LMFDB - Embedded newform 1152.4.d.c.577.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1152,4,Mod(577,1152)] 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("1152.577"); 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([0, 1, 0])) N = Newforms(chi, 4, names="a")

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

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-208] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$67.9702003266$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			577.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1152.577
Dual form			1152.4.d.c.577.1

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

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

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
$5$	22.0000i	1.96774i	0.178885	+	0.983870i	$0.442751\pi$
$5$	22.0000i	1.96774i	−0.178885	+	0.983870i	$0.557249\pi$
$6$	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	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	92.0000i	1.96279i	0.192012	+	0.981393i	$0.438499\pi$
$13$	92.0000i	1.96279i	−0.192012	+	0.981393i	$0.561501\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−104.000	−1.48375	−0.741874	−	0.670540i	$-0.766063\pi$
$17$	−104.000	−1.48375	−0.741874	+	0.670540i	$0.766063\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−359.000	−2.87200
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 130.000i	− 0.832427i	−0.909267	−	0.416214i	$-0.863357\pi$
$29$	− 130.000i	− 0.832427i	0.909267	−	0.416214i	$-0.136643\pi$
$30$	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
$37$	396.000i	1.75951i	0.475424	+	0.879757i	$0.342295\pi$
$37$	396.000i	1.75951i	−0.475424	+	0.879757i	$0.657705\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	472.000	1.79790	0.898951	−	0.438048i	$-0.144330\pi$
$41$	472.000	1.79790	0.898951	+	0.438048i	$0.144330\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−343.000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 518.000i	− 1.34251i	−0.741229	−	0.671253i	$-0.765757\pi$
$53$	− 518.000i	− 1.34251i	0.741229	−	0.671253i	$-0.234243\pi$
$54$	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
$65$	−2024.00	−3.86225
$66$	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
$73$	1098.00	1.76043	0.880214	−	0.474578i	$-0.157399\pi$
$73$	1098.00	1.76043	0.880214	+	0.474578i	$0.157399\pi$
$74$	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
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	− 2288.00i	− 2.91963i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	176.000	0.209618	0.104809	−	0.994492i	$-0.466577\pi$
$89$	176.000	0.209618	0.104809	+	0.994492i	$0.466577\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	594.000	0.621769	0.310884	−	0.950448i	$-0.399375\pi$
$97$	594.000	0.621769	0.310884	+	0.950448i	$0.399375\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 598.000i	− 0.589141i	−0.955630	−	0.294570i	$-0.904823\pi$
$101$	− 598.000i	− 0.589141i	0.955630	−	0.294570i	$-0.0951766\pi$
$102$	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
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	− 1460.00i	− 1.28296i	−0.767140	−	0.641480i	$-0.778321\pi$
$109$	− 1460.00i	− 1.28296i	0.767140	−	0.641480i	$-0.221679\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1328.00	−1.10556	−0.552778	−	0.833329i	$-0.686432\pi$
$113$	−1328.00	−1.10556	−0.552778	+	0.833329i	$0.686432\pi$
$114$	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
$125$	− 5148.00i	− 3.68361i
$126$	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
$131$	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
$137$	−2776.00	−1.73117	−0.865583	−	0.500766i	$-0.833052\pi$
$137$	−2776.00	−1.73117	−0.865583	+	0.500766i	$0.833052\pi$
$138$	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
$145$	2860.00	1.63800
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3514.00i	− 1.93207i	−0.258415	−	0.966034i	$-0.583200\pi$
$149$	− 3514.00i	− 1.93207i	0.258415	−	0.966034i	$-0.416800\pi$
$150$	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
$157$	3924.00i	1.99471i	0.0726920	+	0.997354i	$0.476841\pi$
$157$	3924.00i	1.99471i	−0.0726920	+	0.997354i	$0.523159\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	−6267.00	−2.85253
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4082.00i	1.79392i	0.442108	+	0.896962i	$0.354231\pi$
$173$	4082.00i	1.79392i	−0.442108	+	0.896962i	$0.645769\pi$
$174$	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
$181$	− 2860.00i	− 1.17449i	−0.809410	−	0.587243i	$-0.800213\pi$
$181$	− 2860.00i	− 1.17449i	0.809410	−	0.587243i	$-0.199787\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8712.00	−3.46226
$186$	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	$-0.995726\pi$
$193$	−5362.00	−1.99982	−0.999910	+	0.0134266i	$0.995726\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1174.00i	− 0.424589i	−0.977206	−	0.212295i	$-0.931906\pi$
$197$	− 1174.00i	− 0.424589i	0.977206	−	0.212295i	$-0.0680936\pi$
$198$	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
$205$	10384.0i	3.53780i
$206$	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
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 9568.00i	− 2.91228i
$222$	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
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 2684.00i	− 0.774514i	−0.921972	−	0.387257i	$-0.873423\pi$
$229$	− 2684.00i	− 0.774514i	0.921972	−	0.387257i	$-0.126577\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7088.00	−1.99292	−0.996460	−	0.0840693i	$-0.973208\pi$
$233$	−7088.00	−1.99292	−0.996460	+	0.0840693i	$0.973208\pi$
$234$	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
$241$	−5310.00	−1.41928	−0.709641	−	0.704563i	$-0.751143\pi$
$241$	−5310.00	−1.41928	−0.709641	+	0.704563i	$0.751143\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 7546.00i	− 1.96774i
$246$	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
$257$	8096.00	1.96504	0.982519	−	0.186164i	$-0.0596056\pi$
$257$	8096.00	1.96504	0.982519	+	0.186164i	$0.0596056\pi$
$258$	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
$265$	11396.0	2.64170
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3406.00i	0.771998i	0.922499	+	0.385999i	$0.126143\pi$
$269$	3406.00i	0.771998i	−0.922499	+	0.385999i	$0.873857\pi$
$270$	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
$275$	0	0
$276$	0	0
$277$	1316.00i	0.285454i	0.989762	+	0.142727i	$0.0455871\pi$
$277$	1316.00i	0.285454i	−0.989762	+	0.142727i	$0.954413\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5792.00	1.22961	0.614807	−	0.788677i	$-0.289234\pi$
$281$	5792.00	1.22961	0.614807	+	0.788677i	$0.289234\pi$
$282$	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
$289$	5903.00	1.20151
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9418.00i	1.87783i	0.344143	+	0.938917i	$0.388169\pi$
$293$	9418.00i	1.87783i	−0.344143	+	0.938917i	$0.611831\pi$
$294$	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
$305$	−10296.0	−1.93294
$306$	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
$313$	−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
$317$	− 10274.0i	− 1.82033i	−0.414243	−	0.910166i	$-0.635954\pi$
$317$	− 10274.0i	− 1.82033i	0.414243	−	0.910166i	$-0.364046\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 33028.0i	− 5.63712i
$326$	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
$337$	12366.0	1.99887	0.999435	−	0.0336216i	$-0.0107041\pi$
$337$	12366.0	1.99887	0.999435	+	0.0336216i	$0.0107041\pi$
$338$	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
$353$	−12848.0	−1.93720	−0.968598	−	0.248633i	$-0.920019\pi$
$353$	−12848.0	−1.93720	−0.968598	+	0.248633i	$0.920019\pi$
$354$	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
$365$	24156.0i	3.46406i
$366$	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
$373$	− 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
$377$	11960.0	1.63388
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	374.000i	0.0487469i	0.999703	+	0.0243735i	$0.00775908\pi$
$389$	374.000i	0.0487469i	−0.999703	+	0.0243735i	$0.992241\pi$
$390$	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
$401$	15880.0	1.97758	0.988790	−	0.149315i	$-0.0477068\pi$
$401$	15880.0	1.97758	0.988790	+	0.149315i	$0.0477068\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−7146.00	−0.863929	−0.431964	−	0.901891i	$-0.642179\pi$
$409$	−7146.00	−0.863929	−0.431964	+	0.901891i	$0.642179\pi$
$410$	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
$421$	13412.0i	1.55264i	0.630340	+	0.776319i	$0.282916\pi$
$421$	13412.0i	1.55264i	−0.630340	+	0.776319i	$0.717084\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	37336.0	4.26132
$426$	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	$-0.413040\pi$
$433$	4862.00	0.539614	0.269807	+	0.962914i	$0.413040\pi$
$434$	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
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	3872.00i	0.412473i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10120.0	−1.06368	−0.531840	−	0.846845i	$-0.678499\pi$
$449$	−10120.0	−1.06368	−0.531840	+	0.846845i	$0.678499\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−16506.0	−1.68954	−0.844768	−	0.535132i	$-0.820262\pi$
$457$	−16506.0	−1.68954	−0.844768	+	0.535132i	$0.820262\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 2318.00i	− 0.234187i	−0.993121	−	0.117093i	$-0.962642\pi$
$461$	− 2318.00i	− 0.234187i	0.993121	−	0.117093i	$-0.0373577\pi$
$462$	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
$467$	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
$481$	−36432.0	−3.45355
$482$	0	0
$483$	0	0
$484$	0	0
$485$	13068.0i	1.22348i
$486$	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
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	13520.0i	1.23511i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	13156.0	1.15928
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14270.0i	1.24265i	0.783555	+	0.621323i	$0.213404\pi$
$509$	14270.0i	1.24265i	−0.783555	+	0.621323i	$0.786596\pi$
$510$	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
$521$	−1480.00	−0.124453	−0.0622265	−	0.998062i	$-0.519820\pi$
$521$	−1480.00	−0.124453	−0.0622265	+	0.998062i	$0.519820\pi$
$522$	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
$529$	−12167.0	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	43424.0i	3.52890i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	24460.0i	1.94384i	0.235311	+	0.971920i	$0.424389\pi$
$541$	24460.0i	1.94384i	−0.235311	+	0.971920i	$0.575611\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	32120.0	2.52453
$546$	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
$557$	8626.00i	0.656186i	0.944646	+	0.328093i	$0.106406\pi$
$557$	8626.00i	0.656186i	−0.944646	+	0.328093i	$0.893594\pi$
$558$	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
$565$	− 29216.0i	− 2.17544i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4280.00	0.315337	0.157669	−	0.987492i	$-0.449602\pi$
$569$	4280.00	0.315337	0.157669	+	0.987492i	$0.449602\pi$
$570$	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
$577$	−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
$593$	24368.0	1.68748	0.843738	−	0.536755i	$-0.180350\pi$
$593$	24368.0	1.68748	0.843738	+	0.536755i	$0.180350\pi$
$594$	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	$-0.303862\pi$
$601$	17030.0	1.15585	0.577927	+	0.816089i	$0.303862\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	29282.0i	1.96774i
$606$	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
$613$	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
$617$	−26464.0	−1.72674	−0.863372	−	0.504569i	$-0.831652\pi$
$617$	−26464.0	−1.72674	−0.863372	+	0.504569i	$0.831652\pi$
$618$	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
$625$	68381.0	4.37638
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 41184.0i	− 2.61067i
$630$	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
$637$	− 31556.0i	− 1.96279i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	14872.0	0.916394	0.458197	−	0.888851i	$-0.348495\pi$
$641$	14872.0	0.916394	0.458197	+	0.888851i	$0.348495\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 33358.0i	− 1.99908i	−0.0303236	−	0.999540i	$-0.509654\pi$
$653$	− 33358.0i	− 1.99908i	0.0303236	−	0.999540i	$-0.490346\pi$
$654$	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	$-0.775071\pi$
$661$	− 22068.0i	− 1.29856i	0.760551	−	0.649278i	$-0.224929\pi$
$662$	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
$677$	4054.00i	0.230145i	0.993357	+	0.115072i	$0.0367100\pi$
$677$	4054.00i	0.230145i	−0.993357	+	0.115072i	$0.963290\pi$
$678$	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
$685$	− 61072.0i	− 3.40648i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	47656.0	2.63505
$690$	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
$697$	−49088.0	−2.66763
$698$	0	0
$699$	0	0
$700$	0	0
$701$	20030.0i	1.07920i	0.841920	+	0.539602i	$0.181425\pi$
$701$	20030.0i	1.07920i	−0.841920	+	0.539602i	$0.818575\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	8404.00i	0.445161i	0.974914	+	0.222580i	$0.0714479\pi$
$709$	8404.00i	0.445161i	−0.974914	+	0.222580i	$0.928552\pi$
$710$	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
$725$	46670.0i	2.39073i
$726$	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
$731$	0	0
$732$	0	0
$733$	8732.00i	0.440005i	0.975499	+	0.220003i	$0.0706066\pi$
$733$	8732.00i	0.440005i	−0.975499	+	0.220003i	$0.929393\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	77308.0	3.80181
$746$	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
$757$	22516.0i	1.08105i	0.841327	+	0.540527i	$0.181775\pi$
$757$	22516.0i	1.08105i	−0.841327	+	0.540527i	$0.818225\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−27320.0	−1.30138	−0.650689	−	0.759344i	$-0.725520\pi$
$761$	−27320.0	−1.30138	−0.650689	+	0.759344i	$0.725520\pi$
$762$	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
$773$	− 39542.0i	− 1.83988i	−0.392060	−	0.919940i	$-0.628237\pi$
$773$	− 39542.0i	− 1.83988i	0.392060	−	0.919940i	$-0.371763\pi$
$774$	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
$785$	−86328.0	−3.92507
$786$	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
$793$	−43056.0	−1.92807
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41954.0i	1.86460i	0.361685	+	0.932300i	$0.382202\pi$
$797$	41954.0i	1.86460i	−0.361685	+	0.932300i	$0.617798\pi$
$798$	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
$809$	−39704.0	−1.72549	−0.862743	−	0.505643i	$-0.831255\pi$
$809$	−39704.0	−1.72549	−0.862743	+	0.505643i	$0.831255\pi$
$810$	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
$821$	1850.00i	0.0786424i	0.999227	+	0.0393212i	$0.0125196\pi$
$821$	1850.00i	0.0786424i	−0.999227	+	0.0393212i	$0.987480\pi$
$822$	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
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	41740.0i	1.74872i	0.485276	+	0.874361i	$0.338719\pi$
$829$	41740.0i	1.74872i	−0.485276	+	0.874361i	$0.661281\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	35672.0	1.48375
$834$	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
$841$	7489.00	0.307065
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 137874.i	− 5.61303i
$846$	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	$0.365883\pi$
$853$	45468.0i	1.82508i	−0.408986	+	0.912541i	$0.634117\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20056.0	0.799416	0.399708	−	0.916642i	$-0.369111\pi$
$857$	20056.0	0.799416	0.399708	+	0.916642i	$0.369111\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−89804.0	−3.52997
$866$	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
$881$	−51808.0	−1.98122	−0.990611	−	0.136714i	$-0.956346\pi$
$881$	−51808.0	−1.98122	−0.990611	+	0.136714i	$0.956346\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	0	0
$889$	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
$901$	53872.0i	1.99194i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	62920.0	2.31108
$906$	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
$919$	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
$925$	− 142164.i	− 5.05332i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−47480.0	−1.67682	−0.838411	−	0.545038i	$-0.816515\pi$
$929$	−47480.0	−1.67682	−0.838411	+	0.545038i	$0.816515\pi$
$930$	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	$-0.860542\pi$
$937$	−51946.0	−1.81110	−0.905551	+	0.424238i	$0.860542\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	31378.0i	1.08703i	0.839400	+	0.543514i	$0.182907\pi$
$941$	31378.0i	1.08703i	−0.839400	+	0.543514i	$0.817093\pi$
$942$	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
$949$	101016.i	3.45534i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15512.0	0.527264	0.263632	−	0.964623i	$-0.415079\pi$
$953$	15512.0	0.527264	0.263632	+	0.964623i	$0.415079\pi$
$954$	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
$965$	− 117964.i	− 3.93512i
$966$	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
$971$	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
$977$	−22936.0	−0.751062	−0.375531	−	0.926810i	$-0.622540\pi$
$977$	−22936.0	−0.751062	−0.375531	+	0.926810i	$0.622540\pi$
$978$	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
$985$	25828.0	0.835481
$986$	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
$997$	− 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.d.c.577.2	yes	2
3.2	odd	2		1152.4.d.f.577.1	yes	2
4.3	odd	2	CM	1152.4.d.c.577.2	yes	2
8.3	odd	2	inner	1152.4.d.c.577.1	✓	2
8.5	even	2	inner	1152.4.d.c.577.1	✓	2
12.11	even	2		1152.4.d.f.577.1	yes	2
16.3	odd	4		2304.4.a.o.1.1		1
16.5	even	4		2304.4.a.b.1.1		1
16.11	odd	4		2304.4.a.b.1.1		1
16.13	even	4		2304.4.a.o.1.1		1
24.5	odd	2		1152.4.d.f.577.2	yes	2
24.11	even	2		1152.4.d.f.577.2	yes	2
48.5	odd	4		2304.4.a.p.1.1		1
48.11	even	4		2304.4.a.p.1.1		1
48.29	odd	4		2304.4.a.a.1.1		1
48.35	even	4		2304.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.d.c.577.1	✓	2	8.3	odd	2	inner
1152.4.d.c.577.1	✓	2	8.5	even	2	inner
1152.4.d.c.577.2	yes	2	1.1	even	1	trivial
1152.4.d.c.577.2	yes	2	4.3	odd	2	CM
1152.4.d.f.577.1	yes	2	3.2	odd	2
1152.4.d.f.577.1	yes	2	12.11	even	2
1152.4.d.f.577.2	yes	2	24.5	odd	2
1152.4.d.f.577.2	yes	2	24.11	even	2
2304.4.a.a.1.1		1	48.29	odd	4
2304.4.a.a.1.1		1	48.35	even	4
2304.4.a.b.1.1		1	16.5	even	4
2304.4.a.b.1.1		1	16.11	odd	4
2304.4.a.o.1.1		1	16.3	odd	4
2304.4.a.o.1.1		1	16.13	even	4
2304.4.a.p.1.1		1	48.5	odd	4
2304.4.a.p.1.1		1	48.11	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+22.0000i q^{5} +92.0000i q^{13} -104.000 q^{17} -359.000 q^{25} -130.000i q^{29} +396.000i q^{37} +472.000 q^{41} -343.000 q^{49} -518.000i q^{53} +468.000i q^{61} -2024.00 q^{65} +1098.00 q^{73} -2288.00i q^{85} +176.000 q^{89} +594.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 208 q^{17} - 718 q^{25} + 944 q^{41} - 686 q^{49} - 4048 q^{65} + 2196 q^{73} + 352 q^{89} + 1188 q^{97}+O(q^{100}) \) 2 * q - 208 * q^17 - 718 * q^25 + 944 * q^41 - 686 * q^49 - 4048 * q^65 + 2196 * q^73 + 352 * q^89 + 1188 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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