LMFDB - Embedded newform 576.5.b.c.415.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [576,5,Mod(415,576)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(576, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("576.415");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 576 = 2^{6} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	576.b (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:	$59.5410987363$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			415.4
Root			$-1.22474 + 1.22474i$ of defining polynomial
Character	$\chi$	$=$	576.415
Dual form			576.5.b.c.415.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+19.5959i q^{5} +2.00000i q^{7} +O(q^{10})$	$q+19.5959i q^{5} +2.00000i q^{7} +195.959 q^{11} +241.000 q^{25} -1469.69i q^{29} +478.000i q^{31} -39.1918 q^{35} +2397.00 q^{49} +4605.04i q^{53} +3840.00i q^{55} +1175.76 q^{59} +8158.00 q^{73} +391.918i q^{77} +9118.00i q^{79} -13129.3 q^{83} +17282.0 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 964 q^{25} + 9588 q^{49} + 32632 q^{73} + 69128 q^{97}+O(q^{100}) $ 4 * q + 964 * q^25 + 9588 * q^49 + 32632 * q^73 + 69128 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/576\mathbb{Z}\right)^\times$.

$n$	$65$	$127$	$325$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	19.5959i	0.783837i	0.920000	+	0.391918i	$0.128188\pi$
$5$	19.5959i	0.783837i	−0.920000	+	0.391918i	$0.871812\pi$
$6$	0	0
$7$	2.00000i	0.0408163i	0.999792	+	0.0204082i	$0.00649657\pi$
$7$	2.00000i	0.0408163i	−0.999792	+	0.0204082i	$0.993503\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	195.959	1.61950	0.809749	−	0.586777i	$-0.199603\pi$
$11$	195.959	1.61950	0.809749	+	0.586777i	$0.199603\pi$
$12$	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		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	241.000	0.385600
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 1469.69i	− 1.74756i	−0.486326	−	0.873778i	$-0.661663\pi$
$29$	− 1469.69i	− 1.74756i	0.486326	−	0.873778i	$-0.338337\pi$
$30$	0	0
$31$	478.000i	0.497399i	0.968581	+	0.248699i	$0.0800031\pi$
$31$	478.000i	0.497399i	−0.968581	+	0.248699i	$0.919997\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−39.1918	−0.0319933
$36$	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		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	2397.00	0.998334
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4605.04i	1.63939i	0.572802	+	0.819694i	$0.305857\pi$
$53$	4605.04i	1.63939i	−0.572802	+	0.819694i	$0.694143\pi$
$54$	0	0
$55$	3840.00i	1.26942i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1175.76	0.337764	0.168882	−	0.985636i	$-0.445984\pi$
$59$	1175.76	0.337764	0.168882	+	0.985636i	$0.445984\pi$
$60$	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		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	8158.00	1.53087	0.765434	−	0.643514i	$-0.222524\pi$
$73$	8158.00	1.53087	0.765434	+	0.643514i	$0.222524\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	391.918i	0.0661019i
$78$	0	0
$79$	9118.00i	1.46098i	0.682921	+	0.730492i	$0.260709\pi$
$79$	9118.00i	1.46098i	−0.682921	+	0.730492i	$0.739291\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−13129.3	−1.90583	−0.952915	−	0.303237i	$-0.901933\pi$
$83$	−13129.3	−1.90583	−0.952915	+	0.303237i	$0.901933\pi$
$84$	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
$97$	17282.0	1.83675	0.918376	−	0.395709i	$-0.129501\pi$
$97$	17282.0	1.83675	0.918376	+	0.395709i	$0.129501\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 13031.3i	− 1.27745i	−0.769434	−	0.638726i	$-0.779462\pi$
$101$	− 13031.3i	− 1.27745i	0.769434	−	0.638726i	$-0.220538\pi$
$102$	0	0
$103$	21118.0i	1.99057i	0.0969729	+	0.995287i	$0.469084\pi$
$103$	21118.0i	1.99057i	−0.0969729	+	0.995287i	$0.530916\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16852.5	1.47196	0.735981	−	0.677002i	$-0.236721\pi$
$107$	16852.5	1.47196	0.735981	+	0.677002i	$0.236721\pi$
$108$	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
$121$	23759.0	1.62277
$122$	0	0
$123$	0	0
$124$	0	0
$125$	16970.1i	1.08608i
$126$	0	0
$127$	20642.0i	1.27981i	0.768455	+	0.639903i	$0.221026\pi$
$127$	20642.0i	1.27981i	−0.768455	+	0.639903i	$0.778974\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	19595.9	1.14189	0.570943	−	0.820989i	$-0.306578\pi$
$131$	19595.9	1.14189	0.570943	+	0.820989i	$0.306578\pi$
$132$	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
$145$	28800.0	1.36980
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 19889.9i	− 0.895899i	−0.894059	−	0.447950i	$-0.852154\pi$
$149$	− 19889.9i	− 0.895899i	0.894059	−	0.447950i	$-0.147846\pi$
$150$	0	0
$151$	43202.0i	1.89474i	0.320139	+	0.947371i	$0.396271\pi$
$151$	43202.0i	1.89474i	−0.320139	+	0.947371i	$0.603729\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9366.85	−0.389879
$156$	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		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	28561.0	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 15774.7i	− 0.527071i	−0.964650	−	0.263536i	$-0.915111\pi$
$173$	− 15774.7i	− 0.527071i	0.964650	−	0.263536i	$-0.0848886\pi$
$174$	0	0
$175$	482.000i	0.0157388i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−63098.9	−1.96932	−0.984658	−	0.174495i	$-0.944171\pi$
$179$	−63098.9	−1.96932	−0.984658	+	0.174495i	$0.944171\pi$
$180$	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	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	9602.00	0.257779	0.128889	−	0.991659i	$-0.458859\pi$
$193$	9602.00	0.257779	0.128889	+	0.991659i	$0.458859\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	66528.1i	1.71425i	0.515112	+	0.857123i	$0.327750\pi$
$197$	66528.1i	1.71425i	−0.515112	+	0.857123i	$0.672250\pi$
$198$	0	0
$199$	38398.0i	0.969622i	0.874619	+	0.484811i	$0.161112\pi$
$199$	38398.0i	0.969622i	−0.874619	+	0.484811i	$0.838888\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2939.39	0.0713288
$204$	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
$217$	−956.000	−0.0203020
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 46558.0i	− 0.936234i	−0.883666	−	0.468117i	$-0.844933\pi$
$223$	− 46558.0i	− 0.936234i	0.883666	−	0.468117i	$-0.155067\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	101703.	1.97370	0.986850	−	0.161637i	$-0.0516774\pi$
$227$	101703.	1.97370	0.986850	+	0.161637i	$0.0516774\pi$
$228$	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	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	0	0
$241$	−29762.0	−0.512422	−0.256211	−	0.966621i	$-0.582474\pi$
$241$	−29762.0	−0.512422	−0.256211	+	0.966621i	$0.582474\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	46971.4i	0.782531i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−82890.7	−1.31571	−0.657853	−	0.753147i	$-0.728535\pi$
$251$	−82890.7	−1.31571	−0.657853	+	0.753147i	$0.728535\pi$
$252$	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	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	−90240.0	−1.28501
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 139033.i	− 1.92138i	−0.277622	−	0.960690i	$-0.589546\pi$
$269$	− 139033.i	− 1.92138i	0.277622	−	0.960690i	$-0.410454\pi$
$270$	0	0
$271$	− 143518.i	− 1.95419i	−0.212793	−	0.977097i	$-0.568256\pi$
$271$	− 143518.i	− 1.95419i	0.212793	−	0.977097i	$-0.431744\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	47226.2	0.624478
$276$	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		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−83521.0	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 170191.i	− 1.98244i	−0.132221	−	0.991220i	$-0.542211\pi$
$293$	− 170191.i	− 1.98244i	0.132221	−	0.991220i	$-0.457789\pi$
$294$	0	0
$295$	23040.0i	0.264752i
$296$	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	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	−84962.0	−0.867234	−0.433617	−	0.901097i	$-0.642763\pi$
$313$	−84962.0	−0.867234	−0.433617	+	0.901097i	$0.642763\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	179989.i	1.79113i	0.444934	+	0.895563i	$0.353227\pi$
$317$	179989.i	1.79113i	−0.444934	+	0.895563i	$0.646773\pi$
$318$	0	0
$319$	− 288000.i	− 2.83016i
$320$	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
$337$	−191038.	−1.68213	−0.841066	−	0.540933i	$-0.818071\pi$
$337$	−191038.	−1.68213	−0.841066	+	0.540933i	$0.818071\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	93668.5i	0.805536i
$342$	0	0
$343$	9596.00i	0.0815647i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	72700.9	0.603783	0.301891	−	0.953342i	$-0.402382\pi$
$347$	72700.9	0.603783	0.301891	+	0.953342i	$0.402382\pi$
$348$	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	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	0	0
$361$	−130321.	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	159863.i	1.19995i
$366$	0	0
$367$	− 234722.i	− 1.74270i	−0.490665	−	0.871348i	$-0.663246\pi$
$367$	− 234722.i	− 1.74270i	0.490665	−	0.871348i	$-0.336754\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9210.08	−0.0669138
$372$	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		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	−7680.00	−0.0518131
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 143344.i	− 0.947285i	−0.880717	−	0.473643i	$-0.842939\pi$
$389$	− 143344.i	− 0.947285i	0.880717	−	0.473643i	$-0.157061\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−178676.	−1.14517
$396$	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
$409$	−180962.	−1.08178	−0.540892	−	0.841092i	$-0.681913\pi$
$409$	−180962.	−1.08178	−0.540892	+	0.841092i	$0.681913\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2351.51i	0.0137863i
$414$	0	0
$415$	− 257280.i	− 1.49386i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−300405.	−1.71112	−0.855559	−	0.517706i	$-0.826786\pi$
$419$	−300405.	−1.71112	−0.855559	+	0.517706i	$0.826786\pi$
$420$	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	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	−73922.0	−0.394274	−0.197137	−	0.980376i	$-0.563164\pi$
$433$	−73922.0	−0.394274	−0.197137	+	0.980376i	$0.563164\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	308158.i	1.59899i	0.600676	+	0.799493i	$0.294898\pi$
$439$	308158.i	1.59899i	−0.600676	+	0.799493i	$0.705102\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	75836.2	0.386428	0.193214	−	0.981157i	$-0.438109\pi$
$443$	75836.2	0.386428	0.193214	+	0.981157i	$0.438109\pi$
$444$	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
$457$	136798.	0.655009	0.327505	−	0.944850i	$-0.393792\pi$
$457$	136798.	0.655009	0.327505	+	0.944850i	$0.393792\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 399855.i	− 1.88148i	−0.339124	−	0.940742i	$-0.610131\pi$
$461$	− 399855.i	− 1.88148i	0.339124	−	0.940742i	$-0.389869\pi$
$462$	0	0
$463$	− 366238.i	− 1.70845i	−0.519906	−	0.854223i	$-0.674033\pi$
$463$	− 366238.i	− 1.70845i	0.519906	−	0.854223i	$-0.325967\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	434833.	1.99383	0.996917	−	0.0784588i	$-0.0249999\pi$
$467$	434833.	1.99383	0.996917	+	0.0784588i	$0.0249999\pi$
$468$	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	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	338657.i	1.43971i
$486$	0	0
$487$	466562.i	1.96721i	0.180327	+	0.983607i	$0.442284\pi$
$487$	466562.i	1.96721i	−0.180327	+	0.983607i	$0.557716\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	405244.	1.68094	0.840472	−	0.541855i	$-0.182278\pi$
$491$	405244.	1.68094	0.840472	+	0.541855i	$0.182278\pi$
$492$	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	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	255360.	1.00131
$506$	0	0
$507$	0	0
$508$	0	0
$509$	128647.i	0.496552i	0.968689	+	0.248276i	$0.0798640\pi$
$509$	128647.i	0.496552i	−0.968689	+	0.248276i	$0.920136\pi$
$510$	0	0
$511$	16316.0i	0.0624844i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−413827.	−1.56029
$516$	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
$529$	279841.	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	330240.i	1.15378i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	469714.	1.61680
$540$	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
$553$	−18236.0	−0.0596320
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 582097.i	− 1.87622i	−0.346331	−	0.938112i	$-0.612573\pi$
$557$	− 582097.i	− 1.87622i	0.346331	−	0.938112i	$-0.387427\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−351355.	−1.10848	−0.554242	−	0.832356i	$-0.686992\pi$
$563$	−351355.	−1.10848	−0.554242	+	0.832356i	$0.686992\pi$
$564$	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
$577$	581758.	1.74739	0.873697	−	0.486471i	$-0.161716\pi$
$577$	581758.	1.74739	0.873697	+	0.486471i	$0.161716\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 26258.5i	− 0.0777890i
$582$	0	0
$583$	902400.i	2.65498i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−685073.	−1.98820	−0.994102	−	0.108451i	$-0.965411\pi$
$587$	−685073.	−1.98820	−0.994102	+	0.108451i	$0.965411\pi$
$588$	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	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	712802.	1.97342	0.986711	−	0.162485i	$-0.0519509\pi$
$601$	712802.	1.97342	0.986711	+	0.162485i	$0.0519509\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	465579.i	1.27199i
$606$	0	0
$607$	− 203998.i	− 0.553667i	−0.960918	−	0.276833i	$-0.910715\pi$
$607$	− 203998.i	− 0.553667i	0.960918	−	0.276833i	$-0.0892850\pi$
$608$	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		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−181919.	−0.465713
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	736322.i	1.84931i	0.380809	+	0.924654i	$0.375645\pi$
$631$	736322.i	1.84931i	−0.380809	+	0.924654i	$0.624355\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−404499.	−1.00316
$636$	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	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	230400.	0.547007
$650$	0	0
$651$	0	0
$652$	0	0
$653$	837823.i	1.96484i	0.186696	+	0.982418i	$0.440222\pi$
$653$	837823.i	1.96484i	−0.186696	+	0.982418i	$0.559778\pi$
$654$	0	0
$655$	384000.i	0.895053i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	801081.	1.84461	0.922307	−	0.386457i	$-0.126301\pi$
$659$	801081.	1.84461	0.922307	+	0.386457i	$0.126301\pi$
$660$	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
$673$	869758.	1.92030	0.960148	−	0.279491i	$-0.0901658\pi$
$673$	869758.	1.92030	0.960148	+	0.279491i	$0.0901658\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 197821.i	− 0.431613i	−0.976436	−	0.215807i	$-0.930762\pi$
$677$	− 197821.i	− 0.431613i	0.976436	−	0.215807i	$-0.0692381\pi$
$678$	0	0
$679$	34564.0i	0.0749695i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	392114.	0.840565	0.420282	−	0.907393i	$-0.361931\pi$
$683$	392114.	0.840565	0.420282	+	0.907393i	$0.361931\pi$
$684$	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
$701$	− 70055.4i	− 0.142563i	−0.997456	−	0.0712813i	$-0.977291\pi$
$701$	− 70055.4i	− 0.142563i	0.997456	−	0.0712813i	$-0.0227088\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	26062.6	0.0521409
$708$	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	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	−42236.0	−0.0812479
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 354196.i	− 0.673857i
$726$	0	0
$727$	1.04544e6i	1.97802i	0.147842	+	0.989011i	$0.452767\pi$
$727$	1.04544e6i	1.97802i	−0.147842	+	0.989011i	$0.547233\pi$
$728$	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		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	389760.	0.702239
$746$	0	0
$747$	0	0
$748$	0	0
$749$	33705.0i	0.0600801i
$750$	0	0
$751$	− 837602.i	− 1.48511i	−0.669787	−	0.742554i	$-0.733614\pi$
$751$	− 837602.i	− 1.48511i	0.669787	−	0.742554i	$-0.266386\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−846583.	−1.48517
$756$	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
$769$	−439678.	−0.743502	−0.371751	−	0.928333i	$-0.621242\pi$
$769$	−439678.	−0.743502	−0.371751	+	0.928333i	$0.621242\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 1.18722e6i	− 1.98688i	−0.114353	−	0.993440i	$-0.536479\pi$
$773$	− 1.18722e6i	− 1.98688i	0.114353	−	0.993440i	$-0.463521\pi$
$774$	0	0
$775$	115198.i	0.191797i
$776$	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
$797$	− 1.20446e6i	− 1.89617i	−0.318020	−	0.948084i	$-0.603018\pi$
$797$	− 1.20446e6i	− 1.89617i	0.318020	−	0.948084i	$-0.396982\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.59863e6	2.47924
$804$	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
$821$	1.00439e6i	1.49010i	0.667008	+	0.745050i	$0.267575\pi$
$821$	1.00439e6i	1.49010i	−0.667008	+	0.745050i	$0.732425\pi$
$822$	0	0
$823$	− 1.13376e6i	− 1.67387i	−0.547305	−	0.836933i	$-0.684346\pi$
$823$	− 1.13376e6i	− 1.67387i	0.547305	−	0.836933i	$-0.315654\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−908859.	−1.32888	−0.664439	−	0.747342i	$-0.731330\pi$
$827$	−908859.	−1.32888	−0.664439	+	0.747342i	$0.731330\pi$
$828$	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	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	−1.45272e6	−2.05395
$842$	0	0
$843$	0	0
$844$	0	0
$845$	559679.i	0.783837i
$846$	0	0
$847$	47518.0i	0.0662356i
$848$	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		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	309120.	0.413138
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.78676e6i	2.36606i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−33940.1	−0.0443300
$876$	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		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	−41284.0	−0.0522370
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	− 1.23648e6i	− 1.54362i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	702514.	0.869231
$900$	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	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−2.57280e6	−3.08649
$914$	0	0
$915$	0	0
$916$	0	0
$917$	39191.8i	0.0466076i
$918$	0	0
$919$	− 1.25088e6i	− 1.48110i	−0.672002	−	0.740549i	$-0.734565\pi$
$919$	− 1.25088e6i	− 1.48110i	0.672002	−	0.740549i	$-0.265435\pi$
$920$	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
$937$	−464162.	−0.528677	−0.264338	−	0.964430i	$-0.585154\pi$
$937$	−464162.	−0.528677	−0.264338	+	0.964430i	$0.585154\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 787854.i	− 0.889747i	−0.895593	−	0.444873i	$-0.853249\pi$
$941$	− 787854.i	− 0.889747i	0.895593	−	0.444873i	$-0.146751\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.79146e6	−1.99759	−0.998796	−	0.0490528i	$-0.984380\pi$
$947$	−1.79146e6	−1.99759	−0.998796	+	0.0490528i	$0.984380\pi$
$948$	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
$961$	695037.	0.752595
$962$	0	0
$963$	0	0
$964$	0	0
$965$	188160.i	0.202056i
$966$	0	0
$967$	− 1.77792e6i	− 1.90134i	−0.310204	−	0.950670i	$-0.600397\pi$
$967$	− 1.77792e6i	− 1.90134i	0.310204	−	0.950670i	$-0.399603\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−416021.	−0.441242	−0.220621	−	0.975360i	$-0.570808\pi$
$971$	−416021.	−0.441242	−0.220621	+	0.975360i	$0.570808\pi$
$972$	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	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	−1.30368e6	−1.34369
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	54238.0i	0.0552276i	0.999619	+	0.0276138i	$0.00879087\pi$
$991$	54238.0i	0.0552276i	−0.999619	+	0.0276138i	$0.991209\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−752444.	−0.760025
$996$	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	576.5.b.c.415.4	yes	4
3.2	odd	2	inner	576.5.b.c.415.2	yes	4
4.3	odd	2	inner	576.5.b.c.415.3	yes	4
8.3	odd	2	inner	576.5.b.c.415.1	✓	4
8.5	even	2	inner	576.5.b.c.415.2	yes	4
12.11	even	2	inner	576.5.b.c.415.1	✓	4
24.5	odd	2	CM	576.5.b.c.415.4	yes	4
24.11	even	2	inner	576.5.b.c.415.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.5.b.c.415.1	✓	4	8.3	odd	2	inner
576.5.b.c.415.1	✓	4	12.11	even	2	inner
576.5.b.c.415.2	yes	4	3.2	odd	2	inner
576.5.b.c.415.2	yes	4	8.5	even	2	inner
576.5.b.c.415.3	yes	4	4.3	odd	2	inner
576.5.b.c.415.3	yes	4	24.11	even	2	inner
576.5.b.c.415.4	yes	4	1.1	even	1	trivial
576.5.b.c.415.4	yes	4	24.5	odd	2	CM

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:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	576.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+19.5959i q^{5} +2.00000i q^{7} +O(q^{10})\)	\(q+19.5959i q^{5} +2.00000i q^{7} +195.959 q^{11} +241.000 q^{25} -1469.69i q^{29} +478.000i q^{31} -39.1918 q^{35} +2397.00 q^{49} +4605.04i q^{53} +3840.00i q^{55} +1175.76 q^{59} +8158.00 q^{73} +391.918i q^{77} +9118.00i q^{79} -13129.3 q^{83} +17282.0 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 964 q^{25} + 9588 q^{49} + 32632 q^{73} + 69128 q^{97}+O(q^{100}) \) 4 * q + 964 * q^25 + 9588 * q^49 + 32632 * q^73 + 69128 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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