LMFDB - Embedded newform 864.2.d.b.433.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(433,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("864.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.d (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:	$6.89907473464$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			433.3
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	864.433
Dual form			864.2.d.b.433.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.58579i q^{5} +5.24264 q^{7} +O(q^{10})$	$q+1.58579i q^{5} +5.24264 q^{7} -5.82843i q^{11} +2.48528 q^{25} +2.82843i q^{29} -0.757359 q^{31} +8.31371i q^{35} +20.4853 q^{49} +10.0711i q^{53} +9.24264 q^{55} -11.3137i q^{59} +1.48528 q^{73} -30.5563i q^{77} +10.0000 q^{79} +12.1716i q^{83} -17.9706 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^7	$ 4 q + 4 q^{7} - 24 q^{25} - 20 q^{31} + 48 q^{49} + 20 q^{55} - 28 q^{73} + 40 q^{79} - 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^7 - 24 * q^25 - 20 * q^31 + 48 * q^49 + 20 * q^55 - 28 * q^73 + 40 * q^79 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	1.58579i	0.709185i	0.935021	+	0.354593i	$0.115380\pi$
$5$	1.58579i	0.709185i	−0.935021	+	0.354593i	$0.884620\pi$
$6$	0	0
$7$	5.24264	1.98153	0.990766	−	0.135583i	$-0.0432908\pi$
$7$	5.24264	1.98153	0.990766	+	0.135583i	$0.0432908\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.82843i	− 1.75734i	−0.477432	−	0.878668i	$-0.658432\pi$
$11$	− 5.82843i	− 1.75734i	0.477432	−	0.878668i	$-0.341568\pi$
$12$	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
$25$	2.48528	0.497056
$26$	0	0
$27$	0	0
$28$	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
$31$	−0.757359	−0.136026	−0.0680129	−	0.997684i	$-0.521666\pi$
$31$	−0.757359	−0.136026	−0.0680129	+	0.997684i	$0.521666\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.31371i	1.40527i
$36$	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
$49$	20.4853	2.92647
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.0711i	1.38337i	0.722200	+	0.691684i	$0.243131\pi$
$53$	10.0711i	1.38337i	−0.722200	+	0.691684i	$0.756869\pi$
$54$	0	0
$55$	9.24264	1.24628
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 11.3137i	− 1.47292i	−0.676481	−	0.736460i	$-0.736496\pi$
$59$	− 11.3137i	− 1.47292i	0.676481	−	0.736460i	$-0.263504\pi$
$60$	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
$73$	1.48528	0.173839	0.0869195	−	0.996215i	$-0.472298\pi$
$73$	1.48528	0.173839	0.0869195	+	0.996215i	$0.472298\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 30.5563i	− 3.48222i
$78$	0	0
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.1716i	1.33600i	0.744160	+	0.668002i	$0.232850\pi$
$83$	12.1716i	1.33600i	−0.744160	+	0.668002i	$0.767150\pi$
$84$	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
$97$	−17.9706	−1.82463	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−17.9706	−1.82463	−0.912317	+	0.409484i	$0.865709\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 6.89949i	− 0.686525i	−0.939239	−	0.343263i	$-0.888468\pi$
$101$	− 6.89949i	− 0.686525i	0.939239	−	0.343263i	$-0.111532\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	20.6569i	1.99697i	0.0549930	+	0.998487i	$0.482486\pi$
$107$	20.6569i	1.99697i	−0.0549930	+	0.998487i	$0.517514\pi$
$108$	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
$121$	−22.9706	−2.08823
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.8701i	1.06169i
$126$	0	0
$127$	−6.75736	−0.599619	−0.299809	−	0.953999i	$-0.596923\pi$
$127$	−6.75736	−0.599619	−0.299809	+	0.953999i	$0.596923\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 14.3137i	− 1.25059i	−0.780387	−	0.625297i	$-0.784978\pi$
$131$	− 14.3137i	− 1.25059i	0.780387	−	0.625297i	$-0.215022\pi$
$132$	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
$145$	−4.48528	−0.372482
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.5858i	1.60453i	0.596968	+	0.802265i	$0.296372\pi$
$149$	19.5858i	1.60453i	−0.596968	+	0.802265i	$0.703628\pi$
$150$	0	0
$151$	−20.2132	−1.64493	−0.822464	−	0.568818i	$-0.807401\pi$
$151$	−20.2132	−1.64493	−0.822464	+	0.568818i	$0.807401\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.20101i	− 0.0964675i
$156$	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
$169$	13.0000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 24.8995i	− 1.89307i	−0.322596	−	0.946537i	$-0.604555\pi$
$173$	− 24.8995i	− 1.89307i	0.322596	−	0.946537i	$-0.395445\pi$
$174$	0	0
$175$	13.0294	0.984933
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 15.3431i	− 1.14680i	−0.819275	−	0.573400i	$-0.805624\pi$
$179$	− 15.3431i	− 1.14680i	0.819275	−	0.573400i	$-0.194376\pi$
$180$	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
$193$	−4.51472	−0.324977	−0.162488	−	0.986710i	$-0.551952\pi$
$193$	−4.51472	−0.324977	−0.162488	+	0.986710i	$0.551952\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	28.0711i	1.99998i	0.00438501	+	0.999990i	$0.498604\pi$
$197$	28.0711i	1.99998i	−0.00438501	+	0.999990i	$0.501396\pi$
$198$	0	0
$199$	−14.2132	−1.00755	−0.503774	−	0.863836i	$-0.668055\pi$
$199$	−14.2132	−1.00755	−0.503774	+	0.863836i	$0.668055\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	14.8284i	1.04075i
$204$	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
$217$	−3.97056	−0.269539
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−26.0000	−1.74109	−0.870544	−	0.492090i	$-0.836233\pi$
$223$	−26.0000	−1.74109	−0.870544	+	0.492090i	$0.836233\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 28.2843i	− 1.87729i	−0.344881	−	0.938647i	$-0.612081\pi$
$227$	− 28.2843i	− 1.87729i	0.344881	−	0.938647i	$-0.387919\pi$
$228$	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
$241$	−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
$243$	0	0
$244$	0	0
$245$	32.4853i	2.07541i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.65685i	0.357057i	0.983935	+	0.178529i	$0.0571337\pi$
$251$	5.65685i	0.357057i	−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		−	1.00000i	$-0.5\pi$
$257$	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
$265$	−15.9706	−0.981064
$266$	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
$271$	−32.2132	−1.95681	−0.978406	−	0.206691i	$-0.933731\pi$
$271$	−32.2132	−1.95681	−0.978406	+	0.206691i	$0.933731\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 14.4853i	− 0.873495i
$276$	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
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	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
$295$	17.9411	1.04457
$296$	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
$313$	25.4853	1.44051	0.720257	−	0.693708i	$-0.244024\pi$
$313$	25.4853	1.44051	0.720257	+	0.693708i	$0.244024\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.556349i	0.0312477i	0.999878	+	0.0156238i	$0.00497343\pi$
$317$	0.556349i	0.0312477i	−0.999878	+	0.0156238i	$0.995027\pi$
$318$	0	0
$319$	16.4853	0.922999
$320$	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
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4.41421i	0.239043i
$342$	0	0
$343$	70.6985	3.81736
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.85786i	− 0.368149i	−0.982912	−	0.184075i	$-0.941071\pi$
$347$	− 6.85786i	− 0.368149i	0.982912	−	0.184075i	$-0.0589288\pi$
$348$	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
$361$	19.0000	1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.35534i	0.123284i
$366$	0	0
$367$	23.2426	1.21326	0.606628	−	0.794986i	$-0.292522\pi$
$367$	23.2426	1.21326	0.606628	+	0.794986i	$0.292522\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	52.7990i	2.74119i
$372$	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
$385$	48.4558	2.46954
$386$	0	0
$387$	0	0
$388$	0	0
$389$	36.5563i	1.85348i	0.375703	+	0.926740i	$0.377401\pi$
$389$	36.5563i	1.85348i	−0.375703	+	0.926740i	$0.622599\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	15.8579i	0.797896i
$396$	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
$409$	38.9411	1.92551	0.962757	−	0.270367i	$-0.0871450\pi$
$409$	38.9411	1.92551	0.962757	+	0.270367i	$0.0871450\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 59.3137i	− 2.91864i
$414$	0	0
$415$	−19.3015	−0.947474
$416$	0	0
$417$	0	0
$418$	0	0
$419$	39.5980i	1.93449i	0.253849	+	0.967244i	$0.418303\pi$
$419$	39.5980i	1.93449i	−0.253849	+	0.967244i	$0.581697\pi$
$420$	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
$433$	26.9411	1.29471	0.647354	−	0.762190i	$-0.275876\pi$
$433$	26.9411	1.29471	0.647354	+	0.762190i	$0.275876\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−38.2132	−1.82382	−0.911908	−	0.410394i	$-0.865391\pi$
$439$	−38.2132	−1.82382	−0.911908	+	0.410394i	$0.865391\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 28.2843i	− 1.34383i	−0.740630	−	0.671913i	$-0.765473\pi$
$443$	− 28.2843i	− 1.34383i	0.740630	−	0.671913i	$-0.234527\pi$
$444$	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
$457$	−35.9706	−1.68263	−0.841316	−	0.540544i	$-0.818219\pi$
$457$	−35.9706	−1.68263	−0.841316	+	0.540544i	$0.818219\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 42.8995i	− 1.99803i	−0.0443887	−	0.999014i	$-0.514134\pi$
$461$	− 42.8995i	− 1.99803i	0.0443887	−	0.999014i	$-0.485866\pi$
$462$	0	0
$463$	42.6985	1.98437	0.992183	−	0.124788i	$-0.0398251\pi$
$463$	42.6985	1.98437	0.992183	+	0.124788i	$0.0398251\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4.79899i	− 0.222071i	−0.993816	−	0.111035i	$-0.964583\pi$
$467$	− 4.79899i	− 0.222071i	0.993816	−	0.111035i	$-0.0354167\pi$
$468$	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
$485$	− 28.4975i	− 1.29400i
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.6863i	0.978689i	0.872091	+	0.489344i	$0.162764\pi$
$491$	21.6863i	0.978689i	−0.872091	+	0.489344i	$0.837236\pi$
$492$	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
$505$	10.9411	0.486874
$506$	0	0
$507$	0	0
$508$	0	0
$509$	37.5858i	1.66596i	0.553303	+	0.832980i	$0.313367\pi$
$509$	37.5858i	1.66596i	−0.553303	+	0.832980i	$0.686633\pi$
$510$	0	0
$511$	7.78680	0.344468
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 22.2010i	− 0.978293i
$516$	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
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−32.7574	−1.41622
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 119.397i	− 5.14279i
$540$	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
$553$	52.4264	2.22940
$554$	0	0
$555$	0	0
$556$	0	0
$557$	46.0711i	1.95209i	0.217560	+	0.976047i	$0.430190\pi$
$557$	46.0711i	1.95209i	−0.217560	+	0.976047i	$0.569810\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	47.1421i	1.98680i	0.114684	+	0.993402i	$0.463415\pi$
$563$	47.1421i	1.98680i	−0.114684	+	0.993402i	$0.536585\pi$
$564$	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
$577$	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
$579$	0	0
$580$	0	0
$581$	63.8112i	2.64733i
$582$	0	0
$583$	58.6985	2.43104
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.6274i	1.55305i	0.630087	+	0.776525i	$0.283019\pi$
$587$	37.6274i	1.55305i	−0.630087	+	0.776525i	$0.716981\pi$
$588$	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
$601$	−43.4264	−1.77140	−0.885700	−	0.464258i	$-0.846321\pi$
$601$	−43.4264	−1.77140	−0.885700	+	0.464258i	$0.846321\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 36.4264i	− 1.48094i
$606$	0	0
$607$	22.0000	0.892952	0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000	0.892952	0.446476	+	0.894795i	$0.352679\pi$
$608$	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
$625$	−6.39697	−0.255879
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	29.2426	1.16413	0.582066	−	0.813142i	$-0.302245\pi$
$631$	29.2426	1.16413	0.582066	+	0.813142i	$0.302245\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 10.7157i	− 0.425241i
$636$	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
$649$	−65.9411	−2.58842
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.04163i	0.353826i	0.984226	+	0.176913i	$0.0566112\pi$
$653$	9.04163i	0.353826i	−0.984226	+	0.176913i	$0.943389\pi$
$654$	0	0
$655$	22.6985	0.886903
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.62742i	0.0633952i	0.999498	+	0.0316976i	$0.0100913\pi$
$659$	1.62742i	0.0633952i	−0.999498	+	0.0316976i	$0.989909\pi$
$660$	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
$673$	50.9411	1.96364	0.981818	−	0.189824i	$-0.0607919\pi$
$673$	50.9411	1.96364	0.981818	+	0.189824i	$0.0607919\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.82843i	0.108705i	0.998522	+	0.0543526i	$0.0173095\pi$
$677$	2.82843i	0.108705i	−0.998522	+	0.0543526i	$0.982690\pi$
$678$	0	0
$679$	−94.2132	−3.61557
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.65685i	0.216454i	0.994126	+	0.108227i	$0.0345173\pi$
$683$	5.65685i	0.216454i	−0.994126	+	0.108227i	$0.965483\pi$
$684$	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
$701$	− 51.3848i	− 1.94078i	−0.241551	−	0.970388i	$-0.577656\pi$
$701$	− 51.3848i	− 1.94078i	0.241551	−	0.970388i	$-0.422344\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 36.1716i	− 1.36037i
$708$	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
$721$	−73.3970	−2.73345
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7.02944i	0.261067i
$726$	0	0
$727$	−45.6690	−1.69377	−0.846886	−	0.531775i	$-0.821525\pi$
$727$	−45.6690	−1.69377	−0.846886	+	0.531775i	$0.821525\pi$
$728$	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
$745$	−31.0589	−1.13791
$746$	0	0
$747$	0	0
$748$	0	0
$749$	108.296i	3.95707i
$750$	0	0
$751$	−51.6690	−1.88543	−0.942715	−	0.333599i	$-0.891737\pi$
$751$	−51.6690	−1.88543	−0.942715	+	0.333599i	$0.891737\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 32.0538i	− 1.16656i
$756$	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
$769$	−55.4264	−1.99873	−0.999364	−	0.0356685i	$-0.988644\pi$
$769$	−55.4264	−1.99873	−0.999364	+	0.0356685i	$0.988644\pi$
$770$	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
$775$	−1.88225	−0.0676125
$776$	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
$797$	− 41.8701i	− 1.48311i	−0.670890	−	0.741557i	$-0.734088\pi$
$797$	− 41.8701i	− 1.48311i	0.670890	−	0.741557i	$-0.265912\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 8.65685i	− 0.305494i
$804$	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
$821$	− 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
$823$	6.69848	0.233495	0.116747	−	0.993162i	$-0.462753\pi$
$823$	6.69848	0.233495	0.116747	+	0.993162i	$0.462753\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	56.5685i	1.96708i	0.180688	+	0.983540i	$0.442168\pi$
$827$	56.5685i	1.96708i	−0.180688	+	0.983540i	$0.557832\pi$
$828$	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
$841$	21.0000	0.724138
$842$	0	0
$843$	0	0
$844$	0	0
$845$	20.6152i	0.709185i
$846$	0	0
$847$	−120.426	−4.13790
$848$	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
$865$	39.4853	1.34254
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 58.2843i	− 1.97716i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	62.2304i	2.10377i
$876$	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
$889$	−35.4264	−1.18816
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	24.3310	0.813294
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2.14214i	− 0.0714442i
$900$	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
$913$	70.9411	2.34781
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 75.0416i	− 2.47809i
$918$	0	0
$919$	54.6985	1.80434	0.902168	−	0.431384i	$-0.141975\pi$
$919$	54.6985	1.80434	0.902168	+	0.431384i	$0.141975\pi$
$920$	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
$937$	12.0294	0.392985	0.196492	−	0.980505i	$-0.437045\pi$
$937$	12.0294	0.392985	0.196492	+	0.980505i	$0.437045\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 8.95837i	− 0.292034i	−0.989282	−	0.146017i	$-0.953354\pi$
$941$	− 8.95837i	− 0.292034i	0.989282	−	0.146017i	$-0.0466455\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 49.2843i	− 1.60152i	−0.598983	−	0.800762i	$-0.704428\pi$
$947$	− 49.2843i	− 1.60152i	0.598983	−	0.800762i	$-0.295572\pi$
$948$	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
$961$	−30.4264	−0.981497
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 7.15938i	− 0.230469i
$966$	0	0
$967$	35.2426	1.13333	0.566663	−	0.823949i	$-0.308234\pi$
$967$	35.2426	1.13333	0.566663	+	0.823949i	$0.308234\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	28.1127i	0.902179i	0.892479	+	0.451090i	$0.148965\pi$
$971$	28.1127i	0.902179i	−0.892479	+	0.451090i	$0.851035\pi$
$972$	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
$985$	−44.5147	−1.41836
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−50.2132	−1.59507	−0.797537	−	0.603269i	$-0.793864\pi$
$991$	−50.2132	−1.59507	−0.797537	+	0.603269i	$0.793864\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 22.5391i	− 0.714538i
$996$	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	864.2.d.b.433.3		4
3.2	odd	2	inner	864.2.d.b.433.2		4
4.3	odd	2		216.2.d.a.109.4	yes	4
8.3	odd	2		216.2.d.a.109.1	✓	4
8.5	even	2	inner	864.2.d.b.433.2		4
9.2	odd	6		2592.2.r.o.433.3		8
9.4	even	3		2592.2.r.o.2161.3		8
9.5	odd	6		2592.2.r.o.2161.2		8
9.7	even	3		2592.2.r.o.433.2		8
12.11	even	2		216.2.d.a.109.1	✓	4
16.3	odd	4		6912.2.a.bz.1.1		2
16.5	even	4		6912.2.a.y.1.2		2
16.11	odd	4		6912.2.a.z.1.2		2
16.13	even	4		6912.2.a.by.1.1		2
24.5	odd	2	CM	864.2.d.b.433.3		4
24.11	even	2		216.2.d.a.109.4	yes	4
36.7	odd	6		648.2.n.p.109.3		8
36.11	even	6		648.2.n.p.109.2		8
36.23	even	6		648.2.n.p.541.3		8
36.31	odd	6		648.2.n.p.541.2		8
48.5	odd	4		6912.2.a.by.1.1		2
48.11	even	4		6912.2.a.bz.1.1		2
48.29	odd	4		6912.2.a.y.1.2		2
48.35	even	4		6912.2.a.z.1.2		2
72.5	odd	6		2592.2.r.o.2161.3		8
72.11	even	6		648.2.n.p.109.3		8
72.13	even	6		2592.2.r.o.2161.2		8
72.29	odd	6		2592.2.r.o.433.2		8
72.43	odd	6		648.2.n.p.109.2		8
72.59	even	6		648.2.n.p.541.2		8
72.61	even	6		2592.2.r.o.433.3		8
72.67	odd	6		648.2.n.p.541.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.d.a.109.1	✓	4	8.3	odd	2
216.2.d.a.109.1	✓	4	12.11	even	2
216.2.d.a.109.4	yes	4	4.3	odd	2
216.2.d.a.109.4	yes	4	24.11	even	2
648.2.n.p.109.2		8	36.11	even	6
648.2.n.p.109.2		8	72.43	odd	6
648.2.n.p.109.3		8	36.7	odd	6
648.2.n.p.109.3		8	72.11	even	6
648.2.n.p.541.2		8	36.31	odd	6
648.2.n.p.541.2		8	72.59	even	6
648.2.n.p.541.3		8	36.23	even	6
648.2.n.p.541.3		8	72.67	odd	6
864.2.d.b.433.2		4	3.2	odd	2	inner
864.2.d.b.433.2		4	8.5	even	2	inner
864.2.d.b.433.3		4	1.1	even	1	trivial
864.2.d.b.433.3		4	24.5	odd	2	CM
2592.2.r.o.433.2		8	9.7	even	3
2592.2.r.o.433.2		8	72.29	odd	6
2592.2.r.o.433.3		8	9.2	odd	6
2592.2.r.o.433.3		8	72.61	even	6
2592.2.r.o.2161.2		8	9.5	odd	6
2592.2.r.o.2161.2		8	72.13	even	6
2592.2.r.o.2161.3		8	9.4	even	3
2592.2.r.o.2161.3		8	72.5	odd	6
6912.2.a.y.1.2		2	16.5	even	4
6912.2.a.y.1.2		2	48.29	odd	4
6912.2.a.z.1.2		2	16.11	odd	4
6912.2.a.z.1.2		2	48.35	even	4
6912.2.a.by.1.1		2	16.13	even	4
6912.2.a.by.1.1		2	48.5	odd	4
6912.2.a.bz.1.1		2	16.3	odd	4
6912.2.a.bz.1.1		2	48.11	even	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.58579i q^{5} +5.24264 q^{7} +O(q^{10})\)	\(q+1.58579i q^{5} +5.24264 q^{7} -5.82843i q^{11} +2.48528 q^{25} +2.82843i q^{29} -0.757359 q^{31} +8.31371i q^{35} +20.4853 q^{49} +10.0711i q^{53} +9.24264 q^{55} -11.3137i q^{59} +1.48528 q^{73} -30.5563i q^{77} +10.0000 q^{79} +12.1716i q^{83} -17.9706 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^7	\( 4 q + 4 q^{7} - 24 q^{25} - 20 q^{31} + 48 q^{49} + 20 q^{55} - 28 q^{73} + 40 q^{79} - 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^7 - 24 * q^25 - 20 * q^31 + 48 * q^49 + 20 * q^55 - 28 * q^73 + 40 * q^79 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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