LMFDB - Embedded newform 1008.4.k.a.881.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(881,1008)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1008.881");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.k (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 63)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			881.3
Root			$2.57794i$ of defining polynomial
Character	$\chi$	$=$	1008.881
Dual form			1008.4.k.a.881.4

$q$-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 $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	18.5203	1.00000
$7$	18.5203	1.00000
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 66.7915i	− 1.83076i	−0.402586	−	0.915382i	$-0.631889\pi$
$11$	− 66.7915i	− 1.83076i	0.402586	−	0.915382i	$-0.368111\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 125.124i	− 1.13435i	−0.823597	−	0.567176i	$-0.808036\pi$
$23$	− 125.124i	− 1.13435i	0.823597	−	0.567176i	$-0.191964\pi$
$24$	0	0
$25$	−125.000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	69.7031i	0.446329i	0.974781	+	0.223165i	$0.0716388\pi$
$29$	69.7031i	0.446329i	−0.974781	+	0.223165i	$0.928361\pi$
$30$	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
$37$	10.5830	0.0470226	0.0235113	−	0.999724i	$-0.492515\pi$
$37$	10.5830	0.0470226	0.0235113	+	0.999724i	$0.492515\pi$
$38$	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
$43$	−534.442	−1.89539	−0.947693	−	0.319183i	$-0.896592\pi$
$43$	−534.442	−1.89539	−0.947693	+	0.319183i	$0.896592\pi$
$44$	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
$53$	65.4772i	0.169698i	0.996394	+	0.0848489i	$0.0270408\pi$
$53$	65.4772i	0.169698i	−0.996394	+	0.0848489i	$0.972959\pi$
$54$	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
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−740.000	−1.34933	−0.674667	−	0.738122i	$-0.735713\pi$
$67$	−740.000	−1.34933	−0.674667	+	0.738122i	$0.735713\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 205.717i	− 0.343861i	−0.985109	−	0.171931i	$-0.945000\pi$
$71$	− 205.717i	− 0.343861i	0.985109	−	0.171931i	$-0.0550005\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1237.00i	− 1.83076i
$78$	0	0
$79$	1384.00	1.97104	0.985520	−	0.169559i	$-0.0542343\pi$
$79$	1384.00	1.97104	0.985520	+	0.169559i	$0.0542343\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	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
$95$	0	0
$96$	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
$107$	− 2213.53i	− 1.99991i	−0.00945193	−	0.999955i	$-0.503009\pi$
$107$	− 2213.53i	− 1.99991i	0.00945193	−	0.999955i	$-0.496991\pi$
$108$	0	0
$109$	−2275.35	−1.99944	−0.999718	−	0.0237260i	$-0.992447\pi$
$109$	−2275.35	−1.99944	−0.999718	+	0.0237260i	$0.992447\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2105.12i	− 1.75251i	−0.481849	−	0.876254i	$-0.660035\pi$
$113$	− 2105.12i	− 1.75251i	0.481849	−	0.876254i	$-0.339965\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−3130.11	−2.35170
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2000.00	−1.39741	−0.698706	−	0.715409i	$-0.746240\pi$
$127$	−2000.00	−1.39741	−0.698706	+	0.715409i	$0.746240\pi$
$128$	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
$137$	1645.34i	1.02606i	0.858370	+	0.513031i	$0.171478\pi$
$137$	1645.34i	1.02606i	−0.858370	+	0.513031i	$0.828522\pi$
$138$	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
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3082.50i	− 1.69482i	−0.530941	−	0.847409i	$-0.678161\pi$
$149$	− 3082.50i	− 1.69482i	0.530941	−	0.847409i	$-0.321839\pi$
$150$	0	0
$151$	−2248.89	−1.21200	−0.606000	−	0.795465i	$-0.707227\pi$
$151$	−2248.89	−1.21200	−0.606000	+	0.795465i	$0.707227\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 2317.32i	− 1.13435i
$162$	0	0
$163$	1780.00	0.855340	0.427670	−	0.903935i	$-0.359335\pi$
$163$	1780.00	0.855340	0.427670	+	0.903935i	$0.359335\pi$
$164$	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
$169$	2197.00	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−2315.03	−1.00000
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4523.06i	− 1.88866i	−0.329005	−	0.944328i	$-0.606713\pi$
$179$	− 4523.06i	− 1.88866i	0.329005	−	0.944328i	$-0.393287\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	503.386i	0.190700i	0.995444	+	0.0953502i	$0.0303971\pi$
$191$	503.386i	0.190700i	−0.995444	+	0.0953502i	$0.969603\pi$
$192$	0	0
$193$	2772.75	1.03413	0.517064	−	0.855947i	$-0.327025\pi$
$193$	2772.75	1.03413	0.517064	+	0.855947i	$0.327025\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2021.80i	− 0.731205i	−0.930771	−	0.365603i	$-0.880863\pi$
$197$	− 2021.80i	− 0.731205i	0.930771	−	0.365603i	$-0.119137\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1290.92i	0.446329i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−1772.65	−0.578363	−0.289181	−	0.957274i	$-0.593383\pi$
$211$	−1772.65	−0.578363	−0.289181	+	0.957274i	$0.593383\pi$
$212$	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
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7101.24i	1.99664i	0.0579219	+	0.998321i	$0.481553\pi$
$233$	7101.24i	1.99664i	−0.0579219	+	0.998321i	$0.518447\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 4897.58i	− 1.32551i	−0.748834	−	0.662757i	$-0.769386\pi$
$239$	− 4897.58i	− 1.32551i	0.748834	−	0.662757i	$-0.230614\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	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
$253$	−8357.20	−2.07673
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0		−	1.00000i	$-0.5\pi$
$257$	0	0			1.00000i	$0.5\pi$
$258$	0	0
$259$	196.000	0.0470226
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 8164.84i	− 1.91432i	−0.289561	−	0.957159i	$-0.593509\pi$
$263$	− 8164.84i	− 1.91432i	0.289561	−	0.957159i	$-0.406491\pi$
$264$	0	0
$265$	0	0
$266$	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
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	8348.94i	1.83076i
$276$	0	0
$277$	7310.00	1.58561	0.792807	−	0.609472i	$-0.208619\pi$
$277$	7310.00	1.58561	0.792807	+	0.609472i	$0.208619\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2841.56i	0.603250i	0.953427	+	0.301625i	$0.0975291\pi$
$281$	2841.56i	0.603250i	−0.953427	+	0.301625i	$0.902471\pi$
$282$	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
$289$	−4913.00	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−9898.00	−1.89539
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	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
$317$	− 11207.0i	− 1.98565i	−0.119593	−	0.992823i	$-0.538159\pi$
$317$	− 11207.0i	− 1.98565i	0.119593	−	0.992823i	$-0.461841\pi$
$318$	0	0
$319$	4655.58	0.817124
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−5106.30	−0.847938	−0.423969	−	0.905677i	$-0.639364\pi$
$331$	−5106.30	−0.847938	−0.423969	+	0.905677i	$0.639364\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	11916.5	1.92621	0.963103	−	0.269135i	$-0.0867376\pi$
$337$	11916.5	1.92621	0.963103	+	0.269135i	$0.0867376\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	6352.45	1.00000
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5770.28i	0.892694i	0.894860	+	0.446347i	$0.147275\pi$
$347$	5770.28i	0.892694i	−0.894860	+	0.446347i	$0.852725\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0		−	1.00000i	$-0.5\pi$
$353$	0	0			1.00000i	$0.5\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	13456.9i	1.97835i	0.146729	+	0.989177i	$0.453125\pi$
$359$	13456.9i	1.97835i	−0.146729	+	0.989177i	$0.546875\pi$
$360$	0	0
$361$	6859.00	1.00000
$362$	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
$371$	1212.65i	0.169698i
$372$	0	0
$373$	13970.0	1.93925	0.969624	−	0.244602i	$-0.0786573\pi$
$373$	13970.0	1.93925	0.969624	+	0.244602i	$0.0786573\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	8704.52	1.17974	0.589870	−	0.807498i	$-0.299179\pi$
$379$	8704.52	1.17974	0.589870	+	0.807498i	$0.299179\pi$
$380$	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
$389$	− 451.891i	− 0.0588992i	−0.999566	−	0.0294496i	$-0.990625\pi$
$389$	− 451.891i	− 0.0588992i	0.999566	−	0.0294496i	$-0.00937545\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	12429.8i	1.54791i	0.633239	+	0.773956i	$0.281725\pi$
$401$	12429.8i	1.54791i	−0.633239	+	0.773956i	$0.718275\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 706.855i	− 0.0860873i
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	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		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	−15262.0	−1.76680	−0.883402	−	0.468616i	$-0.844753\pi$
$421$	−15262.0	−1.76680	−0.883402	+	0.468616i	$0.844753\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17180.8i	1.92011i	0.279803	+	0.960057i	$0.409731\pi$
$431$	17180.8i	1.92011i	−0.279803	+	0.960057i	$0.590269\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	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
$443$	14264.3i	1.52983i	0.644129	+	0.764917i	$0.277220\pi$
$443$	14264.3i	1.52983i	−0.644129	+	0.764917i	$0.722780\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11421.0i	1.20043i	0.799841	+	0.600213i	$0.204917\pi$
$449$	11421.0i	1.20043i	−0.799841	+	0.600213i	$0.795083\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	17821.8	1.82422	0.912109	−	0.409947i	$-0.134453\pi$
$457$	17821.8	1.82422	0.912109	+	0.409947i	$0.134453\pi$
$458$	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
$463$	8440.00	0.847171	0.423585	−	0.905856i	$-0.360771\pi$
$463$	8440.00	0.847171	0.423585	+	0.905856i	$0.360771\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	−13705.0	−1.34933
$470$	0	0
$471$	0	0
$472$	0	0
$473$	35696.2i	3.47001i
$474$	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
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3296.61	−0.306742	−0.153371	−	0.988169i	$-0.549013\pi$
$487$	−3296.61	−0.306742	−0.153371	+	0.988169i	$0.549013\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 19811.9i	− 1.82097i	−0.413540	−	0.910486i	$-0.635708\pi$
$491$	− 19811.9i	− 1.82097i	0.413540	−	0.910486i	$-0.364292\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3809.93i	− 0.343861i
$498$	0	0
$499$	21086.6	1.89172	0.945859	−	0.324577i	$-0.105222\pi$
$499$	21086.6	1.89172	0.945859	+	0.324577i	$0.105222\pi$
$500$	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
$505$	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
$527$	0	0
$528$	0	0
$529$	−3488.94	−0.286754
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 22909.5i	− 1.83076i
$540$	0	0
$541$	15878.0	1.26183	0.630914	−	0.775853i	$-0.282680\pi$
$541$	15878.0	1.26183	0.630914	+	0.775853i	$0.282680\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−12980.0	−1.01460	−0.507299	−	0.861770i	$-0.669356\pi$
$547$	−12980.0	−1.01460	−0.507299	+	0.861770i	$0.669356\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	25632.0	1.97104
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 26141.0i	− 1.98856i	−0.106803	−	0.994280i	$-0.534061\pi$
$557$	− 26141.0i	− 1.98856i	0.106803	−	0.994280i	$-0.465939\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21569.7i	1.58919i	0.607138	+	0.794596i	$0.292317\pi$
$569$	21569.7i	1.58919i	−0.607138	+	0.794596i	$0.707683\pi$
$570$	0	0
$571$	−6788.00	−0.497494	−0.248747	−	0.968569i	$-0.580019\pi$
$571$	−6788.00	−0.497494	−0.248747	+	0.968569i	$0.580019\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	15640.5i	1.13435i
$576$	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
$583$	4373.32	0.310677
$584$	0	0
$585$	0	0
$586$	0	0
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6359.56i	0.433797i	0.976194	+	0.216899i	$0.0695941\pi$
$599$	6359.56i	0.433797i	−0.976194	+	0.216899i	$0.930406\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	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
$613$	−15010.0	−0.988986	−0.494493	−	0.869182i	$-0.664646\pi$
$613$	−15010.0	−0.988986	−0.494493	+	0.869182i	$0.664646\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 23368.2i	− 1.52474i	−0.647139	−	0.762372i	$-0.724035\pi$
$617$	− 23368.2i	− 1.52474i	0.647139	−	0.762372i	$-0.275965\pi$
$618$	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
$625$	15625.0	1.00000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−26192.0	−1.65244	−0.826218	−	0.563351i	$-0.809512\pi$
$631$	−26192.0	−1.65244	−0.826218	+	0.563351i	$0.809512\pi$
$632$	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
$641$	15798.1i	0.973458i	0.873553	+	0.486729i	$0.161810\pi$
$641$	15798.1i	0.973458i	−0.873553	+	0.486729i	$0.838190\pi$
$642$	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
$653$	5305.56i	0.317952i	0.987282	+	0.158976i	$0.0508192\pi$
$653$	5305.56i	0.317952i	−0.987282	+	0.158976i	$0.949181\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	25166.1i	1.48761i	0.668399	+	0.743803i	$0.266980\pi$
$659$	25166.1i	1.48761i	−0.668399	+	0.743803i	$0.733020\pi$
$660$	0	0
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8721.51	0.506294
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−9609.37	−0.550392	−0.275196	−	0.961388i	$-0.588743\pi$
$673$	−9609.37	−0.550392	−0.275196	+	0.961388i	$0.588743\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	0	0		−	1.00000i	$-0.5\pi$
$677$	0	0			1.00000i	$0.5\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16522.2i	0.925627i	0.886456	+	0.462813i	$0.153160\pi$
$683$	16522.2i	0.925627i	−0.886456	+	0.462813i	$0.846840\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 29047.8i	− 1.56508i	−0.622602	−	0.782539i	$-0.713924\pi$
$701$	− 29047.8i	− 1.56508i	0.622602	−	0.782539i	$-0.286076\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−35611.8	−1.88636	−0.943180	−	0.332281i	$-0.892182\pi$
$709$	−35611.8	−1.88636	−0.943180	+	0.332281i	$0.892182\pi$
$710$	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
$725$	− 8712.89i	− 0.446329i
$726$	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
$737$	49425.7i	2.47031i
$738$	0	0
$739$	25324.0	1.26057	0.630283	−	0.776365i	$-0.282939\pi$
$739$	25324.0	1.26057	0.630283	+	0.776365i	$0.282939\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 40237.0i	− 1.98675i	−0.114933	−	0.993373i	$-0.536665\pi$
$743$	− 40237.0i	− 1.98675i	0.114933	−	0.993373i	$-0.463335\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 40995.2i	− 1.99991i
$750$	0	0
$751$	41088.5	1.99646	0.998230	−	0.0594732i	$-0.0189421\pi$
$751$	41088.5	1.99646	0.998230	+	0.0594732i	$0.0189421\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−22848.7	−1.09703	−0.548514	−	0.836141i	$-0.684806\pi$
$757$	−22848.7	−1.09703	−0.548514	+	0.836141i	$0.684806\pi$
$758$	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
$763$	−42140.0	−1.99944
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0	0		−	1.00000i	$-0.5\pi$
$773$	0	0			1.00000i	$0.5\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−13740.2	−0.629529
$782$	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
$791$	− 38987.4i	− 1.75251i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	0	0		−	1.00000i	$-0.5\pi$
$797$	0	0			1.00000i	$0.5\pi$
$798$	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
$809$	− 45420.9i	− 1.97393i	−0.160920	−	0.986967i	$-0.551446\pi$
$809$	− 45420.9i	− 1.97393i	0.160920	−	0.986967i	$-0.448554\pi$
$810$	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
$821$	18176.6i	0.772678i	0.922357	+	0.386339i	$0.126261\pi$
$821$	18176.6i	0.772678i	−0.922357	+	0.386339i	$0.873739\pi$
$822$	0	0
$823$	46240.0	1.95848	0.979238	−	0.202716i	$-0.0649768\pi$
$823$	46240.0	1.95848	0.979238	+	0.202716i	$0.0649768\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 12090.1i	− 0.508359i	−0.967157	−	0.254179i	$-0.918195\pi$
$827$	− 12090.1i	− 0.508359i	0.967157	−	0.254179i	$-0.0818054\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	19530.5	0.800790
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−57970.5	−2.35170
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1324.18i	− 0.0533401i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 18601.9i	− 0.733737i	−0.930273	−	0.366868i	$-0.880430\pi$
$863$	− 18601.9i	− 0.733737i	0.930273	−	0.366868i	$-0.119570\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 92439.5i	− 3.60851i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−6550.00	−0.252198	−0.126099	−	0.992018i	$-0.540246\pi$
$877$	−6550.00	−0.252198	−0.126099	+	0.992018i	$0.540246\pi$
$878$	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
$883$	−43014.6	−1.63936	−0.819681	−	0.572820i	$-0.805850\pi$
$883$	−43014.6	−1.63936	−0.819681	+	0.572820i	$0.805850\pi$
$884$	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
$889$	−37040.5	−1.39741
$890$	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
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−14250.0	−0.521680	−0.260840	−	0.965382i	$-0.584000\pi$
$907$	−14250.0	−0.521680	−0.260840	+	0.965382i	$0.584000\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 54982.7i	− 1.99962i	−0.0193736	−	0.999812i	$-0.506167\pi$
$911$	− 54982.7i	− 1.99962i	0.0193736	−	0.999812i	$-0.493833\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−51301.1	−1.84142	−0.920711	−	0.390244i	$-0.872391\pi$
$919$	−51301.1	−1.84142	−0.920711	+	0.390244i	$0.872391\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−1322.88	−0.0470226
$926$	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
$931$	0	0
$932$	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
$947$	12007.4i	0.412025i	0.978549	+	0.206013i	$0.0660488\pi$
$947$	12007.4i	0.412025i	−0.978549	+	0.206013i	$0.933951\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15373.4i	0.522553i	0.965264	+	0.261276i	$0.0841434\pi$
$953$	15373.4i	0.522553i	−0.965264	+	0.261276i	$0.915857\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	30472.1i	1.02606i
$960$	0	0
$961$	29791.0	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−52040.0	−1.73060	−0.865302	−	0.501251i	$-0.832873\pi$
$967$	−52040.0	−1.73060	−0.865302	+	0.501251i	$0.832873\pi$
$968$	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
$977$	− 7584.55i	− 0.248364i	−0.992259	−	0.124182i	$-0.960369\pi$
$977$	− 7584.55i	− 0.248364i	0.992259	−	0.124182i	$-0.0396306\pi$
$978$	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
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	66871.3i	2.15003i
$990$	0	0
$991$	−24155.7	−0.774300	−0.387150	−	0.922017i	$-0.626540\pi$
$991$	−24155.7	−0.774300	−0.387150	+	0.922017i	$0.626540\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	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	1008.4.k.a.881.3		4
3.2	odd	2	inner	1008.4.k.a.881.4		4
4.3	odd	2		63.4.c.a.62.4	yes	4
7.6	odd	2	CM	1008.4.k.a.881.3		4
12.11	even	2		63.4.c.a.62.1	✓	4
21.20	even	2	inner	1008.4.k.a.881.4		4
28.3	even	6		441.4.p.b.215.1		8
28.11	odd	6		441.4.p.b.215.1		8
28.19	even	6		441.4.p.b.80.4		8
28.23	odd	6		441.4.p.b.80.4		8
28.27	even	2		63.4.c.a.62.4	yes	4
84.11	even	6		441.4.p.b.215.4		8
84.23	even	6		441.4.p.b.80.1		8
84.47	odd	6		441.4.p.b.80.1		8
84.59	odd	6		441.4.p.b.215.4		8
84.83	odd	2		63.4.c.a.62.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.c.a.62.1	✓	4	12.11	even	2
63.4.c.a.62.1	✓	4	84.83	odd	2
63.4.c.a.62.4	yes	4	4.3	odd	2
63.4.c.a.62.4	yes	4	28.27	even	2
441.4.p.b.80.1		8	84.23	even	6
441.4.p.b.80.1		8	84.47	odd	6
441.4.p.b.80.4		8	28.19	even	6
441.4.p.b.80.4		8	28.23	odd	6
441.4.p.b.215.1		8	28.3	even	6
441.4.p.b.215.1		8	28.11	odd	6
441.4.p.b.215.4		8	84.11	even	6
441.4.p.b.215.4		8	84.59	odd	6
1008.4.k.a.881.3		4	1.1	even	1	trivial
1008.4.k.a.881.3		4	7.6	odd	2	CM
1008.4.k.a.881.4		4	3.2	odd	2	inner
1008.4.k.a.881.4		4	21.20	even	2	inner

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+18.5203 q^{7} +O(q^{10})\)	\(q+18.5203 q^{7} -66.7915i q^{11} -125.124i q^{23} -125.000 q^{25} +69.7031i q^{29} +10.5830 q^{37} -534.442 q^{43} +343.000 q^{49} +65.4772i q^{53} -740.000 q^{67} -205.717i q^{71} -1237.00i q^{77} +1384.00 q^{79} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 500 q^{25} + 1372 q^{49} - 2960 q^{67} + 5536 q^{79}+O(q^{100}) \) 4 * q - 500 * q^25 + 1372 * q^49 - 2960 * q^67 + 5536 * q^79

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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