LMFDB - Embedded newform 3240.1.bh.c.269.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,1,Mod(269,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 1, 3]))

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

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

chi := DirichletCharacter("3240.269");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3240 = 2^{3} \cdot 3^{4} \cdot 5$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3240.bh (of order $6$ , degree $2$ , 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:	$1.61697064093$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 1080)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.1080.1
Artin image:	$C_6\times S_3$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			269.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3240.269
Dual form			3240.1.bh.c.1349.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} -1.00000 q^{8} -1.00000 q^{10} +(0.500000 - 0.866025i) q^{11} +(-0.500000 - 0.866025i) q^{13} +(-0.500000 + 0.866025i) q^{16} +1.00000 q^{17} +(-0.500000 + 0.866025i) q^{20} +(-0.500000 - 0.866025i) q^{22} +(-0.500000 - 0.866025i) q^{23} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{26} +(0.500000 - 0.866025i) q^{29} +(0.500000 + 0.866025i) q^{31} +(0.500000 + 0.866025i) q^{32} +(0.500000 - 0.866025i) q^{34} -2.00000 q^{37} +(0.500000 + 0.866025i) q^{40} +(-0.500000 + 0.866025i) q^{43} -1.00000 q^{44} -1.00000 q^{46} +(-0.500000 + 0.866025i) q^{47} +(-0.500000 - 0.866025i) q^{49} +(0.500000 + 0.866025i) q^{50} +(-0.500000 + 0.866025i) q^{52} -1.00000 q^{55} +(-0.500000 - 0.866025i) q^{58} +(-1.00000 - 1.73205i) q^{59} +1.00000 q^{62} +1.00000 q^{64} +(-0.500000 + 0.866025i) q^{65} +(1.00000 + 1.73205i) q^{67} +(-0.500000 - 0.866025i) q^{68} +(-1.00000 + 1.73205i) q^{74} +(0.500000 - 0.866025i) q^{79} +1.00000 q^{80} +(-0.500000 - 0.866025i) q^{85} +(0.500000 + 0.866025i) q^{86} +(-0.500000 + 0.866025i) q^{88} +(-0.500000 + 0.866025i) q^{92} +(0.500000 + 0.866025i) q^{94} -1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{2} - q^{4} - q^{5} - 2 q^{8} - 2 q^{10} + q^{11} - q^{13} - q^{16} + 2 q^{17} - q^{20} - q^{22} - q^{23} - q^{25} - 2 q^{26} + q^{29} + q^{31} + q^{32} + q^{34} - 4 q^{37} + q^{40}+ \cdots - 2 q^{98}+O(q^{100})$ 2 * q + q^2 - q^4 - q^5 - 2 * q^8 - 2 * q^10 + q^11 - q^13 - q^16 + 2 * q^17 - q^20 - q^22 - q^23 - q^25 - 2 * q^26 + q^29 + q^31 + q^32 + q^34 - 4 * q^37 + q^40 - q^43 - 2 * q^44 - 2 * q^46 - q^47 - q^49 + q^50 - q^52 - 2 * q^55 - q^58 - 2 * q^59 + 2 * q^62 + 2 * q^64 - q^65 + 2 * q^67 - q^68 - 2 * q^74 + q^79 + 2 * q^80 - q^85 + q^86 - q^88 - q^92 + q^94 - 2 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$ .

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$-1$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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

)

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

$2$	0.500000	−	0.866025i	0.500000	−	0.866025i
$2$	0.500000	−	0.866025i	0.500000	−	0.866025i
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$6$		0			0
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	−1.00000			−1.00000
$9$		0			0
$10$	−1.00000			−1.00000
$11$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$13$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$17$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$21$		0			0
$22$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$23$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$23$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$	−1.00000			−1.00000
$27$		0			0
$28$		0			0
$29$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$29$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$30$		0			0
$31$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$31$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$32$	0.500000	+	0.866025i	0.500000	+	0.866025i
$33$		0			0
$34$	0.500000	−	0.866025i	0.500000	−	0.866025i
$35$		0			0
$36$		0			0
$37$	−2.00000			−2.00000			−1.00000			$\pi$
$37$	−2.00000			−2.00000			−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	0.500000	+	0.866025i	0.500000	+	0.866025i
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		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			$\pi$
$44$	−1.00000			−1.00000
$45$		0			0
$46$	−1.00000			−1.00000
$47$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$47$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$48$		0			0
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$	0.500000	+	0.866025i	0.500000	+	0.866025i
$51$		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
$55$	−1.00000			−1.00000
$56$		0			0
$57$		0			0
$58$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$59$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$59$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$	1.00000			1.00000
$63$		0			0
$64$	1.00000			1.00000
$65$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$66$		0			0
$67$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$67$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$68$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$79$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$80$	1.00000			1.00000
$81$		0			0
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$86$	0.500000	+	0.866025i	0.500000	+	0.866025i
$87$		0			0
$88$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$93$		0			0
$94$	0.500000	+	0.866025i	0.500000	+	0.866025i
$95$		0			0
$96$		0			0
$97$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$97$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$98$	−1.00000			−1.00000
$99$		0			0
$100$	1.00000			1.00000
$101$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$101$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$102$		0			0
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$	0.500000	+	0.866025i	0.500000	+	0.866025i
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$111$		0			0
$112$		0			0
$113$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$113$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$114$		0			0
$115$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$116$	−1.00000			−1.00000
$117$		0			0
$118$	−2.00000			−2.00000
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	0.500000	−	0.866025i	0.500000	−	0.866025i
$125$	1.00000			1.00000
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	0.500000	−	0.866025i	0.500000	−	0.866025i
$129$		0			0
$130$	0.500000	+	0.866025i	0.500000	+	0.866025i
$131$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$131$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$	2.00000			2.00000
$135$		0			0
$136$	−1.00000			−1.00000
$137$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$137$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.00000			−1.00000
$144$		0			0
$145$	−1.00000			−1.00000
$146$		0			0
$147$		0			0
$148$	1.00000	+	1.73205i	1.00000	+	1.73205i
$149$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$149$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$151$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.500000	−	0.866025i	0.500000	−	0.866025i
$156$		0			0
$157$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$157$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$158$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$159$		0			0
$160$	0.500000	−	0.866025i	0.500000	−	0.866025i
$161$		0			0
$162$		0			0
$163$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$163$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$167$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$		0			0
$170$	−1.00000			−1.00000
$171$		0			0
$172$	1.00000			1.00000
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$	0.500000	+	0.866025i	0.500000	+	0.866025i
$177$		0			0
$178$		0			0
$179$	2.00000			2.00000			1.00000			$0$
$179$	2.00000			2.00000			1.00000			$0$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	0.500000	+	0.866025i	0.500000	+	0.866025i
$185$	1.00000	+	1.73205i	1.00000	+	1.73205i
$186$		0			0
$187$	0.500000	−	0.866025i	0.500000	−	0.866025i
$188$	1.00000			1.00000
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$199$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$200$	0.500000	−	0.866025i	0.500000	−	0.866025i
$201$		0			0
$202$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.00000			1.00000
$209$		0			0
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.00000			1.00000
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	0.500000	+	0.866025i	0.500000	+	0.866025i
$221$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$		0			0
$226$	−1.00000			−1.00000
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$	0.500000	+	0.866025i	0.500000	+	0.866025i
$231$		0			0
$232$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$233$	−2.00000			−2.00000			−1.00000			$\pi$
$233$	−2.00000			−2.00000			−1.00000			$\pi$
$234$		0			0
$235$	1.00000			1.00000
$236$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$246$		0			0
$247$		0			0
$248$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$249$		0			0
$250$	0.500000	−	0.866025i	0.500000	−	0.866025i
$251$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$251$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$252$		0			0
$253$	−1.00000			−1.00000
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$257$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$	1.00000			1.00000
$261$		0			0
$262$	1.00000			1.00000
$263$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$263$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.00000	−	1.73205i	1.00000	−	1.73205i
$269$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$269$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	2.00000			2.00000			1.00000			$0$
$271$	2.00000			2.00000			1.00000			$0$
$272$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$273$		0			0
$274$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$275$	0.500000	+	0.866025i	0.500000	+	0.866025i
$276$		0			0
$277$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$277$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		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
$286$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$287$		0			0
$288$		0			0
$289$		0			0
$290$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$291$		0			0
$292$		0			0
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$296$	2.00000			2.00000
$297$		0			0
$298$	1.00000			1.00000
$299$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$300$		0			0
$301$		0			0
$302$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$307$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$308$		0			0
$309$		0			0
$310$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	−1.00000			−1.00000
$315$		0			0
$316$	−1.00000			−1.00000
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$320$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.00000			1.00000
$326$	0.500000	−	0.866025i	0.500000	−	0.866025i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$	2.00000			2.00000
$335$	1.00000	−	1.73205i	1.00000	−	1.73205i
$336$		0			0
$337$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$337$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$338$		0			0
$339$		0			0
$340$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$341$	1.00000			1.00000
$342$		0			0
$343$		0			0
$344$	0.500000	−	0.866025i	0.500000	−	0.866025i
$345$		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			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$	1.00000			1.00000
$353$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$353$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.00000	−	1.73205i	1.00000	−	1.73205i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	1.00000			1.00000
$369$		0			0
$370$	2.00000			2.00000
$371$		0			0
$372$		0			0
$373$	−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			$\pi$
$374$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$375$		0			0
$376$	0.500000	−	0.866025i	0.500000	−	0.866025i
$377$	−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
$382$		0			0
$383$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$383$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$389$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$390$		0			0
$391$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$392$	0.500000	+	0.866025i	0.500000	+	0.866025i
$393$		0			0
$394$		0			0
$395$	−1.00000			−1.00000
$396$		0			0
$397$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$397$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$398$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$399$		0			0
$400$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	0.500000	−	0.866025i	0.500000	−	0.866025i
$404$	−1.00000			−1.00000
$405$		0			0
$406$		0			0
$407$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$408$		0			0
$409$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$409$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.500000	−	0.866025i	0.500000	−	0.866025i
$417$		0			0
$418$		0			0
$419$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$419$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	0.500000	−	0.866025i	0.500000	−	0.866025i
$431$		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			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$439$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$440$	1.00000			1.00000
$441$		0			0
$442$	−1.00000			−1.00000
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$		0			0
$460$	1.00000			1.00000
$461$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$461$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	0.500000	+	0.866025i	0.500000	+	0.866025i
$465$		0			0
$466$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$	0.500000	−	0.866025i	0.500000	−	0.866025i
$471$		0			0
$472$	1.00000	+	1.73205i	1.00000	+	1.73205i
$473$	0.500000	+	0.866025i	0.500000	+	0.866025i
$474$		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
$482$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$	0.500000	+	0.866025i	0.500000	+	0.866025i
$491$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$492$		0			0
$493$	0.500000	−	0.866025i	0.500000	−	0.866025i
$494$		0			0
$495$		0			0
$496$	−1.00000			−1.00000
$497$		0			0
$498$		0			0
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$501$		0			0
$502$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$503$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$503$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$	−1.00000			−1.00000
$506$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$507$		0			0
$508$		0			0
$509$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$509$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$		0			0
$514$	−1.00000			−1.00000
$515$		0			0
$516$		0			0
$517$	0.500000	+	0.866025i	0.500000	+	0.866025i
$518$		0			0
$519$		0			0
$520$	0.500000	−	0.866025i	0.500000	−	0.866025i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$523$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$524$	0.500000	−	0.866025i	0.500000	−	0.866025i
$525$		0			0
$526$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$527$	0.500000	+	0.866025i	0.500000	+	0.866025i
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$537$		0			0
$538$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$539$	−1.00000			−1.00000
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	1.00000	−	1.73205i	1.00000	−	1.73205i
$543$		0			0
$544$	0.500000	+	0.866025i	0.500000	+	0.866025i
$545$		0			0
$546$		0			0
$547$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$547$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$548$	−2.00000			−2.00000
$549$		0			0
$550$	1.00000			1.00000
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	1.00000			1.00000
$560$		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
$565$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$566$	2.00000			2.00000
$567$		0			0
$568$		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
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$	0.500000	+	0.866025i	0.500000	+	0.866025i
$573$		0			0
$574$		0			0
$575$	1.00000			1.00000
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$	0.500000	+	0.866025i	0.500000	+	0.866025i
$581$		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
$589$		0			0
$590$	1.00000	+	1.73205i	1.00000	+	1.73205i
$591$		0			0
$592$	1.00000	−	1.73205i	1.00000	−	1.73205i
$593$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$593$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$	0.500000	−	0.866025i	0.500000	−	0.866025i
$597$		0			0
$598$	0.500000	+	0.866025i	0.500000	+	0.866025i
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		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
$603$		0			0
$604$	−1.00000			−1.00000
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.00000			1.00000
$612$		0			0
$613$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$613$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$614$	0.500000	−	0.866025i	0.500000	−	0.866025i
$615$		0			0
$616$		0			0
$617$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$617$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$	−1.00000			−1.00000
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$629$	−2.00000			−2.00000
$630$		0			0
$631$	2.00000			2.00000			1.00000			$0$
$631$	2.00000			2.00000			1.00000			$0$
$632$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$638$	−1.00000			−1.00000
$639$		0			0
$640$	−1.00000			−1.00000
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		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
$647$	−2.00000			−2.00000			−1.00000			$\pi$
$647$	−2.00000			−2.00000			−1.00000			$\pi$
$648$		0			0
$649$	−2.00000			−2.00000
$650$	0.500000	−	0.866025i	0.500000	−	0.866025i
$651$		0			0
$652$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$	0.500000	−	0.866025i	0.500000	−	0.866025i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$659$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.00000			−1.00000
$668$	1.00000	−	1.73205i	1.00000	−	1.73205i
$669$		0			0
$670$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$671$		0			0
$672$		0			0
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$	0.500000	+	0.866025i	0.500000	+	0.866025i
$681$		0			0
$682$	0.500000	−	0.866025i	0.500000	−	0.866025i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	−2.00000			−2.00000
$686$		0			0
$687$		0			0
$688$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−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
$700$		0			0
$701$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$701$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$702$		0			0
$703$		0			0
$704$	0.500000	−	0.866025i	0.500000	−	0.866025i
$705$		0			0
$706$	0.500000	+	0.866025i	0.500000	+	0.866025i
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.500000	−	0.866025i	0.500000	−	0.866025i
$714$		0			0
$715$	0.500000	+	0.866025i	0.500000	+	0.866025i
$716$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$	0.500000	−	0.866025i	0.500000	−	0.866025i
$723$		0			0
$724$		0			0
$725$	0.500000	+	0.866025i	0.500000	+	0.866025i
$726$		0			0
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$732$		0			0
$733$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$733$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$	0.500000	−	0.866025i	0.500000	−	0.866025i
$737$	2.00000			2.00000
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	1.00000	−	1.73205i	1.00000	−	1.73205i
$741$		0			0
$742$		0			0
$743$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$743$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$744$		0			0
$745$	0.500000	−	0.866025i	0.500000	−	0.866025i
$746$	−1.00000			−1.00000
$747$		0			0
$748$	−1.00000			−1.00000
$749$		0			0
$750$		0			0
$751$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$751$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$752$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$753$		0			0
$754$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$755$	−1.00000			−1.00000
$756$		0			0
$757$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$757$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$758$		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
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−1.00000			−1.00000
$767$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$768$		0			0
$769$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$769$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−1.00000			−1.00000
$776$		0			0
$777$		0			0
$778$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−1.00000			−1.00000
$783$		0			0
$784$	1.00000			1.00000
$785$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$786$		0			0
$787$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$787$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$788$		0			0
$789$		0			0
$790$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$791$		0			0
$792$		0			0
$793$		0			0
$794$	0.500000	−	0.866025i	0.500000	−	0.866025i
$795$		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
$799$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$800$	−1.00000			−1.00000
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$807$		0			0
$808$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$	1.00000	+	1.73205i	1.00000	+	1.73205i
$815$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$816$		0			0
$817$		0			0
$818$	1.00000			1.00000
$819$		0			0
$820$		0			0
$821$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$821$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		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			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$833$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$834$		0			0
$835$	1.00000	−	1.73205i	1.00000	−	1.73205i
$836$		0			0
$837$		0			0
$838$	1.00000			1.00000
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$		0			0
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$	0.500000	+	0.866025i	0.500000	+	0.866025i
$851$	1.00000	+	1.73205i	1.00000	+	1.73205i
$852$		0			0
$853$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$853$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$857$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$861$		0			0
$862$		0			0
$863$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$863$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$870$		0			0
$871$	1.00000	−	1.73205i	1.00000	−	1.73205i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−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			$\pi$
$878$	1.00000	+	1.73205i	1.00000	+	1.73205i
$879$		0			0
$880$	0.500000	−	0.866025i	0.500000	−	0.866025i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−2.00000			−2.00000			−1.00000			$\pi$
$883$	−2.00000			−2.00000			−1.00000			$\pi$
$884$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$885$		0			0
$886$		0			0
$887$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$887$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.00000			1.00000
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.500000	+	0.866025i	0.500000	+	0.866025i
$905$		0			0
$906$		0			0
$907$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$907$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$919$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$920$	0.500000	−	0.866025i	0.500000	−	0.866025i
$921$		0			0
$922$	1.00000	+	1.73205i	1.00000	+	1.73205i
$923$		0			0
$924$		0			0
$925$	1.00000	−	1.73205i	1.00000	−	1.73205i
$926$		0			0
$927$		0			0
$928$	1.00000			1.00000
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$		0			0
$932$	1.00000	+	1.73205i	1.00000	+	1.73205i
$933$		0			0
$934$		0			0
$935$	−1.00000			−1.00000
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$941$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$941$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$	2.00000			2.00000
$945$		0			0
$946$	1.00000			1.00000
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$953$	1.00000			1.00000			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
$961$		0			0
$962$	2.00000			2.00000
$963$		0			0
$964$	−1.00000			−1.00000
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$971$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$977$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	1.00000			1.00000
$981$		0			0
$982$	−2.00000			−2.00000
$983$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$983$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$987$		0			0
$988$		0			0
$989$	1.00000			1.00000
$990$		0			0
$991$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$992$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$993$		0			0
$994$		0			0
$995$	0.500000	+	0.866025i	0.500000	+	0.866025i
$996$		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
$999$		0			0

Display

a_p

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p

up to: 50 250 1000 (See

a_n

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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	3240.1.bh.c.269.1		2
3.2	odd	2		3240.1.bh.b.269.1		2
5.4	even	2		3240.1.bh.a.269.1		2
8.5	even	2		3240.1.bh.d.269.1		2
9.2	odd	6		1080.1.i.c.269.1	yes	1
9.4	even	3	inner	3240.1.bh.c.1349.1		2
9.5	odd	6		3240.1.bh.b.1349.1		2
9.7	even	3		1080.1.i.b.269.1	yes	1
15.14	odd	2		3240.1.bh.d.269.1		2
24.5	odd	2		3240.1.bh.a.269.1		2
40.29	even	2		3240.1.bh.b.269.1		2
45.4	even	6		3240.1.bh.a.1349.1		2
45.14	odd	6		3240.1.bh.d.1349.1		2
45.29	odd	6		1080.1.i.a.269.1	✓	1
45.34	even	6		1080.1.i.d.269.1	yes	1
72.5	odd	6		3240.1.bh.a.1349.1		2
72.13	even	6		3240.1.bh.d.1349.1		2
72.29	odd	6		1080.1.i.d.269.1	yes	1
72.61	even	6		1080.1.i.a.269.1	✓	1
120.29	odd	2	CM	3240.1.bh.c.269.1		2
360.29	odd	6		1080.1.i.b.269.1	yes	1
360.149	odd	6	inner	3240.1.bh.c.1349.1		2
360.229	even	6		3240.1.bh.b.1349.1		2
360.349	even	6		1080.1.i.c.269.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.1.i.a.269.1	✓	1	45.29	odd	6
1080.1.i.a.269.1	✓	1	72.61	even	6
1080.1.i.b.269.1	yes	1	9.7	even	3
1080.1.i.b.269.1	yes	1	360.29	odd	6
1080.1.i.c.269.1	yes	1	9.2	odd	6
1080.1.i.c.269.1	yes	1	360.349	even	6
1080.1.i.d.269.1	yes	1	45.34	even	6
1080.1.i.d.269.1	yes	1	72.29	odd	6
3240.1.bh.a.269.1		2	5.4	even	2
3240.1.bh.a.269.1		2	24.5	odd	2
3240.1.bh.a.1349.1		2	45.4	even	6
3240.1.bh.a.1349.1		2	72.5	odd	6
3240.1.bh.b.269.1		2	3.2	odd	2
3240.1.bh.b.269.1		2	40.29	even	2
3240.1.bh.b.1349.1		2	9.5	odd	6
3240.1.bh.b.1349.1		2	360.229	even	6
3240.1.bh.c.269.1		2	1.1	even	1	trivial
3240.1.bh.c.269.1		2	120.29	odd	2	CM
3240.1.bh.c.1349.1		2	9.4	even	3	inner
3240.1.bh.c.1349.1		2	360.149	odd	6	inner
3240.1.bh.d.269.1		2	8.5	even	2
3240.1.bh.d.269.1		2	15.14	odd	2
3240.1.bh.d.1349.1		2	45.14	odd	6
3240.1.bh.d.1349.1		2	72.13	even	6

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

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

Label	3240.1.bh.c.269.1

Level	$3240$
Weight	$1$
Character	3240.269
Analytic conductor	$1.617$
Analytic rank	$0$
Dimension	$2$
Projective image	$D_{3}$
CM discriminant	-120
Inner twists	$4$