LMFDB - Embedded newform 4032.2.p.h.1567.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4032,2,Mod(1567,4032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4032, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4032.1567");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4032 = 2^{6} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4032.p (of order $2$, degree $1$, 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:	$32.1956820950$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.629407744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 $ x^8 - 2x^6 + 2x^4 - 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 448)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			1567.2
Root			$0.767178 - 1.18804i$ of defining polynomial
Character	$\chi$	$=$	4032.1567
Dual form			4032.2.p.h.1567.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.91044 q^{5} +2.64575i q^{7} +O(q^{10})$	$q-3.91044 q^{5} +2.64575i q^{7} -5.59388 q^{13} -8.66259i q^{19} -6.00000i q^{23} +10.2915 q^{25} -10.3460i q^{35} -7.00000 q^{49} +5.29570i q^{59} +0.543544 q^{61} +21.8745 q^{65} +15.8745i q^{71} +5.29150i q^{79} +18.1669i q^{83} -14.8000i q^{91} +33.8745i q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 40 q^{25} - 56 q^{49} + 48 q^{65}+O(q^{100}) $ 8 * q + 40 * q^25 - 56 * q^49 + 48 * q^65

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$1793$	$3781$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$5$	−3.91044	−1.74880	−0.874400	−	0.485206i	$-0.838745\pi$
$5$	−3.91044	−1.74880	−0.874400	+	0.485206i	$0.838745\pi$
$6$	0	0
$7$	2.64575i	1.00000i
$7$	2.64575i	1.00000i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−5.59388	−1.55146	−0.775732	−	0.631063i	$-0.782619\pi$
$13$	−5.59388	−1.55146	−0.775732	+	0.631063i	$0.782619\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	0	0
$19$	− 8.66259i	− 1.98734i	−0.112360	−	0.993668i	$-0.535841\pi$
$19$	− 8.66259i	− 1.98734i	0.112360	−	0.993668i	$-0.464159\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 6.00000i	− 1.25109i	−0.780189	−	0.625543i	$-0.784877\pi$
$23$	− 6.00000i	− 1.25109i	0.780189	−	0.625543i	$-0.215123\pi$
$24$	0	0
$25$	10.2915	2.05830
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	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
$35$	− 10.3460i	− 1.74880i
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	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
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	5.29570i	0.689442i	0.938705	+	0.344721i	$0.112026\pi$
$59$	5.29570i	0.689442i	−0.938705	+	0.344721i	$0.887974\pi$
$60$	0	0
$61$	0.543544	0.0695936	0.0347968	−	0.999394i	$-0.488922\pi$
$61$	0.543544	0.0695936	0.0347968	+	0.999394i	$0.488922\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	21.8745	2.71320
$66$	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
$71$	15.8745i	1.88396i	0.335673	+	0.941979i	$0.391036\pi$
$71$	15.8745i	1.88396i	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.29150i	0.595341i	0.954669	+	0.297670i	$0.0962096\pi$
$79$	5.29150i	0.595341i	−0.954669	+	0.297670i	$0.903790\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	18.1669i	1.99408i	0.0769020	+	0.997039i	$0.475497\pi$
$83$	18.1669i	1.99408i	−0.0769020	+	0.997039i	$0.524503\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	− 14.8000i	− 1.55146i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	33.8745i	3.47545i
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	8.96077	0.891630	0.445815	−	0.895125i	$-0.352914\pi$
$101$	8.96077	0.891630	0.445815	+	0.895125i	$0.352914\pi$
$102$	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		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.8745	1.49335	0.746674	−	0.665190i	$-0.231650\pi$
$113$	15.8745	1.49335	0.746674	+	0.665190i	$0.231650\pi$
$114$	0	0
$115$	23.4626i	2.18790i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−20.6921	−1.85076
$126$	0	0
$127$	− 2.00000i	− 0.177471i	−0.996055	−	0.0887357i	$-0.971717\pi$
$127$	− 2.00000i	− 0.177471i	0.996055	−	0.0887357i	$-0.0282826\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 13.1166i	− 1.14600i	−0.819555	−	0.573000i	$-0.805779\pi$
$131$	− 13.1166i	− 1.14600i	0.819555	−	0.573000i	$-0.194221\pi$
$132$	0	0
$133$	22.9191	1.98734
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	13.7129i	1.16312i	0.813505	+	0.581558i	$0.197557\pi$
$139$	13.7129i	1.16312i	−0.813505	+	0.581558i	$0.802443\pi$
$140$	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
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	− 10.0000i	− 0.813788i	−0.913475	−	0.406894i	$-0.866612\pi$
$151$	− 10.0000i	− 0.813788i	0.913475	−	0.406894i	$-0.133388\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−17.8687	−1.42608	−0.713040	−	0.701123i	$-0.752682\pi$
$157$	−17.8687	−1.42608	−0.713040	+	0.701123i	$0.752682\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	15.8745	1.25109
$162$	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
$169$	18.2915	1.40704
$170$	0	0
$171$	0	0
$172$	0	0
$173$	24.6025	1.87049	0.935247	−	0.353995i	$-0.115177\pi$
$173$	24.6025	1.87049	0.935247	+	0.353995i	$0.115177\pi$
$174$	0	0
$175$	27.2288i	2.05830i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	10.6442	0.791178	0.395589	−	0.918428i	$-0.370540\pi$
$181$	10.6442	0.791178	0.395589	+	0.918428i	$0.370540\pi$
$182$	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
$191$	15.8745i	1.14864i	0.818631	+	0.574320i	$0.194733\pi$
$191$	15.8745i	1.14864i	−0.818631	+	0.574320i	$0.805267\pi$
$192$	0	0
$193$	26.4575	1.90445	0.952227	−	0.305392i	$-0.0987875\pi$
$193$	26.4575	1.90445	0.952227	+	0.305392i	$0.0987875\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	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
$205$	0	0
$206$	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
$221$	0	0
$222$	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
$227$	15.3964i	1.02189i	0.859612	+	0.510947i	$0.170705\pi$
$227$	15.3964i	1.02189i	−0.859612	+	0.510947i	$0.829295\pi$
$228$	0	0
$229$	29.0565	1.92011	0.960053	−	0.279817i	$-0.0902736\pi$
$229$	29.0565	1.92011	0.960053	+	0.279817i	$0.0902736\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 30.0000i	− 1.94054i	−0.242028	−	0.970269i	$-0.577812\pi$
$239$	− 30.0000i	− 1.94054i	0.242028	−	0.970269i	$-0.422188\pi$
$240$	0	0
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	27.3730	1.74880
$246$	0	0
$247$	48.4575i	3.08328i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 23.2172i	− 1.46546i	−0.680520	−	0.732730i	$-0.738246\pi$
$251$	− 23.2172i	− 1.46546i	0.680520	−	0.732730i	$-0.261754\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	15.8745i	0.978864i	0.872041	+	0.489432i	$0.162796\pi$
$263$	15.8745i	0.978864i	−0.872041	+	0.489432i	$0.837204\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−32.4234	−1.97689	−0.988444	−	0.151585i	$-0.951562\pi$
$269$	−32.4234	−1.97689	−0.988444	+	0.151585i	$0.951562\pi$
$270$	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
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	9.74968i	0.579558i	0.957094	+	0.289779i	$0.0935819\pi$
$283$	9.74968i	0.579558i	−0.957094	+	0.289779i	$0.906418\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	17.0000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	14.5018	0.847206	0.423603	−	0.905848i	$-0.360765\pi$
$293$	14.5018	0.847206	0.423603	+	0.905848i	$0.360765\pi$
$294$	0	0
$295$	− 20.7085i	− 1.20570i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	33.5633i	1.94101i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.12549	−0.121705
$306$	0	0
$307$	32.1252i	1.83348i	0.399482	+	0.916741i	$0.369190\pi$
$307$	32.1252i	1.83348i	−0.399482	+	0.916741i	$0.630810\pi$
$308$	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
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−57.5694	−3.19338
$326$	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
$337$	26.4575	1.44123	0.720616	−	0.693334i	$-0.243859\pi$
$337$	26.4575	1.44123	0.720616	+	0.693334i	$0.243859\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 18.5203i	− 1.00000i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	−24.0062	−1.28502	−0.642510	−	0.766277i	$-0.722107\pi$
$349$	−24.0062	−1.28502	−0.642510	+	0.766277i	$0.722107\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	− 62.0762i	− 3.29466i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 6.00000i	− 0.316668i	−0.987386	−	0.158334i	$-0.949388\pi$
$359$	− 6.00000i	− 0.316668i	0.987386	−	0.158334i	$-0.0506123\pi$
$360$	0	0
$361$	−56.0405	−2.94950
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	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
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	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
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 20.6921i	− 1.04113i
$396$	0	0
$397$	12.8184	0.643337	0.321668	−	0.946852i	$-0.395756\pi$
$397$	12.8184	0.643337	0.321668	+	0.946852i	$0.395756\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	15.8745	0.792735	0.396368	−	0.918092i	$-0.370271\pi$
$401$	15.8745	0.792735	0.396368	+	0.918092i	$0.370271\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.0111	−0.689442
$414$	0	0
$415$	− 71.0405i	− 3.48724i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 38.8590i	− 1.89839i	−0.314695	−	0.949193i	$-0.601902\pi$
$419$	− 38.8590i	− 1.89839i	0.314695	−	0.949193i	$-0.398098\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.43808i	0.0695936i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	18.0000i	0.867029i	0.901146	+	0.433515i	$0.142727\pi$
$431$	18.0000i	0.867029i	−0.901146	+	0.433515i	$0.857273\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−51.9756	−2.48633
$438$	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		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	57.8745i	2.71320i
$456$	0	0
$457$	5.29150	0.247526	0.123763	−	0.992312i	$-0.460504\pi$
$457$	5.29150	0.247526	0.123763	+	0.992312i	$0.460504\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.19024	0.288308	0.144154	−	0.989555i	$-0.453954\pi$
$461$	6.19024	0.288308	0.144154	+	0.989555i	$0.453954\pi$
$462$	0	0
$463$	26.4575i	1.22958i	0.788689	+	0.614792i	$0.210760\pi$
$463$	26.4575i	1.22958i	−0.788689	+	0.614792i	$0.789240\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 41.6295i	− 1.92638i	−0.268814	−	0.963192i	$-0.586632\pi$
$467$	− 41.6295i	− 1.92638i	0.268814	−	0.963192i	$-0.413368\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	− 89.1511i	− 4.09053i
$476$	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
$487$	38.0000i	1.72194i	0.508652	+	0.860972i	$0.330144\pi$
$487$	38.0000i	1.72194i	−0.508652	+	0.860972i	$0.669856\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−42.0000	−1.88396
$498$	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
$505$	−35.0405	−1.55928
$506$	0	0
$507$	0	0
$508$	0	0
$509$	37.4737	1.66099	0.830497	−	0.557024i	$-0.188057\pi$
$509$	37.4737	1.66099	0.830497	+	0.557024i	$0.188057\pi$
$510$	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	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	38.2626i	1.67311i	0.547885	+	0.836554i	$0.315433\pi$
$523$	38.2626i	1.67311i	−0.547885	+	0.836554i	$0.684567\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−13.0000	−0.565217
$530$	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
$539$	0	0
$540$	0	0
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	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
$553$	−14.0000	−0.595341
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	46.6799i	1.96732i	0.180032	+	0.983661i	$0.442380\pi$
$563$	46.6799i	1.96732i	−0.180032	+	0.983661i	$0.557620\pi$
$564$	0	0
$565$	−62.0762	−2.61157
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−47.6235	−1.99648	−0.998241	−	0.0592869i	$-0.981117\pi$
$569$	−47.6235	−1.99648	−0.998241	+	0.0592869i	$0.981117\pi$
$570$	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
$575$	− 61.7490i	− 2.57511i
$576$	0	0
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−48.0651	−1.99408
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.5792i	1.50978i	0.655849	+	0.754892i	$0.272311\pi$
$587$	36.5792i	1.50978i	−0.655849	+	0.754892i	$0.727689\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 47.6235i	− 1.94584i	−0.231133	−	0.972922i	$-0.574243\pi$
$599$	− 47.6235i	− 1.94584i	0.231133	−	0.972922i	$-0.425757\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	43.0148	1.74880
$606$	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
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−47.6235	−1.91725	−0.958625	−	0.284670i	$-0.908116\pi$
$617$	−47.6235	−1.91725	−0.958625	+	0.284670i	$0.908116\pi$
$618$	0	0
$619$	42.2259i	1.69720i	0.529034	+	0.848601i	$0.322554\pi$
$619$	42.2259i	1.69720i	−0.529034	+	0.848601i	$0.677446\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	29.4575	1.17830
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	− 37.0405i	− 1.47456i	−0.675587	−	0.737280i	$-0.736110\pi$
$631$	− 37.0405i	− 1.47456i	0.675587	−	0.737280i	$-0.263890\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.82087i	0.310362i
$636$	0	0
$637$	39.1572	1.55146
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.8745	0.627005	0.313503	−	0.949587i	$-0.398498\pi$
$641$	15.8745	0.627005	0.313503	+	0.949587i	$0.398498\pi$
$642$	0	0
$643$	48.3633i	1.90726i	0.300978	+	0.953631i	$0.402687\pi$
$643$	48.3633i	1.90726i	−0.300978	+	0.953631i	$0.597313\pi$
$644$	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
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	51.2915i	2.00412i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	41.3313	1.60760	0.803801	−	0.594898i	$-0.202807\pi$
$661$	41.3313	1.60760	0.803801	+	0.594898i	$0.202807\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−89.6235	−3.47545
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.45150	−0.363251	−0.181625	−	0.983368i	$-0.558136\pi$
$677$	−9.45150	−0.363251	−0.181625	+	0.983368i	$0.558136\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	70.3878	2.68938
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	− 4.69934i	− 0.178771i	−0.995997	−	0.0893857i	$-0.971510\pi$
$691$	− 4.69934i	− 0.178771i	0.995997	−	0.0893857i	$-0.0284904\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 53.6235i	− 2.03406i
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	23.7080i	0.891630i
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	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
$725$	0	0
$726$	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
$733$	33.0197	1.21961	0.609806	−	0.792551i	$-0.291247\pi$
$733$	33.0197	1.21961	0.609806	+	0.792551i	$0.291247\pi$
$734$	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
$743$	− 54.0000i	− 1.98107i	−0.137268	−	0.990534i	$-0.543832\pi$
$743$	− 54.0000i	− 1.98107i	0.137268	−	0.990534i	$-0.456168\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 50.0000i	− 1.82453i	−0.409605	−	0.912263i	$-0.634333\pi$
$751$	− 50.0000i	− 1.82453i	0.409605	−	0.912263i	$-0.365667\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	39.1044i	1.42315i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 29.6235i	− 1.06964i
$768$	0	0
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	53.1155	1.91043	0.955215	−	0.295912i	$-0.0956236\pi$
$773$	53.1155	1.91043	0.955215	+	0.295912i	$0.0956236\pi$
$774$	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
$785$	69.8745	2.49393
$786$	0	0
$787$	− 33.2123i	− 1.18389i	−0.805978	−	0.591945i	$-0.798360\pi$
$787$	− 33.2123i	− 1.18389i	0.805978	−	0.591945i	$-0.201640\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	42.0000i	1.49335i
$792$	0	0
$793$	−3.04052	−0.107972
$794$	0	0
$795$	0	0
$796$	0	0
$797$	55.8860	1.97958	0.989792	−	0.142521i	$-0.0455210\pi$
$797$	55.8860	1.97958	0.989792	+	0.142521i	$0.0455210\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−62.0762	−2.18790
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−47.6235	−1.67435	−0.837177	−	0.546932i	$-0.815796\pi$
$809$	−47.6235	−1.67435	−0.837177	+	0.546932i	$0.815796\pi$
$810$	0	0
$811$	29.9510i	1.05172i	0.850570	+	0.525861i	$0.176257\pi$
$811$	29.9510i	1.05172i	−0.850570	+	0.525861i	$0.823743\pi$
$812$	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
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	26.4575i	0.922251i	0.887335	+	0.461125i	$0.152554\pi$
$823$	26.4575i	0.922251i	−0.887335	+	0.461125i	$0.847446\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	57.5694	1.99947	0.999735	−	0.0230361i	$-0.00733328\pi$
$829$	57.5694	1.99947	0.999735	+	0.0230361i	$0.00733328\pi$
$830$	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
$841$	29.0000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−71.5277	−2.46063
$846$	0	0
$847$	− 29.1033i	− 1.00000i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−7.76806	−0.265973	−0.132987	−	0.991118i	$-0.542457\pi$
$853$	−7.76806	−0.265973	−0.132987	+	0.991118i	$0.542457\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	22.0245i	0.751467i	0.926728	+	0.375734i	$0.122609\pi$
$859$	22.0245i	0.751467i	−0.926728	+	0.375734i	$0.877391\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	15.8745i	0.540375i	0.962808	+	0.270187i	$0.0870856\pi$
$863$	15.8745i	0.540375i	−0.962808	+	0.270187i	$0.912914\pi$
$864$	0	0
$865$	−96.2065	−3.27112
$866$	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
$875$	− 54.7461i	− 1.85076i
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	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
$889$	5.29150	0.177471
$890$	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
$905$	−41.6235	−1.38361
$906$	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
$911$	− 30.0000i	− 0.993944i	−0.867766	−	0.496972i	$-0.834445\pi$
$911$	− 30.0000i	− 0.993944i	0.867766	−	0.496972i	$-0.165555\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	34.7032	1.14600
$918$	0	0
$919$	− 58.2065i	− 1.92006i	−0.279904	−	0.960028i	$-0.590303\pi$
$919$	− 58.2065i	− 1.92006i	0.279904	−	0.960028i	$-0.409697\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 88.8001i	− 2.92289i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	60.6382i	1.98734i
$932$	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
$941$	19.0614	0.621385	0.310693	−	0.950510i	$-0.399439\pi$
$941$	19.0614	0.621385	0.310693	+	0.950510i	$0.399439\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	0	0		−	1.00000i	$-0.5\pi$
$947$	0	0			1.00000i	$0.5\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	0	0
$955$	− 62.0762i	− 2.00874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 47.6235i	− 1.53784i
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−103.460	−3.33051
$966$	0	0
$967$	− 58.0000i	− 1.86515i	−0.360971	−	0.932577i	$-0.617555\pi$
$967$	− 58.0000i	− 1.86515i	0.360971	−	0.932577i	$-0.382445\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	62.3216i	2.00000i	0.00218468	+	0.999998i	$0.499305\pi$
$971$	62.3216i	2.00000i	−0.00218468	+	0.999998i	$0.500695\pi$
$972$	0	0
$973$	−36.2810	−1.16312
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	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
$989$	0	0
$990$	0	0
$991$	− 37.0405i	− 1.17663i	−0.808632	−	0.588315i	$-0.799791\pi$
$991$	− 37.0405i	− 1.17663i	0.808632	−	0.588315i	$-0.200209\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	47.4688	1.50335	0.751675	−	0.659533i	$-0.229246\pi$
$997$	47.4688	1.50335	0.751675	+	0.659533i	$0.229246\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4032.2.p.h.1567.2		8
3.2	odd	2		448.2.e.a.223.6	yes	8
4.3	odd	2	inner	4032.2.p.h.1567.1		8
7.6	odd	2	inner	4032.2.p.h.1567.8		8
8.3	odd	2	inner	4032.2.p.h.1567.7		8
8.5	even	2	inner	4032.2.p.h.1567.8		8
12.11	even	2		448.2.e.a.223.4	yes	8
21.20	even	2		448.2.e.a.223.3	✓	8
24.5	odd	2		448.2.e.a.223.3	✓	8
24.11	even	2		448.2.e.a.223.5	yes	8
28.27	even	2	inner	4032.2.p.h.1567.7		8
48.5	odd	4		1792.2.f.k.1791.6		8
48.11	even	4		1792.2.f.k.1791.4		8
48.29	odd	4		1792.2.f.k.1791.3		8
48.35	even	4		1792.2.f.k.1791.5		8
56.13	odd	2	CM	4032.2.p.h.1567.2		8
56.27	even	2	inner	4032.2.p.h.1567.1		8
84.83	odd	2		448.2.e.a.223.5	yes	8
168.83	odd	2		448.2.e.a.223.4	yes	8
168.125	even	2		448.2.e.a.223.6	yes	8
336.83	odd	4		1792.2.f.k.1791.4		8
336.125	even	4		1792.2.f.k.1791.6		8
336.251	odd	4		1792.2.f.k.1791.5		8
336.293	even	4		1792.2.f.k.1791.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.2.e.a.223.3	✓	8	21.20	even	2
448.2.e.a.223.3	✓	8	24.5	odd	2
448.2.e.a.223.4	yes	8	12.11	even	2
448.2.e.a.223.4	yes	8	168.83	odd	2
448.2.e.a.223.5	yes	8	24.11	even	2
448.2.e.a.223.5	yes	8	84.83	odd	2
448.2.e.a.223.6	yes	8	3.2	odd	2
448.2.e.a.223.6	yes	8	168.125	even	2
1792.2.f.k.1791.3		8	48.29	odd	4
1792.2.f.k.1791.3		8	336.293	even	4
1792.2.f.k.1791.4		8	48.11	even	4
1792.2.f.k.1791.4		8	336.83	odd	4
1792.2.f.k.1791.5		8	48.35	even	4
1792.2.f.k.1791.5		8	336.251	odd	4
1792.2.f.k.1791.6		8	48.5	odd	4
1792.2.f.k.1791.6		8	336.125	even	4
4032.2.p.h.1567.1		8	4.3	odd	2	inner
4032.2.p.h.1567.1		8	56.27	even	2	inner
4032.2.p.h.1567.2		8	1.1	even	1	trivial
4032.2.p.h.1567.2		8	56.13	odd	2	CM
4032.2.p.h.1567.7		8	8.3	odd	2	inner
4032.2.p.h.1567.7		8	28.27	even	2	inner
4032.2.p.h.1567.8		8	7.6	odd	2	inner
4032.2.p.h.1567.8		8	8.5	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:	\( N \)	\(=\)	\( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4032.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.91044 q^{5} +2.64575i q^{7} +O(q^{10})\)	\(q-3.91044 q^{5} +2.64575i q^{7} -5.59388 q^{13} -8.66259i q^{19} -6.00000i q^{23} +10.2915 q^{25} -10.3460i q^{35} -7.00000 q^{49} +5.29570i q^{59} +0.543544 q^{61} +21.8745 q^{65} +15.8745i q^{71} +5.29150i q^{79} +18.1669i q^{83} -14.8000i q^{91} +33.8745i q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 40 q^{25} - 56 q^{49} + 48 q^{65}+O(q^{100}) \) 8 * q + 40 * q^25 - 56 * q^49 + 48 * q^65

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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