LMFDB - Embedded newform 1539.1.c.d.892.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1539,1,Mod(892,1539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1539, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1539.892");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1539 = 3^{4} \cdot 19$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1539.c (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:	$0.768061054442$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 171)
Projective image:	$A_{4}$
Projective field:	Galois closure of 4.0.29241.1
Artin image:	$\SL(2,3):C_2$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

Embedding invariants

Embedding label			892.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1539.892
Dual form			1539.1.c.d.892.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{5} +1.00000 q^{7} +1.00000i q^{8} +1.00000i q^{10} -1.00000 q^{11} +1.00000i q^{13} +1.00000i q^{14} -1.00000 q^{16} -1.00000i q^{19} -1.00000i q^{22} -1.00000 q^{23} -1.00000 q^{26} -1.00000i q^{29} -1.00000i q^{31} +1.00000 q^{35} +1.00000 q^{38} +1.00000i q^{40} +1.00000i q^{41} +1.00000 q^{43} -1.00000i q^{46} +1.00000 q^{47} -1.00000 q^{55} +1.00000i q^{56} +1.00000 q^{58} -1.00000i q^{59} -1.00000 q^{61} +1.00000 q^{62} -1.00000 q^{64} +1.00000i q^{65} -1.00000i q^{67} +1.00000i q^{70} -1.00000 q^{77} -1.00000i q^{79} -1.00000 q^{80} -1.00000 q^{82} -1.00000 q^{83} +1.00000i q^{86} -1.00000i q^{88} +1.00000i q^{91} +1.00000i q^{94} -1.00000i q^{95} +1.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{5} + 2 q^{7} - 2 q^{11} - 2 q^{16} - 2 q^{23} - 2 q^{26} + 2 q^{35} + 2 q^{38} + 2 q^{43} + 2 q^{47} - 2 q^{55} + 2 q^{58} - 2 q^{61} + 2 q^{62} - 2 q^{64} - 2 q^{77} - 2 q^{80} - 2 q^{82} - 2 q^{83}+O(q^{100})$ 2 * q + 2 * q^5 + 2 * q^7 - 2 * q^11 - 2 * q^16 - 2 * q^23 - 2 * q^26 + 2 * q^35 + 2 * q^38 + 2 * q^43 + 2 * q^47 - 2 * q^55 + 2 * q^58 - 2 * q^61 + 2 * q^62 - 2 * q^64 - 2 * q^77 - 2 * q^80 - 2 * q^82 - 2 * q^83

Character values

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

$n$	$325$	$1217$
$\chi(n)$	$-1$	$1$

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	0.866025	+	0.500000i	$0.166667\pi$
$2$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	0	0
$7$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$7$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$8$	1.00000i	1.00000i
$9$	0	0
$10$	1.00000i	1.00000i
$11$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$12$	0	0
$13$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$13$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$14$	1.00000i	1.00000i
$15$	0	0
$16$	−1.00000	−1.00000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	− 1.00000i	− 1.00000i
$19$	− 1.00000i	− 1.00000i
$20$	0	0
$21$	0	0
$22$	− 1.00000i	− 1.00000i
$23$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$23$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$24$	0	0
$25$	0	0
$26$	−1.00000	−1.00000
$27$	0	0
$28$	0	0
$29$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$29$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$30$	0	0
$31$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$31$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000	1.00000
$36$	0	0
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	1.00000	1.00000
$39$	0	0
$40$	1.00000i	1.00000i
$41$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$41$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$42$	0	0
$43$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$43$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$44$	0	0
$45$	0	0
$46$	− 1.00000i	− 1.00000i
$47$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$47$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	−1.00000	−1.00000
$56$	1.00000i	1.00000i
$57$	0	0
$58$	1.00000	1.00000
$59$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$59$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$60$	0	0
$61$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$61$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$62$	1.00000	1.00000
$63$	0	0
$64$	−1.00000	−1.00000
$65$	1.00000i	1.00000i
$66$	0	0
$67$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$67$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$68$	0	0
$69$	0	0
$70$	1.00000i	1.00000i
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.00000	−1.00000
$78$	0	0
$79$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$79$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$80$	−1.00000	−1.00000
$81$	0	0
$82$	−1.00000	−1.00000
$83$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$83$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$84$	0	0
$85$	0	0
$86$	1.00000i	1.00000i
$87$	0	0
$88$	− 1.00000i	− 1.00000i
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	1.00000i	1.00000i
$92$	0	0
$93$	0	0
$94$	1.00000i	1.00000i
$95$	− 1.00000i	− 1.00000i
$96$	0	0
$97$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$97$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$101$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$102$	0	0
$103$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$103$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$104$	−1.00000	−1.00000
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	0	0
$109$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$109$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$110$	− 1.00000i	− 1.00000i
$111$	0	0
$112$	−1.00000	−1.00000
$113$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$113$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$114$	0	0
$115$	−1.00000	−1.00000
$116$	0	0
$117$	0	0
$118$	1.00000	1.00000
$119$	0	0
$120$	0	0
$121$	0	0
$122$	− 1.00000i	− 1.00000i
$123$	0	0
$124$	0	0
$125$	−1.00000	−1.00000
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 1.00000i	− 1.00000i
$129$	0	0
$130$	−1.00000	−1.00000
$131$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$131$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$132$	0	0
$133$	− 1.00000i	− 1.00000i
$134$	1.00000	1.00000
$135$	0	0
$136$	0	0
$137$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$137$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$138$	0	0
$139$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$139$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 1.00000i	− 1.00000i
$144$	0	0
$145$	− 1.00000i	− 1.00000i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$149$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$150$	0	0
$151$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$151$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$152$	1.00000	1.00000
$153$	0	0
$154$	− 1.00000i	− 1.00000i
$155$	− 1.00000i	− 1.00000i
$156$	0	0
$157$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$157$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$158$	1.00000	1.00000
$159$	0	0
$160$	0	0
$161$	−1.00000	−1.00000
$162$	0	0
$163$	0	0		−	1.00000i	$-0.5\pi$
$163$	0	0			1.00000i	$0.5\pi$
$164$	0	0
$165$	0	0
$166$	− 1.00000i	− 1.00000i
$167$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$167$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$173$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$174$	0	0
$175$	0	0
$176$	1.00000	1.00000
$177$	0	0
$178$	0	0
$179$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$179$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	−1.00000	−1.00000
$183$	0	0
$184$	− 1.00000i	− 1.00000i
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	1.00000	1.00000
$191$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$191$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$192$	0	0
$193$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$193$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$194$	−1.00000	−1.00000
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	0	0
$202$	1.00000i	1.00000i
$203$	− 1.00000i	− 1.00000i
$204$	0	0
$205$	1.00000i	1.00000i
$206$	−1.00000	−1.00000
$207$	0	0
$208$	− 1.00000i	− 1.00000i
$209$	1.00000i	1.00000i
$210$	0	0
$211$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$211$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.00000	1.00000
$216$	0	0
$217$	− 1.00000i	− 1.00000i
$218$	−2.00000	−2.00000
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$223$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$224$	0	0
$225$	0	0
$226$	1.00000	1.00000
$227$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$227$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$228$	0	0
$229$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$229$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$230$	− 1.00000i	− 1.00000i
$231$	0	0
$232$	1.00000	1.00000
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	1.00000	1.00000
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$239$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$240$	0	0
$241$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$241$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1.00000	1.00000
$248$	1.00000	1.00000
$249$	0	0
$250$	− 1.00000i	− 1.00000i
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	1.00000	1.00000
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$257$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	1.00000i	1.00000i
$263$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$263$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$264$	0	0
$265$	0	0
$266$	1.00000	1.00000
$267$	0	0
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	− 1.00000i	− 1.00000i
$275$	0	0
$276$	0	0
$277$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$277$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$278$	− 1.00000i	− 1.00000i
$279$	0	0
$280$	1.00000i	1.00000i
$281$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$281$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$282$	0	0
$283$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$283$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$284$	0	0
$285$	0	0
$286$	1.00000	1.00000
$287$	1.00000i	1.00000i
$288$	0	0
$289$	−1.00000	−1.00000
$290$	1.00000	1.00000
$291$	0	0
$292$	0	0
$293$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$293$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$294$	0	0
$295$	− 1.00000i	− 1.00000i
$296$	0	0
$297$	0	0
$298$	1.00000i	1.00000i
$299$	− 1.00000i	− 1.00000i
$300$	0	0
$301$	1.00000	1.00000
$302$	−1.00000	−1.00000
$303$	0	0
$304$	1.00000i	1.00000i
$305$	−1.00000	−1.00000
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	0	0
$310$	1.00000	1.00000
$311$	−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	$0.666667\pi$
$314$	1.00000i	1.00000i
$315$	0	0
$316$	0	0
$317$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$317$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$318$	0	0
$319$	1.00000i	1.00000i
$320$	−1.00000	−1.00000
$321$	0	0
$322$	− 1.00000i	− 1.00000i
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	−1.00000	−1.00000
$329$	1.00000	1.00000
$330$	0	0
$331$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$331$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$332$	0	0
$333$	0	0
$334$	−1.00000	−1.00000
$335$	− 1.00000i	− 1.00000i
$336$	0	0
$337$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$337$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.00000i	1.00000i
$342$	0	0
$343$	−1.00000	−1.00000
$344$	1.00000i	1.00000i
$345$	0	0
$346$	1.00000	1.00000
$347$	−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
$349$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$349$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$353$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	2.00000	2.00000
$359$	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	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$367$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$368$	1.00000	1.00000
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$373$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$374$	0	0
$375$	0	0
$376$	1.00000i	1.00000i
$377$	1.00000	1.00000
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	0	0
$382$	1.00000i	1.00000i
$383$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$383$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$384$	0	0
$385$	−1.00000	−1.00000
$386$	1.00000	1.00000
$387$	0	0
$388$	0	0
$389$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$389$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.00000i	− 1.00000i
$396$	0	0
$397$	0	0		−	1.00000i	$-0.5\pi$
$397$	0	0			1.00000i	$0.5\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$401$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$402$	0	0
$403$	1.00000	1.00000
$404$	0	0
$405$	0	0
$406$	1.00000	1.00000
$407$	0	0
$408$	0	0
$409$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$409$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$410$	−1.00000	−1.00000
$411$	0	0
$412$	0	0
$413$	− 1.00000i	− 1.00000i
$414$	0	0
$415$	−1.00000	−1.00000
$416$	0	0
$417$	0	0
$418$	−1.00000	−1.00000
$419$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$419$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$420$	0	0
$421$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$421$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$422$	−1.00000	−1.00000
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−1.00000	−1.00000
$428$	0	0
$429$	0	0
$430$	1.00000i	1.00000i
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0	0
$433$	0	0	1.00000			$0$
$433$	0	0	−1.00000			$\pi$
$434$	1.00000	1.00000
$435$	0	0
$436$	0	0
$437$	1.00000i	1.00000i
$438$	0	0
$439$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$439$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$440$	− 1.00000i	− 1.00000i
$441$	0	0
$442$	0	0
$443$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$443$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$444$	0	0
$445$	0	0
$446$	−1.00000	−1.00000
$447$	0	0
$448$	−1.00000	−1.00000
$449$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$449$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$450$	0	0
$451$	− 1.00000i	− 1.00000i
$452$	0	0
$453$	0	0
$454$	1.00000	1.00000
$455$	1.00000i	1.00000i
$456$	0	0
$457$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$457$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$458$	− 1.00000i	− 1.00000i
$459$	0	0
$460$	0	0
$461$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$461$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$462$	0	0
$463$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$463$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$464$	1.00000i	1.00000i
$465$	0	0
$466$	0	0
$467$	−2.00000	−2.00000	−1.00000			$\pi$
$467$	−2.00000	−2.00000	−1.00000			$\pi$
$468$	0	0
$469$	− 1.00000i	− 1.00000i
$470$	1.00000i	1.00000i
$471$	0	0
$472$	1.00000	1.00000
$473$	−1.00000	−1.00000
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	− 1.00000i	− 1.00000i
$479$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$480$	0	0
$481$	0	0
$482$	1.00000	1.00000
$483$	0	0
$484$	0	0
$485$	1.00000i	1.00000i
$486$	0	0
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	− 1.00000i	− 1.00000i
$489$	0	0
$490$	0	0
$491$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$491$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$492$	0	0
$493$	0	0
$494$	1.00000i	1.00000i
$495$	0	0
$496$	1.00000i	1.00000i
$497$	0	0
$498$	0	0
$499$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$499$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	1.00000	1.00000
$506$	1.00000i	1.00000i
$507$	0	0
$508$	0	0
$509$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$509$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$510$	0	0
$511$	0	0
$512$	− 1.00000i	− 1.00000i
$513$	0	0
$514$	−1.00000	−1.00000
$515$	1.00000i	1.00000i
$516$	0	0
$517$	−1.00000	−1.00000
$518$	0	0
$519$	0	0
$520$	−1.00000	−1.00000
$521$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$521$	2.00000i	2.00000i			1.00000i	$0.5\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	1.00000i	1.00000i
$527$	0	0
$528$	0	0
$529$	0	0
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.00000	−1.00000
$534$	0	0
$535$	0	0
$536$	1.00000	1.00000
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	0	0		−	1.00000i	$-0.5\pi$
$541$	0	0			1.00000i	$0.5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.00000i	2.00000i
$546$	0	0
$547$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$547$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.00000	−1.00000
$552$	0	0
$553$	− 1.00000i	− 1.00000i
$554$	1.00000i	1.00000i
$555$	0	0
$556$	0	0
$557$	2.00000	2.00000	1.00000			$0$
$557$	2.00000	2.00000	1.00000			$0$
$558$	0	0
$559$	1.00000i	1.00000i
$560$	−1.00000	−1.00000
$561$	0	0
$562$	−1.00000	−1.00000
$563$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$563$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$564$	0	0
$565$	− 1.00000i	− 1.00000i
$566$	1.00000i	1.00000i
$567$	0	0
$568$	0	0
$569$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$569$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$570$	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
$574$	−1.00000	−1.00000
$575$	0	0
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	− 1.00000i	− 1.00000i
$579$	0	0
$580$	0	0
$581$	−1.00000	−1.00000
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	1.00000	1.00000
$587$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$587$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$588$	0	0
$589$	−1.00000	−1.00000
$590$	1.00000	1.00000
$591$	0	0
$592$	0	0
$593$	0	0		−	1.00000i	$-0.5\pi$
$593$	0	0			1.00000i	$0.5\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	1.00000	1.00000
$599$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$599$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$600$	0	0
$601$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$601$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$602$	1.00000i	1.00000i
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$607$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$608$	0	0
$609$	0	0
$610$	− 1.00000i	− 1.00000i
$611$	1.00000i	1.00000i
$612$	0	0
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	0	0
$615$	0	0
$616$	− 1.00000i	− 1.00000i
$617$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$617$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$618$	0	0
$619$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$619$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$620$	0	0
$621$	0	0
$622$	− 1.00000i	− 1.00000i
$623$	0	0
$624$	0	0
$625$	−1.00000	−1.00000
$626$	− 1.00000i	− 1.00000i
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	0	0		−	1.00000i	$-0.5\pi$
$631$	0	0			1.00000i	$0.5\pi$
$632$	1.00000	1.00000
$633$	0	0
$634$	−1.00000	−1.00000
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−1.00000	−1.00000
$639$	0	0
$640$	− 1.00000i	− 1.00000i
$641$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$641$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$642$	0	0
$643$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$643$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	1.00000i	1.00000i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$653$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$654$	0	0
$655$	1.00000	1.00000
$656$	− 1.00000i	− 1.00000i
$657$	0	0
$658$	1.00000i	1.00000i
$659$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$659$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$660$	0	0
$661$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$661$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$662$	1.00000	1.00000
$663$	0	0
$664$	− 1.00000i	− 1.00000i
$665$	− 1.00000i	− 1.00000i
$666$	0	0
$667$	1.00000i	1.00000i
$668$	0	0
$669$	0	0
$670$	1.00000	1.00000
$671$	1.00000	1.00000
$672$	0	0
$673$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$673$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$674$	−1.00000	−1.00000
$675$	0	0
$676$	0	0
$677$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$677$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$678$	0	0
$679$	1.00000i	1.00000i
$680$	0	0
$681$	0	0
$682$	−1.00000	−1.00000
$683$	0	0	1.00000			$0$
$683$	0	0	−1.00000			$\pi$
$684$	0	0
$685$	−1.00000	−1.00000
$686$	− 1.00000i	− 1.00000i
$687$	0	0
$688$	−1.00000	−1.00000
$689$	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
$694$	− 1.00000i	− 1.00000i
$695$	−1.00000	−1.00000
$696$	0	0
$697$	0	0
$698$	− 1.00000i	− 1.00000i
$699$	0	0
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	0	0
$703$	0	0
$704$	1.00000	1.00000
$705$	0	0
$706$	1.00000i	1.00000i
$707$	1.00000	1.00000
$708$	0	0
$709$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$709$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1.00000i	1.00000i
$714$	0	0
$715$	− 1.00000i	− 1.00000i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	1.00000i	1.00000i
$722$	− 1.00000i	− 1.00000i
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$727$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$728$	−1.00000	−1.00000
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$733$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$734$	− 1.00000i	− 1.00000i
$735$	0	0
$736$	0	0
$737$	1.00000i	1.00000i
$738$	0	0
$739$	2.00000	2.00000	1.00000			$0$
$739$	2.00000	2.00000	1.00000			$0$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$743$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$744$	0	0
$745$	1.00000	1.00000
$746$	1.00000	1.00000
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$751$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$752$	−1.00000	−1.00000
$753$	0	0
$754$	1.00000i	1.00000i
$755$	1.00000i	1.00000i
$756$	0	0
$757$	0	0		−	1.00000i	$-0.5\pi$
$757$	0	0			1.00000i	$0.5\pi$
$758$	0	0
$759$	0	0
$760$	1.00000	1.00000
$761$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$761$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$762$	0	0
$763$	2.00000i	2.00000i
$764$	0	0
$765$	0	0
$766$	1.00000	1.00000
$767$	1.00000	1.00000
$768$	0	0
$769$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$769$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$770$	− 1.00000i	− 1.00000i
$771$	0	0
$772$	0	0
$773$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$773$	− 2.00000i	− 2.00000i		−	1.00000i	$-0.5\pi$
$774$	0	0
$775$	0	0
$776$	−1.00000	−1.00000
$777$	0	0
$778$	− 1.00000i	− 1.00000i
$779$	1.00000	1.00000
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.00000	1.00000
$786$	0	0
$787$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$787$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$788$	0	0
$789$	0	0
$790$	1.00000	1.00000
$791$	− 1.00000i	− 1.00000i
$792$	0	0
$793$	− 1.00000i	− 1.00000i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$797$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	−1.00000	−1.00000
$803$	0	0
$804$	0	0
$805$	−1.00000	−1.00000
$806$	1.00000i	1.00000i
$807$	0	0
$808$	1.00000i	1.00000i
$809$	2.00000	2.00000	1.00000			$0$
$809$	2.00000	2.00000	1.00000			$0$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.00000i	− 1.00000i
$818$	1.00000	1.00000
$819$	0	0
$820$	0	0
$821$	−1.00000	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$821$	−1.00000	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$822$	0	0
$823$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$823$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$824$	−1.00000	−1.00000
$825$	0	0
$826$	1.00000	1.00000
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	0	0	1.00000			$0$
$829$	0	0	−1.00000			$\pi$
$830$	− 1.00000i	− 1.00000i
$831$	0	0
$832$	− 1.00000i	− 1.00000i
$833$	0	0
$834$	0	0
$835$	1.00000i	1.00000i
$836$	0	0
$837$	0	0
$838$	1.00000i	1.00000i
$839$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$839$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$840$	0	0
$841$	0	0
$842$	−1.00000	−1.00000
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$853$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$854$	− 1.00000i	− 1.00000i
$855$	0	0
$856$	0	0
$857$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$857$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$858$	0	0
$859$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$859$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	− 1.00000i	− 1.00000i
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.00000i	1.00000i
$870$	0	0
$871$	1.00000	1.00000
$872$	−2.00000	−2.00000
$873$	0	0
$874$	−1.00000	−1.00000
$875$	−1.00000	−1.00000
$876$	0	0
$877$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$877$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$878$	−1.00000	−1.00000
$879$	0	0
$880$	1.00000	1.00000
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	0	0
$883$	0	0		−	1.00000i	$-0.5\pi$
$883$	0	0			1.00000i	$0.5\pi$
$884$	0	0
$885$	0	0
$886$	1.00000i	1.00000i
$887$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$887$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.00000i	− 1.00000i
$894$	0	0
$895$	− 2.00000i	− 2.00000i
$896$	− 1.00000i	− 1.00000i
$897$	0	0
$898$	−2.00000	−2.00000
$899$	−1.00000	−1.00000
$900$	0	0
$901$	0	0
$902$	1.00000	1.00000
$903$	0	0
$904$	1.00000	1.00000
$905$	0	0
$906$	0	0
$907$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$907$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$908$	0	0
$909$	0	0
$910$	−1.00000	−1.00000
$911$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$911$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$912$	0	0
$913$	1.00000	1.00000
$914$	1.00000i	1.00000i
$915$	0	0
$916$	0	0
$917$	1.00000	1.00000
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	− 1.00000i	− 1.00000i
$921$	0	0
$922$	1.00000i	1.00000i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	1.00000i	1.00000i
$927$	0	0
$928$	0	0
$929$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$929$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	− 2.00000i	− 2.00000i
$935$	0	0
$936$	0	0
$937$	0	0		−	1.00000i	$-0.5\pi$
$937$	0	0			1.00000i	$0.5\pi$
$938$	1.00000	1.00000
$939$	0	0
$940$	0	0
$941$	− 1.00000i	− 1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$941$	− 1.00000i	− 1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$942$	0	0
$943$	− 1.00000i	− 1.00000i
$944$	1.00000i	1.00000i
$945$	0	0
$946$	− 1.00000i	− 1.00000i
$947$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$947$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$948$	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	0
$955$	1.00000	1.00000
$956$	0	0
$957$	0	0
$958$	− 1.00000i	− 1.00000i
$959$	−1.00000	−1.00000
$960$	0	0
$961$	0	0
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.00000i	− 1.00000i
$966$	0	0
$967$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$967$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$968$	0	0
$969$	0	0
$970$	−1.00000	−1.00000
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	−1.00000	−1.00000
$974$	0	0
$975$	0	0
$976$	1.00000	1.00000
$977$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$977$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	1.00000i	1.00000i
$983$	1.00000i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$983$	1.00000i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.00000	−1.00000
$990$	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
$997$	1.00000	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$997$	1.00000	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$998$	− 1.00000i	− 1.00000i
$999$	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.c.d.892.2		2
3.2	odd	2		1539.1.c.c.892.1		2
9.2	odd	6		513.1.o.a.37.1		4
9.4	even	3		171.1.o.a.151.1	yes	4
9.5	odd	6		513.1.o.a.208.2		4
9.7	even	3		171.1.o.a.94.2	yes	4
19.18	odd	2	inner	1539.1.c.d.892.1		2
36.7	odd	6		2736.1.bs.a.1633.2		4
36.31	odd	6		2736.1.bs.a.2545.2		4
57.56	even	2		1539.1.c.c.892.2		2
171.4	even	9		3249.1.be.a.2473.1		12
171.7	even	3		3249.1.i.a.3181.2		4
171.13	odd	18		3249.1.be.a.3004.2		12
171.16	even	9		3249.1.be.a.1777.2		12
171.22	odd	18		3249.1.bc.a.2860.1		12
171.25	even	9		3249.1.bc.a.1921.1		12
171.31	odd	6		3249.1.s.a.1015.1		4
171.34	odd	18		3249.1.bc.a.1390.1		12
171.40	odd	18		3249.1.bc.a.2428.1		12
171.43	even	9		3249.1.be.a.1210.2		12
171.49	even	3		3249.1.i.a.430.2		4
171.52	odd	18		3249.1.be.a.1210.1		12
171.56	even	6		513.1.o.a.37.2		4
171.61	even	9		3249.1.bc.a.1390.2		12
171.67	odd	18		3249.1.be.a.2104.1		12
171.70	odd	18		3249.1.bc.a.1921.2		12
171.79	odd	18		3249.1.be.a.1777.1		12
171.85	even	9		3249.1.be.a.2104.2		12
171.88	odd	6		3249.1.i.a.3181.1		4
171.94	odd	6		171.1.o.a.151.2	yes	4
171.97	odd	18		3249.1.be.a.1345.2		12
171.103	odd	6		3249.1.i.a.430.1		4
171.106	even	3		3249.1.s.a.2596.1		4
171.112	even	9		3249.1.bc.a.2428.2		12
171.113	even	6		513.1.o.a.208.1		4
171.121	even	3		3249.1.s.a.1015.2		4
171.124	odd	18		3249.1.bc.a.1021.2		12
171.130	even	9		3249.1.bc.a.2860.2		12
171.139	even	9		3249.1.be.a.3004.1		12
171.142	even	9		3249.1.bc.a.1021.1		12
171.148	odd	18		3249.1.be.a.2473.2		12
171.151	odd	6		171.1.o.a.94.1	✓	4
171.157	even	9		3249.1.bc.a.2293.1		12
171.160	odd	6		3249.1.s.a.2596.2		4
171.166	odd	18		3249.1.bc.a.2293.2		12
171.169	even	9		3249.1.be.a.1345.1		12
684.151	even	6		2736.1.bs.a.1633.1		4
684.607	even	6		2736.1.bs.a.2545.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.1.o.a.94.1	✓	4	171.151	odd	6
171.1.o.a.94.2	yes	4	9.7	even	3
171.1.o.a.151.1	yes	4	9.4	even	3
171.1.o.a.151.2	yes	4	171.94	odd	6
513.1.o.a.37.1		4	9.2	odd	6
513.1.o.a.37.2		4	171.56	even	6
513.1.o.a.208.1		4	171.113	even	6
513.1.o.a.208.2		4	9.5	odd	6
1539.1.c.c.892.1		2	3.2	odd	2
1539.1.c.c.892.2		2	57.56	even	2
1539.1.c.d.892.1		2	19.18	odd	2	inner
1539.1.c.d.892.2		2	1.1	even	1	trivial
2736.1.bs.a.1633.1		4	684.151	even	6
2736.1.bs.a.1633.2		4	36.7	odd	6
2736.1.bs.a.2545.1		4	684.607	even	6
2736.1.bs.a.2545.2		4	36.31	odd	6
3249.1.i.a.430.1		4	171.103	odd	6
3249.1.i.a.430.2		4	171.49	even	3
3249.1.i.a.3181.1		4	171.88	odd	6
3249.1.i.a.3181.2		4	171.7	even	3
3249.1.s.a.1015.1		4	171.31	odd	6
3249.1.s.a.1015.2		4	171.121	even	3
3249.1.s.a.2596.1		4	171.106	even	3
3249.1.s.a.2596.2		4	171.160	odd	6
3249.1.bc.a.1021.1		12	171.142	even	9
3249.1.bc.a.1021.2		12	171.124	odd	18
3249.1.bc.a.1390.1		12	171.34	odd	18
3249.1.bc.a.1390.2		12	171.61	even	9
3249.1.bc.a.1921.1		12	171.25	even	9
3249.1.bc.a.1921.2		12	171.70	odd	18
3249.1.bc.a.2293.1		12	171.157	even	9
3249.1.bc.a.2293.2		12	171.166	odd	18
3249.1.bc.a.2428.1		12	171.40	odd	18
3249.1.bc.a.2428.2		12	171.112	even	9
3249.1.bc.a.2860.1		12	171.22	odd	18
3249.1.bc.a.2860.2		12	171.130	even	9
3249.1.be.a.1210.1		12	171.52	odd	18
3249.1.be.a.1210.2		12	171.43	even	9
3249.1.be.a.1345.1		12	171.169	even	9
3249.1.be.a.1345.2		12	171.97	odd	18
3249.1.be.a.1777.1		12	171.79	odd	18
3249.1.be.a.1777.2		12	171.16	even	9
3249.1.be.a.2104.1		12	171.67	odd	18
3249.1.be.a.2104.2		12	171.85	even	9
3249.1.be.a.2473.1		12	171.4	even	9
3249.1.be.a.2473.2		12	171.148	odd	18
3249.1.be.a.3004.1		12	171.139	even	9
3249.1.be.a.3004.2		12	171.13	odd	18

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.c.d.892.2

Level	$1539$
Weight	$1$
Character	1539.892
Analytic conductor	$0.768$
Analytic rank	$0$
Dimension	$2$
Projective image	$A_{4}$
CM/RM	no
Inner twists	$2$