LMFDB - Embedded newform 1764.1.h.a.1763.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1764,1,Mod(1763,1764)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1764, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("1764.1763");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1764.h (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:	$0.880350682285$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Projective image:	$D_{8}$
Projective field:	Galois closure of 8.0.38423222208.1

Embedding invariants

Embedding label			1763.7
Root			$0.923880 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	1764.1763
Dual form			1764.1.h.a.1763.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +0.765367 q^{5} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +0.765367 q^{5} -1.00000i q^{8} +0.765367i q^{10} -1.84776i q^{13} +1.00000 q^{16} +1.84776 q^{17} -0.765367 q^{20} -0.414214 q^{25} +1.84776 q^{26} +1.00000i q^{32} +1.84776i q^{34} -1.41421 q^{37} -0.765367i q^{40} +1.84776 q^{41} -0.414214i q^{50} +1.84776i q^{52} +1.41421i q^{53} +0.765367i q^{61} -1.00000 q^{64} -1.41421i q^{65} -1.84776 q^{68} -0.765367i q^{73} -1.41421i q^{74} +0.765367 q^{80} +1.84776i q^{82} +1.41421 q^{85} -0.765367 q^{89} +0.765367i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{4}+O(q^{10}) $ 8 * q - 8 * q^4	$ 8 q - 8 q^{4} + 8 q^{16} + 8 q^{25} - 8 q^{64}+O(q^{100}) $ 8 * q - 8 * q^4 + 8 * q^16 + 8 * q^25 - 8 * q^64

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	1.00000i	1.00000i
$2$	1.00000i	1.00000i
$3$	0	0
$3$	0	0
$4$	−1.00000	−1.00000
$5$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$5$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	− 1.00000i	− 1.00000i
$9$	0	0
$10$	0.765367i	0.765367i
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$13$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	1.00000
$17$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$17$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−0.765367	−0.765367
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−0.414214	−0.414214
$26$	1.84776	1.84776
$27$	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	$0.5\pi$
$32$	1.00000i	1.00000i
$33$	0	0
$34$	1.84776i	1.84776i
$35$	0	0
$36$	0	0
$37$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$37$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$38$	0	0
$39$	0	0
$40$	− 0.765367i	− 0.765367i
$41$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$41$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	0	0
$50$	− 0.414214i	− 0.414214i
$51$	0	0
$52$	1.84776i	1.84776i
$53$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$53$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$54$	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
$61$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$61$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−1.00000
$65$	− 1.41421i	− 1.41421i
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−1.84776	−1.84776
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$73$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$74$	− 1.41421i	− 1.41421i
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	0.765367	0.765367
$81$	0	0
$82$	1.84776i	1.84776i
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	1.41421	1.41421
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$89$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$97$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$98$	0	0
$99$	0	0
$100$	0.414214	0.414214
$101$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$101$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−1.84776	−1.84776
$105$	0	0
$106$	−1.41421	−1.41421
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$109$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$113$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.00000	−1.00000
$122$	−0.765367	−0.765367
$123$	0	0
$124$	0	0
$125$	−1.08239	−1.08239
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 1.00000i	− 1.00000i
$129$	0	0
$130$	1.41421	1.41421
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	− 1.84776i	− 1.84776i
$137$	0	0	1.00000			$0$
$137$	0	0	−1.00000			$\pi$
$138$	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
$145$	0	0
$146$	0.765367	0.765367
$147$	0	0
$148$	1.41421	1.41421
$149$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$149$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$150$	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
$157$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$157$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$158$	0	0
$159$	0	0
$160$	0.765367i	0.765367i
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−1.84776	−1.84776
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	−2.41421	−2.41421
$170$	1.41421i	1.41421i
$171$	0	0
$172$	0	0
$173$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$173$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	− 0.765367i	− 0.765367i
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$181$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.08239	−1.08239
$186$	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
$193$	0	0		−	1.00000i	$-0.5\pi$
$193$	0	0			1.00000i	$0.5\pi$
$194$	−0.765367	−0.765367
$195$	0	0
$196$	0	0
$197$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$197$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0.414214i	0.414214i
$201$	0	0
$202$	− 0.765367i	− 0.765367i
$203$	0	0
$204$	0	0
$205$	1.41421	1.41421
$206$	0	0
$207$	0	0
$208$	− 1.84776i	− 1.84776i
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	− 1.41421i	− 1.41421i
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	1.41421i	1.41421i
$219$	0	0
$220$	0	0
$221$	− 3.41421i	− 3.41421i
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	0	0
$226$	−1.41421	−1.41421
$227$	0	0	1.00000			$0$
$227$	0	0	−1.00000			$\pi$
$228$	0	0
$229$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$229$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$233$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$234$	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
$241$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$241$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$242$	− 1.00000i	− 1.00000i
$243$	0	0
$244$	− 0.765367i	− 0.765367i
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	− 1.08239i	− 1.08239i
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	1.00000	1.00000
$257$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$257$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$258$	0	0
$259$	0	0
$260$	1.41421i	1.41421i
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	1.08239i	1.08239i
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$269$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	1.84776	1.84776
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	0	0		−	1.00000i	$-0.5\pi$
$277$	0	0			1.00000i	$0.5\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$281$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$282$	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
$289$	2.41421	2.41421
$290$	0	0
$291$	0	0
$292$	0.765367i	0.765367i
$293$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$293$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$294$	0	0
$295$	0	0
$296$	1.41421i	1.41421i
$297$	0	0
$298$	−1.41421	−1.41421
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.585786i	0.585786i
$306$	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	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	0	0
$313$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$313$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$314$	1.84776	1.84776
$315$	0	0
$316$	0	0
$317$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$317$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$318$	0	0
$319$	0	0
$320$	−0.765367	−0.765367
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0.765367i	0.765367i
$326$	0	0
$327$	0	0
$328$	− 1.84776i	− 1.84776i
$329$	0	0
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$337$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$338$	− 2.41421i	− 2.41421i
$339$	0	0
$340$	−1.41421	−1.41421
$341$	0	0
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	− 1.84776i	− 1.84776i
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$349$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$353$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$354$	0	0
$355$	0	0
$356$	0.765367	0.765367
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−1.00000	−1.00000
$362$	0.765367	0.765367
$363$	0	0
$364$	0	0
$365$	− 0.585786i	− 0.585786i
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0	0
$370$	− 1.08239i	− 1.08239i
$371$	0	0
$372$	0	0
$373$	0	0		−	1.00000i	$-0.5\pi$
$373$	0	0			1.00000i	$0.5\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	− 0.765367i	− 0.765367i
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	1.41421	1.41421
$395$	0	0
$396$	0	0
$397$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$397$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$398$	0	0
$399$	0	0
$400$	−0.414214	−0.414214
$401$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$401$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$402$	0	0
$403$	0	0
$404$	0.765367	0.765367
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$409$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$410$	1.41421i	1.41421i
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	1.84776	1.84776
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0		−	1.00000i	$-0.5\pi$
$421$	0	0			1.00000i	$0.5\pi$
$422$	0	0
$423$	0	0
$424$	1.41421	1.41421
$425$	−0.765367	−0.765367
$426$	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
$433$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$433$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$434$	0	0
$435$	0	0
$436$	−1.41421	−1.41421
$437$	0	0
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	0	0
$442$	3.41421	3.41421
$443$	0	0		−	1.00000i	$-0.5\pi$
$443$	0	0			1.00000i	$0.5\pi$
$444$	0	0
$445$	−0.585786	−0.585786
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$449$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$450$	0	0
$451$	0	0
$452$	− 1.41421i	− 1.41421i
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.00000	2.00000	1.00000			$0$
$457$	2.00000	2.00000	1.00000			$0$
$458$	0.765367	0.765367
$459$	0	0
$460$	0	0
$461$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$461$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	2.00000	2.00000
$467$	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	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	2.61313i	2.61313i
$482$	−1.84776	−1.84776
$483$	0	0
$484$	1.00000	1.00000
$485$	0.585786i	0.585786i
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	0.765367	0.765367
$489$	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
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	1.08239	1.08239
$501$	0	0
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	−0.585786	−0.585786
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$509$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$510$	0	0
$511$	0	0
$512$	1.00000i	1.00000i
$513$	0	0
$514$	1.84776i	1.84776i
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	−1.41421	−1.41421
$521$	−0.765367	−0.765367	−0.382683	−	0.923880i	$-0.625000\pi$
$521$	−0.765367	−0.765367	−0.382683	+	0.923880i	$0.625000\pi$
$522$	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
$529$	−1.00000	−1.00000
$530$	−1.08239	−1.08239
$531$	0	0
$532$	0	0
$533$	− 3.41421i	− 3.41421i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	− 1.84776i	− 1.84776i
$539$	0	0
$540$	0	0
$541$	−2.00000	−2.00000	−1.00000			$\pi$
$541$	−2.00000	−2.00000	−1.00000			$\pi$
$542$	0	0
$543$	0	0
$544$	1.84776i	1.84776i
$545$	1.08239	1.08239
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$557$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−2.00000	−2.00000
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	1.08239i	1.08239i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	0	0	1.00000			$0$
$571$	0	0	−1.00000			$\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$577$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$578$	2.41421i	2.41421i
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	−0.765367	−0.765367
$585$	0	0
$586$	− 0.765367i	− 0.765367i
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−1.41421	−1.41421
$593$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$593$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$594$	0	0
$595$	0	0
$596$	− 1.41421i	− 1.41421i
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$601$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−0.765367	−0.765367
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	−0.585786	−0.585786
$611$	0	0
$612$	0	0
$613$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$613$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	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
$625$	−0.414214	−0.414214
$626$	1.84776	1.84776
$627$	0	0
$628$	1.84776i	1.84776i
$629$	−2.61313	−2.61313
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	0	0
$633$	0	0
$634$	−1.41421	−1.41421
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	− 0.765367i	− 0.765367i
$641$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$641$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$642$	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	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	−0.765367	−0.765367
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	1.84776	1.84776
$657$	0	0
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	1.84776i	1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$661$	1.84776i	1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$662$	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
$673$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$673$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$674$	− 1.41421i	− 1.41421i
$675$	0	0
$676$	2.41421	2.41421
$677$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$677$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$678$	0	0
$679$	0	0
$680$	− 1.41421i	− 1.41421i
$681$	0	0
$682$	0	0
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.61313	2.61313
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	1.84776	1.84776
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.41421	3.41421
$698$	−1.84776	−1.84776
$699$	0	0
$700$	0	0
$701$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$701$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0.765367i	0.765367i
$707$	0	0
$708$	0	0
$709$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$709$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$710$	0	0
$711$	0	0
$712$	0.765367i	0.765367i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	0	0
$721$	0	0
$722$	− 1.00000i	− 1.00000i
$723$	0	0
$724$	0.765367i	0.765367i
$725$	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
$730$	0.585786	0.585786
$731$	0	0
$732$	0	0
$733$	− 0.765367i	− 0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$733$	− 0.765367i	− 0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	1.08239	1.08239
$741$	0	0
$742$	0	0
$743$	0	0		−	1.00000i	$-0.5\pi$
$743$	0	0			1.00000i	$0.5\pi$
$744$	0	0
$745$	1.08239i	1.08239i
$746$	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
$757$	−1.41421	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$757$	−1.41421	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$761$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	0.765367i	0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$769$	0.765367i	0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$773$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$774$	0	0
$775$	0	0
$776$	0.765367	0.765367
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.41421i	− 1.41421i
$786$	0	0
$787$	0	0		−	1.00000i	$-0.5\pi$
$787$	0	0			1.00000i	$0.5\pi$
$788$	1.41421i	1.41421i
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1.41421	1.41421
$794$	0.765367	0.765367
$795$	0	0
$796$	0	0
$797$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$797$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$798$	0	0
$799$	0	0
$800$	− 0.414214i	− 0.414214i
$801$	0	0
$802$	−2.00000	−2.00000
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0.765367i	0.765367i
$809$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$809$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$810$	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
$818$	−0.765367	−0.765367
$819$	0	0
$820$	−1.41421	−1.41421
$821$	− 1.41421i	− 1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$821$	− 1.41421i	− 1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$822$	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		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$829$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$830$	0	0
$831$	0	0
$832$	1.84776i	1.84776i
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.84776	−1.84776
$846$	0	0
$847$	0	0
$848$	1.41421i	1.41421i
$849$	0	0
$850$	− 0.765367i	− 0.765367i
$851$	0	0
$852$	0	0
$853$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$853$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$857$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$858$	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
$865$	−1.41421	−1.41421
$866$	−0.765367	−0.765367
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	− 1.41421i	− 1.41421i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.41421	1.41421	0.707107	−	0.707107i	$-0.250000\pi$
$877$	1.41421	1.41421	0.707107	+	0.707107i	$0.250000\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.84776	1.84776	0.923880	−	0.382683i	$-0.125000\pi$
$881$	1.84776	1.84776	0.923880	+	0.382683i	$0.125000\pi$
$882$	0	0
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	3.41421i	3.41421i
$885$	0	0
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	− 0.585786i	− 0.585786i
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	1.41421	1.41421
$899$	0	0
$900$	0	0
$901$	2.61313i	2.61313i
$902$	0	0
$903$	0	0
$904$	1.41421	1.41421
$905$	− 0.585786i	− 0.585786i
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	2.00000i	2.00000i
$915$	0	0
$916$	0.765367i	0.765367i
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	− 1.84776i	− 1.84776i
$923$	0	0
$924$	0	0
$925$	0.585786	0.585786
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0.765367	0.765367	0.382683	−	0.923880i	$-0.375000\pi$
$929$	0.765367	0.765367	0.382683	+	0.923880i	$0.375000\pi$
$930$	0	0
$931$	0	0
$932$	2.00000i	2.00000i
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$937$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.84776	−1.84776	−0.923880	−	0.382683i	$-0.875000\pi$
$941$	−1.84776	−1.84776	−0.923880	+	0.382683i	$0.875000\pi$
$942$	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
$949$	−1.41421	−1.41421
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.41421i	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$953$	1.41421i	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−1.00000	−1.00000
$962$	−2.61313	−2.61313
$963$	0	0
$964$	− 1.84776i	− 1.84776i
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	1.00000i	1.00000i
$969$	0	0
$970$	−0.585786	−0.585786
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0.765367i	0.765367i
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	− 1.08239i	− 1.08239i
$986$	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
$997$	− 1.84776i	− 1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$997$	− 1.84776i	− 1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.1.h.a.1763.7	yes	8
3.2	odd	2	inner	1764.1.h.a.1763.2	✓	8
4.3	odd	2	CM	1764.1.h.a.1763.7	yes	8
7.2	even	3		1764.1.q.b.1403.6		16
7.3	odd	6		1764.1.q.b.215.3		16
7.4	even	3		1764.1.q.b.215.2		16
7.5	odd	6		1764.1.q.b.1403.7		16
7.6	odd	2	inner	1764.1.h.a.1763.6	yes	8
12.11	even	2	inner	1764.1.h.a.1763.2	✓	8
21.2	odd	6		1764.1.q.b.1403.3		16
21.5	even	6		1764.1.q.b.1403.2		16
21.11	odd	6		1764.1.q.b.215.7		16
21.17	even	6		1764.1.q.b.215.6		16
21.20	even	2	inner	1764.1.h.a.1763.3	yes	8
28.3	even	6		1764.1.q.b.215.3		16
28.11	odd	6		1764.1.q.b.215.2		16
28.19	even	6		1764.1.q.b.1403.7		16
28.23	odd	6		1764.1.q.b.1403.6		16
28.27	even	2	inner	1764.1.h.a.1763.6	yes	8
84.11	even	6		1764.1.q.b.215.7		16
84.23	even	6		1764.1.q.b.1403.3		16
84.47	odd	6		1764.1.q.b.1403.2		16
84.59	odd	6		1764.1.q.b.215.6		16
84.83	odd	2	inner	1764.1.h.a.1763.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1764.1.h.a.1763.2	✓	8	3.2	odd	2	inner
1764.1.h.a.1763.2	✓	8	12.11	even	2	inner
1764.1.h.a.1763.3	yes	8	21.20	even	2	inner
1764.1.h.a.1763.3	yes	8	84.83	odd	2	inner
1764.1.h.a.1763.6	yes	8	7.6	odd	2	inner
1764.1.h.a.1763.6	yes	8	28.27	even	2	inner
1764.1.h.a.1763.7	yes	8	1.1	even	1	trivial
1764.1.h.a.1763.7	yes	8	4.3	odd	2	CM
1764.1.q.b.215.2		16	7.4	even	3
1764.1.q.b.215.2		16	28.11	odd	6
1764.1.q.b.215.3		16	7.3	odd	6
1764.1.q.b.215.3		16	28.3	even	6
1764.1.q.b.215.6		16	21.17	even	6
1764.1.q.b.215.6		16	84.59	odd	6
1764.1.q.b.215.7		16	21.11	odd	6
1764.1.q.b.215.7		16	84.11	even	6
1764.1.q.b.1403.2		16	21.5	even	6
1764.1.q.b.1403.2		16	84.47	odd	6
1764.1.q.b.1403.3		16	21.2	odd	6
1764.1.q.b.1403.3		16	84.23	even	6
1764.1.q.b.1403.6		16	7.2	even	3
1764.1.q.b.1403.6		16	28.23	odd	6
1764.1.q.b.1403.7		16	7.5	odd	6
1764.1.q.b.1403.7		16	28.19	even	6

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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1764.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.765367 q^{5} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +0.765367 q^{5} -1.00000i q^{8} +0.765367i q^{10} -1.84776i q^{13} +1.00000 q^{16} +1.84776 q^{17} -0.765367 q^{20} -0.414214 q^{25} +1.84776 q^{26} +1.00000i q^{32} +1.84776i q^{34} -1.41421 q^{37} -0.765367i q^{40} +1.84776 q^{41} -0.414214i q^{50} +1.84776i q^{52} +1.41421i q^{53} +0.765367i q^{61} -1.00000 q^{64} -1.41421i q^{65} -1.84776 q^{68} -0.765367i q^{73} -1.41421i q^{74} +0.765367 q^{80} +1.84776i q^{82} +1.41421 q^{85} -0.765367 q^{89} +0.765367i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{4}+O(q^{10}) \) 8 * q - 8 * q^4	\( 8 q - 8 q^{4} + 8 q^{16} + 8 q^{25} - 8 q^{64}+O(q^{100}) \) 8 * q - 8 * q^4 + 8 * q^16 + 8 * q^25 - 8 * q^64

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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