LMFDB - Embedded newform 728.1.ce.c.237.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [728,1,Mod(237,728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 2]))

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

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

chi := DirichletCharacter("728.237");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$728 = 2^{3} \cdot 7 \cdot 13$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	728.ce (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:	$0.363319329197$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{2} + 1$ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.0.626971072.1

Embedding invariants

Embedding label			237.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	728.237
Dual form			728.1.ce.c.685.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.866025 - 1.50000i) q^{3} +(-0.500000 + 0.866025i) q^{4} +1.73205 q^{5} +(0.866025 - 1.50000i) q^{6} +(0.500000 - 0.866025i) q^{7} -1.00000 q^{8} +(-1.00000 + 1.73205i) q^{9} +(0.866025 + 1.50000i) q^{10} +1.73205 q^{12} +(-0.866025 - 0.500000i) q^{13} +1.00000 q^{14} +(-1.50000 - 2.59808i) q^{15} +(-0.500000 - 0.866025i) q^{16} -2.00000 q^{18} +(-0.866025 + 1.50000i) q^{20} -1.73205 q^{21} +(0.500000 + 0.866025i) q^{23} +(0.866025 + 1.50000i) q^{24} +2.00000 q^{25} -1.00000i q^{26} +1.73205 q^{27} +(0.500000 + 0.866025i) q^{28} +(1.50000 - 2.59808i) q^{30} +(0.500000 - 0.866025i) q^{32} +(0.866025 - 1.50000i) q^{35} +(-1.00000 - 1.73205i) q^{36} +1.73205i q^{39} -1.73205 q^{40} +(-0.866025 - 1.50000i) q^{42} +(-1.73205 + 3.00000i) q^{45} +(-0.500000 + 0.866025i) q^{46} +(-0.866025 + 1.50000i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(1.00000 + 1.73205i) q^{50} +(0.866025 - 0.500000i) q^{52} +(0.866025 + 1.50000i) q^{54} +(-0.500000 + 0.866025i) q^{56} +(-0.866025 + 1.50000i) q^{59} +3.00000 q^{60} +(-0.866025 + 1.50000i) q^{61} +(1.00000 + 1.73205i) q^{63} +1.00000 q^{64} +(-1.50000 - 0.866025i) q^{65} +(0.866025 - 1.50000i) q^{69} +1.73205 q^{70} +(-0.500000 + 0.866025i) q^{71} +(1.00000 - 1.73205i) q^{72} +(-1.73205 - 3.00000i) q^{75} +(-1.50000 + 0.866025i) q^{78} -2.00000 q^{79} +(-0.866025 - 1.50000i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(0.866025 - 1.50000i) q^{84} -3.46410 q^{90} +(-0.866025 + 0.500000i) q^{91} -1.00000 q^{92} -1.73205 q^{96} +(0.500000 - 0.866025i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{2} - 2 q^{4} + 2 q^{7} - 4 q^{8} - 4 q^{9} + 4 q^{14} - 6 q^{15} - 2 q^{16} - 8 q^{18} + 2 q^{23} + 8 q^{25} + 2 q^{28} + 6 q^{30} + 2 q^{32} - 4 q^{36} - 2 q^{46} - 2 q^{49} + 4 q^{50} - 2 q^{56}+ \cdots + 2 q^{98}+O(q^{100})$ 4 * q + 2 * q^2 - 2 * q^4 + 2 * q^7 - 4 * q^8 - 4 * q^9 + 4 * q^14 - 6 * q^15 - 2 * q^16 - 8 * q^18 + 2 * q^23 + 8 * q^25 + 2 * q^28 + 6 * q^30 + 2 * q^32 - 4 * q^36 - 2 * q^46 - 2 * q^49 + 4 * q^50 - 2 * q^56 + 12 * q^60 + 4 * q^63 + 4 * q^64 - 6 * q^65 - 2 * q^71 + 4 * q^72 - 6 * q^78 - 8 * q^79 - 2 * q^81 - 4 * q^92 + 2 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/728\mathbb{Z}\right)^\times$ .

$n$	$183$	$365$	$521$	$561$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$

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
$3$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$3$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$4$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$5$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$5$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$6$	0.866025	−	1.50000i	0.866025	−	1.50000i
$7$	0.500000	−	0.866025i	0.500000	−	0.866025i
$7$	0.500000	−	0.866025i	0.500000	−	0.866025i
$8$	−1.00000			−1.00000
$9$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$10$	0.866025	+	1.50000i	0.866025	+	1.50000i
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	1.73205			1.73205
$13$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$13$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$14$	1.00000			1.00000
$15$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	−2.00000			−2.00000
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$21$	−1.73205			−1.73205
$22$		0			0
$23$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$23$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$24$	0.866025	+	1.50000i	0.866025	+	1.50000i
$25$	2.00000			2.00000
$26$		−	1.00000i		−	1.00000i
$27$	1.73205			1.73205
$28$	0.500000	+	0.866025i	0.500000	+	0.866025i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$	1.50000	−	2.59808i	1.50000	−	2.59808i
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.500000	−	0.866025i	0.500000	−	0.866025i
$33$		0			0
$34$		0			0
$35$	0.866025	−	1.50000i	0.866025	−	1.50000i
$36$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$			1.73205i			1.73205i
$40$	−1.73205			−1.73205
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$	−1.73205	+	3.00000i	−1.73205	+	3.00000i
$46$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$	1.00000	+	1.73205i	1.00000	+	1.73205i
$51$		0			0
$52$	0.866025	−	0.500000i	0.866025	−	0.500000i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	0.866025	+	1.50000i	0.866025	+	1.50000i
$55$		0			0
$56$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$57$		0			0
$58$		0			0
$59$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$59$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$60$	3.00000			3.00000
$61$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$61$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$	1.00000	+	1.73205i	1.00000	+	1.73205i
$64$	1.00000			1.00000
$65$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$	0.866025	−	1.50000i	0.866025	−	1.50000i
$70$	1.73205			1.73205
$71$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$71$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$72$	1.00000	−	1.73205i	1.00000	−	1.73205i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$76$		0			0
$77$		0			0
$78$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$79$	−2.00000			−2.00000			−1.00000			$\pi$
$79$	−2.00000			−2.00000			−1.00000			$\pi$
$80$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	0.866025	−	1.50000i	0.866025	−	1.50000i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$	−3.46410			−3.46410
$91$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$92$	−1.00000			−1.00000
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−1.73205			−1.73205
$97$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$97$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$98$	0.500000	−	0.866025i	0.500000	−	0.866025i
$99$		0			0
$100$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	0.866025	+	0.500000i	0.866025	+	0.500000i
$105$	−3.00000			−3.00000
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−1.00000			−1.00000
$113$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$113$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$114$		0			0
$115$	0.866025	+	1.50000i	0.866025	+	1.50000i
$116$		0			0
$117$	1.73205	−	1.00000i	1.73205	−	1.00000i
$118$	−1.73205			−1.73205
$119$		0			0
$120$	1.50000	+	2.59808i	1.50000	+	2.59808i
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$	−1.73205			−1.73205
$123$		0			0
$124$		0			0
$125$	1.73205			1.73205
$126$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$127$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$127$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$128$	0.500000	+	0.866025i	0.500000	+	0.866025i
$129$		0			0
$130$		−	1.73205i		−	1.73205i
$131$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$131$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	3.00000			3.00000
$136$		0			0
$137$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$137$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$138$	1.73205			1.73205
$139$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$139$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$140$	0.866025	+	1.50000i	0.866025	+	1.50000i
$141$		0			0
$142$	−1.00000			−1.00000
$143$		0			0
$144$	2.00000			2.00000
$145$		0			0
$146$		0			0
$147$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	1.73205	−	3.00000i	1.73205	−	3.00000i
$151$	−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
$156$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$159$		0			0
$160$	0.866025	−	1.50000i	0.866025	−	1.50000i
$161$	1.00000			1.00000
$162$	0.500000	−	0.866025i	0.500000	−	0.866025i
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$	1.73205			1.73205
$169$	0.500000	+	0.866025i	0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$173$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	1.00000	−	1.73205i	1.00000	−	1.73205i
$176$		0			0
$177$	3.00000			3.00000
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$181$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$181$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$182$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$183$	3.00000			3.00000
$184$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.866025	−	1.50000i	0.866025	−	1.50000i
$190$		0			0
$191$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$191$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$192$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$193$	−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
$195$			3.00000i			3.00000i
$196$	1.00000			1.00000
$197$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$197$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	−2.00000			−2.00000
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−2.00000			−2.00000
$208$			1.00000i			1.00000i
$209$		0			0
$210$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$	1.73205			1.73205
$214$		0			0
$215$		0			0
$216$	−1.73205			−1.73205
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$225$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$226$	2.00000			2.00000
$227$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$227$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$231$		0			0
$232$		0			0
$233$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$233$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$234$	1.73205	+	1.00000i	1.73205	+	1.00000i
$235$		0			0
$236$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$237$	1.73205	+	3.00000i	1.73205	+	3.00000i
$238$		0			0
$239$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$239$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$240$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−1.00000			−1.00000
$243$		0			0
$244$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$245$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.866025	+	1.50000i	0.866025	+	1.50000i
$251$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$251$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$252$	−2.00000			−2.00000
$253$		0			0
$254$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$	1.50000	−	0.866025i	1.50000	−	0.866025i
$261$		0			0
$262$	0.866025	+	1.50000i	0.866025	+	1.50000i
$263$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$263$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$269$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$270$	1.50000	+	2.59808i	1.50000	+	2.59808i
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$	1.50000	+	0.866025i	1.50000	+	0.866025i
$274$	1.00000			1.00000
$275$		0			0
$276$	0.866025	+	1.50000i	0.866025	+	1.50000i
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$281$	2.00000			2.00000			1.00000			$0$
$281$	2.00000			2.00000			1.00000			$0$
$282$		0			0
$283$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$283$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$284$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.00000	+	1.73205i	1.00000	+	1.73205i
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$293$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$294$	−1.73205			−1.73205
$295$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	1.00000i		−	1.00000i
$300$	3.46410			3.46410
$301$		0			0
$302$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$303$		0			0
$304$		0			0
$305$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$306$		0			0
$307$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$307$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		−	1.73205i		−	1.73205i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	1.73205	+	3.00000i	1.73205	+	3.00000i
$316$	1.00000	−	1.73205i	1.00000	−	1.73205i
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$	1.73205			1.73205
$321$		0			0
$322$	0.500000	+	0.866025i	0.500000	+	0.866025i
$323$		0			0
$324$	1.00000			1.00000
$325$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$326$		0			0
$327$		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
$334$		0			0
$335$		0			0
$336$	0.866025	+	1.50000i	0.866025	+	1.50000i
$337$	−2.00000			−2.00000			−1.00000			$\pi$
$337$	−2.00000			−2.00000			−1.00000			$\pi$
$338$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$339$	−3.46410			−3.46410
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.00000			−1.00000
$344$		0			0
$345$	1.50000	−	2.59808i	1.50000	−	2.59808i
$346$	−1.73205			−1.73205
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$349$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$350$	2.00000			2.00000
$351$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$	1.50000	+	2.59808i	1.50000	+	2.59808i
$355$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$359$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$360$	1.73205	−	3.00000i	1.73205	−	3.00000i
$361$	0.500000	+	0.866025i	0.500000	+	0.866025i
$362$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$363$	1.73205			1.73205
$364$		−	1.00000i		−	1.00000i
$365$		0			0
$366$	1.50000	+	2.59808i	1.50000	+	2.59808i
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$	0.500000	−	0.866025i	0.500000	−	0.866025i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$376$		0			0
$377$		0			0
$378$	1.73205			1.73205
$379$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$379$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$380$		0			0
$381$	0.866025	−	1.50000i	0.866025	−	1.50000i
$382$	2.00000			2.00000
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$	0.866025	−	1.50000i	0.866025	−	1.50000i
$385$		0			0
$386$	0.500000	−	0.866025i	0.500000	−	0.866025i
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$	−2.59808	+	1.50000i	−2.59808	+	1.50000i
$391$		0			0
$392$	0.500000	+	0.866025i	0.500000	+	0.866025i
$393$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$394$		0			0
$395$	−3.46410			−3.46410
$396$		0			0
$397$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$397$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$398$		0			0
$399$		0			0
$400$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$401$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$401$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$	−1.73205			−1.73205
$412$		0			0
$413$	0.866025	+	1.50000i	0.866025	+	1.50000i
$414$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$415$		0			0
$416$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$417$		0			0
$418$		0			0
$419$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$419$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$420$	1.50000	−	2.59808i	1.50000	−	2.59808i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	0.866025	+	1.50000i	0.866025	+	1.50000i
$427$	0.866025	+	1.50000i	0.866025	+	1.50000i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$431$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$432$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$433$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$433$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	2.00000			2.00000
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.500000	−	0.866025i	0.500000	−	0.866025i
$449$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$449$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$450$	−4.00000			−4.00000
$451$		0			0
$452$	1.00000	+	1.73205i	1.00000	+	1.73205i
$453$	0.866025	+	1.50000i	0.866025	+	1.50000i
$454$	1.73205			1.73205
$455$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$456$		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
$460$	−1.73205			−1.73205
$461$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$461$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$462$		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
$466$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$467$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$467$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$468$			2.00000i			2.00000i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.866025	−	1.50000i	0.866025	−	1.50000i
$473$		0			0
$474$	−1.73205	+	3.00000i	−1.73205	+	3.00000i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$	−3.00000			−3.00000
$481$		0			0
$482$		0			0
$483$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$484$	−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	$-0.666667\pi$
$487$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$488$	0.866025	−	1.50000i	0.866025	−	1.50000i
$489$		0			0
$490$	0.866025	−	1.50000i	0.866025	−	1.50000i
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	0.500000	+	0.866025i	0.500000	+	0.866025i
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$501$		0			0
$502$	1.73205			1.73205
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$505$		0			0
$506$		0			0
$507$	0.866025	−	1.50000i	0.866025	−	1.50000i
$508$	−1.00000			−1.00000
$509$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$509$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	3.00000			3.00000
$520$	1.50000	+	0.866025i	1.50000	+	0.866025i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$523$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$524$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$525$	−3.46410			−3.46410
$526$	0.500000	−	0.866025i	0.500000	−	0.866025i
$527$		0			0
$528$		0			0
$529$		0			0
$530$		0			0
$531$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	1.73205			1.73205
$539$		0			0
$540$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	1.50000	+	2.59808i	1.50000	+	2.59808i
$544$		0			0
$545$		0			0
$546$			1.73205i			1.73205i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	0.500000	+	0.866025i	0.500000	+	0.866025i
$549$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$550$		0			0
$551$		0			0
$552$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$553$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$	−1.73205			−1.73205
$561$		0			0
$562$	1.00000	+	1.73205i	1.00000	+	1.73205i
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$	1.73205	−	3.00000i	1.73205	−	3.00000i
$566$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$567$	−1.00000			−1.00000
$568$	0.500000	−	0.866025i	0.500000	−	0.866025i
$569$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$569$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	−3.46410			−3.46410
$574$		0			0
$575$	1.00000	+	1.73205i	1.00000	+	1.73205i
$576$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.500000	−	0.866025i	0.500000	−	0.866025i
$579$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	3.00000	−	1.73205i	3.00000	−	1.73205i
$586$		0			0
$587$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$587$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$588$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$589$		0			0
$590$	−3.00000			−3.00000
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	0.866025	−	0.500000i	0.866025	−	0.500000i
$599$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$599$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$600$	1.73205	+	3.00000i	1.73205	+	3.00000i
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$		0			0
$604$	0.500000	−	0.866025i	0.500000	−	0.866025i
$605$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$	−3.00000			−3.00000
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	0.866025	+	1.50000i	0.866025	+	1.50000i
$615$		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
$619$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$619$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$620$		0			0
$621$	0.866025	+	1.50000i	0.866025	+	1.50000i
$622$		0			0
$623$		0			0
$624$	1.50000	−	0.866025i	1.50000	−	0.866025i
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−1.73205	+	3.00000i	−1.73205	+	3.00000i
$631$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$631$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$632$	2.00000			2.00000
$633$		0			0
$634$		0			0
$635$	0.866025	+	1.50000i	0.866025	+	1.50000i
$636$		0			0
$637$			1.00000i			1.00000i
$638$		0			0
$639$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$640$	0.866025	+	1.50000i	0.866025	+	1.50000i
$641$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$641$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$642$		0			0
$643$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$643$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$644$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$	0.500000	+	0.866025i	0.500000	+	0.866025i
$649$		0			0
$650$		−	2.00000i		−	2.00000i
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$	3.00000			3.00000
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$661$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$673$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$673$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$674$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$675$	3.46410			3.46410
$676$	−1.00000			−1.00000
$677$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$677$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$678$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$679$		0			0
$680$		0			0
$681$	−3.00000			−3.00000
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$	0.866025	−	1.50000i	0.866025	−	1.50000i
$686$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$687$		0			0
$688$		0			0
$689$		0			0
$690$	3.00000			3.00000
$691$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$691$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$692$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	0.866025	−	1.50000i	0.866025	−	1.50000i
$699$	0.866025	+	1.50000i	0.866025	+	1.50000i
$700$	1.00000	+	1.73205i	1.00000	+	1.73205i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		−	1.73205i		−	1.73205i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$	−1.73205			−1.73205
$711$	2.00000	−	3.46410i	2.00000	−	3.46410i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	0.866025	+	1.50000i	0.866025	+	1.50000i
$718$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$	3.46410			3.46410
$721$		0			0
$722$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$723$		0			0
$724$	0.866025	−	1.50000i	0.866025	−	1.50000i
$725$		0			0
$726$	0.866025	+	1.50000i	0.866025	+	1.50000i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$	0.866025	−	0.500000i	0.866025	−	0.500000i
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$733$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$733$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$736$	1.00000			1.00000
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−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
$750$	1.50000	−	2.59808i	1.50000	−	2.59808i
$751$	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			0
$753$	−3.00000			−3.00000
$754$		0			0
$755$	−1.73205			−1.73205
$756$	0.866025	+	1.50000i	0.866025	+	1.50000i
$757$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$757$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$	1.73205			1.73205
$763$		0			0
$764$	1.00000	+	1.73205i	1.00000	+	1.73205i
$765$		0			0
$766$		0			0
$767$	1.50000	−	0.866025i	1.50000	−	0.866025i
$768$	1.73205			1.73205
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$		0			0
$772$	1.00000			1.00000
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−2.59808	−	1.50000i	−2.59808	−	1.50000i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$785$		0			0
$786$	1.50000	−	2.59808i	1.50000	−	2.59808i
$787$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$787$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$788$		0			0
$789$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$790$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$791$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$792$		0			0
$793$	1.50000	−	0.866025i	1.50000	−	0.866025i
$794$	1.73205			1.73205
$795$		0			0
$796$		0			0
$797$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$797$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$	1.00000	−	1.73205i	1.00000	−	1.73205i
$801$		0			0
$802$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$803$		0			0
$804$		0			0
$805$	1.73205			1.73205
$806$		0			0
$807$	−3.00000			−3.00000
$808$		0			0
$809$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$809$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$810$	0.866025	−	1.50000i	0.866025	−	1.50000i
$811$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$811$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		−	2.00000i		−	2.00000i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$823$	−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
$826$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	1.00000	−	1.73205i	1.00000	−	1.73205i
$829$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$829$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$830$		0			0
$831$		0			0
$832$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$	3.00000			3.00000
$841$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$842$		0			0
$843$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$844$		0			0
$845$	0.866025	+	1.50000i	0.866025	+	1.50000i
$846$		0			0
$847$	0.500000	+	0.866025i	0.500000	+	0.866025i
$848$		0			0
$849$	1.50000	−	2.59808i	1.50000	−	2.59808i
$850$		0			0
$851$		0			0
$852$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$853$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$853$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$854$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$		0			0
$861$		0			0
$862$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$863$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$863$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$864$	0.866025	−	1.50000i	0.866025	−	1.50000i
$865$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$866$		0			0
$867$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.866025	−	1.50000i	0.866025	−	1.50000i
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$881$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$882$	1.00000	+	1.73205i	1.00000	+	1.73205i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	5.19615			5.19615
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	1.00000			1.00000
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	1.00000			1.00000
$897$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$898$	1.00000			1.00000
$899$		0			0
$900$	−2.00000	−	3.46410i	−2.00000	−	3.46410i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$905$	−3.00000			−3.00000
$906$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$	0.866025	+	1.50000i	0.866025	+	1.50000i
$909$		0			0
$910$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$911$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$911$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$912$		0			0
$913$		0			0
$914$	0.500000	−	0.866025i	0.500000	−	0.866025i
$915$	5.19615			5.19615
$916$		0			0
$917$	0.866025	−	1.50000i	0.866025	−	1.50000i
$918$		0			0
$919$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$919$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$920$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$921$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$922$	1.73205			1.73205
$923$	0.866025	−	0.500000i	0.866025	−	0.500000i
$924$		0			0
$925$		0			0
$926$	0.500000	+	0.866025i	0.500000	+	0.866025i
$927$		0			0
$928$		0			0
$929$		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
$932$	0.500000	−	0.866025i	0.500000	−	0.866025i
$933$		0			0
$934$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$935$		0			0
$936$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$941$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$	1.73205			1.73205
$945$	1.50000	−	2.59808i	1.50000	−	2.59808i
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$	−3.46410			−3.46410
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$953$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$954$		0			0
$955$	1.73205	−	3.00000i	1.73205	−	3.00000i
$956$	0.500000	−	0.866025i	0.500000	−	0.866025i
$957$		0			0
$958$		0			0
$959$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$960$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$961$	1.00000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$966$	0.866025	−	1.50000i	0.866025	−	1.50000i
$967$	2.00000			2.00000			1.00000			$0$
$967$	2.00000			2.00000			1.00000			$0$
$968$	0.500000	−	0.866025i	0.500000	−	0.866025i
$969$		0			0
$970$		0			0
$971$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$971$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$		0			0
$974$	1.00000			1.00000
$975$			3.46410i			3.46410i
$976$	1.73205			1.73205
$977$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$977$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$	1.73205			1.73205
$981$		0			0
$982$		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			0
$989$		0			0
$990$		0			0
$991$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$991$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$995$		0			0
$996$		0			0
$997$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$997$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$998$		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

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) (See only

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)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	728.1.ce.c.237.1	✓	4
4.3	odd	2		2912.1.cu.c.1329.2		4
7.6	odd	2	inner	728.1.ce.c.237.2	yes	4
8.3	odd	2		2912.1.cu.c.1329.1		4
8.5	even	2	inner	728.1.ce.c.237.2	yes	4
13.9	even	3	inner	728.1.ce.c.685.1	yes	4
28.27	even	2		2912.1.cu.c.1329.1		4
52.35	odd	6		2912.1.cu.c.1777.2		4
56.13	odd	2	CM	728.1.ce.c.237.1	✓	4
56.27	even	2		2912.1.cu.c.1329.2		4
91.48	odd	6	inner	728.1.ce.c.685.2	yes	4
104.35	odd	6		2912.1.cu.c.1777.1		4
104.61	even	6	inner	728.1.ce.c.685.2	yes	4
364.139	even	6		2912.1.cu.c.1777.1		4
728.139	even	6		2912.1.cu.c.1777.2		4
728.685	odd	6	inner	728.1.ce.c.685.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.1.ce.c.237.1	✓	4	1.1	even	1	trivial
728.1.ce.c.237.1	✓	4	56.13	odd	2	CM
728.1.ce.c.237.2	yes	4	7.6	odd	2	inner
728.1.ce.c.237.2	yes	4	8.5	even	2	inner
728.1.ce.c.685.1	yes	4	13.9	even	3	inner
728.1.ce.c.685.1	yes	4	728.685	odd	6	inner
728.1.ce.c.685.2	yes	4	91.48	odd	6	inner
728.1.ce.c.685.2	yes	4	104.61	even	6	inner
2912.1.cu.c.1329.1		4	8.3	odd	2
2912.1.cu.c.1329.1		4	28.27	even	2
2912.1.cu.c.1329.2		4	4.3	odd	2
2912.1.cu.c.1329.2		4	56.27	even	2
2912.1.cu.c.1777.1		4	104.35	odd	6
2912.1.cu.c.1777.1		4	364.139	even	6
2912.1.cu.c.1777.2		4	52.35	odd	6
2912.1.cu.c.1777.2		4	728.139	even	6

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	728.1.ce.c.237.1

Level	$728$
Weight	$1$
Character	728.237
Analytic conductor	$0.363$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{6}$
CM discriminant	-56
Inner twists	$8$