LMFDB - Embedded newform 1759.1.b.b.1758.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1759,1,Mod(1758,1759)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1759, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1759.1758");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1759$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1759.b (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:	yes
Analytic conductor:	$0.877855357221$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - 3x - 1$ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.9573337234561.1
Artin image:	$D_9$
Artin field:	Galois closure of 9.1.9573337234561.1

Embedding invariants

Embedding label			1758.3
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	1759.1758

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.53209 q^{5} +1.00000 q^{8} +1.00000 q^{9} -1.53209 q^{10} +1.53209 q^{11} -1.87939 q^{13} -1.00000 q^{16} +1.53209 q^{17} -1.00000 q^{18} -1.53209 q^{22} +0.347296 q^{23} +1.34730 q^{25} +1.87939 q^{26} -1.00000 q^{31} -1.53209 q^{34} +1.53209 q^{40} -1.87939 q^{41} -1.87939 q^{43} +1.53209 q^{45} -0.347296 q^{46} -1.87939 q^{47} +1.00000 q^{49} -1.34730 q^{50} +0.347296 q^{53} +2.34730 q^{55} +1.00000 q^{62} +1.00000 q^{64} -2.87939 q^{65} +0.347296 q^{71} +1.00000 q^{72} -1.53209 q^{80} +1.00000 q^{81} +1.87939 q^{82} +2.34730 q^{85} +1.87939 q^{86} +1.53209 q^{88} +2.00000 q^{89} -1.53209 q^{90} +1.87939 q^{94} -1.00000 q^{98} +1.53209 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 3 q^{2} + 3 q^{8} + 3 q^{9} - 3 q^{16} - 3 q^{18} + 3 q^{25} - 3 q^{31} + 3 q^{49} - 3 q^{50} + 6 q^{55} + 3 q^{62} + 3 q^{64} - 3 q^{65} + 3 q^{72} + 3 q^{81} + 6 q^{85} + 6 q^{89} - 3 q^{98}+O(q^{100})$ 3 * q - 3 * q^2 + 3 * q^8 + 3 * q^9 - 3 * q^16 - 3 * q^18 + 3 * q^25 - 3 * q^31 + 3 * q^49 - 3 * q^50 + 6 * q^55 + 3 * q^62 + 3 * q^64 - 3 * q^65 + 3 * q^72 + 3 * q^81 + 6 * q^85 + 6 * q^89 - 3 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1759\mathbb{Z}\right)^\times$ .

$n$	$6$
$\chi(n)$	$-1$

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	0	0	1.00000			$0$
$3$	0	0	−1.00000			$\pi$
$4$	0	0
$5$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$5$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	1.00000	1.00000
$9$	1.00000	1.00000
$10$	−1.53209	−1.53209
$11$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$11$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$12$	0	0
$13$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$13$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−1.00000
$17$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$17$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$18$	−1.00000	−1.00000
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	0	0
$22$	−1.53209	−1.53209
$23$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$23$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$24$	0	0
$25$	1.34730	1.34730
$26$	1.87939	1.87939
$27$	0	0
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$31$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$32$	0	0
$33$	0	0
$34$	−1.53209	−1.53209
$35$	0	0
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	0	0
$40$	1.53209	1.53209
$41$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$41$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$42$	0	0
$43$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$43$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$44$	0	0
$45$	1.53209	1.53209
$46$	−0.347296	−0.347296
$47$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$47$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$48$	0	0
$49$	1.00000	1.00000
$50$	−1.34730	−1.34730
$51$	0	0
$52$	0	0
$53$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$53$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$54$	0	0
$55$	2.34730	2.34730
$56$	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	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	1.00000	1.00000
$63$	0	0
$64$	1.00000	1.00000
$65$	−2.87939	−2.87939
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$71$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$72$	1.00000	1.00000
$73$	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
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	−1.53209	−1.53209
$81$	1.00000	1.00000
$82$	1.87939	1.87939
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	2.34730	2.34730
$86$	1.87939	1.87939
$87$	0	0
$88$	1.53209	1.53209
$89$	2.00000	2.00000	1.00000			$0$
$89$	2.00000	2.00000	1.00000			$0$
$90$	−1.53209	−1.53209
$91$	0	0
$92$	0	0
$93$	0	0
$94$	1.87939	1.87939
$95$	0	0
$96$	0	0
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	−1.00000	−1.00000
$99$	1.53209	1.53209
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	−1.87939	−1.87939
$105$	0	0
$106$	−0.347296	−0.347296
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$109$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$110$	−2.34730	−2.34730
$111$	0	0
$112$	0	0
$113$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$113$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$114$	0	0
$115$	0.532089	0.532089
$116$	0	0
$117$	−1.87939	−1.87939
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.34730	1.34730
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0.532089	0.532089
$126$	0	0
$127$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$127$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$128$	−1.00000	−1.00000
$129$	0	0
$130$	2.87939	2.87939
$131$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$131$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	1.53209	1.53209
$137$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$137$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$138$	0	0
$139$	0	0	1.00000			$0$
$139$	0	0	−1.00000			$\pi$
$140$	0	0
$141$	0	0
$142$	−0.347296	−0.347296
$143$	−2.87939	−2.87939
$144$	−1.00000	−1.00000
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$149$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	1.53209	1.53209
$154$	0	0
$155$	−1.53209	−1.53209
$156$	0	0
$157$	0	0	1.00000			$0$
$157$	0	0	−1.00000			$\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	−1.00000	−1.00000
$163$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$163$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	0	0
$169$	2.53209	2.53209
$170$	−2.34730	−2.34730
$171$	0	0
$172$	0	0
$173$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$173$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$174$	0	0
$175$	0	0
$176$	−1.53209	−1.53209
$177$	0	0
$178$	−2.00000	−2.00000
$179$	2.00000	2.00000	1.00000			$0$
$179$	2.00000	2.00000	1.00000			$0$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0.347296	0.347296
$185$	0	0
$186$	0	0
$187$	2.34730	2.34730
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	0	0
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	−1.53209	−1.53209
$199$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$199$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$200$	1.34730	1.34730
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−2.87939	−2.87939
$206$	0	0
$207$	0.347296	0.347296
$208$	1.87939	1.87939
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−2.87939	−2.87939
$216$	0	0
$217$	0	0
$218$	1.87939	1.87939
$219$	0	0
$220$	0	0
$221$	−2.87939	−2.87939
$222$	0	0
$223$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$223$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$224$	0	0
$225$	1.34730	1.34730
$226$	1.00000	1.00000
$227$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$227$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	−0.532089	−0.532089
$231$	0	0
$232$	0	0
$233$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$233$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$234$	1.87939	1.87939
$235$	−2.87939	−2.87939
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$239$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$240$	0	0
$241$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$242$	−1.34730	−1.34730
$243$	0	0
$244$	0	0
$245$	1.53209	1.53209
$246$	0	0
$247$	0	0
$248$	−1.00000	−1.00000
$249$	0	0
$250$	−0.532089	−0.532089
$251$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$251$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$252$	0	0
$253$	0.532089	0.532089
$254$	−1.53209	−1.53209
$255$	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
$262$	−0.347296	−0.347296
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0.532089	0.532089
$266$	0	0
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$271$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$272$	−1.53209	−1.53209
$273$	0	0
$274$	−0.347296	−0.347296
$275$	2.06418	2.06418
$276$	0	0
$277$	0	0	1.00000			$0$
$277$	0	0	−1.00000			$\pi$
$278$	0	0
$279$	−1.00000	−1.00000
$280$	0	0
$281$	0	0	1.00000			$0$
$281$	0	0	−1.00000			$\pi$
$282$	0	0
$283$	0	0	1.00000			$0$
$283$	0	0	−1.00000			$\pi$
$284$	0	0
$285$	0	0
$286$	2.87939	2.87939
$287$	0	0
$288$	0	0
$289$	1.34730	1.34730
$290$	0	0
$291$	0	0
$292$	0	0
$293$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$293$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	1.00000	1.00000
$299$	−0.652704	−0.652704
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	−1.53209	−1.53209
$307$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$307$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$308$	0	0
$309$	0	0
$310$	1.53209	1.53209
$311$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$311$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$312$	0	0
$313$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$313$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$314$	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
$320$	1.53209	1.53209
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−2.53209	−2.53209
$326$	1.87939	1.87939
$327$	0	0
$328$	−1.87939	−1.87939
$329$	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
$337$	2.00000	2.00000	1.00000			$0$
$337$	2.00000	2.00000	1.00000			$0$
$338$	−2.53209	−2.53209
$339$	0	0
$340$	0	0
$341$	−1.53209	−1.53209
$342$	0	0
$343$	0	0
$344$	−1.87939	−1.87939
$345$	0	0
$346$	−1.53209	−1.53209
$347$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$347$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$348$	0	0
$349$	2.00000	2.00000	1.00000			$0$
$349$	2.00000	2.00000	1.00000			$0$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0	0	1.00000			$0$
$353$	0	0	−1.00000			$\pi$
$354$	0	0
$355$	0.532089	0.532089
$356$	0	0
$357$	0	0
$358$	−2.00000	−2.00000
$359$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$359$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$360$	1.53209	1.53209
$361$	1.00000	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			$\pi$
$368$	−0.347296	−0.347296
$369$	−1.87939	−1.87939
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$373$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$374$	−2.34730	−2.34730
$375$	0	0
$376$	−1.87939	−1.87939
$377$	0	0
$378$	0	0
$379$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$379$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$380$	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
$387$	−1.87939	−1.87939
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0.532089	0.532089
$392$	1.00000	1.00000
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$397$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$398$	1.87939	1.87939
$399$	0	0
$400$	−1.34730	−1.34730
$401$	0	0	1.00000			$0$
$401$	0	0	−1.00000			$\pi$
$402$	0	0
$403$	1.87939	1.87939
$404$	0	0
$405$	1.53209	1.53209
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$409$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$410$	2.87939	2.87939
$411$	0	0
$412$	0	0
$413$	0	0
$414$	−0.347296	−0.347296
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$419$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$420$	0	0
$421$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$421$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$422$	0	0
$423$	−1.87939	−1.87939
$424$	0.347296	0.347296
$425$	2.06418	2.06418
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	2.87939	2.87939
$431$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$431$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$432$	0	0
$433$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$433$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$439$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$440$	2.34730	2.34730
$441$	1.00000	1.00000
$442$	2.87939	2.87939
$443$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$443$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$444$	0	0
$445$	3.06418	3.06418
$446$	1.00000	1.00000
$447$	0	0
$448$	0	0
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	−1.34730	−1.34730
$451$	−2.87939	−2.87939
$452$	0	0
$453$	0	0
$454$	1.00000	1.00000
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$461$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	−0.347296	−0.347296
$467$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$467$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$468$	0	0
$469$	0	0
$470$	2.87939	2.87939
$471$	0	0
$472$	0	0
$473$	−2.87939	−2.87939
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.347296	0.347296
$478$	−1.53209	−1.53209
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	1.00000	1.00000
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$487$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$488$	0	0
$489$	0	0
$490$	−1.53209	−1.53209
$491$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$491$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	2.34730	2.34730
$496$	1.00000	1.00000
$497$	0	0
$498$	0	0
$499$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$499$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$500$	0	0
$501$	0	0
$502$	1.87939	1.87939
$503$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$503$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$504$	0	0
$505$	0	0
$506$	−0.532089	−0.532089
$507$	0	0
$508$	0	0
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	1.00000
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−2.87939	−2.87939
$518$	0	0
$519$	0	0
$520$	−2.87939	−2.87939
$521$	2.00000	2.00000	1.00000			$0$
$521$	2.00000	2.00000	1.00000			$0$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.53209	−1.53209
$528$	0	0
$529$	−0.879385	−0.879385
$530$	−0.532089	−0.532089
$531$	0	0
$532$	0	0
$533$	3.53209	3.53209
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.53209	1.53209
$540$	0	0
$541$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$541$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$542$	1.87939	1.87939
$543$	0	0
$544$	0	0
$545$	−2.87939	−2.87939
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0	0
$549$	0	0
$550$	−2.06418	−2.06418
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$557$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$558$	1.00000	1.00000
$559$	3.53209	3.53209
$560$	0	0
$561$	0	0
$562$	0	0
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	0	0
$565$	−1.53209	−1.53209
$566$	0	0
$567$	0	0
$568$	0.347296	0.347296
$569$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$569$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$570$	0	0
$571$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0.467911	0.467911
$576$	1.00000	1.00000
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	−1.34730	−1.34730
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0.532089	0.532089
$584$	0	0
$585$	−2.87939	−2.87939
$586$	−0.347296	−0.347296
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0.652704	0.652704
$599$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$599$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$600$	0	0
$601$	0	0	1.00000			$0$
$601$	0	0	−1.00000			$\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.06418	2.06418
$606$	0	0
$607$	0	0	1.00000			$0$
$607$	0	0	−1.00000			$\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.53209	3.53209
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	−0.347296	−0.347296
$615$	0	0
$616$	0	0
$617$	0	0	1.00000			$0$
$617$	0	0	−1.00000			$\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	1.00000	1.00000
$623$	0	0
$624$	0	0
$625$	−0.532089	−0.532089
$626$	1.00000	1.00000
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$631$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.34730	2.34730
$636$	0	0
$637$	−1.87939	−1.87939
$638$	0	0
$639$	0.347296	0.347296
$640$	−1.53209	−1.53209
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$647$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$648$	1.00000	1.00000
$649$	0	0
$650$	2.53209	2.53209
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0.532089	0.532089
$656$	1.87939	1.87939
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	0	0
$661$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$661$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$662$	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
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	−2.00000	−2.00000
$675$	0	0
$676$	0	0
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	2.34730	2.34730
$681$	0	0
$682$	1.53209	1.53209
$683$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$683$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$684$	0	0
$685$	0.532089	0.532089
$686$	0	0
$687$	0	0
$688$	1.87939	1.87939
$689$	−0.652704	−0.652704
$690$	0	0
$691$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$691$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$692$	0	0
$693$	0	0
$694$	1.00000	1.00000
$695$	0	0
$696$	0	0
$697$	−2.87939	−2.87939
$698$	−2.00000	−2.00000
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	0	0
$703$	0	0
$704$	1.53209	1.53209
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	−0.532089	−0.532089
$711$	0	0
$712$	2.00000	2.00000
$713$	−0.347296	−0.347296
$714$	0	0
$715$	−4.41147	−4.41147
$716$	0	0
$717$	0	0
$718$	−1.53209	−1.53209
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−1.53209	−1.53209
$721$	0	0
$722$	−1.00000	−1.00000
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$727$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$728$	0	0
$729$	1.00000	1.00000
$730$	0	0
$731$	−2.87939	−2.87939
$732$	0	0
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	1.87939	1.87939
$739$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$739$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	−1.53209	−1.53209
$746$	1.00000	1.00000
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$751$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$752$	1.87939	1.87939
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	−0.347296	−0.347296
$759$	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
$765$	2.34730	2.34730
$766$	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
$773$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$773$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$774$	1.87939	1.87939
$775$	−1.34730	−1.34730
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0.532089	0.532089
$782$	−0.532089	−0.532089
$783$	0	0
$784$	−1.00000	−1.00000
$785$	0	0
$786$	0	0
$787$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$787$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	1.53209	1.53209
$793$	0	0
$794$	1.87939	1.87939
$795$	0	0
$796$	0	0
$797$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$797$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$798$	0	0
$799$	−2.87939	−2.87939
$800$	0	0
$801$	2.00000	2.00000
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	−1.87939	−1.87939
$807$	0	0
$808$	0	0
$809$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$809$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$810$	−1.53209	−1.53209
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.87939	−2.87939
$816$	0	0
$817$	0	0
$818$	−0.347296	−0.347296
$819$	0	0
$820$	0	0
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$829$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$830$	0	0
$831$	0	0
$832$	−1.87939	−1.87939
$833$	1.53209	1.53209
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	−0.347296	−0.347296
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	−1.53209	−1.53209
$843$	0	0
$844$	0	0
$845$	3.87939	3.87939
$846$	1.87939	1.87939
$847$	0	0
$848$	−0.347296	−0.347296
$849$	0	0
$850$	−2.06418	−2.06418
$851$	0	0
$852$	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	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	0	0	1.00000			$0$
$859$	0	0	−1.00000			$\pi$
$860$	0	0
$861$	0	0
$862$	1.87939	1.87939
$863$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$863$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$864$	0	0
$865$	2.34730	2.34730
$866$	1.00000	1.00000
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	−1.87939	−1.87939
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	−0.347296	−0.347296
$879$	0	0
$880$	−2.34730	−2.34730
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	−1.00000	−1.00000
$883$	0	0	1.00000			$0$
$883$	0	0	−1.00000			$\pi$
$884$	0	0
$885$	0	0
$886$	1.00000	1.00000
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	−3.06418	−3.06418
$891$	1.53209	1.53209
$892$	0	0
$893$	0	0
$894$	0	0
$895$	3.06418	3.06418
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0.532089	0.532089
$902$	2.87939	2.87939
$903$	0	0
$904$	−1.00000	−1.00000
$905$	0	0
$906$	0	0
$907$	0.347296	0.347296	0.173648	−	0.984808i	$-0.444444\pi$
$907$	0.347296	0.347296	0.173648	+	0.984808i	$0.444444\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0.532089	0.532089
$921$	0	0
$922$	−0.347296	−0.347296
$923$	−0.652704	−0.652704
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$929$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	−1.53209	−1.53209
$935$	3.59627	3.59627
$936$	−1.87939	−1.87939
$937$	−1.87939	−1.87939	−0.939693	−	0.342020i	$-0.888889\pi$
$937$	−1.87939	−1.87939	−0.939693	+	0.342020i	$0.888889\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	−0.652704	−0.652704
$944$	0	0
$945$	0	0
$946$	2.87939	2.87939
$947$	0	0	1.00000			$0$
$947$	0	0	−1.00000			$\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	−0.347296	−0.347296
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	0	0
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	1.34730	1.34730
$969$	0	0
$970$	0	0
$971$	1.53209	1.53209	0.766044	−	0.642788i	$-0.222222\pi$
$971$	1.53209	1.53209	0.766044	+	0.642788i	$0.222222\pi$
$972$	0	0
$973$	0	0
$974$	−1.53209	−1.53209
$975$	0	0
$976$	0	0
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	3.06418	3.06418
$980$	0	0
$981$	−1.87939	−1.87939
$982$	1.87939	1.87939
$983$	2.00000	2.00000	1.00000			$0$
$983$	2.00000	2.00000	1.00000			$0$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.652704	−0.652704
$990$	−2.34730	−2.34730
$991$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.87939	−2.87939
$996$	0	0
$997$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$997$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$998$	−1.53209	−1.53209
$999$	0	0

Display

a_p

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p

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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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1759.1.b.b.1758.3	✓	3
1759.1758	odd	2	CM	1759.1.b.b.1758.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1759.1.b.b.1758.3	✓	3	1.1	even	1	trivial
1759.1.b.b.1758.3	✓	3	1759.1758	odd	2	CM

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1759.1.b.b.1758.3

Level	$1759$
Weight	$1$
Character	1759.1758
Self dual	yes
Analytic conductor	$0.878$
Analytic rank	$0$
Dimension	$3$
Projective image	$D_{9}$
CM discriminant	-1759
Inner twists	$2$