LMFDB - Embedded newform 1008.4.k.a.881.1

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			881.1
Root			$-1.16372i$ of defining polynomial
Character	$\chi$	$=$	1008.881
Dual form			1008.4.k.a.881.2

$q$-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
$11$	− 29.3750i	− 0.805172i	−0.915382	−	0.402586i	$-0.868111\pi$
$11$	− 29.3750i	− 0.805172i	0.915382	−	0.402586i	$-0.131889\pi$
$12$	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
$23$	181.692i	1.64719i	0.567176	+	0.823597i	$0.308036\pi$
$23$	181.692i	1.64719i	−0.567176	+	0.823597i	$0.691964\pi$
$24$	0	0
$25$	−125.000	−1.00000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 304.463i	− 1.94956i	−0.223165	−	0.974781i	$-0.571639\pi$
$29$	− 304.463i	− 1.94956i	0.223165	−	0.974781i	$-0.428361\pi$
$30$	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	$-0.507485\pi$
$37$	−10.5830	−0.0470226	−0.0235113	+	0.999724i	$0.507485\pi$
$38$	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	$-0.103408\pi$
$43$	534.442	1.89539	0.947693	+	0.319183i	$0.103408\pi$
$44$	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
$53$	768.909i	1.99279i	0.0848489	+	0.996394i	$0.472959\pi$
$53$	768.909i	1.99279i	−0.0848489	+	0.996394i	$0.527041\pi$
$54$	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
$71$	1178.70i	1.97022i	0.171931	+	0.985109i	$0.445000\pi$
$71$	1178.70i	1.97022i	−0.171931	+	0.985109i	$0.555000\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	544.032i	0.805172i
$78$	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
$107$	− 20.9231i	− 0.0189039i	−0.999955	−	0.00945193i	$-0.996991\pi$
$107$	− 20.9231i	− 0.0189039i	0.999955	−	0.00945193i	$-0.00300869\pi$
$108$	0	0
$109$	2275.35	1.99944	0.999718	−	0.0237260i	$-0.00755292\pi$
$109$	2275.35	1.99944	0.999718	+	0.0237260i	$0.00755292\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1157.60i	0.963699i	0.876254	+	0.481849i	$0.160035\pi$
$113$	1157.60i	0.963699i	−0.876254	+	0.481849i	$0.839965\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	468.111	0.351699
$122$	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
$137$	2752.87i	1.71674i	0.513031	+	0.858370i	$0.328522\pi$
$137$	2752.87i	1.71674i	−0.513031	+	0.858370i	$0.671478\pi$
$138$	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
$149$	1931.33i	1.06188i	0.847409	+	0.530941i	$0.178161\pi$
$149$	1931.33i	1.06188i	−0.847409	+	0.530941i	$0.821839\pi$
$150$	0	0
$151$	2248.89	1.21200	0.606000	−	0.795465i	$-0.292773\pi$
$151$	2248.89	1.21200	0.606000	+	0.795465i	$0.292773\pi$
$152$	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
$161$	− 3364.99i	− 1.64719i
$162$	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
$179$	1575.84i	0.658010i	0.944328	+	0.329005i	$0.106713\pi$
$179$	1575.84i	0.658010i	−0.944328	+	0.329005i	$0.893287\pi$
$180$	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
$191$	5255.29i	1.99089i	0.0953502	+	0.995444i	$0.469603\pi$
$191$	5255.29i	1.99089i	−0.0953502	+	0.995444i	$0.530397\pi$
$192$	0	0
$193$	−2772.75	−1.03413	−0.517064	−	0.855947i	$-0.672975\pi$
$193$	−2772.75	−1.03413	−0.517064	+	0.855947i	$0.672975\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5147.21i	1.86154i	0.365603	+	0.930771i	$0.380863\pi$
$197$	5147.21i	1.86154i	−0.365603	+	0.930771i	$0.619137\pi$
$198$	0	0
$199$	0	0	1.00000			$0$
$199$	0	0	−1.00000			$\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	5638.73i	1.94956i
$204$	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	$-0.406617\pi$
$211$	1772.65	0.578363	0.289181	+	0.957274i	$0.406617\pi$
$212$	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
$233$	− 412.009i	− 0.115844i	−0.998321	−	0.0579219i	$-0.981553\pi$
$233$	− 412.009i	− 0.115844i	0.998321	−	0.0579219i	$-0.0184474\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 5533.66i	− 1.49767i	−0.662757	−	0.748834i	$-0.730614\pi$
$239$	− 5533.66i	− 1.49767i	0.662757	−	0.748834i	$-0.269386\pi$
$240$	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
$253$	5337.20	1.32627
$254$	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
$263$	− 2470.04i	− 0.579123i	−0.957159	−	0.289561i	$-0.906491\pi$
$263$	− 2470.04i	− 0.579123i	0.957159	−	0.289561i	$-0.0935094\pi$
$264$	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
$275$	3671.87i	0.805172i
$276$	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
$281$	− 8982.08i	− 1.90685i	−0.301625	−	0.953427i	$-0.597529\pi$
$281$	− 8982.08i	− 1.90685i	0.301625	−	0.953427i	$-0.402471\pi$
$282$	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
$317$	1349.97i	0.239185i	0.992823	+	0.119593i	$0.0381588\pi$
$317$	1349.97i	0.239185i	−0.992823	+	0.119593i	$0.961841\pi$
$318$	0	0
$319$	−8943.58	−1.56973
$320$	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	$-0.360636\pi$
$331$	5106.30	0.847938	0.423969	+	0.905677i	$0.360636\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−11916.5	−1.92621	−0.963103	−	0.269135i	$-0.913262\pi$
$337$	−11916.5	−1.92621	−0.963103	+	0.269135i	$0.913262\pi$
$338$	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
$347$	− 11568.6i	− 1.78972i	−0.446347	−	0.894860i	$-0.647275\pi$
$347$	− 11568.6i	− 1.78972i	0.446347	−	0.894860i	$-0.352725\pi$
$348$	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
$359$	− 1996.13i	− 0.293459i	−0.989177	−	0.146729i	$-0.953125\pi$
$359$	− 1996.13i	− 0.293459i	0.989177	−	0.146729i	$-0.0468746\pi$
$360$	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
$371$	− 14240.4i	− 1.99279i
$372$	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	$-0.700821\pi$
$379$	−8704.52	−1.17974	−0.589870	+	0.807498i	$0.700821\pi$
$380$	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
$389$	15337.9i	1.99913i	0.0294496	+	0.999566i	$0.490625\pi$
$389$	15337.9i	1.99913i	−0.0294496	+	0.999566i	$0.509375\pi$
$390$	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
$401$	− 10169.8i	− 1.26648i	−0.773956	−	0.633239i	$-0.781725\pi$
$401$	− 10169.8i	− 1.26648i	0.773956	−	0.633239i	$-0.218275\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	310.876i	0.0378612i
$408$	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
$431$	− 5007.24i	− 0.559606i	−0.960057	−	0.279803i	$-0.909731\pi$
$431$	− 5007.24i	− 0.559606i	0.960057	−	0.279803i	$-0.0902692\pi$
$432$	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
$443$	12011.8i	1.28826i	0.764917	+	0.644129i	$0.222780\pi$
$443$	12011.8i	1.28826i	−0.764917	+	0.644129i	$0.777220\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 15219.6i	− 1.59968i	−0.600213	−	0.799841i	$-0.704917\pi$
$449$	− 15219.6i	− 1.59968i	0.600213	−	0.799841i	$-0.295083\pi$
$450$	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	$-0.865547\pi$
$457$	−17821.8	−1.82422	−0.912109	+	0.409947i	$0.865547\pi$
$458$	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
$473$	− 15699.2i	− 1.52611i
$474$	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	$-0.450987\pi$
$487$	3296.61	0.306742	0.153371	+	0.988169i	$0.450987\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 8998.48i	− 0.827079i	−0.910486	−	0.413540i	$-0.864292\pi$
$491$	− 8998.48i	− 0.827079i	0.910486	−	0.413540i	$-0.135708\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 21829.8i	− 1.97022i
$498$	0	0
$499$	−21086.6	−1.89172	−0.945859	−	0.324577i	$-0.894778\pi$
$499$	−21086.6	−1.89172	−0.945859	+	0.324577i	$0.894778\pi$
$500$	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
$529$	−20845.1	−1.71325
$530$	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
$539$	− 10075.6i	− 0.805172i
$540$	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
$557$	− 2807.99i	− 0.213606i	−0.994280	−	0.106803i	$-0.965939\pi$
$557$	− 2807.99i	− 0.213606i	0.994280	−	0.106803i	$-0.0340613\pi$
$558$	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
$569$	16481.1i	1.21428i	0.794596	+	0.607138i	$0.207683\pi$
$569$	16481.1i	1.21428i	−0.794596	+	0.607138i	$0.792317\pi$
$570$	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
$575$	− 22711.5i	− 1.64719i
$576$	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
$583$	22586.7	1.60454
$584$	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
$599$	28622.4i	1.95239i	0.216899	+	0.976194i	$0.430406\pi$
$599$	28622.4i	1.95239i	−0.216899	+	0.976194i	$0.569594\pi$
$600$	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
$617$	− 19836.0i	− 1.29428i	−0.762372	−	0.647139i	$-0.775965\pi$
$617$	− 19836.0i	− 1.29428i	0.762372	−	0.647139i	$-0.224035\pi$
$618$	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
$641$	− 28353.5i	− 1.74711i	−0.486729	−	0.873553i	$-0.661810\pi$
$641$	− 28353.5i	− 1.74711i	0.486729	−	0.873553i	$-0.338190\pi$
$642$	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
$653$	32948.9i	1.97456i	0.158976	+	0.987282i	$0.449181\pi$
$653$	32948.9i	1.97456i	−0.158976	+	0.987282i	$0.550819\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 22614.9i	− 1.33680i	−0.743803	−	0.668399i	$-0.766980\pi$
$659$	− 22614.9i	− 1.33680i	0.743803	−	0.668399i	$-0.233020\pi$
$660$	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
$667$	55318.5	3.21130
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	9609.37	0.550392	0.275196	−	0.961388i	$-0.411257\pi$
$673$	9609.37	0.550392	0.275196	+	0.961388i	$0.411257\pi$
$674$	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
$683$	31645.9i	1.77291i	0.462813	+	0.886456i	$0.346840\pi$
$683$	31645.9i	1.77291i	−0.462813	+	0.886456i	$0.653160\pi$
$684$	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
$701$	23110.9i	1.24520i	0.782539	+	0.622602i	$0.213924\pi$
$701$	23110.9i	1.24520i	−0.782539	+	0.622602i	$0.786076\pi$
$702$	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	$-0.107818\pi$
$709$	35611.8	1.88636	0.943180	+	0.332281i	$0.107818\pi$
$710$	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
$725$	38057.8i	1.94956i
$726$	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
$737$	21737.5i	1.08645i
$738$	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
$743$	4655.40i	0.229865i	0.993373	+	0.114933i	$0.0366652\pi$
$743$	4655.40i	0.229865i	−0.993373	+	0.114933i	$0.963335\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	387.501i	0.0189039i
$750$	0	0
$751$	−41088.5	−1.99646	−0.998230	−	0.0594732i	$-0.981058\pi$
$751$	−41088.5	−1.99646	−0.998230	+	0.0594732i	$0.981058\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	22848.7	1.09703	0.548514	−	0.836141i	$-0.315194\pi$
$757$	22848.7	1.09703	0.548514	+	0.836141i	$0.315194\pi$
$758$	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
$781$	34624.2	1.58636
$782$	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
$791$	− 21439.1i	− 0.963699i
$792$	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
$809$	− 7405.65i	− 0.321840i	−0.986967	−	0.160920i	$-0.948554\pi$
$809$	− 7405.65i	− 0.321840i	0.986967	−	0.160920i	$-0.0514461\pi$
$810$	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
$821$	43395.4i	1.84471i	0.386339	+	0.922357i	$0.373739\pi$
$821$	43395.4i	1.84471i	−0.386339	+	0.922357i	$0.626261\pi$
$822$	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
$827$	46002.9i	1.93431i	0.254179	+	0.967157i	$0.418195\pi$
$827$	46002.9i	1.93431i	−0.254179	+	0.967157i	$0.581805\pi$
$828$	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
$841$	−68308.5	−2.80079
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−8669.54	−0.351699
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1922.85i	− 0.0774553i
$852$	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
$863$	47169.0i	1.86055i	0.366868	+	0.930273i	$0.380430\pi$
$863$	47169.0i	1.86055i	−0.366868	+	0.930273i	$0.619570\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 40655.0i	− 1.58703i
$870$	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	$-0.194150\pi$
$883$	43014.6	1.63936	0.819681	+	0.572820i	$0.194150\pi$
$884$	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	$-0.416000\pi$
$907$	14250.0	0.521680	0.260840	+	0.965382i	$0.416000\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1065.41i	− 0.0387473i	−0.999812	−	0.0193736i	$-0.993833\pi$
$911$	− 1065.41i	− 0.0387473i	0.999812	−	0.0193736i	$-0.00616720\pi$
$912$	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	$-0.127609\pi$
$919$	51301.1	1.84142	0.920711	+	0.390244i	$0.127609\pi$
$920$	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
$947$	57034.5i	1.95710i	0.206013	+	0.978549i	$0.433951\pi$
$947$	57034.5i	1.95710i	−0.206013	+	0.978549i	$0.566049\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 56795.7i	− 1.93053i	−0.261276	−	0.965264i	$-0.584143\pi$
$953$	− 56795.7i	− 1.93053i	0.261276	−	0.965264i	$-0.415857\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 50983.8i	− 1.71674i
$960$	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
$977$	60603.4i	1.98452i	0.124182	+	0.992259i	$0.460369\pi$
$977$	60603.4i	1.98452i	−0.124182	+	0.992259i	$0.539631\pi$
$978$	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
$989$	97103.9i	3.12207i
$990$	0	0
$991$	24155.7	0.774300	0.387150	−	0.922017i	$-0.373460\pi$
$991$	24155.7	0.774300	0.387150	+	0.922017i	$0.373460\pi$
$992$	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.881.1		4
3.2	odd	2	inner	1008.4.k.a.881.2		4
4.3	odd	2		63.4.c.a.62.3	yes	4
7.6	odd	2	CM	1008.4.k.a.881.1		4
12.11	even	2		63.4.c.a.62.2	✓	4
21.20	even	2	inner	1008.4.k.a.881.2		4
28.3	even	6		441.4.p.b.215.2		8
28.11	odd	6		441.4.p.b.215.2		8
28.19	even	6		441.4.p.b.80.3		8
28.23	odd	6		441.4.p.b.80.3		8
28.27	even	2		63.4.c.a.62.3	yes	4
84.11	even	6		441.4.p.b.215.3		8
84.23	even	6		441.4.p.b.80.2		8
84.47	odd	6		441.4.p.b.80.2		8
84.59	odd	6		441.4.p.b.215.3		8
84.83	odd	2		63.4.c.a.62.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.4.c.a.62.2	✓	4	12.11	even	2
63.4.c.a.62.2	✓	4	84.83	odd	2
63.4.c.a.62.3	yes	4	4.3	odd	2
63.4.c.a.62.3	yes	4	28.27	even	2
441.4.p.b.80.2		8	84.23	even	6
441.4.p.b.80.2		8	84.47	odd	6
441.4.p.b.80.3		8	28.19	even	6
441.4.p.b.80.3		8	28.23	odd	6
441.4.p.b.215.2		8	28.3	even	6
441.4.p.b.215.2		8	28.11	odd	6
441.4.p.b.215.3		8	84.11	even	6
441.4.p.b.215.3		8	84.59	odd	6
1008.4.k.a.881.1		4	1.1	even	1	trivial
1008.4.k.a.881.1		4	7.6	odd	2	CM
1008.4.k.a.881.2		4	3.2	odd	2	inner
1008.4.k.a.881.2		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)

\(f(q)\)	\(=\)	\(q-18.5203 q^{7} +O(q^{10})\)	\(q-18.5203 q^{7} -29.3750i q^{11} +181.692i q^{23} -125.000 q^{25} -304.463i q^{29} -10.5830 q^{37} +534.442 q^{43} +343.000 q^{49} +768.909i q^{53} -740.000 q^{67} +1178.70i q^{71} +544.032i q^{77} +1384.00 q^{79} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 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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