LMFDB - Embedded newform 1152.4.f.b.575.3

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 = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,148] 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			575.3
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1152.575
Dual form			1152.4.f.b.575.4

$q$-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
$5$	12.7279	1.13842	0.569210	−	0.822192i	$-0.307249\pi$
$5$	12.7279	1.13842	0.569210	+	0.822192i	$0.307249\pi$
$6$	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	$-0.938499\pi$
$13$	− 18.0000i	− 0.384023i	0.981393	−	0.192012i	$-0.0615011\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.07107i	− 0.100882i	−0.998727	−	0.0504408i	$-0.983937\pi$
$17$	− 7.07107i	− 0.100882i	0.998727	−	0.0504408i	$-0.0160626\pi$
$18$	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
$25$	37.0000	0.296000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	292.742	1.87451	0.937256	−	0.348641i	$-0.113357\pi$
$29$	292.742	1.87451	0.937256	+	0.348641i	$0.113357\pi$
$30$	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
$41$	− 171.120i	− 0.651815i	−0.945402	−	0.325908i	$-0.894330\pi$
$41$	− 171.120i	− 0.651815i	0.945402	−	0.325908i	$-0.105670\pi$
$42$	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
$53$	−38.1838	−0.0989612	−0.0494806	−	0.998775i	$-0.515757\pi$
$53$	−38.1838	−0.0989612	−0.0494806	+	0.998775i	$0.515757\pi$
$54$	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
$65$	− 229.103i	− 0.437180i
$66$	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	$-0.342601\pi$
$73$	592.000	0.949156	0.474578	+	0.880214i	$0.342601\pi$
$74$	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
$85$	− 90.0000i	− 0.114846i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 1305.32i	− 1.55465i	−0.629101	−	0.777323i	$-0.716577\pi$
$89$	− 1305.32i	− 1.55465i	0.629101	−	0.777323i	$-0.283423\pi$
$90$	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	$-0.100625\pi$
$97$	1816.00	1.90090	0.950448	+	0.310884i	$0.100625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1794.64	−1.76805	−0.884025	−	0.467440i	$-0.845177\pi$
$101$	−1794.64	−1.76805	−0.884025	+	0.467440i	$0.845177\pi$
$102$	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	$0.278321\pi$
$109$	1746.00i	1.53428i	−0.641480	+	0.767140i	$0.721679\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 476.590i	− 0.396759i	−0.980125	−	0.198380i	$-0.936432\pi$
$113$	− 476.590i	− 0.396759i	0.980125	−	0.198380i	$-0.0635679\pi$
$114$	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
$125$	−1120.06	−0.801448
$126$	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
$137$	− 3098.54i	− 1.93231i	−0.257965	−	0.966154i	$-0.583052\pi$
$137$	− 3098.54i	− 1.93231i	0.257965	−	0.966154i	$-0.416948\pi$
$138$	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
$145$	3726.00	2.13398
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1820.09	1.00072	0.500362	−	0.865816i	$-0.333200\pi$
$149$	1820.09	1.00072	0.500362	+	0.865816i	$0.333200\pi$
$150$	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
$173$	1463.71	0.643259	0.321630	−	0.946866i	$-0.395769\pi$
$173$	1463.71	0.643259	0.321630	+	0.946866i	$0.395769\pi$
$174$	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	$0.300213\pi$
$181$	3942.00i	1.61882i	−0.587243	+	0.809410i	$0.699787\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 5040.26i	− 2.00307i
$186$	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
$197$	2991.06	1.08175	0.540874	−	0.841104i	$-0.318094\pi$
$197$	2991.06	1.08175	0.540874	+	0.841104i	$0.318094\pi$
$198$	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
$205$	− 2178.00i	− 0.742040i
$206$	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
$221$	−127.279	−0.0387408
$222$	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	$-0.626577\pi$
$229$	− 6390.00i	− 1.84394i	0.387257	−	0.921972i	$-0.373423\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 4589.12i	− 1.29032i	−0.764050	−	0.645158i	$-0.776792\pi$
$233$	− 4589.12i	− 1.29032i	0.764050	−	0.645158i	$-0.223208\pi$
$234$	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	$-0.251143\pi$
$241$	5272.00	1.40913	0.704563	+	0.709641i	$0.251143\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4365.68	1.13842
$246$	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
$257$	4640.03i	1.12622i	0.826383	+	0.563108i	$0.190394\pi$
$257$	4640.03i	1.12622i	−0.826383	+	0.563108i	$0.809606\pi$
$258$	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
$265$	−486.000	−0.112659
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3347.44	−0.758726	−0.379363	−	0.925248i	$-0.623857\pi$
$269$	−3347.44	−0.758726	−0.379363	+	0.925248i	$0.623857\pi$
$270$	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	$0.454413\pi$
$277$	9126.00i	1.97952i	−0.142727	+	0.989762i	$0.545587\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 9349.37i	− 1.98483i	−0.122945	−	0.992414i	$-0.539234\pi$
$281$	− 9349.37i	− 1.98483i	0.122945	−	0.992414i	$-0.460766\pi$
$282$	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
$289$	4863.00	0.989823
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9100.46	−1.81452	−0.907261	−	0.420569i	$-0.861831\pi$
$293$	−9100.46	−1.81452	−0.907261	+	0.420569i	$0.861831\pi$
$294$	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
$305$	5956.67i	1.11829i
$306$	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
$317$	−3958.38	−0.701341	−0.350670	−	0.936499i	$-0.614046\pi$
$317$	−3958.38	−0.701341	−0.350670	+	0.936499i	$0.614046\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 666.000i	− 0.113671i
$326$	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	$-0.489296\pi$
$337$	416.000	0.0672432	0.0336216	+	0.999435i	$0.489296\pi$
$338$	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
$353$	6752.87i	1.01818i	0.860712	+	0.509092i	$0.170019\pi$
$353$	6752.87i	1.01818i	−0.860712	+	0.509092i	$0.829981\pi$
$354$	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
$365$	7534.93	1.08054
$366$	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
$377$	− 5269.36i	− 0.719856i
$378$	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
$389$	−11111.5	−1.44826	−0.724131	−	0.689662i	$-0.757759\pi$
$389$	−11111.5	−1.44826	−0.724131	+	0.689662i	$0.757759\pi$
$390$	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
$401$	12924.5i	1.60952i	0.593598	+	0.804761i	$0.297707\pi$
$401$	12924.5i	1.60952i	−0.593598	+	0.804761i	$0.702293\pi$
$402$	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	$-0.857821\pi$
$409$	−14920.0	−1.80378	−0.901891	+	0.431964i	$0.857821\pi$
$410$	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	$-0.782916\pi$
$421$	− 10890.0i	− 1.26068i	0.776319	−	0.630340i	$-0.217084\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 261.630i	− 0.0298609i
$426$	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
$445$	− 16614.0i	− 1.76984i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	18550.2i	1.94975i	0.222742	+	0.974877i	$0.428499\pi$
$449$	18550.2i	1.94975i	−0.222742	+	0.974877i	$0.571501\pi$
$450$	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	$-0.679738\pi$
$457$	−10456.0	−1.07026	−0.535132	+	0.844768i	$0.679738\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	15540.8	1.57008	0.785040	−	0.619445i	$-0.212642\pi$
$461$	15540.8	1.57008	0.785040	+	0.619445i	$0.212642\pi$
$462$	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
$485$	23113.9	2.16402
$486$	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
$493$	− 2070.00i	− 0.189104i
$494$	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
$505$	−22842.0	−2.01278
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2634.68	0.229431	0.114715	−	0.993398i	$-0.463404\pi$
$509$	2634.68	0.229431	0.114715	+	0.993398i	$0.463404\pi$
$510$	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
$521$	15738.8i	1.32347i	0.749737	+	0.661736i	$0.230180\pi$
$521$	15738.8i	1.32347i	−0.749737	+	0.661736i	$0.769820\pi$
$522$	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
$533$	−3080.16	−0.250312
$534$	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	$0.0756109\pi$
$541$	5922.00i	0.470622i	−0.971920	+	0.235311i	$0.924389\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	22223.0i	1.74665i
$546$	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
$557$	−23661.2	−1.79992	−0.899962	−	0.435969i	$-0.856406\pi$
$557$	−23661.2	−1.79992	−0.899962	+	0.435969i	$0.856406\pi$
$558$	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
$565$	− 6066.00i	− 0.451679i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 15928.3i	− 1.17355i	−0.809751	−	0.586774i	$-0.800398\pi$
$569$	− 15928.3i	− 1.17355i	0.809751	−	0.586774i	$-0.199602\pi$
$570$	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
$593$	6269.21i	0.434141i	0.976156	+	0.217070i	$0.0696501\pi$
$593$	6269.21i	0.434141i	−0.976156	+	0.217070i	$0.930350\pi$
$594$	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
$605$	16940.9	1.13842
$606$	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
$617$	29649.0i	1.93456i	0.253712	+	0.967280i	$0.418348\pi$
$617$	29649.0i	1.93456i	−0.253712	+	0.967280i	$0.581652\pi$
$618$	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
$625$	−18881.0	−1.20838
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−2800.14	−0.177502
$630$	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
$641$	− 9883.94i	− 0.609036i	−0.952507	−	0.304518i	$-0.901505\pi$
$641$	− 9883.94i	− 0.609036i	0.952507	−	0.304518i	$-0.0984954\pi$
$642$	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
$653$	−22872.1	−1.37068	−0.685340	−	0.728224i	$-0.740346\pi$
$653$	−22872.1	−1.37068	−0.685340	+	0.728224i	$0.740346\pi$
$654$	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
$677$	−21879.3	−1.24208	−0.621041	−	0.783778i	$-0.713290\pi$
$677$	−21879.3	−1.24208	−0.621041	+	0.783778i	$0.713290\pi$
$678$	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
$685$	− 39438.0i	− 2.19978i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	687.308i	0.0380034i
$690$	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
$697$	−1210.00	−0.0657561
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−36261.8	−1.95377	−0.976884	−	0.213771i	$-0.931425\pi$
$701$	−36261.8	−1.95377	−0.976884	+	0.213771i	$0.931425\pi$
$702$	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	$0.428552\pi$
$709$	36810.0i	1.94983i	−0.222580	+	0.974914i	$0.571448\pi$
$710$	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
$725$	10831.5	0.554856
$726$	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	$0.429393\pi$
$733$	38718.0i	1.95100i	−0.220003	+	0.975499i	$0.570607\pi$
$734$	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
$745$	23166.0	1.13924
$746$	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	$-0.681775\pi$
$757$	− 35046.0i	− 1.68265i	0.540527	−	0.841327i	$-0.318225\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 3225.82i	− 0.153661i	−0.997044	−	0.0768304i	$-0.975520\pi$
$761$	− 3225.82i	− 0.153661i	0.997044	−	0.0768304i	$-0.0244800\pi$
$762$	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
$773$	39876.6	1.85545	0.927724	−	0.373268i	$-0.121763\pi$
$773$	39876.6	1.85545	0.927724	+	0.373268i	$0.121763\pi$
$774$	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
$785$	− 49944.4i	− 2.27082i
$786$	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
$797$	41174.8	1.82997	0.914986	−	0.403486i	$-0.132202\pi$
$797$	41174.8	1.82997	0.914986	+	0.403486i	$0.132202\pi$
$798$	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
$809$	44529.3i	1.93519i	0.252508	+	0.967595i	$0.418745\pi$
$809$	44529.3i	1.93519i	−0.252508	+	0.967595i	$0.581255\pi$
$810$	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
$821$	−31934.4	−1.35751	−0.678756	−	0.734364i	$-0.737480\pi$
$821$	−31934.4	−1.35751	−0.678756	+	0.734364i	$0.737480\pi$
$822$	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	$-0.838719\pi$
$829$	− 23166.0i	− 0.970553i	0.874361	−	0.485276i	$-0.161281\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2425.38i	− 0.100882i
$834$	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
$841$	61309.0	2.51380
$842$	0	0
$843$	0	0
$844$	0	0
$845$	23839.4	0.970533
$846$	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
$857$	− 46704.4i	− 1.86160i	−0.365528	−	0.930800i	$-0.619111\pi$
$857$	− 46704.4i	− 1.86160i	0.365528	−	0.930800i	$-0.380889\pi$
$858$	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
$865$	18630.0	0.732299
$866$	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
$881$	− 41689.6i	− 1.59428i	−0.603796	−	0.797139i	$-0.706346\pi$
$881$	− 41689.6i	− 1.59428i	0.603796	−	0.797139i	$-0.293654\pi$
$882$	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
$901$	270.000i	0.00998336i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	50173.5i	1.84290i
$906$	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
$925$	− 14652.0i	− 0.520816i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 55399.0i	− 1.95649i	−0.207446	−	0.978246i	$-0.566515\pi$
$929$	− 55399.0i	− 1.95649i	0.207446	−	0.978246i	$-0.433485\pi$
$930$	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
$941$	12078.8	0.418446	0.209223	−	0.977868i	$-0.432907\pi$
$941$	12078.8	0.418446	0.209223	+	0.977868i	$0.432907\pi$
$942$	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
$953$	− 29165.3i	− 0.991351i	−0.868508	−	0.495676i	$-0.834921\pi$
$953$	− 29165.3i	− 0.991351i	0.868508	−	0.495676i	$-0.165079\pi$
$954$	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
$965$	68247.1	2.27663
$966$	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
$977$	− 23808.3i	− 0.779626i	−0.920894	−	0.389813i	$-0.872540\pi$
$977$	− 23808.3i	− 0.779626i	0.920894	−	0.389813i	$-0.127460\pi$
$978$	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
$985$	38070.0	1.23148
$986$	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.b.575.3	yes	4
3.2	odd	2	inner	1152.4.f.b.575.1	✓	4
4.3	odd	2	CM	1152.4.f.b.575.3	yes	4
8.3	odd	2	inner	1152.4.f.b.575.2	yes	4
8.5	even	2	inner	1152.4.f.b.575.2	yes	4
12.11	even	2	inner	1152.4.f.b.575.1	✓	4
16.3	odd	4		2304.4.c.d.2303.1		2
16.5	even	4		2304.4.c.b.2303.2		2
16.11	odd	4		2304.4.c.b.2303.2		2
16.13	even	4		2304.4.c.d.2303.1		2
24.5	odd	2	inner	1152.4.f.b.575.4	yes	4
24.11	even	2	inner	1152.4.f.b.575.4	yes	4
48.5	odd	4		2304.4.c.b.2303.1		2
48.11	even	4		2304.4.c.b.2303.1		2
48.29	odd	4		2304.4.c.d.2303.2		2
48.35	even	4		2304.4.c.d.2303.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1152.4.f.b.575.1	✓	4	3.2	odd	2	inner
1152.4.f.b.575.1	✓	4	12.11	even	2	inner
1152.4.f.b.575.2	yes	4	8.3	odd	2	inner
1152.4.f.b.575.2	yes	4	8.5	even	2	inner
1152.4.f.b.575.3	yes	4	1.1	even	1	trivial
1152.4.f.b.575.3	yes	4	4.3	odd	2	CM
1152.4.f.b.575.4	yes	4	24.5	odd	2	inner
1152.4.f.b.575.4	yes	4	24.11	even	2	inner
2304.4.c.b.2303.1		2	48.5	odd	4
2304.4.c.b.2303.1		2	48.11	even	4
2304.4.c.b.2303.2		2	16.5	even	4
2304.4.c.b.2303.2		2	16.11	odd	4
2304.4.c.d.2303.1		2	16.3	odd	4
2304.4.c.d.2303.1		2	16.13	even	4
2304.4.c.d.2303.2		2	48.29	odd	4
2304.4.c.d.2303.2		2	48.35	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.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+12.7279 q^{5} -18.0000i q^{13} -7.07107i q^{17} +37.0000 q^{25} +292.742 q^{29} -396.000i q^{37} -171.120i q^{41} +343.000 q^{49} -38.1838 q^{53} +468.000i q^{61} -229.103i q^{65} +592.000 q^{73} -90.0000i q^{85} -1305.32i q^{89} +1816.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 148 q^{25} + 1372 q^{49} + 2368 q^{73} + 7264 q^{97}+O(q^{100}) \) 4 * q + 148 * q^25 + 1372 * q^49 + 2368 * q^73 + 7264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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