LMFDB - Embedded newform 975.1.ba.b.776.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,1,Mod(776,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 4]))

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

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

chi := DirichletCharacter("975.776");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$975 = 3 \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	975.ba (of order $6$ , degree $2$ , 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:	$0.486588387317$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 1$ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.12675.1
Artin image:	$C_6\times S_3$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			776.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	975.776
Dual form			975.1.ba.b.926.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{9} -1.00000 q^{12} -1.00000 q^{13} +(-0.500000 + 0.866025i) q^{16} +(0.500000 + 0.866025i) q^{19} -1.00000 q^{21} -1.00000 q^{27} +(-0.500000 + 0.866025i) q^{28} +2.00000 q^{31} +(-0.500000 + 0.866025i) q^{36} +(1.00000 - 1.73205i) q^{37} +(-0.500000 + 0.866025i) q^{39} +(-0.500000 - 0.866025i) q^{43} +(0.500000 + 0.866025i) q^{48} +(0.500000 + 0.866025i) q^{52} +1.00000 q^{57} +(-1.00000 - 1.73205i) q^{61} +(-0.500000 + 0.866025i) q^{63} +1.00000 q^{64} +(-0.500000 + 0.866025i) q^{67} +1.00000 q^{73} +(0.500000 - 0.866025i) q^{76} -1.00000 q^{79} +(-0.500000 + 0.866025i) q^{81} +(0.500000 + 0.866025i) q^{84} +(0.500000 + 0.866025i) q^{91} +(1.00000 - 1.73205i) q^{93} +(1.00000 + 1.73205i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{3} - q^{4} - q^{7} - q^{9} - 2 q^{12} - 2 q^{13} - q^{16} + q^{19} - 2 q^{21} - 2 q^{27} - q^{28} + 4 q^{31} - q^{36} + 2 q^{37} - q^{39} - q^{43} + q^{48} + q^{52} + 2 q^{57} - 2 q^{61}+ \cdots + 2 q^{97}+O(q^{100})$ 2 * q + q^3 - q^4 - q^7 - q^9 - 2 * q^12 - 2 * q^13 - q^16 + q^19 - 2 * q^21 - 2 * q^27 - q^28 + 4 * q^31 - q^36 + 2 * q^37 - q^39 - q^43 + q^48 + q^52 + 2 * q^57 - 2 * q^61 - q^63 + 2 * q^64 - q^67 + 2 * q^73 + q^76 - 2 * q^79 - q^81 + q^84 + q^91 + 2 * q^93 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$ .

$n$	$301$	$326$	$352$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	0.500000	−	0.866025i	0.500000	−	0.866025i
$3$	0.500000	−	0.866025i	0.500000	−	0.866025i
$4$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$7$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	−1.00000			−1.00000
$13$	−1.00000			−1.00000
$13$	−1.00000			−1.00000
$14$		0			0
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$19$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$	−1.00000			−1.00000
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−1.00000			−1.00000
$28$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	2.00000			2.00000			1.00000			$0$
$31$	2.00000			2.00000			1.00000			$0$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$37$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$37$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$38$		0			0
$39$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$43$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	0.500000	+	0.866025i	0.500000	+	0.866025i
$49$		0			0
$50$		0			0
$51$		0			0
$52$	0.500000	+	0.866025i	0.500000	+	0.866025i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.00000			1.00000
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$61$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$62$		0			0
$63$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$64$	1.00000			1.00000
$65$		0			0
$66$		0			0
$67$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$67$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$		0			0
$73$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$73$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$		0			0
$76$	0.500000	−	0.866025i	0.500000	−	0.866025i
$77$		0			0
$78$		0			0
$79$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$79$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	0.500000	+	0.866025i	0.500000	+	0.866025i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	0.500000	+	0.866025i	0.500000	+	0.866025i
$92$		0			0
$93$	1.00000	−	1.73205i	1.00000	−	1.73205i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$97$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$103$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$104$		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	$0.666667\pi$
$108$	0.500000	+	0.866025i	0.500000	+	0.866025i
$109$	2.00000			2.00000			1.00000			$0$
$109$	2.00000			2.00000			1.00000			$0$
$110$		0			0
$111$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$112$	1.00000			1.00000
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.500000	+	0.866025i	0.500000	+	0.866025i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$125$		0			0
$126$		0			0
$127$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$127$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$128$		0			0
$129$	−1.00000			−1.00000
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	0.500000	−	0.866025i	0.500000	−	0.866025i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$139$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1.00000			1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−2.00000			−2.00000
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$151$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.00000			1.00000
$157$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$157$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$163$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	1.00000			1.00000
$170$		0			0
$171$	0.500000	−	0.866025i	0.500000	−	0.866025i
$172$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$181$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$	−2.00000			−2.00000
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.500000	+	0.866025i	0.500000	+	0.866025i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	0.500000	−	0.866025i	0.500000	−	0.866025i
$193$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$193$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$199$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$200$		0			0
$201$	0.500000	+	0.866025i	0.500000	+	0.866025i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.500000	−	0.866025i	0.500000	−	0.866025i
$209$		0			0
$210$		0			0
$211$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$211$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$218$		0			0
$219$	0.500000	−	0.866025i	0.500000	−	0.866025i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$223$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$224$		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	$0.333333\pi$
$228$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$229$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$229$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$241$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$	0.500000	+	0.866025i	0.500000	+	0.866025i
$244$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$245$		0			0
$246$		0			0
$247$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$	1.00000			1.00000
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	−2.00000			−2.00000
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.00000			1.00000
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$271$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$272$		0			0
$273$	1.00000			1.00000
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$277$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$278$		0			0
$279$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$283$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$	2.00000			2.00000
$292$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$302$		0			0
$303$		0			0
$304$	−1.00000			−1.00000
$305$		0			0
$306$		0			0
$307$	−2.00000			−2.00000			−1.00000			$\pi$
$307$	−2.00000			−2.00000			−1.00000			$\pi$
$308$		0			0
$309$	0.500000	−	0.866025i	0.500000	−	0.866025i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$313$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$	0.500000	+	0.866025i	0.500000	+	0.866025i
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.00000			1.00000
$325$		0			0
$326$		0			0
$327$	1.00000	−	1.73205i	1.00000	−	1.73205i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$331$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	−2.00000			−2.00000
$334$		0			0
$335$		0			0
$336$	0.500000	−	0.866025i	0.500000	−	0.866025i
$337$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$337$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.00000			−1.00000
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$349$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$350$		0			0
$351$	1.00000			1.00000
$352$		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		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$	−1.00000			−1.00000
$364$	0.500000	−	0.866025i	0.500000	−	0.866025i
$365$		0			0
$366$		0			0
$367$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$367$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−2.00000			−2.00000
$373$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$373$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$379$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$380$		0			0
$381$	0.500000	+	0.866025i	0.500000	+	0.866025i
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$388$	1.00000	−	1.73205i	1.00000	−	1.73205i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$397$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$398$		0			0
$399$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$400$		0			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−2.00000			−2.00000
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$409$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$410$		0			0
$411$		0			0
$412$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.00000			1.00000
$418$		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
$421$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$421$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	0.500000	−	0.866025i	0.500000	−	0.866025i
$433$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$433$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$434$		0			0
$435$		0			0
$436$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$437$		0			0
$438$		0			0
$439$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$439$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$449$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$449$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$457$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		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	$0.333333\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	0.500000	−	0.866025i	0.500000	−	0.866025i
$469$	1.00000			1.00000
$470$		0			0
$471$	0.500000	−	0.866025i	0.500000	−	0.866025i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$482$		0			0
$483$		0			0
$484$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$487$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$488$		0			0
$489$	2.00000			2.00000
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$497$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.500000	−	0.866025i	0.500000	−	0.866025i
$508$	1.00000			1.00000
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$512$		0			0
$513$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$514$		0			0
$515$		0			0
$516$	0.500000	+	0.866025i	0.500000	+	0.866025i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$523$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−1.00000			−1.00000
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	2.00000			2.00000			1.00000			$0$
$541$	2.00000			2.00000			1.00000			$0$
$542$		0			0
$543$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$547$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$548$		0			0
$549$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.500000	+	0.866025i	0.500000	+	0.866025i
$554$		0			0
$555$		0			0
$556$	0.500000	−	0.866025i	0.500000	−	0.866025i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	0.500000	+	0.866025i	0.500000	+	0.866025i
$560$		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
$567$	1.00000			1.00000
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		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			0
$576$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$577$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$577$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$578$		0			0
$579$	0.500000	+	0.866025i	0.500000	+	0.866025i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$	1.00000	+	1.73205i	1.00000	+	1.73205i
$590$		0			0
$591$		0			0
$592$	1.00000	+	1.73205i	1.00000	+	1.73205i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.00000			1.00000
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$601$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$602$		0			0
$603$	1.00000			1.00000
$604$	0.500000	+	0.866025i	0.500000	+	0.866025i
$605$		0			0
$606$		0			0
$607$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$607$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$613$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		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
$622$		0			0
$623$		0			0
$624$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$625$		0			0
$626$		0			0
$627$		0			0
$628$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$629$		0			0
$630$		0			0
$631$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$631$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$	1.00000	+	1.73205i	1.00000	+	1.73205i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$643$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−2.00000			−2.00000
$652$	1.00000	−	1.73205i	1.00000	−	1.73205i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	0.500000	+	0.866025i	0.500000	+	0.866025i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$673$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	1.00000	−	1.73205i	1.00000	−	1.73205i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	−1.00000			−1.00000
$685$		0			0
$686$		0			0
$687$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$688$	1.00000			1.00000
$689$		0			0
$690$		0			0
$691$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$691$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	2.00000			2.00000
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$709$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$	0.500000	+	0.866025i	0.500000	+	0.866025i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$722$		0			0
$723$	1.00000			1.00000
$724$	0.500000	+	0.866025i	0.500000	+	0.866025i
$725$		0			0
$726$		0			0
$727$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$727$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$	1.00000	+	1.73205i	1.00000	+	1.73205i
$733$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$733$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$739$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$	−1.00000			−1.00000
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$751$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.500000	−	0.866025i	0.500000	−	0.866025i
$757$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$757$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−1.00000			−1.00000
$769$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$769$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$770$		0			0
$771$		0			0
$772$	1.00000			1.00000
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$787$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.00000	+	1.73205i	1.00000	+	1.73205i
$794$		0			0
$795$		0			0
$796$	0.500000	−	0.866025i	0.500000	−	0.866025i
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.500000	−	0.866025i	0.500000	−	0.866025i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$811$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$812$		0			0
$813$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	0.500000	−	0.866025i	0.500000	−	0.866025i
$818$		0			0
$819$	0.500000	−	0.866025i	0.500000	−	0.866025i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$823$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−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
$829$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$829$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$830$		0			0
$831$	−1.00000			−1.00000
$832$	−1.00000			−1.00000
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−2.00000			−2.00000
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	2.00000			2.00000
$845$		0			0
$846$		0			0
$847$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$848$		0			0
$849$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−2.00000			−2.00000			−1.00000			$\pi$
$853$	−2.00000			−2.00000			−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
$859$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$859$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.500000	+	0.866025i	0.500000	+	0.866025i
$868$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$869$		0			0
$870$		0			0
$871$	0.500000	−	0.866025i	0.500000	−	0.866025i
$872$		0			0
$873$	1.00000	−	1.73205i	1.00000	−	1.73205i
$874$		0			0
$875$		0			0
$876$	−1.00000			−1.00000
$877$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$877$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$878$		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
$883$	−2.00000			−2.00000			−1.00000			$\pi$
$883$	−2.00000			−2.00000			−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	1.00000			1.00000
$890$		0			0
$891$		0			0
$892$	1.00000			1.00000
$893$		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
$903$	0.500000	+	0.866025i	0.500000	+	0.866025i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$907$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.500000	+	0.866025i	0.500000	+	0.866025i
$917$		0			0
$918$		0			0
$919$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$919$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$937$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$938$		0			0
$939$	0.500000	−	0.866025i	0.500000	−	0.866025i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	1.00000			1.00000
$949$	−1.00000			−1.00000
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$953$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	3.00000			3.00000
$962$		0			0
$963$		0			0
$964$	0.500000	−	0.866025i	0.500000	−	0.866025i
$965$		0			0
$966$		0			0
$967$	−2.00000			−2.00000			−1.00000			$\pi$
$967$	−2.00000			−2.00000			−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	0.500000	−	0.866025i	0.500000	−	0.866025i
$973$	0.500000	−	0.866025i	0.500000	−	0.866025i
$974$		0			0
$975$		0			0
$976$	2.00000			2.00000
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$989$		0			0
$990$		0			0
$991$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$991$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$992$		0			0
$993$	1.00000			1.00000
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$997$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$998$		0			0
$999$	−1.00000	+	1.73205i	−1.00000	+	1.73205i

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	975.1.ba.b.776.1	yes	2
3.2	odd	2	CM	975.1.ba.b.776.1	yes	2
5.2	odd	4		975.1.y.a.74.1		4
5.3	odd	4		975.1.y.a.74.2		4
5.4	even	2		975.1.ba.a.776.1	✓	2
13.3	even	3	inner	975.1.ba.b.926.1	yes	2
15.2	even	4		975.1.y.a.74.1		4
15.8	even	4		975.1.y.a.74.2		4
15.14	odd	2		975.1.ba.a.776.1	✓	2
39.29	odd	6	inner	975.1.ba.b.926.1	yes	2
65.3	odd	12		975.1.y.a.224.1		4
65.29	even	6		975.1.ba.a.926.1	yes	2
65.42	odd	12		975.1.y.a.224.2		4
195.29	odd	6		975.1.ba.a.926.1	yes	2
195.68	even	12		975.1.y.a.224.1		4
195.107	even	12		975.1.y.a.224.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.1.y.a.74.1		4	5.2	odd	4
975.1.y.a.74.1		4	15.2	even	4
975.1.y.a.74.2		4	5.3	odd	4
975.1.y.a.74.2		4	15.8	even	4
975.1.y.a.224.1		4	65.3	odd	12
975.1.y.a.224.1		4	195.68	even	12
975.1.y.a.224.2		4	65.42	odd	12
975.1.y.a.224.2		4	195.107	even	12
975.1.ba.a.776.1	✓	2	5.4	even	2
975.1.ba.a.776.1	✓	2	15.14	odd	2
975.1.ba.a.926.1	yes	2	65.29	even	6
975.1.ba.a.926.1	yes	2	195.29	odd	6
975.1.ba.b.776.1	yes	2	1.1	even	1	trivial
975.1.ba.b.776.1	yes	2	3.2	odd	2	CM
975.1.ba.b.926.1	yes	2	13.3	even	3	inner
975.1.ba.b.926.1	yes	2	39.29	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	975.1.ba.b.776.1

Level	$975$
Weight	$1$
Character	975.776
Analytic conductor	$0.487$
Analytic rank	$0$
Dimension	$2$
Projective image	$D_{3}$
CM discriminant	-3
Inner twists	$4$