LMFDB - Embedded newform 720.4.f.d.289.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [720,4,Mod(289,720)] 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("720.289"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 720 = 2^{4} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	720.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-328] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$42.4813752041$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-5}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 5 $ x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			289.2
Root			$-2.23607i$ of defining polynomial
Character	$\chi$	$=$	720.289
Dual form			720.4.f.d.289.1

$q$-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 $\left(\mathbb{Z}/720\mathbb{Z}\right)^\times$.

$n$	$181$	$271$	$577$	$641$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	11.1803i	1.00000i
$5$	11.1803i	1.00000i
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 138.636i	− 1.97790i	−0.148265	−	0.988948i	$-0.547369\pi$
$17$	− 138.636i	− 1.97790i	0.148265	−	0.988948i	$-0.452631\pi$
$18$	0	0
$19$	−164.000	−1.98022	−0.990110	−	0.140293i	$-0.955195\pi$
$19$	−164.000	−1.98022	−0.990110	+	0.140293i	$0.955195\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	98.3870i	0.891961i	0.895043	+	0.445981i	$0.147145\pi$
$23$	98.3870i	0.891961i	−0.895043	+	0.445981i	$0.852855\pi$
$24$	0	0
$25$	−125.000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	232.000	1.34414	0.672071	−	0.740486i	$-0.265405\pi$
$31$	232.000	1.34414	0.672071	+	0.740486i	$0.265405\pi$
$32$	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
$41$	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
$47$	− 545.601i	− 1.69328i	−0.532168	−	0.846639i	$-0.678623\pi$
$47$	− 545.601i	− 1.69328i	0.532168	−	0.846639i	$-0.321377\pi$
$48$	0	0
$49$	343.000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 621.627i	− 1.61108i	−0.592544	−	0.805538i	$-0.701876\pi$
$53$	− 621.627i	− 1.61108i	0.592544	−	0.805538i	$-0.298124\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−358.000	−0.751430	−0.375715	−	0.926735i	$-0.622603\pi$
$61$	−358.000	−0.751430	−0.375715	+	0.926735i	$0.622603\pi$
$62$	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
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	304.000	0.432945	0.216473	−	0.976289i	$-0.430545\pi$
$79$	304.000	0.432945	0.216473	+	0.976289i	$0.430545\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 1270.09i	− 1.67964i	−0.542865	−	0.839820i	$-0.682660\pi$
$83$	− 1270.09i	− 1.67964i	0.542865	−	0.839820i	$-0.317340\pi$
$84$	0	0
$85$	1550.00	1.97790
$86$	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
$95$	− 1833.58i	− 1.98022i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	17.8885i	0.0161622i	0.999967	+	0.00808108i	$0.00257232\pi$
$107$	17.8885i	0.0161622i	−0.999967	+	0.00808108i	$0.997428\pi$
$108$	0	0
$109$	−1834.00	−1.61161	−0.805804	−	0.592182i	$-0.798267\pi$
$109$	−1834.00	−1.61161	−0.805804	+	0.592182i	$0.798267\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1426.61i	− 1.18765i	−0.804595	−	0.593824i	$-0.797617\pi$
$113$	− 1426.61i	− 1.18765i	0.804595	−	0.593824i	$-0.202383\pi$
$114$	0	0
$115$	−1100.00	−0.891961
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1331.00	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1397.54i	− 1.00000i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1990.10i	− 1.24106i	−0.784181	−	0.620532i	$-0.786917\pi$
$137$	− 1990.10i	− 1.24106i	0.784181	−	0.620532i	$-0.213083\pi$
$138$	0	0
$139$	−1604.00	−0.978773	−0.489387	−	0.872067i	$-0.662779\pi$
$139$	−1604.00	−0.978773	−0.489387	+	0.872067i	$0.662779\pi$
$140$	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		−	1.00000i	$-0.5\pi$
$149$	0	0			1.00000i	$0.5\pi$
$150$	0	0
$151$	3112.00	1.67716	0.838579	−	0.544779i	$-0.183387\pi$
$151$	3112.00	1.67716	0.838579	+	0.544779i	$0.183387\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2593.84i	1.34414i
$156$	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
$167$	− 3765.54i	− 1.74483i	−0.488769	−	0.872414i	$-0.662554\pi$
$167$	− 3765.54i	− 1.74483i	0.488769	−	0.872414i	$-0.337446\pi$
$168$	0	0
$169$	2197.00	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	424.853i	0.186711i	0.995633	+	0.0933554i	$0.0297593\pi$
$173$	424.853i	0.186711i	−0.995633	+	0.0933554i	$0.970241\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	1298.00	0.533036	0.266518	−	0.963830i	$-0.414127\pi$
$181$	1298.00	0.533036	0.266518	+	0.963830i	$0.414127\pi$
$182$	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
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2553.59i	− 0.923532i	−0.887002	−	0.461766i	$-0.847216\pi$
$197$	− 2553.59i	− 0.923532i	0.887002	−	0.461766i	$-0.152784\pi$
$198$	0	0
$199$	−5456.00	−1.94355	−0.971773	−	0.235919i	$-0.924190\pi$
$199$	−5456.00	−1.94355	−0.971773	+	0.235919i	$0.924190\pi$
$200$	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
$209$	0	0
$210$	0	0
$211$	4228.00	1.37947	0.689733	−	0.724063i	$-0.257728\pi$
$211$	4228.00	1.37947	0.689733	+	0.724063i	$0.257728\pi$
$212$	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	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6457.76i	1.88818i	0.329688	+	0.944090i	$0.393057\pi$
$227$	6457.76i	1.88818i	−0.329688	+	0.944090i	$0.606943\pi$
$228$	0	0
$229$	−286.000	−0.0825302	−0.0412651	−	0.999148i	$-0.513139\pi$
$229$	−286.000	−0.0825302	−0.0412651	+	0.999148i	$0.513139\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4449.78i	1.25114i	0.780170	+	0.625568i	$0.215133\pi$
$233$	4449.78i	1.25114i	−0.780170	+	0.625568i	$0.784867\pi$
$234$	0	0
$235$	6100.00	1.69328
$236$	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
$241$	−1438.00	−0.384356	−0.192178	−	0.981360i	$-0.561555\pi$
$241$	−1438.00	−0.384356	−0.192178	+	0.981360i	$0.561555\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3834.86i	1.00000i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6542.73i	1.58803i	0.607896	+	0.794017i	$0.292014\pi$
$257$	6542.73i	1.58803i	−0.607896	+	0.794017i	$0.707986\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 6341.49i	− 1.48682i	−0.668837	−	0.743409i	$-0.733208\pi$
$263$	− 6341.49i	− 1.48682i	0.668837	−	0.743409i	$-0.266792\pi$
$264$	0	0
$265$	6950.00	1.61108
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−6752.00	−1.51349	−0.756743	−	0.653712i	$-0.773211\pi$
$271$	−6752.00	−1.51349	−0.756743	+	0.653712i	$0.773211\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	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
$289$	−14307.0	−2.91207
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3161.80i	0.630424i	0.949021	+	0.315212i	$0.102076\pi$
$293$	3161.80i	0.630424i	−0.949021	+	0.315212i	$0.897924\pi$
$294$	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
$305$	− 4002.56i	− 0.751430i
$306$	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
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10084.7i	1.78679i	0.449276	+	0.893393i	$0.351682\pi$
$317$	10084.7i	1.78679i	−0.449276	+	0.893393i	$0.648318\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	22736.3i	3.91667i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−5852.00	−0.971767	−0.485884	−	0.874023i	$-0.661502\pi$
$331$	−5852.00	−0.971767	−0.485884	+	0.874023i	$0.661502\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	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
$347$	8550.72i	1.32284i	0.750014	+	0.661422i	$0.230047\pi$
$347$	8550.72i	1.32284i	−0.750014	+	0.661422i	$0.769953\pi$
$348$	0	0
$349$	−3706.00	−0.568417	−0.284209	−	0.958762i	$-0.591731\pi$
$349$	−3706.00	−0.568417	−0.284209	+	0.958762i	$0.591731\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2473.09i	− 0.372888i	−0.982466	−	0.186444i	$-0.940304\pi$
$353$	− 2473.09i	− 0.372888i	0.982466	−	0.186444i	$-0.0596962\pi$
$354$	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
$361$	20037.0	2.92127
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	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
$379$	−4484.00	−0.607725	−0.303862	−	0.952716i	$-0.598276\pi$
$379$	−4484.00	−0.607725	−0.303862	+	0.952716i	$0.598276\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 9561.43i	− 1.27563i	−0.770190	−	0.637815i	$-0.779839\pi$
$383$	− 9561.43i	− 1.27563i	0.770190	−	0.637815i	$-0.220161\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	13640.0	1.76421
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3398.82i	0.432945i
$396$	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
$409$	−14326.0	−1.73197	−0.865984	−	0.500071i	$-0.833307\pi$
$409$	−14326.0	−1.73197	−0.865984	+	0.500071i	$0.833307\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	14200.0	1.67964
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	6878.00	0.796231	0.398115	−	0.917335i	$-0.369664\pi$
$421$	6878.00	0.796231	0.398115	+	0.917335i	$0.369664\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	17329.5i	1.97790i
$426$	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
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 16135.5i	− 1.76628i
$438$	0	0
$439$	−16976.0	−1.84560	−0.922802	−	0.385274i	$-0.874107\pi$
$439$	−16976.0	−1.84560	−0.922802	+	0.385274i	$0.874107\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	18049.5i	1.93580i	0.251335	+	0.967900i	$0.419130\pi$
$443$	18049.5i	1.93580i	−0.251335	+	0.967900i	$0.580870\pi$
$444$	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
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0	0		−	1.00000i	$-0.5\pi$
$461$	0	0			1.00000i	$0.5\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4329.03i	− 0.428958i	−0.976729	−	0.214479i	$-0.931195\pi$
$467$	− 4329.03i	− 0.428958i	0.976729	−	0.214479i	$-0.0688054\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	20500.0	1.98022
$476$	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	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−19316.0	−1.73287	−0.866436	−	0.499289i	$-0.833595\pi$
$499$	−19316.0	−1.73287	−0.866436	+	0.499289i	$0.833595\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 21958.2i	− 1.94646i	−0.229843	−	0.973228i	$-0.573821\pi$
$503$	− 21958.2i	− 1.94646i	0.229843	−	0.973228i	$-0.426179\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	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
$521$	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
$527$	− 32163.6i	− 2.65857i
$528$	0	0
$529$	2487.00	0.204405
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−200.000	−0.0161622
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−3238.00	−0.257324	−0.128662	−	0.991688i	$-0.541068\pi$
$541$	−3238.00	−0.257324	−0.128662	+	0.991688i	$0.541068\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 20504.7i	− 1.61161i
$546$	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
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26184.4i	1.99186i	0.0901226	+	0.995931i	$0.471274\pi$
$557$	26184.4i	1.99186i	−0.0901226	+	0.995931i	$0.528726\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 14632.8i	− 1.09538i	−0.836681	−	0.547691i	$-0.815507\pi$
$563$	− 14632.8i	− 1.09538i	0.836681	−	0.547691i	$-0.184493\pi$
$564$	0	0
$565$	15950.0	1.18765
$566$	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
$571$	−26012.0	−1.90642	−0.953212	−	0.302302i	$-0.902245\pi$
$571$	−26012.0	−1.90642	−0.953212	+	0.302302i	$0.902245\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 12298.4i	− 0.891961i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	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
$587$	25777.4i	1.81252i	0.422725	+	0.906258i	$0.361074\pi$
$587$	25777.4i	1.81252i	−0.422725	+	0.906258i	$0.638926\pi$
$588$	0	0
$589$	−38048.0	−2.66170
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 7866.49i	− 0.544752i	−0.962191	−	0.272376i	$-0.912191\pi$
$593$	− 7866.49i	− 0.544752i	0.962191	−	0.272376i	$-0.0878094\pi$
$594$	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
$601$	−13642.0	−0.925905	−0.462952	−	0.886383i	$-0.653210\pi$
$601$	−13642.0	−0.925905	−0.462952	+	0.886383i	$0.653210\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 14881.0i	− 1.00000i
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	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
$617$	− 27991.1i	− 1.82638i	−0.407529	−	0.913192i	$-0.633609\pi$
$617$	− 27991.1i	− 1.82638i	0.407529	−	0.913192i	$-0.366391\pi$
$618$	0	0
$619$	−3476.00	−0.225706	−0.112853	−	0.993612i	$-0.535999\pi$
$619$	−3476.00	−0.225706	−0.112853	+	0.993612i	$0.535999\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	27808.0	1.75439	0.877194	−	0.480136i	$-0.159413\pi$
$631$	27808.0	1.75439	0.877194	+	0.480136i	$0.159413\pi$
$632$	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
$641$	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
$647$	− 2638.56i	− 0.160328i	−0.996782	−	0.0801642i	$-0.974456\pi$
$647$	− 2638.56i	− 0.160328i	0.996782	−	0.0801642i	$-0.0255445\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 22678.2i	− 1.35906i	−0.733647	−	0.679530i	$-0.762184\pi$
$653$	− 22678.2i	− 1.35906i	0.733647	−	0.679530i	$-0.237816\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	32978.0	1.94054	0.970269	−	0.242029i	$-0.0778130\pi$
$661$	32978.0	1.94054	0.970269	+	0.242029i	$0.0778130\pi$
$662$	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
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 32901.5i	− 1.86781i	−0.357521	−	0.933905i	$-0.616378\pi$
$677$	− 32901.5i	− 1.86781i	0.357521	−	0.933905i	$-0.383622\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 27512.6i	− 1.54135i	−0.637231	−	0.770673i	$-0.719920\pi$
$683$	− 27512.6i	− 1.54135i	0.637231	−	0.770673i	$-0.280080\pi$
$684$	0	0
$685$	22250.0	1.24106
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	32812.0	1.80641	0.903204	−	0.429212i	$-0.141209\pi$
$691$	32812.0	1.80641	0.903204	+	0.429212i	$0.141209\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 17933.3i	− 0.978773i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−37726.0	−1.99835	−0.999175	−	0.0406201i	$-0.987067\pi$
$709$	−37726.0	−1.99835	−0.999175	+	0.0406201i	$0.987067\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	22825.8i	1.19892i
$714$	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	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	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
$739$	−8804.00	−0.438241	−0.219121	−	0.975698i	$-0.570319\pi$
$739$	−8804.00	−0.438241	−0.219121	+	0.975698i	$0.570319\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	10241.2i	0.505670i	0.967509	+	0.252835i	$0.0813630\pi$
$743$	10241.2i	0.505670i	−0.967509	+	0.252835i	$0.918637\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	17512.0	0.850895	0.425447	−	0.904983i	$-0.360117\pi$
$751$	17512.0	0.850895	0.425447	+	0.904983i	$0.360117\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	34793.2i	1.67716i
$756$	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
$769$	8786.00	0.412004	0.206002	−	0.978552i	$-0.433955\pi$
$769$	8786.00	0.412004	0.206002	+	0.978552i	$0.433955\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 10281.4i	− 0.478393i	−0.970971	−	0.239196i	$-0.923116\pi$
$773$	− 10281.4i	− 0.478393i	0.970971	−	0.239196i	$-0.0768840\pi$
$774$	0	0
$775$	−29000.0	−1.34414
$776$	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
$797$	− 1426.61i	− 0.0634042i	−0.999497	−	0.0317021i	$-0.989907\pi$
$797$	− 1426.61i	− 0.0634042i	0.999497	−	0.0317021i	$-0.0100928\pi$
$798$	0	0
$799$	−75640.0	−3.34912
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	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
$811$	14092.0	0.610157	0.305078	−	0.952327i	$-0.401317\pi$
$811$	14092.0	0.610157	0.305078	+	0.952327i	$0.401317\pi$
$812$	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
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 42968.3i	− 1.80672i	−0.428888	−	0.903358i	$-0.641095\pi$
$827$	− 42968.3i	− 1.80672i	0.428888	−	0.903358i	$-0.358905\pi$
$828$	0	0
$829$	45254.0	1.89594	0.947971	−	0.318356i	$-0.103131\pi$
$829$	45254.0	1.89594	0.947971	+	0.318356i	$0.103131\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 47552.2i	− 1.97790i
$834$	0	0
$835$	42100.0	1.74483
$836$	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
$841$	−24389.0	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	24563.2i	1.00000i
$846$	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
$857$	47033.5i	1.87472i	0.348367	+	0.937358i	$0.386736\pi$
$857$	47033.5i	1.87472i	−0.348367	+	0.937358i	$0.613264\pi$
$858$	0	0
$859$	28204.0	1.12027	0.560133	−	0.828403i	$-0.310750\pi$
$859$	28204.0	1.12027	0.560133	+	0.828403i	$0.310750\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	26340.9i	1.03900i	0.854472	+	0.519498i	$0.173881\pi$
$863$	26340.9i	1.03900i	−0.854472	+	0.519498i	$0.826119\pi$
$864$	0	0
$865$	−4750.00	−0.186711
$866$	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
$881$	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
$887$	− 23085.2i	− 0.873871i	−0.899493	−	0.436936i	$-0.856064\pi$
$887$	− 23085.2i	− 0.873871i	0.899493	−	0.436936i	$-0.143936\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	89478.5i	3.35306i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−86180.0	−3.18654
$902$	0	0
$903$	0	0
$904$	0	0
$905$	14512.1i	0.533036i
$906$	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
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−21224.0	−0.761823	−0.380911	−	0.924612i	$-0.624390\pi$
$919$	−21224.0	−0.761823	−0.380911	+	0.924612i	$0.624390\pi$
$920$	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
$931$	−56252.0	−1.98022
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0	0	1.00000			$0$
$937$	0	0	−1.00000			$\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0		−	1.00000i	$-0.5\pi$
$941$	0	0			1.00000i	$0.5\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	45097.0i	1.54747i	0.633508	+	0.773736i	$0.281614\pi$
$947$	45097.0i	1.54747i	−0.633508	+	0.773736i	$0.718386\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30289.8i	1.02957i	0.857319	+	0.514786i	$0.172129\pi$
$953$	30289.8i	1.02957i	−0.857319	+	0.514786i	$0.827871\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	24033.0	0.806720
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 13018.4i	− 0.426300i	−0.977019	−	0.213150i	$-0.931628\pi$
$977$	− 13018.4i	− 0.426300i	0.977019	−	0.213150i	$-0.0683723\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	60794.2i	1.97257i	0.165057	+	0.986284i	$0.447219\pi$
$983$	60794.2i	1.97257i	−0.165057	+	0.986284i	$0.552781\pi$
$984$	0	0
$985$	28550.0	0.923532
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	62368.0	1.99918	0.999589	−	0.0286779i	$-0.00912971\pi$
$991$	62368.0	1.99918	0.999589	+	0.0286779i	$0.00912971\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 60999.9i	− 1.94355i
$996$	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	720.4.f.d.289.2		2
3.2	odd	2	inner	720.4.f.d.289.1		2
4.3	odd	2		45.4.b.a.19.2	yes	2
5.4	even	2	inner	720.4.f.d.289.1		2
12.11	even	2		45.4.b.a.19.1	✓	2
15.14	odd	2	CM	720.4.f.d.289.2		2
20.3	even	4		225.4.a.k.1.2		2
20.7	even	4		225.4.a.k.1.1		2
20.19	odd	2		45.4.b.a.19.1	✓	2
60.23	odd	4		225.4.a.k.1.1		2
60.47	odd	4		225.4.a.k.1.2		2
60.59	even	2		45.4.b.a.19.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.b.a.19.1	✓	2	12.11	even	2
45.4.b.a.19.1	✓	2	20.19	odd	2
45.4.b.a.19.2	yes	2	4.3	odd	2
45.4.b.a.19.2	yes	2	60.59	even	2
225.4.a.k.1.1		2	20.7	even	4
225.4.a.k.1.1		2	60.23	odd	4
225.4.a.k.1.2		2	20.3	even	4
225.4.a.k.1.2		2	60.47	odd	4
720.4.f.d.289.1		2	3.2	odd	2	inner
720.4.f.d.289.1		2	5.4	even	2	inner
720.4.f.d.289.2		2	1.1	even	1	trivial
720.4.f.d.289.2		2	15.14	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:	\( N \)	\(=\)	\( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	720.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+11.1803i q^{5} -138.636i q^{17} -164.000 q^{19} +98.3870i q^{23} -125.000 q^{25} +232.000 q^{31} -545.601i q^{47} +343.000 q^{49} -621.627i q^{53} -358.000 q^{61} +304.000 q^{79} -1270.09i q^{83} +1550.00 q^{85} -1833.58i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 328 q^{19} - 250 q^{25} + 464 q^{31} + 686 q^{49} - 716 q^{61} + 608 q^{79} + 3100 q^{85}+O(q^{100}) \) 2 * q - 328 * q^19 - 250 * q^25 + 464 * q^31 + 686 * q^49 - 716 * q^61 + 608 * q^79 + 3100 * q^85

Introduction

L-functions

Modular forms

Varieties

Fields

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Database

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