LMFDB - Embedded newform 1152.4.f.c.575.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,4,Mod(575,1152)] 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("1152.575"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1152 = 2^{7} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1152.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,852] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$67.9702003266$
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:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			575.4
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.575
Dual form			1152.4.f.c.575.3

$q$-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 $\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times$.

$n$	$127$	$641$	$901$
$\chi(n)$	$-1$	$-1$	$-1$

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
$5$	18.3848	1.64438	0.822192	−	0.569210i	$-0.192751\pi$
$5$	18.3848	1.64438	0.822192	+	0.569210i	$0.192751\pi$
$6$	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	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	18.0000i	0.384023i	0.981393	+	0.192012i	$0.0615011\pi$
$13$	18.0000i	0.384023i	−0.981393	+	0.192012i	$0.938499\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 140.007i	− 1.99745i	−0.0504408	−	0.998727i	$-0.516063\pi$
$17$	− 140.007i	− 1.99745i	0.0504408	−	0.998727i	$-0.483937\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	213.000	1.70400
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−108.894	−0.697282	−0.348641	−	0.937256i	$-0.613357\pi$
$29$	−108.894	−0.697282	−0.348641	+	0.937256i	$0.613357\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 396.000i	− 1.75951i	−0.475424	−	0.879757i	$-0.657705\pi$
$37$	− 396.000i	− 1.75951i	0.475424	−	0.879757i	$-0.342295\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 496.389i	− 1.89080i	−0.325908	−	0.945402i	$-0.605670\pi$
$41$	− 496.389i	− 1.89080i	0.325908	−	0.945402i	$-0.394330\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	343.000	1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	770.746	1.99755	0.998775	−	0.0494806i	$-0.0157566\pi$
$53$	770.746	1.99755	0.998775	+	0.0494806i	$0.0157566\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	468.000i	0.982316i	0.871071	+	0.491158i	$0.163426\pi$
$61$	468.000i	0.982316i	−0.871071	+	0.491158i	$0.836574\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	330.926i	0.631482i
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−592.000	−0.949156	−0.474578	−	0.880214i	$-0.657399\pi$
$73$	−592.000	−0.949156	−0.474578	+	0.880214i	$0.657399\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	− 2574.00i	− 3.28458i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1056.42i	1.25820i	0.777323	+	0.629101i	$0.216577\pi$
$89$	1056.42i	1.25820i	−0.777323	+	0.629101i	$0.783423\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1816.00	−1.90090	−0.950448	−	0.310884i	$-0.899375\pi$
$97$	−1816.00	−1.90090	−0.950448	+	0.310884i	$0.899375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	948.937	0.934879	0.467440	−	0.884025i	$-0.345177\pi$
$101$	948.937	0.934879	0.467440	+	0.884025i	$0.345177\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	− 1746.00i	− 1.53428i	−0.641480	−	0.767140i	$-0.721679\pi$
$109$	− 1746.00i	− 1.53428i	0.641480	−	0.767140i	$-0.278321\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2354.67i	1.96025i	0.198380	+	0.980125i	$0.436432\pi$
$113$	2354.67i	1.96025i	−0.198380	+	0.980125i	$0.563568\pi$
$114$	0	0
$115$	0	0
$116$	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
$125$	1617.86	1.15765
$126$	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	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 827.315i	− 0.515929i	−0.966154	−	0.257965i	$-0.916948\pi$
$137$	− 827.315i	− 0.515929i	0.966154	−	0.257965i	$-0.0830518\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2002.00	−1.14660
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3149.45	1.73163	0.865816	−	0.500362i	$-0.166800\pi$
$149$	3149.45	1.73163	0.865816	+	0.500362i	$0.166800\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 3924.00i	− 1.99471i	−0.0726920	−	0.997354i	$-0.523159\pi$
$157$	− 3924.00i	− 1.99471i	0.0726920	−	0.997354i	$-0.476841\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	1873.00	0.852526
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4309.11	1.89373	0.946866	−	0.321630i	$-0.104231\pi$
$173$	4309.11	1.89373	0.946866	+	0.321630i	$0.104231\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	0	0
$181$	− 3942.00i	− 1.61882i	−0.587243	−	0.809410i	$-0.699787\pi$
$181$	− 3942.00i	− 1.61882i	0.587243	−	0.809410i	$-0.300213\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 7280.37i	− 2.89332i
$186$	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
$193$	5362.00	1.99982	0.999910	−	0.0134266i	$-0.00427395\pi$
$193$	5362.00	1.99982	0.999910	+	0.0134266i	$0.00427395\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4651.35	−1.68221	−0.841104	−	0.540874i	$-0.818094\pi$
$197$	−4651.35	−1.68221	−0.841104	+	0.540874i	$0.818094\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	− 9126.00i	− 3.10921i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2520.13	0.767069
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	6390.00i	1.84394i	0.387257	+	0.921972i	$0.373423\pi$
$229$	6390.00i	1.84394i	−0.387257	+	0.921972i	$0.626577\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 5434.82i	− 1.52810i	−0.645158	−	0.764050i	$-0.723208\pi$
$233$	− 5434.82i	− 1.52810i	0.645158	−	0.764050i	$-0.276792\pi$
$234$	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
$241$	−5272.00	−1.40913	−0.704563	−	0.709641i	$-0.748857\pi$
$241$	−5272.00	−1.40913	−0.704563	+	0.709641i	$0.748857\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6305.98	1.64438
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6809.44i	1.65277i	0.563108	+	0.826383i	$0.309606\pi$
$257$	6809.44i	1.65277i	−0.563108	+	0.826383i	$0.690394\pi$
$258$	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
$265$	14170.0	3.28474
$266$	0	0
$267$	0	0
$268$	0	0
$269$	8164.25	1.85050	0.925248	−	0.379363i	$-0.123857\pi$
$269$	8164.25	1.85050	0.925248	+	0.379363i	$0.123857\pi$
$270$	0	0
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 9126.00i	− 1.97952i	−0.142727	−	0.989762i	$-0.545587\pi$
$277$	− 9126.00i	− 1.97952i	0.142727	−	0.989762i	$-0.454413\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1158.24i	0.245889i	0.992414	+	0.122945i	$0.0392337\pi$
$281$	1158.24i	0.245889i	−0.992414	+	0.122945i	$0.960766\pi$
$282$	0	0
$283$	0	0		−	1.00000i	$-0.5\pi$
$283$	0	0			1.00000i	$0.5\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−14689.0	−2.98982
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4218.60	−0.841137	−0.420569	−	0.907261i	$-0.638169\pi$
$293$	−4218.60	−0.841137	−0.420569	+	0.907261i	$0.638169\pi$
$294$	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
$305$	8604.08i	1.61530i
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−6838.00	−1.23485	−0.617423	−	0.786632i	$-0.711823\pi$
$313$	−6838.00	−1.23485	−0.617423	+	0.786632i	$0.711823\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10571.2	−1.87300	−0.936499	−	0.350670i	$-0.885954\pi$
$317$	−10571.2	−1.87300	−0.936499	+	0.350670i	$0.885954\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3834.00i	0.654376i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−416.000	−0.0672432	−0.0336216	−	0.999435i	$-0.510704\pi$
$337$	−416.000	−0.0672432	−0.0336216	+	0.999435i	$0.510704\pi$
$338$	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
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	8964.00i	1.37488i	0.726243	+	0.687438i	$0.241265\pi$
$349$	8964.00i	1.37488i	−0.726243	+	0.687438i	$0.758735\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11416.9i	1.72142i	0.509092	+	0.860712i	$0.329981\pi$
$353$	11416.9i	1.72142i	−0.509092	+	0.860712i	$0.670019\pi$
$354$	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
$361$	−6859.00	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10883.8	−1.56078
$366$	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
$373$	− 6372.00i	− 0.884530i	−0.896884	−	0.442265i	$-0.854175\pi$
$373$	− 6372.00i	− 0.884530i	0.896884	−	0.442265i	$-0.145825\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1960.10i	− 0.267773i
$378$	0	0
$379$	0	0		−	1.00000i	$-0.5\pi$
$379$	0	0			1.00000i	$0.5\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10582.6	1.37932	0.689662	−	0.724131i	$-0.257759\pi$
$389$	10582.6	1.37932	0.689662	+	0.724131i	$0.257759\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	12564.0i	1.58834i	0.607699	+	0.794168i	$0.292093\pi$
$397$	12564.0i	1.58834i	−0.607699	+	0.794168i	$0.707907\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9533.21i	1.18720i	0.804761	+	0.593598i	$0.202293\pi$
$401$	9533.21i	1.18720i	−0.804761	+	0.593598i	$0.797707\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14920.0	1.80378	0.901891	−	0.431964i	$-0.142179\pi$
$409$	14920.0	1.80378	0.901891	+	0.431964i	$0.142179\pi$
$410$	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
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	10890.0i	1.26068i	0.776319	+	0.630340i	$0.217084\pi$
$421$	10890.0i	1.26068i	−0.776319	+	0.630340i	$0.782916\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 29821.5i	− 3.40366i
$426$	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
$433$	−4862.00	−0.539614	−0.269807	−	0.962914i	$-0.586960\pi$
$433$	−4862.00	−0.539614	−0.269807	+	0.962914i	$0.586960\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	19422.0i	2.06897i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 4238.40i	− 0.445484i	−0.974877	−	0.222742i	$-0.928499\pi$
$449$	− 4238.40i	− 0.445484i	0.974877	−	0.222742i	$-0.0715008\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10456.0	1.07026	0.535132	−	0.844768i	$-0.320262\pi$
$457$	10456.0	1.07026	0.535132	+	0.844768i	$0.320262\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−12262.6	−1.23889	−0.619445	−	0.785040i	$-0.712642\pi$
$461$	−12262.6	−1.23889	−0.619445	+	0.785040i	$0.712642\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	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
$481$	7128.00	0.675694
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−33386.8	−3.12580
$486$	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	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	15246.0i	1.39279i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	17446.0	1.53730
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−22815.5	−1.98680	−0.993398	−	0.114715i	$-0.963404\pi$
$509$	−22815.5	−1.98680	−0.993398	+	0.114715i	$0.963404\pi$
$510$	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
$521$	− 17831.8i	− 1.49947i	−0.661736	−	0.749737i	$-0.730180\pi$
$521$	− 17831.8i	− 1.49947i	0.661736	−	0.749737i	$-0.269820\pi$
$522$	0	0
$523$	0	0		−	1.00000i	$-0.5\pi$
$523$	0	0			1.00000i	$0.5\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	8935.00	0.726112
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 5922.00i	− 0.470622i	−0.971920	−	0.235311i	$-0.924389\pi$
$541$	− 5922.00i	− 0.470622i	0.971920	−	0.235311i	$-0.0756109\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 32099.8i	− 2.52295i
$546$	0	0
$547$	0	0		−	1.00000i	$-0.5\pi$
$547$	0	0			1.00000i	$0.5\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11462.2	0.871937	0.435969	−	0.899962i	$-0.356406\pi$
$557$	11462.2	0.871937	0.435969	+	0.899962i	$0.356406\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	43290.0i	3.22341i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21981.1i	1.61950i	0.586774	+	0.809751i	$0.300398\pi$
$569$	21981.1i	1.61950i	−0.586774	+	0.809751i	$0.699602\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3454.00	−0.249206	−0.124603	−	0.992207i	$-0.539766\pi$
$577$	−3454.00	−0.249206	−0.124603	+	0.992207i	$0.539766\pi$
$578$	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
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28192.3i	1.95231i	0.217070	+	0.976156i	$0.430350\pi$
$593$	28192.3i	1.95231i	−0.217070	+	0.976156i	$0.569650\pi$
$594$	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
$601$	−17030.0	−1.15585	−0.577927	−	0.816089i	$-0.696138\pi$
$601$	−17030.0	−1.15585	−0.577927	+	0.816089i	$0.696138\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	24470.1	1.64438
$606$	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
$613$	19548.0i	1.28799i	0.765031	+	0.643994i	$0.222724\pi$
$613$	19548.0i	1.28799i	−0.765031	+	0.643994i	$0.777276\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7776.76i	0.507424i	0.967280	+	0.253712i	$0.0816515\pi$
$617$	7776.76i	0.507424i	−0.967280	+	0.253712i	$0.918348\pi$
$618$	0	0
$619$	0	0		−	1.00000i	$-0.5\pi$
$619$	0	0			1.00000i	$0.5\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	3119.00	0.199616
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−55442.8	−3.51455
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6174.00i	0.384023i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	30916.1i	1.90501i	0.304518	+	0.952507i	$0.401505\pi$
$641$	30916.1i	1.90501i	−0.304518	+	0.952507i	$0.598495\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−24303.3	−1.45645	−0.728224	−	0.685340i	$-0.759654\pi$
$653$	−24303.3	−1.45645	−0.728224	+	0.685340i	$0.759654\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	22068.0i	1.29856i	0.760551	+	0.649278i	$0.224929\pi$
$661$	22068.0i	1.29856i	−0.760551	+	0.649278i	$0.775071\pi$
$662$	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
$673$	4462.00	0.255568	0.127784	−	0.991802i	$-0.459214\pi$
$673$	4462.00	0.255568	0.127784	+	0.991802i	$0.459214\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	27612.5	1.56756	0.783778	−	0.621041i	$-0.213290\pi$
$677$	27612.5	1.56756	0.783778	+	0.621041i	$0.213290\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	− 15210.0i	− 0.848386i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	13873.4i	0.767106i
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−69498.0	−3.77679
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7935.15	0.427541	0.213771	−	0.976884i	$-0.431425\pi$
$701$	7935.15	0.427541	0.213771	+	0.976884i	$0.431425\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 36810.0i	− 1.94983i	−0.222580	−	0.974914i	$-0.571448\pi$
$709$	− 36810.0i	− 1.94983i	0.222580	−	0.974914i	$-0.428552\pi$
$710$	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
$725$	−23194.5	−1.18817
$726$	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
$733$	− 38718.0i	− 1.95100i	−0.220003	−	0.975499i	$-0.570607\pi$
$733$	− 38718.0i	− 1.95100i	0.220003	−	0.975499i	$-0.429393\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	57902.0	2.84747
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0	0	1.00000			$0$
$751$	0	0	−1.00000			$\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35046.0i	1.68265i	0.540527	+	0.841327i	$0.318225\pi$
$757$	35046.0i	1.68265i	−0.540527	+	0.841327i	$0.681775\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	41862.1i	1.99409i	0.0768304	+	0.997044i	$0.475520\pi$
$761$	41862.1i	1.99409i	−0.0768304	+	0.997044i	$0.524480\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−9650.00	−0.452520	−0.226260	−	0.974067i	$-0.572650\pi$
$769$	−9650.00	−0.452520	−0.226260	+	0.974067i	$0.572650\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	16044.3	0.746535	0.373268	−	0.927724i	$-0.378237\pi$
$773$	16044.3	0.746535	0.373268	+	0.927724i	$0.378237\pi$
$774$	0	0
$775$	0	0
$776$	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
$785$	− 72141.9i	− 3.28007i
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−8424.00	−0.377232
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18157.1	0.806972	0.403486	−	0.914986i	$-0.367798\pi$
$797$	18157.1	0.806972	0.403486	+	0.914986i	$0.367798\pi$
$798$	0	0
$799$	0	0
$800$	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
$809$	11620.6i	0.505016i	0.967595	+	0.252508i	$0.0812555\pi$
$809$	11620.6i	0.505016i	−0.967595	+	0.252508i	$0.918745\pi$
$810$	0	0
$811$	0	0		−	1.00000i	$-0.5\pi$
$811$	0	0			1.00000i	$0.5\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	34550.7	1.46873	0.734364	−	0.678756i	$-0.237480\pi$
$821$	34550.7	1.46873	0.734364	+	0.678756i	$0.237480\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	23166.0i	0.970553i	0.874361	+	0.485276i	$0.161281\pi$
$829$	23166.0i	0.970553i	−0.874361	+	0.485276i	$0.838719\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 48022.4i	− 1.99745i
$834$	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
$841$	−12531.0	−0.513797
$842$	0	0
$843$	0	0
$844$	0	0
$845$	34434.7	1.40188
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	− 45468.0i	− 1.82508i	−0.408986	−	0.912541i	$-0.634117\pi$
$853$	− 45468.0i	− 1.82508i	0.408986	−	0.912541i	$-0.365883\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18340.9i	0.731055i	0.930800	+	0.365528i	$0.119111\pi$
$857$	18340.9i	0.731055i	−0.930800	+	0.365528i	$0.880889\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	79222.0	3.11402
$866$	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
$877$	29844.0i	1.14910i	0.818470	+	0.574550i	$0.194823\pi$
$877$	29844.0i	1.14910i	−0.818470	+	0.574550i	$0.805177\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 31578.0i	− 1.20759i	−0.797139	−	0.603796i	$-0.793654\pi$
$881$	− 31578.0i	− 1.20759i	0.797139	−	0.603796i	$-0.206346\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	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
$901$	− 107910.i	− 3.99001i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 72472.8i	− 2.66196i
$906$	0	0
$907$	0	0		−	1.00000i	$-0.5\pi$
$907$	0	0			1.00000i	$0.5\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	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
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 84348.0i	− 2.99821i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 11747.9i	− 0.414893i	−0.978246	−	0.207446i	$-0.933485\pi$
$929$	− 11747.9i	− 0.414893i	0.978246	−	0.207446i	$-0.0665152\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	51946.0	1.81110	0.905551	−	0.424238i	$-0.139458\pi$
$937$	51946.0	1.81110	0.905551	+	0.424238i	$0.139458\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−56454.0	−1.95574	−0.977868	−	0.209223i	$-0.932907\pi$
$941$	−56454.0	−1.95574	−0.977868	+	0.209223i	$0.932907\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	− 10656.0i	− 0.364498i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	51102.6i	1.73702i	0.495676	+	0.868508i	$0.334921\pi$
$953$	51102.6i	1.73702i	−0.495676	+	0.868508i	$0.665079\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	29791.0	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	98579.2	3.28847
$966$	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	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	56244.7i	1.84179i	0.389813	+	0.920894i	$0.372540\pi$
$977$	56244.7i	1.84179i	−0.389813	+	0.920894i	$0.627460\pi$
$978$	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
$985$	−85514.0	−2.76620
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 34164.0i	− 1.08524i	−0.839978	−	0.542620i	$-0.817432\pi$
$997$	− 34164.0i	− 1.08524i	0.839978	−	0.542620i	$-0.182568\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1152.4.f.c.575.4	yes	4
3.2	odd	2	inner	1152.4.f.c.575.2	yes	4
4.3	odd	2	CM	1152.4.f.c.575.4	yes	4
8.3	odd	2	inner	1152.4.f.c.575.1	✓	4
8.5	even	2	inner	1152.4.f.c.575.1	✓	4
12.11	even	2	inner	1152.4.f.c.575.2	yes	4
16.3	odd	4		2304.4.c.a.2303.1		2
16.5	even	4		2304.4.c.c.2303.2		2
16.11	odd	4		2304.4.c.c.2303.2		2
16.13	even	4		2304.4.c.a.2303.1		2
24.5	odd	2	inner	1152.4.f.c.575.3	yes	4
24.11	even	2	inner	1152.4.f.c.575.3	yes	4
48.5	odd	4		2304.4.c.c.2303.1		2
48.11	even	4		2304.4.c.c.2303.1		2
48.29	odd	4		2304.4.c.a.2303.2		2
48.35	even	4		2304.4.c.a.2303.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.f.c.575.1	✓	4	8.3	odd	2	inner
1152.4.f.c.575.1	✓	4	8.5	even	2	inner
1152.4.f.c.575.2	yes	4	3.2	odd	2	inner
1152.4.f.c.575.2	yes	4	12.11	even	2	inner
1152.4.f.c.575.3	yes	4	24.5	odd	2	inner
1152.4.f.c.575.3	yes	4	24.11	even	2	inner
1152.4.f.c.575.4	yes	4	1.1	even	1	trivial
1152.4.f.c.575.4	yes	4	4.3	odd	2	CM
2304.4.c.a.2303.1		2	16.3	odd	4
2304.4.c.a.2303.1		2	16.13	even	4
2304.4.c.a.2303.2		2	48.29	odd	4
2304.4.c.a.2303.2		2	48.35	even	4
2304.4.c.c.2303.1		2	48.5	odd	4
2304.4.c.c.2303.1		2	48.11	even	4
2304.4.c.c.2303.2		2	16.5	even	4
2304.4.c.c.2303.2		2	16.11	odd	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1152 = 2^{7} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1152.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+18.3848 q^{5} +18.0000i q^{13} -140.007i q^{17} +213.000 q^{25} -108.894 q^{29} -396.000i q^{37} -496.389i q^{41} +343.000 q^{49} +770.746 q^{53} +468.000i q^{61} +330.926i q^{65} -592.000 q^{73} -2574.00i q^{85} +1056.42i q^{89} -1816.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 852 q^{25} + 1372 q^{49} - 2368 q^{73} - 7264 q^{97}+O(q^{100}) \) 4 * q + 852 * q^25 + 1372 * q^49 - 2368 * q^73 - 7264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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