LMFDB - Embedded newform 5184.2.d.l.2593.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5184,2,Mod(2593,5184)] 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("5184.2593"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5184, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,20,0,0,0,0, 0,0,0,0,0,0,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(41)] == traces)

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

Self dual:	no
Analytic conductor:	$41.3944484078$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			2593.3
Root			$-1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	5184.2593
Dual form			5184.2.d.l.2593.2

$q$-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 $\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$-1$	$1$	$1$

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
$5$	0	0	1.00000			$0$
$5$	0	0	−1.00000			$\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.550510i	0.165985i	0.996550	+	0.0829925i	$0.0264478\pi$
$11$	0.550510i	0.165985i	−0.996550	+	0.0829925i	$0.973552\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	7.89898	1.91578	0.957892	−	0.287129i	$-0.0927008\pi$
$17$	7.89898	1.91578	0.957892	+	0.287129i	$0.0927008\pi$
$18$	0	0
$19$	6.34847i	1.45644i	0.685344	+	0.728219i	$0.259652\pi$
$19$	6.34847i	1.45644i	−0.685344	+	0.728219i	$0.740348\pi$
$20$	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
$25$	5.00000	1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.79796	−1.06166	−0.530831	−	0.847477i	$-0.678120\pi$
$41$	−6.79796	−1.06166	−0.530831	+	0.847477i	$0.678120\pi$
$42$	0	0
$43$	− 12.3485i	− 1.88312i	−0.336840	−	0.941562i	$-0.609358\pi$
$43$	− 12.3485i	− 1.88312i	0.336840	−	0.941562i	$-0.390642\pi$
$44$	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
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	15.2474i	1.98505i	0.122047	+	0.992524i	$0.461054\pi$
$59$	15.2474i	1.98505i	−0.122047	+	0.992524i	$0.538946\pi$
$60$	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
$67$	− 0.348469i	− 0.0425723i	−0.999773	−	0.0212861i	$-0.993224\pi$
$67$	− 0.348469i	− 0.0425723i	0.999773	−	0.0212861i	$-0.00677610\pi$
$68$	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
$73$	−15.6969	−1.83719	−0.918594	−	0.395203i	$-0.870674\pi$
$73$	−15.6969	−1.83719	−0.918594	+	0.395203i	$0.870674\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	18.0000i	1.97576i	0.155230	+	0.987878i	$0.450388\pi$
$83$	18.0000i	1.97576i	−0.155230	+	0.987878i	$0.549612\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	9.69694	0.984575	0.492287	−	0.870433i	$-0.336161\pi$
$97$	9.69694	0.984575	0.492287	+	0.870433i	$0.336161\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 14.1464i	− 1.36759i	−0.729676	−	0.683793i	$-0.760329\pi$
$107$	− 14.1464i	− 1.36759i	0.729676	−	0.683793i	$-0.239671\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	10.6969	0.972449
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.0000i	1.57267i	0.617802	+	0.786334i	$0.288023\pi$
$131$	18.0000i	1.57267i	−0.617802	+	0.786334i	$0.711977\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.5959	1.93050	0.965250	−	0.261329i	$-0.0841608\pi$
$137$	22.5959	1.93050	0.965250	+	0.261329i	$0.0841608\pi$
$138$	0	0
$139$	18.3485i	1.55630i	0.628080	+	0.778148i	$0.283841\pi$
$139$	18.3485i	1.55630i	−0.628080	+	0.778148i	$0.716159\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0		−	1.00000i	$-0.5\pi$
$151$	0	0			1.00000i	$0.5\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	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
$163$	2.00000i	0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	2.00000i	0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$	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
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	18.0000i	1.34538i	0.739923	+	0.672692i	$0.234862\pi$
$179$	18.0000i	1.34538i	−0.739923	+	0.672692i	$0.765138\pi$
$180$	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
$187$	4.34847i	0.317992i
$188$	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
$193$	−3.69694	−0.266111	−0.133056	−	0.991109i	$-0.542479\pi$
$193$	−3.69694	−0.266111	−0.133056	+	0.991109i	$0.542479\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.49490	−0.241747
$210$	0	0
$211$	− 14.0000i	− 0.963800i	−0.876226	−	0.481900i	$-0.839947\pi$
$211$	− 14.0000i	− 0.963800i	0.876226	−	0.481900i	$-0.160053\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	17.4495i	1.15816i	0.815270	+	0.579082i	$0.196589\pi$
$227$	17.4495i	1.15816i	−0.815270	+	0.579082i	$0.803411\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.1010	0.661740	0.330870	−	0.943676i	$-0.392658\pi$
$233$	10.1010	0.661740	0.330870	+	0.943676i	$0.392658\pi$
$234$	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
$241$	−27.6969	−1.78412	−0.892058	−	0.451920i	$-0.850739\pi$
$241$	−27.6969	−1.78412	−0.892058	+	0.451920i	$0.850739\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	29.9444i	1.89007i	0.326965	+	0.945036i	$0.393974\pi$
$251$	29.9444i	1.89007i	−0.326965	+	0.945036i	$0.606026\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.7980	1.54685	0.773427	−	0.633885i	$-0.218541\pi$
$257$	24.7980	1.54685	0.773427	+	0.633885i	$0.218541\pi$
$258$	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
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.75255i	0.165985i
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.0000	1.07379	0.536895	−	0.843649i	$-0.319597\pi$
$281$	18.0000	1.07379	0.536895	+	0.843649i	$0.319597\pi$
$282$	0	0
$283$	− 22.0000i	− 1.30776i	−0.756596	−	0.653882i	$-0.773139\pi$
$283$	− 22.0000i	− 1.30776i	0.756596	−	0.653882i	$-0.226861\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	45.3939	2.67023
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	0	0
$296$	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
$307$	− 24.3485i	− 1.38964i	−0.719183	−	0.694820i	$-0.755484\pi$
$307$	− 24.3485i	− 1.38964i	0.719183	−	0.694820i	$-0.244516\pi$
$308$	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
$313$	−34.3939	−1.94406	−0.972028	−	0.234863i	$-0.924536\pi$
$313$	−34.3939	−1.94406	−0.972028	+	0.234863i	$0.924536\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	50.1464i	2.79022i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	26.0000i	1.42909i	0.699590	+	0.714545i	$0.253366\pi$
$331$	26.0000i	1.42909i	−0.699590	+	0.714545i	$0.746634\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−22.3939	−1.21987	−0.609936	−	0.792451i	$-0.708805\pi$
$337$	−22.3939	−1.21987	−0.609936	+	0.792451i	$0.708805\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 28.8434i	− 1.54839i	−0.632945	−	0.774197i	$-0.718154\pi$
$347$	− 28.8434i	− 1.54839i	0.632945	−	0.774197i	$-0.281846\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−4.59592	−0.244616	−0.122308	−	0.992492i	$-0.539030\pi$
$353$	−4.59592	−0.244616	−0.122308	+	0.992492i	$0.539030\pi$
$354$	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
$361$	−21.3031	−1.12121
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	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
$379$	11.6515i	0.598499i	0.954175	+	0.299249i	$0.0967363\pi$
$379$	11.6515i	0.598499i	−0.954175	+	0.299249i	$0.903264\pi$
$380$	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
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	37.2929	1.86232	0.931158	−	0.364615i	$-0.118800\pi$
$401$	37.2929	1.86232	0.931158	+	0.364615i	$0.118800\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	40.3939	1.99735	0.998674	−	0.0514740i	$-0.0163919\pi$
$409$	40.3939	1.99735	0.998674	+	0.0514740i	$0.0163919\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	18.0000i	0.879358i	0.898155	+	0.439679i	$0.144908\pi$
$419$	18.0000i	0.879358i	−0.898155	+	0.439679i	$0.855092\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	39.4949	1.91578
$426$	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
$433$	33.6969	1.61937	0.809686	−	0.586864i	$-0.199638\pi$
$433$	33.6969	1.61937	0.809686	+	0.586864i	$0.199638\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 18.5505i	− 0.881361i	−0.897664	−	0.440681i	$-0.854737\pi$
$443$	− 18.5505i	− 0.881361i	0.897664	−	0.440681i	$-0.145263\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−25.8990	−1.22225	−0.611124	−	0.791535i	$-0.709282\pi$
$449$	−25.8990	−1.22225	−0.611124	+	0.791535i	$0.709282\pi$
$450$	0	0
$451$	− 3.74235i	− 0.176220i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	16.3939	0.766873	0.383437	−	0.923567i	$-0.374740\pi$
$457$	16.3939	0.766873	0.383437	+	0.923567i	$0.374740\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0		−	1.00000i	$-0.5\pi$
$463$	0	0			1.00000i	$0.5\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 11.9444i	− 0.552720i	−0.961054	−	0.276360i	$-0.910872\pi$
$467$	− 11.9444i	− 0.552720i	0.961054	−	0.276360i	$-0.0891283\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.79796	0.312570
$474$	0	0
$475$	31.7423i	1.45644i
$476$	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
$485$	0	0
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 33.2474i	− 1.50044i	−0.661190	−	0.750218i	$-0.729948\pi$
$491$	− 33.2474i	− 1.50044i	0.661190	−	0.750218i	$-0.270052\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	43.7423i	1.95818i	0.203436	+	0.979088i	$0.434789\pi$
$499$	43.7423i	1.95818i	−0.203436	+	0.979088i	$0.565211\pi$
$500$	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
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−36.1918	−1.58559	−0.792797	−	0.609486i	$-0.791376\pi$
$521$	−36.1918	−1.58559	−0.792797	+	0.609486i	$0.791376\pi$
$522$	0	0
$523$	− 38.0000i	− 1.66162i	−0.556553	−	0.830812i	$-0.687876\pi$
$523$	− 38.0000i	− 1.66162i	0.556553	−	0.830812i	$-0.312124\pi$
$524$	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
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 3.85357i	− 0.165985i
$540$	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
$547$	30.3485i	1.29761i	0.760956	+	0.648803i	$0.224730\pi$
$547$	30.3485i	1.29761i	−0.760956	+	0.648803i	$0.775270\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	46.8434i	1.97421i	0.160066	+	0.987106i	$0.448829\pi$
$563$	46.8434i	1.97421i	−0.160066	+	0.987106i	$0.551171\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−40.5959	−1.70187	−0.850935	−	0.525271i	$-0.823964\pi$
$569$	−40.5959	−1.70187	−0.850935	+	0.525271i	$0.823964\pi$
$570$	0	0
$571$	− 25.7423i	− 1.07728i	−0.842535	−	0.538642i	$-0.818938\pi$
$571$	− 25.7423i	− 1.07728i	0.842535	−	0.538642i	$-0.181062\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−46.3939	−1.93140	−0.965701	−	0.259656i	$-0.916391\pi$
$577$	−46.3939	−1.93140	−0.965701	+	0.259656i	$0.916391\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44.6413i	1.84254i	0.388918	+	0.921272i	$0.372849\pi$
$587$	44.6413i	1.84254i	−0.388918	+	0.921272i	$0.627151\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	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
$601$	8.30306	0.338689	0.169344	−	0.985557i	$-0.445835\pi$
$601$	8.30306	0.338689	0.169344	+	0.985557i	$0.445835\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	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
$617$	−19.2929	−0.776701	−0.388351	−	0.921512i	$-0.626955\pi$
$617$	−19.2929	−0.776701	−0.388351	+	0.921512i	$0.626955\pi$
$618$	0	0
$619$	− 49.7423i	− 1.99931i	−0.0261952	−	0.999657i	$-0.508339\pi$
$619$	− 49.7423i	− 1.99931i	0.0261952	−	0.999657i	$-0.491661\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	25.0000	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3.49490	0.138040	0.0690201	−	0.997615i	$-0.478013\pi$
$641$	3.49490	0.138040	0.0690201	+	0.997615i	$0.478013\pi$
$642$	0	0
$643$	− 17.6515i	− 0.696108i	−0.937474	−	0.348054i	$-0.886843\pi$
$643$	− 17.6515i	− 0.696108i	0.937474	−	0.348054i	$-0.113157\pi$
$644$	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
$649$	−8.39388	−0.329488
$650$	0	0
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	18.0000i	0.701180i	0.936529	+	0.350590i	$0.114019\pi$
$659$	18.0000i	0.701180i	−0.936529	+	0.350590i	$0.885981\pi$
$660$	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
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 47.9444i	− 1.83454i	−0.398265	−	0.917270i	$-0.630387\pi$
$683$	− 47.9444i	− 1.83454i	0.398265	−	0.917270i	$-0.369613\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 46.0000i	− 1.74992i	−0.484193	−	0.874961i	$-0.660887\pi$
$691$	− 46.0000i	− 1.74992i	0.484193	−	0.874961i	$-0.339113\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−53.6969	−2.03392
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	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
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	0	0		−	1.00000i	$-0.5\pi$
$727$	0	0			1.00000i	$0.5\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 97.5403i	− 3.60766i
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.191836	0.00706637
$738$	0	0
$739$	19.7423i	0.726234i	0.931744	+	0.363117i	$0.118287\pi$
$739$	19.7423i	0.726234i	−0.931744	+	0.363117i	$0.881713\pi$
$740$	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
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0		−	1.00000i	$-0.5\pi$
$751$	0	0			1.00000i	$0.5\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 43.1566i	− 1.54625i
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	50.0000i	1.78231i	0.453701	+	0.891154i	$0.350103\pi$
$787$	50.0000i	1.78231i	−0.453701	+	0.891154i	$0.649897\pi$
$788$	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
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 8.64133i	− 0.304946i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	51.9898	1.82786	0.913932	−	0.405868i	$-0.133031\pi$
$809$	51.9898	1.82786	0.913932	+	0.405868i	$0.133031\pi$
$810$	0	0
$811$	55.7423i	1.95738i	0.205347	+	0.978689i	$0.434168\pi$
$811$	55.7423i	1.95738i	−0.205347	+	0.978689i	$0.565832\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	78.3939	2.74265
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 54.0000i	− 1.87776i	−0.344239	−	0.938882i	$-0.611863\pi$
$827$	− 54.0000i	− 1.87776i	0.344239	−	0.938882i	$-0.388137\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−55.2929	−1.91578
$834$	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
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	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
$857$	−54.0000	−1.84460	−0.922302	−	0.386469i	$-0.873695\pi$
$857$	−54.0000	−1.84460	−0.922302	+	0.386469i	$0.873695\pi$
$858$	0	0
$859$	− 36.3485i	− 1.24019i	−0.784525	−	0.620097i	$-0.787093\pi$
$859$	− 36.3485i	− 1.24019i	0.784525	−	0.620097i	$-0.212907\pi$
$860$	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
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	18.0000	0.606435	0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000	0.606435	0.303218	+	0.952921i	$0.401939\pi$
$882$	0	0
$883$	50.4393i	1.69742i	0.528861	+	0.848709i	$0.322619\pi$
$883$	50.4393i	1.69742i	−0.528861	+	0.848709i	$0.677381\pi$
$884$	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
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 56.4393i	− 1.87404i	−0.349281	−	0.937018i	$-0.613574\pi$
$907$	− 56.4393i	− 1.87404i	0.349281	−	0.937018i	$-0.386426\pi$
$908$	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
$913$	−9.90918	−0.327946
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	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
$929$	−54.0000	−1.77168	−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000	−1.77168	−0.885841	+	0.463988i	$0.846418\pi$
$930$	0	0
$931$	− 44.4393i	− 1.45644i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	34.0000	1.11073	0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000	1.11073	0.555366	+	0.831606i	$0.312578\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	61.5403i	1.99979i	0.0144491	+	0.999896i	$0.495401\pi$
$947$	61.5403i	1.99979i	−0.0144491	+	0.999896i	$0.504599\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	18.1918	0.589291	0.294646	−	0.955607i	$-0.404798\pi$
$953$	18.1918	0.589291	0.294646	+	0.955607i	$0.404798\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0		−	1.00000i	$-0.5\pi$
$967$	0	0			1.00000i	$0.5\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 54.0000i	− 1.73294i	−0.499227	−	0.866471i	$-0.666383\pi$
$971$	− 54.0000i	− 1.73294i	0.499227	−	0.866471i	$-0.333617\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.8888	−1.62808	−0.814038	−	0.580812i	$-0.802735\pi$
$977$	−50.8888	−1.62808	−0.814038	+	0.580812i	$0.802735\pi$
$978$	0	0
$979$	9.90918i	0.316699i
$980$	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
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	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	5184.2.d.l.2593.3		4
3.2	odd	2		5184.2.d.e.2593.2		4
4.3	odd	2	inner	5184.2.d.l.2593.2		4
8.3	odd	2	CM	5184.2.d.l.2593.3		4
8.5	even	2	inner	5184.2.d.l.2593.2		4
9.2	odd	6		1728.2.r.c.1441.2		8
9.4	even	3		576.2.r.c.97.1	✓	8
9.5	odd	6		1728.2.r.c.289.3		8
9.7	even	3		576.2.r.c.481.4	yes	8
12.11	even	2		5184.2.d.e.2593.3		4
24.5	odd	2		5184.2.d.e.2593.3		4
24.11	even	2		5184.2.d.e.2593.2		4
36.7	odd	6		576.2.r.c.481.1	yes	8
36.11	even	6		1728.2.r.c.1441.3		8
36.23	even	6		1728.2.r.c.289.2		8
36.31	odd	6		576.2.r.c.97.4	yes	8
72.5	odd	6		1728.2.r.c.289.2		8
72.11	even	6		1728.2.r.c.1441.2		8
72.13	even	6		576.2.r.c.97.4	yes	8
72.29	odd	6		1728.2.r.c.1441.3		8
72.43	odd	6		576.2.r.c.481.4	yes	8
72.59	even	6		1728.2.r.c.289.3		8
72.61	even	6		576.2.r.c.481.1	yes	8
72.67	odd	6		576.2.r.c.97.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.r.c.97.1	✓	8	9.4	even	3
576.2.r.c.97.1	✓	8	72.67	odd	6
576.2.r.c.97.4	yes	8	36.31	odd	6
576.2.r.c.97.4	yes	8	72.13	even	6
576.2.r.c.481.1	yes	8	36.7	odd	6
576.2.r.c.481.1	yes	8	72.61	even	6
576.2.r.c.481.4	yes	8	9.7	even	3
576.2.r.c.481.4	yes	8	72.43	odd	6
1728.2.r.c.289.2		8	36.23	even	6
1728.2.r.c.289.2		8	72.5	odd	6
1728.2.r.c.289.3		8	9.5	odd	6
1728.2.r.c.289.3		8	72.59	even	6
1728.2.r.c.1441.2		8	9.2	odd	6
1728.2.r.c.1441.2		8	72.11	even	6
1728.2.r.c.1441.3		8	36.11	even	6
1728.2.r.c.1441.3		8	72.29	odd	6
5184.2.d.e.2593.2		4	3.2	odd	2
5184.2.d.e.2593.2		4	24.11	even	2
5184.2.d.e.2593.3		4	12.11	even	2
5184.2.d.e.2593.3		4	24.5	odd	2
5184.2.d.l.2593.2		4	4.3	odd	2	inner
5184.2.d.l.2593.2		4	8.5	even	2	inner
5184.2.d.l.2593.3		4	1.1	even	1	trivial
5184.2.d.l.2593.3		4	8.3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.550510i q^{11} +7.89898 q^{17} +6.34847i q^{19} +5.00000 q^{25} -6.79796 q^{41} -12.3485i q^{43} -7.00000 q^{49} +15.2474i q^{59} -0.348469i q^{67} -15.6969 q^{73} +18.0000i q^{83} +18.0000 q^{89} +9.69694 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{17} + 20 q^{25} + 12 q^{41} - 28 q^{49} - 4 q^{73} + 72 q^{89} - 20 q^{97}+O(q^{100}) \) 4 * q + 12 * q^17 + 20 * q^25 + 12 * q^41 - 28 * q^49 - 4 * q^73 + 72 * q^89 - 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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