.311135\pi$
$937$	32074.0	1.11826	0.559131	+	0.829079i	$0.311135\pi$
$938$	0	0
$939$	0	0
$940$	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
$945$	0	0
$946$	0	0
$947$	−20308.1	−0.696858	−0.348429	−	0.937335i	$-0.613285\pi$
$947$	−20308.1	−0.696858	−0.348429	+	0.937335i	$0.613285\pi$
$948$	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
$965$	26209.4	0.874311
$966$	0	0
$967$	14094.4	0.468712	0.234356	−	0.972151i	$-0.424702\pi$
$967$	14094.4	0.468712	0.234356	+	0.972151i	$0.424702\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	59674.2	1.97223	0.986115	−	0.166066i	$-0.0531063\pi$
$971$	59674.2	1.97223	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		−	1.00000i	$-0.5\pi$
$977$	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
$995$	763.675	0.0243318
$996$	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	2304.4.a.ca.1.2		4
3.2	odd	2	inner	2304.4.a.ca.1.4		4
4.3	odd	2	inner	2304.4.a.ca.1.1		4
8.3	odd	2	inner	2304.4.a.ca.1.3		4
8.5	even	2	inner	2304.4.a.ca.1.4		4
12.11	even	2	inner	2304.4.a.ca.1.3		4
16.3	odd	4		1152.4.d.l.577.4	yes	4
16.5	even	4		1152.4.d.l.577.1	✓	4
16.11	odd	4		1152.4.d.l.577.2	yes	4
16.13	even	4		1152.4.d.l.577.3	yes	4
24.5	odd	2	CM	2304.4.a.ca.1.2		4
24.11	even	2	inner	2304.4.a.ca.1.1		4
48.5	odd	4		1152.4.d.l.577.3	yes	4
48.11	even	4		1152.4.d.l.577.4	yes	4
48.29	odd	4		1152.4.d.l.577.1	✓	4
48.35	even	4		1152.4.d.l.577.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.d.l.577.1	✓	4	16.5	even	4
1152.4.d.l.577.1	✓	4	48.29	odd	4
1152.4.d.l.577.2	yes	4	16.11	odd	4
1152.4.d.l.577.2	yes	4	48.35	even	4
1152.4.d.l.577.3	yes	4	16.13	even	4
1152.4.d.l.577.3	yes	4	48.5	odd	4
1152.4.d.l.577.4	yes	4	16.3	odd	4
1152.4.d.l.577.4	yes	4	48.11	even	4
2304.4.a.ca.1.1		4	4.3	odd	2	inner
2304.4.a.ca.1.1		4	24.11	even	2	inner
2304.4.a.ca.1.2		4	1.1	even	1	trivial
2304.4.a.ca.1.2		4	24.5	odd	2	CM
2304.4.a.ca.1.3		4	8.3	odd	2	inner
2304.4.a.ca.1.3		4	12.11	even	2	inner
2304.4.a.ca.1.4		4	3.2	odd	2	inner
2304.4.a.ca.1.4		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2304 = 2^{8} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2304.a (trivial)

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

Related objects

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