LMFDB - Embedded newform 1539.1.cf.a.1106.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1539,1,Mod(188,1539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1539, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([3, 10]))

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

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

chi := DirichletCharacter("1539.188");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1539 = 3^{4} \cdot 19$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1539.cf (of order $18$ , degree $6$ , 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:	$0.768061054442$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{3} + 1$ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 513)
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.1002862414008009.1

Embedding invariants

Embedding label			1106.1
Root			$-0.173648 + 0.984808i$ of defining polynomial
Character	$\chi$	$=$	1539.1106
Dual form			1539.1.cf.a.917.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.939693 + 0.342020i) q^{4} +1.53209 q^{7} +(0.266044 + 0.223238i) q^{13} +(0.766044 - 0.642788i) q^{16} +(-0.500000 - 0.866025i) q^{19} +(-0.939693 + 0.342020i) q^{25} +(-1.43969 + 0.524005i) q^{28} +(0.939693 + 1.62760i) q^{31} +1.53209 q^{37} +(1.76604 + 0.642788i) q^{43} +1.34730 q^{49} +(-0.326352 - 0.118782i) q^{52} +(0.0603074 - 0.342020i) q^{61} +(-0.500000 + 0.866025i) q^{64} +(0.266044 - 1.50881i) q^{67} +(-0.326352 - 1.85083i) q^{73} +(0.766044 + 0.642788i) q^{76} +(-1.43969 + 1.20805i) q^{79} +(0.407604 + 0.342020i) q^{91} +(0.347296 + 1.96962i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 3 q^{13} - 3 q^{19} - 3 q^{28} + 6 q^{43} + 6 q^{49} - 3 q^{52} + 6 q^{61} - 3 q^{64} - 3 q^{67} - 3 q^{73} - 3 q^{79} + 6 q^{91}+O(q^{100})$ 6 * q - 3 * q^13 - 3 * q^19 - 3 * q^28 + 6 * q^43 + 6 * q^49 - 3 * q^52 + 6 * q^61 - 3 * q^64 - 3 * q^67 - 3 * q^73 - 3 * q^79 + 6 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1539\mathbb{Z}\right)^\times$ .

$n$	$325$	$1217$
$\chi(n)$	$e\left(\frac{1}{9}\right)$	$e\left(\frac{5}{6}\right)$

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		−0.173648	−	0.984808i	$-0.555556\pi$
$2$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$3$		0			0
$3$		0			0
$4$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$5$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$5$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$6$		0			0
$7$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$7$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	0.266044	+	0.223238i	0.266044	+	0.223238i	0.766044	−	0.642788i	$-0.222222\pi$
$13$	0.266044	+	0.223238i	0.266044	+	0.223238i	−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$	0.766044	−	0.642788i	0.766044	−	0.642788i
$17$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$17$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$18$		0			0
$19$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$19$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$23$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$24$		0			0
$25$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$26$		0			0
$27$		0			0
$28$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$29$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$29$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$30$		0			0
$31$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$31$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$37$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$41$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$42$		0			0
$43$	1.76604	+	0.642788i	1.76604	+	0.642788i	1.00000			$0$
$43$	1.76604	+	0.642788i	1.76604	+	0.642788i	0.766044	+	0.642788i	$0.222222\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$47$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$48$		0			0
$49$	1.34730			1.34730
$50$		0			0
$51$		0			0
$52$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$53$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$53$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$59$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$60$		0			0
$61$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	−0.939693	−	0.342020i	$-0.888889\pi$
$61$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	1.00000			$0$
$62$		0			0
$63$		0			0
$64$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$	0.266044	−	1.50881i	0.266044	−	1.50881i	−0.500000	−	0.866025i	$-0.666667\pi$
$67$	0.266044	−	1.50881i	0.266044	−	1.50881i	0.766044	−	0.642788i	$-0.222222\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$71$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$72$		0			0
$73$	−0.326352	−	1.85083i	−0.326352	−	1.85083i	−0.500000	−	0.866025i	$-0.666667\pi$
$73$	−0.326352	−	1.85083i	−0.326352	−	1.85083i	0.173648	−	0.984808i	$-0.444444\pi$
$74$		0			0
$75$		0			0
$76$	0.766044	+	0.642788i	0.766044	+	0.642788i
$77$		0			0
$78$		0			0
$79$	−1.43969	+	1.20805i	−1.43969	+	1.20805i	−0.500000	+	0.866025i	$0.666667\pi$
$79$	−1.43969	+	1.20805i	−1.43969	+	1.20805i	−0.939693	+	0.342020i	$0.888889\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$89$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$90$		0			0
$91$	0.407604	+	0.342020i	0.407604	+	0.342020i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$97$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$98$		0			0
$99$		0			0
$100$	0.766044	−	0.642788i	0.766044	−	0.642788i
$101$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$101$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$102$		0			0
$103$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$103$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$109$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$110$		0			0
$111$		0			0
$112$	1.17365	−	0.984808i	1.17365	−	0.984808i
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$125$		0			0
$126$		0			0
$127$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$127$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$131$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$132$		0			0
$133$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$137$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$138$		0			0
$139$	−1.43969	−	1.20805i	−1.43969	−	1.20805i	−0.939693	−	0.342020i	$-0.888889\pi$
$139$	−1.43969	−	1.20805i	−1.43969	−	1.20805i	−0.500000	−	0.866025i	$-0.666667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$149$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$149$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$150$		0			0
$151$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$151$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.0603074	+	0.342020i	0.0603074	+	0.342020i	1.00000			$0$
$157$	0.0603074	+	0.342020i	0.0603074	+	0.342020i	−0.939693	+	0.342020i	$0.888889\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$163$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$167$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$168$		0			0
$169$	−0.152704	−	0.866025i	−0.152704	−	0.866025i
$170$		0			0
$171$		0			0
$172$	−1.87939			−1.87939
$173$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$173$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$174$		0			0
$175$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$181$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$182$		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
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	−0.326352	−	0.118782i	−0.326352	−	0.118782i	0.173648	−	0.984808i	$-0.444444\pi$
$193$	−0.326352	−	0.118782i	−0.326352	−	0.118782i	−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$	−1.26604	+	0.460802i	−1.26604	+	0.460802i
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	−0.500000	−	0.866025i	$-0.666667\pi$
$199$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	0.173648	+	0.984808i	$0.444444\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.347296			0.347296
$209$		0			0
$210$		0			0
$211$	0.266044	−	0.223238i	0.266044	−	0.223238i	−0.500000	−	0.866025i	$-0.666667\pi$
$211$	0.266044	−	0.223238i	0.266044	−	0.223238i	0.766044	+	0.642788i	$0.222222\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.43969	+	2.49362i	1.43969	+	2.49362i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.326352	−	0.118782i	−0.326352	−	0.118782i	0.173648	−	0.984808i	$-0.444444\pi$
$223$	−0.326352	−	0.118782i	−0.326352	−	0.118782i	−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$229$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$233$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.939693	−	0.342020i	$-0.888889\pi$
$241$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$		0			0
$244$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$245$		0			0
$246$		0			0
$247$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$251$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.173648	−	0.984808i	0.173648	−	0.984808i
$257$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$257$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$258$		0			0
$259$	2.34730			2.34730
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$263$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.266044	+	1.50881i	0.266044	+	1.50881i
$269$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$269$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$270$		0			0
$271$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$271$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$277$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$281$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$282$		0			0
$283$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$283$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.766044	+	0.642788i	0.766044	+	0.642788i
$290$		0			0
$291$		0			0
$292$	0.939693	+	1.62760i	0.939693	+	1.62760i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	2.70574	+	0.984808i	2.70574	+	0.984808i
$302$		0			0
$303$		0			0
$304$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$305$		0			0
$306$		0			0
$307$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$307$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$313$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$314$		0			0
$315$		0			0
$316$	0.939693	−	1.62760i	0.939693	−	1.62760i
$317$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$317$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$331$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.326352	−	1.85083i	−0.326352	−	1.85083i	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	−0.326352	−	1.85083i	−0.326352	−	1.85083i	0.173648	−	0.984808i	$-0.444444\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.532089			0.532089
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$347$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$348$		0			0
$349$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$349$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$359$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$360$		0			0
$361$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$362$		0			0
$363$		0			0
$364$	−0.500000	−	0.181985i	−0.500000	−	0.181985i
$365$		0			0
$366$		0			0
$367$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.939693	−	0.342020i	$-0.888889\pi$
$367$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$373$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$379$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$383$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$389$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$389$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.939693	+	0.342020i	$0.888889\pi$
$398$		0			0
$399$		0			0
$400$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$401$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$401$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$402$		0			0
$403$	−0.113341	+	0.642788i	−0.113341	+	0.642788i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$409$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$410$		0			0
$411$		0			0
$412$	1.76604	−	0.642788i	1.76604	−	0.642788i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$421$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.0923963	−	0.524005i	0.0923963	−	0.524005i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$431$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$432$		0			0
$433$	0.266044	+	0.223238i	0.266044	+	0.223238i	0.766044	−	0.642788i	$-0.222222\pi$
$433$	0.266044	+	0.223238i	0.266044	+	0.223238i	−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$		0			0
$436$	0.500000	−	0.866025i	0.500000	−	0.866025i
$437$		0			0
$438$		0			0
$439$	0.266044	+	1.50881i	0.266044	+	1.50881i	0.766044	+	0.642788i	$0.222222\pi$
$439$	0.266044	+	1.50881i	0.266044	+	1.50881i	−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$443$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$449$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$449$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$457$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$461$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$462$		0			0
$463$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$463$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$	0.407604	−	2.31164i	0.407604	−	2.31164i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.766044	+	0.642788i	0.766044	+	0.642788i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$479$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$480$		0			0
$481$	0.407604	+	0.342020i	0.407604	+	0.342020i
$482$		0			0
$483$		0			0
$484$	0.766044	+	0.642788i	0.766044	+	0.642788i
$485$		0			0
$486$		0			0
$487$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$487$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$491$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.76604	+	0.642788i	1.76604	+	0.642788i
$497$		0			0
$498$		0			0
$499$	0.266044	+	1.50881i	0.266044	+	1.50881i	0.766044	+	0.642788i	$0.222222\pi$
$499$	0.266044	+	1.50881i	0.266044	+	1.50881i	−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$503$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$509$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$509$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$510$		0			0
$511$	−0.500000	−	2.83564i	−0.500000	−	2.83564i
$512$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.939693	−	0.342020i	$-0.888889\pi$
$523$	−1.43969	+	0.524005i	−1.43969	+	0.524005i	−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.766044	−	0.642788i	0.766044	−	0.642788i
$530$		0			0
$531$		0			0
$532$	1.17365	+	0.984808i	1.17365	+	0.984808i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.76604	−	0.642788i	1.76604	−	0.642788i	0.766044	−	0.642788i	$-0.222222\pi$
$541$	1.76604	−	0.642788i	1.76604	−	0.642788i	1.00000			$0$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	0.173648	+	0.984808i	$0.444444\pi$
$547$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	−0.500000	+	0.866025i	$0.666667\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.20574	+	1.85083i	−2.20574	+	1.85083i
$554$		0			0
$555$		0			0
$556$	1.76604	+	0.642788i	1.76604	+	0.642788i
$557$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$557$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$558$		0			0
$559$	0.326352	+	0.565258i	0.326352	+	0.565258i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$571$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$577$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$587$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$588$		0			0
$589$	0.939693	−	1.62760i	0.939693	−	1.62760i
$590$		0			0
$591$		0			0
$592$	1.17365	−	0.984808i	1.17365	−	0.984808i
$593$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$593$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$599$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$600$		0			0
$601$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$601$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$602$		0			0
$603$		0			0
$604$	1.53209	+	1.28558i	1.53209	+	1.28558i
$605$		0			0
$606$		0			0
$607$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$607$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$613$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$617$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$618$		0			0
$619$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$619$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.766044	−	0.642788i	0.766044	−	0.642788i
$626$		0			0
$627$		0			0
$628$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$629$		0			0
$630$		0			0
$631$	1.17365	+	0.984808i	1.17365	+	0.984808i	1.00000			$0$
$631$	1.17365	+	0.984808i	1.17365	+	0.984808i	0.173648	+	0.984808i	$0.444444\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.358441	+	0.300767i	0.358441	+	0.300767i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$641$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$642$		0			0
$643$	−0.326352	−	0.118782i	−0.326352	−	0.118782i	0.173648	−	0.984808i	$-0.444444\pi$
$643$	−0.326352	−	0.118782i	−0.326352	−	0.118782i	−0.500000	+	0.866025i	$0.666667\pi$
$644$		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			0
$651$		0			0
$652$	0.266044	+	0.223238i	0.266044	+	0.223238i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$659$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$660$		0			0
$661$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$661$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$662$		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
$673$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$673$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$674$		0			0
$675$		0			0
$676$	0.439693	+	0.761570i	0.439693	+	0.761570i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	0.532089	+	3.01763i	0.532089	+	3.01763i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	1.76604	−	0.642788i	1.76604	−	0.642788i
$689$		0			0
$690$		0			0
$691$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$691$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.17365	−	0.984808i	1.17365	−	0.984808i
$701$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$701$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$702$		0			0
$703$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.76604	−	0.642788i	1.76604	−	0.642788i	0.766044	−	0.642788i	$-0.222222\pi$
$709$	1.76604	−	0.642788i	1.76604	−	0.642788i	1.00000			$0$
$710$		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		−0.173648	−	0.984808i	$-0.555556\pi$
$719$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$720$		0			0
$721$	−2.87939			−2.87939
$722$		0			0
$723$		0			0
$724$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$725$		0			0
$726$		0			0
$727$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.500000	−	0.866025i	$-0.666667\pi$
$727$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.939693	+	0.342020i	$0.888889\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$733$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.326352	−	0.118782i	−0.326352	−	0.118782i	0.173648	−	0.984808i	$-0.444444\pi$
$739$	−0.326352	−	0.118782i	−0.326352	−	0.118782i	−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$743$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	−0.939693	−	0.342020i	$-0.888889\pi$
$751$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	1.00000			$0$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.266044	+	1.50881i	0.266044	+	1.50881i	0.766044	+	0.642788i	$0.222222\pi$
$757$	0.266044	+	1.50881i	0.266044	+	1.50881i	−0.500000	+	0.866025i	$0.666667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	−1.17365	+	0.984808i	−1.17365	+	0.984808i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	−0.939693	−	0.342020i	$-0.888889\pi$
$769$	0.0603074	−	0.342020i	0.0603074	−	0.342020i	1.00000			$0$
$770$		0			0
$771$		0			0
$772$	0.347296			0.347296
$773$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$773$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$774$		0			0
$775$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.03209	−	0.866025i	1.03209	−	0.866025i
$785$		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
$792$		0			0
$793$	0.0923963	−	0.0775297i	0.0923963	−	0.0775297i
$794$		0			0
$795$		0			0
$796$	0.266044	−	0.223238i	0.266044	−	0.223238i
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		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
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$811$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$821$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$822$		0			0
$823$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$823$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$827$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$828$		0			0
$829$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$829$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$830$		0			0
$831$		0			0
$832$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$839$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$840$		0			0
$841$	0.173648	+	0.984808i	0.173648	+	0.984808i
$842$		0			0
$843$		0			0
$844$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$845$		0			0
$846$		0			0
$847$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.0603074	+	0.342020i	0.0603074	+	0.342020i	1.00000			$0$
$853$	0.0603074	+	0.342020i	0.0603074	+	0.342020i	−0.939693	+	0.342020i	$0.888889\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$857$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$858$		0			0
$859$	1.76604	−	0.642788i	1.76604	−	0.642788i	0.766044	−	0.642788i	$-0.222222\pi$
$859$	1.76604	−	0.642788i	1.76604	−	0.642788i	1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	−2.20574	−	1.85083i	−2.20574	−	1.85083i
$869$		0			0
$870$		0			0
$871$	0.407604	−	0.342020i	0.407604	−	0.342020i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.76604	+	0.642788i	1.76604	+	0.642788i	1.00000			$0$
$877$	1.76604	+	0.642788i	1.76604	+	0.642788i	0.766044	+	0.642788i	$0.222222\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$881$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$	1.76604	−	0.642788i	1.76604	−	0.642788i	0.766044	−	0.642788i	$-0.222222\pi$
$883$	1.76604	−	0.642788i	1.76604	−	0.642788i	1.00000			$0$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$887$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$888$		0			0
$889$	−0.266044	+	1.50881i	−0.266044	+	1.50881i
$890$		0			0
$891$		0			0
$892$	0.347296			0.347296
$893$		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
$907$	−1.87939	−	0.684040i	−1.87939	−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$907$	−1.87939	−	0.684040i	−1.87939	−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$917$		0			0
$918$		0			0
$919$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$919$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$929$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$930$		0			0
$931$	−0.673648	−	1.16679i	−0.673648	−	1.16679i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	0.173648	+	0.984808i	$0.444444\pi$
$937$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	−0.500000	+	0.866025i	$0.666667\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$941$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$947$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$948$		0			0
$949$	0.326352	−	0.565258i	0.326352	−	0.565258i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$953$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.26604	+	2.19285i	−1.26604	+	2.19285i
$962$		0			0
$963$		0			0
$964$	1.17365	−	0.984808i	1.17365	−	0.984808i
$965$		0			0
$966$		0			0
$967$	0.266044	−	0.223238i	0.266044	−	0.223238i	−0.500000	−	0.866025i	$-0.666667\pi$
$967$	0.266044	−	0.223238i	0.266044	−	0.223238i	0.766044	+	0.642788i	$0.222222\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$971$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$972$		0			0
$973$	−2.20574	−	1.85083i	−2.20574	−	1.85083i
$974$		0			0
$975$		0			0
$976$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$977$		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		−0.939693	−	0.342020i	$-0.888889\pi$
$983$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$989$		0			0
$990$		0			0
$991$	0.0603074	+	0.342020i	0.0603074	+	0.342020i	1.00000			$0$
$991$	0.0603074	+	0.342020i	0.0603074	+	0.342020i	−0.939693	+	0.342020i	$0.888889\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.43969	−	1.20805i	−1.43969	−	1.20805i	−0.939693	−	0.342020i	$-0.888889\pi$
$997$	−1.43969	−	1.20805i	−1.43969	−	1.20805i	−0.500000	−	0.866025i	$-0.666667\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1539.1.cf.a.1106.1		6
3.2	odd	2	CM	1539.1.cf.a.1106.1		6
9.2	odd	6		1539.1.bq.a.593.1		6
9.4	even	3		513.1.bs.a.80.1	✓	6
9.5	odd	6		513.1.bs.a.80.1	✓	6
9.7	even	3		1539.1.bq.a.593.1		6
19.5	even	9		1539.1.bq.a.1430.1		6
57.5	odd	18		1539.1.bq.a.1430.1		6
171.5	odd	18		513.1.bs.a.404.1	yes	6
171.43	even	9	inner	1539.1.cf.a.917.1		6
171.119	odd	18	inner	1539.1.cf.a.917.1		6
171.157	even	9		513.1.bs.a.404.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.1.bs.a.80.1	✓	6	9.4	even	3
513.1.bs.a.80.1	✓	6	9.5	odd	6
513.1.bs.a.404.1	yes	6	171.5	odd	18
513.1.bs.a.404.1	yes	6	171.157	even	9
1539.1.bq.a.593.1		6	9.2	odd	6
1539.1.bq.a.593.1		6	9.7	even	3
1539.1.bq.a.1430.1		6	19.5	even	9
1539.1.bq.a.1430.1		6	57.5	odd	18
1539.1.cf.a.917.1		6	171.43	even	9	inner
1539.1.cf.a.917.1		6	171.119	odd	18	inner
1539.1.cf.a.1106.1		6	1.1	even	1	trivial
1539.1.cf.a.1106.1		6	3.2	odd	2	CM

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	1539.1.cf.a.1106.1

Level	$1539$
Weight	$1$
Character	1539.1106
Analytic conductor	$0.768$
Analytic rank	$0$
Dimension	$6$
Projective image	$D_{9}$
CM discriminant	-3
Inner twists	$4$