LMFDB - Embedded newform 864.5.h.b.593.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [864,5,Mod(593,864)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 5, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("864.593"); S:= CuspForms(chi, 5); N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	864.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,46] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$89.3116481044$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			593.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	864.593

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+39.9706 q^{5} +85.8528 q^{7} -240.706 q^{11} +972.646 q^{25} +818.000 q^{29} +1373.20 q^{31} +3431.59 q^{35} +4969.71 q^{49} +2379.08 q^{53} -9621.14 q^{55} +6862.00 q^{59} -1860.70 q^{73} -20665.3 q^{77} +9118.00 q^{79} -9281.28 q^{83} -15089.8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 46 q^{5} + 2 q^{7} - 142 q^{11} + 384 q^{25} + 1636 q^{29} - 478 q^{31} + 2926 q^{35} + 9600 q^{49} - 3218 q^{53} - 9026 q^{55} + 13724 q^{59} + 8158 q^{73} - 28942 q^{77} + 18236 q^{79} + 4178 q^{83}+ \cdots - 17282 q^{97}+O(q^{100}) $ 2 * q + 46 * q^5 + 2 * q^7 - 142 * q^11 + 384 * q^25 + 1636 * q^29 - 478 * q^31 + 2926 * q^35 + 9600 * q^49 - 3218 * q^53 - 9026 * q^55 + 13724 * q^59 + 8158 * q^73 - 28942 * q^77 + 18236 * q^79 + 4178 * q^83 - 17282 * q^97

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
$5$	39.9706	1.59882	0.799411	−	0.600784i	$-0.205145\pi$
$5$	39.9706	1.59882	0.799411	+	0.600784i	$0.205145\pi$
$6$	0	0
$7$	85.8528	1.75210	0.876049	−	0.482222i	$-0.160170\pi$
$7$	85.8528	1.75210	0.876049	+	0.482222i	$0.160170\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−240.706	−1.98930	−0.994651	−	0.103289i	$-0.967063\pi$
$11$	−240.706	−1.98930	−0.994651	+	0.103289i	$0.967063\pi$
$12$	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	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	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	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	972.646	1.55623
$26$	0	0
$27$	0	0
$28$	0	0
$29$	818.000	0.972652	0.486326	−	0.873778i	$-0.338337\pi$
$29$	818.000	0.972652	0.486326	+	0.873778i	$0.338337\pi$
$30$	0	0
$31$	1373.20	1.42893	0.714466	−	0.699670i	$-0.246670\pi$
$31$	1373.20	1.42893	0.714466	+	0.699670i	$0.246670\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3431.59	2.80129
$36$	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	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	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	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	4969.71	2.06985
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2379.08	0.846950	0.423475	−	0.905908i	$-0.360810\pi$
$53$	2379.08	0.846950	0.423475	+	0.905908i	$0.360810\pi$
$54$	0	0
$55$	−9621.14	−3.18054
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6862.00	1.97127	0.985636	−	0.168882i	$-0.0540156\pi$
$59$	6862.00	1.97127	0.985636	+	0.168882i	$0.0540156\pi$
$60$	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	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−1860.70	−0.349164	−0.174582	−	0.984643i	$-0.555857\pi$
$73$	−1860.70	−0.349164	−0.174582	+	0.984643i	$0.555857\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−20665.3	−3.48545
$78$	0	0
$79$	9118.00	1.46098	0.730492	−	0.682921i	$-0.239291\pi$
$79$	9118.00	1.46098	0.730492	+	0.682921i	$0.239291\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9281.28	−1.34726	−0.673630	−	0.739069i	$-0.735266\pi$
$83$	−9281.28	−1.34726	−0.673630	+	0.739069i	$0.735266\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15089.8	−1.60376	−0.801882	−	0.597483i	$-0.796168\pi$
$97$	−15089.8	−1.60376	−0.801882	+	0.597483i	$0.796168\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−19134.4	−1.87574	−0.937870	−	0.346987i	$-0.887205\pi$
$101$	−19134.4	−1.87574	−0.937870	+	0.346987i	$0.887205\pi$
$102$	0	0
$103$	21118.0	1.99057	0.995287	−	0.0969729i	$-0.0309160\pi$
$103$	21118.0	1.99057	0.995287	+	0.0969729i	$0.0309160\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6843.68	0.597754	0.298877	−	0.954292i	$-0.403388\pi$
$107$	6843.68	0.597754	0.298877	+	0.954292i	$0.403388\pi$
$108$	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	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	43298.2	2.95733
$122$	0	0
$123$	0	0
$124$	0	0
$125$	13895.6	0.889319
$126$	0	0
$127$	31788.8	1.97091	0.985454	−	0.169945i	$-0.0543589\pi$
$127$	31788.8	1.97091	0.985454	+	0.169945i	$0.0543589\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2881.56	−0.167913	−0.0839567	−	0.996469i	$-0.526756\pi$
$131$	−2881.56	−0.167913	−0.0839567	+	0.996469i	$0.526756\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	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
$145$	32695.9	1.55510
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2623.88	−0.118187	−0.0590937	−	0.998252i	$-0.518821\pi$
$149$	−2623.88	−0.118187	−0.0590937	+	0.998252i	$0.518821\pi$
$150$	0	0
$151$	8957.93	0.392874	0.196437	−	0.980516i	$-0.437063\pi$
$151$	8957.93	0.392874	0.196437	+	0.980516i	$0.437063\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	54887.7	2.28461
$156$	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	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	28561.0	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	42532.3	1.42111	0.710553	−	0.703643i	$-0.248445\pi$
$173$	42532.3	1.42111	0.710553	+	0.703643i	$0.248445\pi$
$174$	0	0
$175$	83504.4	2.72667
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−60236.2	−1.87997	−0.939986	−	0.341212i	$-0.889163\pi$
$179$	−60236.2	−1.87997	−0.939986	+	0.341212i	$0.889163\pi$
$180$	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
$193$	−68780.0	−1.84649	−0.923247	−	0.384208i	$-0.874475\pi$
$193$	−68780.0	−1.84649	−0.923247	+	0.384208i	$0.874475\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−37624.1	−0.969467	−0.484734	−	0.874662i	$-0.661083\pi$
$197$	−37624.1	−0.969467	−0.484734	+	0.874662i	$0.661083\pi$
$198$	0	0
$199$	−79189.9	−1.99970	−0.999848	−	0.0174508i	$-0.994445\pi$
$199$	−79189.9	−1.99970	−0.999848	+	0.0174508i	$0.994445\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	70227.6	1.70418
$204$	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
$217$	117893.	2.50363
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	46558.0	0.936234	0.468117	−	0.883666i	$-0.344933\pi$
$223$	46558.0	0.936234	0.468117	+	0.883666i	$0.344933\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16658.0	−0.323274	−0.161637	−	0.986850i	$-0.551677\pi$
$227$	−16658.0	−0.323274	−0.161637	+	0.986850i	$0.551677\pi$
$228$	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	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	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	$-0.417526\pi$
$241$	29762.0	0.512422	0.256211	+	0.966621i	$0.417526\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	198642.	3.30932
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−94898.0	−1.50629	−0.753147	−	0.657853i	$-0.771465\pi$
$251$	−94898.0	−1.50629	−0.753147	+	0.657853i	$0.771465\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	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
$265$	95093.3	1.35412
$266$	0	0
$267$	0	0
$268$	0	0
$269$	40178.0	0.555244	0.277622	−	0.960690i	$-0.410454\pi$
$269$	40178.0	0.555244	0.277622	+	0.960690i	$0.410454\pi$
$270$	0	0
$271$	−44691.0	−0.608529	−0.304264	−	0.952588i	$-0.598411\pi$
$271$	−44691.0	−0.608529	−0.304264	+	0.952588i	$0.598411\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−234121.	−3.09582
$276$	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	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	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
$289$	83521.0	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−22702.0	−0.264441	−0.132221	−	0.991220i	$-0.542211\pi$
$293$	−22702.0	−0.264441	−0.132221	+	0.991220i	$0.542211\pi$
$294$	0	0
$295$	274278.	3.15172
$296$	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	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	110424.	1.12713	0.563565	−	0.826072i	$-0.309429\pi$
$313$	110424.	1.12713	0.563565	+	0.826072i	$0.309429\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−111164.	−1.10623	−0.553113	−	0.833106i	$-0.686560\pi$
$317$	−111164.	−1.10623	−0.553113	+	0.833106i	$0.686560\pi$
$318$	0	0
$319$	−196897.	−1.93490
$320$	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
$337$	−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
$341$	−330538.	−2.84258
$342$	0	0
$343$	220531.	1.87448
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−51830.2	−0.430451	−0.215226	−	0.976564i	$-0.569049\pi$
$347$	−51830.2	−0.430451	−0.215226	+	0.976564i	$0.569049\pi$
$348$	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	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	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
$365$	−74373.1	−0.558252
$366$	0	0
$367$	2894.55	0.0214907	0.0107453	−	0.999942i	$-0.496580\pi$
$367$	2894.55	0.0214907	0.0107453	+	0.999942i	$0.496580\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	204251.	1.48394
$372$	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	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	−826002.	−5.57262
$386$	0	0
$387$	0	0
$388$	0	0
$389$	257411.	1.70109	0.850545	−	0.525902i	$-0.176272\pi$
$389$	257411.	1.70109	0.850545	+	0.525902i	$0.176272\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	364452.	2.33585
$396$	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	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−334178.	−1.99771	−0.998853	−	0.0478805i	$-0.984753\pi$
$409$	−334178.	−1.99771	−0.998853	+	0.0478805i	$0.984753\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	589122.	3.45386
$414$	0	0
$415$	−370978.	−2.15403
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−181778.	−1.03541	−0.517706	−	0.855559i	$-0.673214\pi$
$419$	−181778.	−1.03541	−0.517706	+	0.855559i	$0.673214\pi$
$420$	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
$433$	281407.	1.50092	0.750462	−	0.660914i	$-0.229831\pi$
$433$	281407.	1.50092	0.750462	+	0.660914i	$0.229831\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−354586.	−1.83989	−0.919947	−	0.392043i	$-0.871769\pi$
$439$	−354586.	−1.83989	−0.919947	+	0.392043i	$0.871769\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	385102.	1.96231	0.981157	−	0.193214i	$-0.0618912\pi$
$443$	385102.	1.96231	0.981157	+	0.193214i	$0.0618912\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	410186.	1.96403	0.982016	−	0.188798i	$-0.0604590\pi$
$457$	410186.	1.96403	0.982016	+	0.188798i	$0.0604590\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−274213.	−1.29029	−0.645144	−	0.764061i	$-0.723203\pi$
$461$	−274213.	−1.29029	−0.645144	+	0.764061i	$0.723203\pi$
$462$	0	0
$463$	9921.15	0.0462807	0.0231404	−	0.999732i	$-0.492634\pi$
$463$	9921.15	0.0462807	0.0231404	+	0.999732i	$0.492634\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	359466.	1.64825	0.824126	−	0.566406i	$-0.191667\pi$
$467$	359466.	1.64825	0.824126	+	0.566406i	$0.191667\pi$
$468$	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
$485$	−603148.	−2.56413
$486$	0	0
$487$	−466562.	−1.96721	−0.983607	−	0.180327i	$-0.942284\pi$
$487$	−466562.	−1.96721	−0.983607	+	0.180327i	$0.942284\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−481582.	−1.99760	−0.998798	−	0.0490245i	$-0.984389\pi$
$491$	−481582.	−1.99760	−0.998798	+	0.0490245i	$0.984389\pi$
$492$	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	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−764814.	−2.99898
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−139557.	−0.538663	−0.269331	−	0.963048i	$-0.586803\pi$
$509$	−139557.	−0.538663	−0.269331	+	0.963048i	$0.586803\pi$
$510$	0	0
$511$	−159746.	−0.611770
$512$	0	0
$513$	0	0
$514$	0	0
$515$	844098.	3.18257
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	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
$529$	279841.	1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	273546.	0.955702
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.19624e6	−4.11755
$540$	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
$553$	782806.	2.55979
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−611560.	−1.97119	−0.985595	−	0.169124i	$-0.945906\pi$
$557$	−611560.	−1.97119	−0.985595	+	0.169124i	$0.945906\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	40451.2	0.127619	0.0638094	−	0.997962i	$-0.479675\pi$
$563$	40451.2	0.127619	0.0638094	+	0.997962i	$0.479675\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	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
$577$	−581758.	−1.74739	−0.873697	−	0.486471i	$-0.838284\pi$
$577$	−581758.	−1.74739	−0.873697	+	0.486471i	$0.838284\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−796824.	−2.36053
$582$	0	0
$583$	−572658.	−1.68484
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−555922.	−1.61338	−0.806692	−	0.590973i	$-0.798744\pi$
$587$	−555922.	−1.61338	−0.806692	+	0.590973i	$0.798744\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	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
$601$	−458055.	−1.26814	−0.634072	−	0.773274i	$-0.718618\pi$
$601$	−458055.	−1.26814	−0.634072	+	0.773274i	$0.718618\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.73065e6	4.72824
$606$	0	0
$607$	203998.	0.553667	0.276833	−	0.960918i	$-0.410715\pi$
$607$	203998.	0.553667	0.276833	+	0.960918i	$0.410715\pi$
$608$	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	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	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
$625$	−52488.7	−0.134371
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	105542.	0.265073	0.132536	−	0.991178i	$-0.457688\pi$
$631$	105542.	0.265073	0.132536	+	0.991178i	$0.457688\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.27061e6	3.15113
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	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	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	−1.65172e6	−3.92146
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−805185.	−1.88829	−0.944147	−	0.329525i	$-0.893111\pi$
$653$	−805185.	−1.88829	−0.944147	+	0.329525i	$0.893111\pi$
$654$	0	0
$655$	−115178.	−0.268464
$656$	0	0
$657$	0	0
$658$	0	0
$659$	525926.	1.21103	0.605513	−	0.795835i	$-0.292968\pi$
$659$	525926.	1.21103	0.605513	+	0.795835i	$0.292968\pi$
$660$	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
$673$	215619.	0.476055	0.238028	−	0.971258i	$-0.423499\pi$
$673$	215619.	0.476055	0.238028	+	0.971258i	$0.423499\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	895058.	1.95287	0.976436	−	0.215807i	$-0.0692381\pi$
$677$	895058.	1.95287	0.976436	+	0.215807i	$0.0692381\pi$
$678$	0	0
$679$	−1.29550e6	−2.80995
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−846578.	−1.81479	−0.907393	−	0.420282i	$-0.861931\pi$
$683$	−846578.	−1.81479	−0.907393	+	0.420282i	$0.861931\pi$
$684$	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
$701$	429481.	0.873993	0.436997	−	0.899463i	$-0.356042\pi$
$701$	429481.	0.873993	0.436997	+	0.899463i	$0.356042\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.64274e6	−3.28648
$708$	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
$721$	1.81304e6	3.48768
$722$	0	0
$723$	0	0
$724$	0	0
$725$	795624.	1.51367
$726$	0	0
$727$	387381.	0.732941	0.366471	−	0.930430i	$-0.380566\pi$
$727$	387381.	0.732941	0.366471	+	0.930430i	$0.380566\pi$
$728$	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	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−104878.	−0.188961
$746$	0	0
$747$	0	0
$748$	0	0
$749$	587550.	1.04732
$750$	0	0
$751$	1.07310e6	1.90266	0.951329	−	0.308177i	$-0.0997189\pi$
$751$	1.07310e6	1.90266	0.951329	+	0.308177i	$0.0997189\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	358054.	0.628137
$756$	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	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−731022.	−1.23617	−0.618084	−	0.786112i	$-0.712091\pi$
$769$	−731022.	−1.23617	−0.618084	+	0.786112i	$0.712091\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	136658.	0.228705	0.114353	−	0.993440i	$-0.463521\pi$
$773$	136658.	0.228705	0.114353	+	0.993440i	$0.463521\pi$
$774$	0	0
$775$	1.33564e6	2.22375
$776$	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
$797$	841087.	1.32411	0.662055	−	0.749455i	$-0.269684\pi$
$797$	841087.	1.32411	0.662055	+	0.749455i	$0.269684\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	447880.	0.694594
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	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
$821$	−899182.	−1.33402	−0.667008	−	0.745050i	$-0.732425\pi$
$821$	−899182.	−1.33402	−0.667008	+	0.745050i	$0.732425\pi$
$822$	0	0
$823$	75202.2	0.111028	0.0555138	−	0.998458i	$-0.482320\pi$
$823$	75202.2	0.111028	0.0555138	+	0.998458i	$0.482320\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1.02226e6	−1.49468	−0.747342	−	0.664439i	$-0.768670\pi$
$827$	−1.02226e6	−1.49468	−0.747342	+	0.664439i	$0.768670\pi$
$828$	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
$841$	−38157.0	−0.0539489
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.14160e6	1.59882
$846$	0	0
$847$	3.71727e6	5.18152
$848$	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	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	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	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	1.70004e6	2.27210
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−2.19475e6	−2.90634
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.19298e6	1.55817
$876$	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	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	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	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	2.72915e6	3.45322
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−2.40768e6	−3.00574
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.12328e6	1.38985
$900$	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	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	2.23406e6	2.68011
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−247390.	−0.294201
$918$	0	0
$919$	−1.60846e6	−1.90449	−0.952246	−	0.305333i	$-0.901232\pi$
$919$	−1.60846e6	−1.90449	−0.952246	+	0.305333i	$0.901232\pi$
$920$	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	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−1.69868e6	−1.93478	−0.967390	−	0.253291i	$-0.918487\pi$
$937$	−1.69868e6	−1.93478	−0.967390	+	0.253291i	$0.918487\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	110730.	0.125050	0.0625251	−	0.998043i	$-0.480085\pi$
$941$	110730.	0.125050	0.0625251	+	0.998043i	$0.480085\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.59544e6	−1.77902	−0.889509	−	0.456917i	$-0.848954\pi$
$947$	−1.59544e6	−1.77902	−0.889509	+	0.456917i	$0.848954\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	962167.	1.04185
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2.74918e6	−2.95221
$966$	0	0
$967$	386547.	0.413380	0.206690	−	0.978406i	$-0.433731\pi$
$967$	386547.	0.413380	0.206690	+	0.978406i	$0.433731\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.27989e6	1.35749	0.678743	−	0.734376i	$-0.262525\pi$
$971$	1.27989e6	1.35749	0.678743	+	0.734376i	$0.262525\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	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
$985$	−1.50385e6	−1.55001
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−1.72748e6	−1.75900	−0.879502	−	0.475895i	$-0.842124\pi$
$991$	−1.72748e6	−1.75900	−0.879502	+	0.475895i	$0.842124\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.16527e6	−3.19716
$996$	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	864.5.h.b.593.2		2
3.2	odd	2		864.5.h.a.593.1		2
4.3	odd	2		216.5.h.b.53.2	yes	2
8.3	odd	2		216.5.h.a.53.1	✓	2
8.5	even	2		864.5.h.a.593.1		2
12.11	even	2		216.5.h.a.53.1	✓	2
24.5	odd	2	CM	864.5.h.b.593.2		2
24.11	even	2		216.5.h.b.53.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.5.h.a.53.1	✓	2	8.3	odd	2
216.5.h.a.53.1	✓	2	12.11	even	2
216.5.h.b.53.2	yes	2	4.3	odd	2
216.5.h.b.53.2	yes	2	24.11	even	2
864.5.h.a.593.1		2	3.2	odd	2
864.5.h.a.593.1		2	8.5	even	2
864.5.h.b.593.2		2	1.1	even	1	trivial
864.5.h.b.593.2		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	864.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+39.9706 q^{5} +85.8528 q^{7} -240.706 q^{11} +972.646 q^{25} +818.000 q^{29} +1373.20 q^{31} +3431.59 q^{35} +4969.71 q^{49} +2379.08 q^{53} -9621.14 q^{55} +6862.00 q^{59} -1860.70 q^{73} -20665.3 q^{77} +9118.00 q^{79} -9281.28 q^{83} -15089.8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 46 q^{5} + 2 q^{7} - 142 q^{11} + 384 q^{25} + 1636 q^{29} - 478 q^{31} + 2926 q^{35} + 9600 q^{49} - 3218 q^{53} - 9026 q^{55} + 13724 q^{59} + 8158 q^{73} - 28942 q^{77} + 18236 q^{79} + 4178 q^{83}+ \cdots - 17282 q^{97}+O(q^{100}) \) 2 * q + 46 * q^5 + 2 * q^7 - 142 * q^11 + 384 * q^25 + 1636 * q^29 - 478 * q^31 + 2926 * q^35 + 9600 * q^49 - 3218 * q^53 - 9026 * q^55 + 13724 * q^59 + 8158 * q^73 - 28942 * q^77 + 18236 * q^79 + 4178 * q^83 - 17282 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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