LMFDB - Embedded newform 476.1.o.c.67.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [476,1,Mod(67,476)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(476, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("476.67");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$476 = 2^{2} \cdot 7 \cdot 17$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	476.o (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:	$0.237554946013$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.188737808.1

Embedding invariants

Embedding label			67.1
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	476.67
Dual form			476.1.o.c.135.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.866025 + 1.50000i) q^{3} +(-0.500000 + 0.866025i) q^{4} -1.73205 q^{6} +1.00000i q^{7} -1.00000 q^{8} +(-1.00000 - 1.73205i) q^{9} +(0.866025 - 1.50000i) q^{11} +(-0.866025 - 1.50000i) q^{12} +1.00000 q^{13} +(-0.866025 + 0.500000i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(1.00000 - 1.73205i) q^{18} +(-1.50000 - 0.866025i) q^{21} +1.73205 q^{22} +(0.866025 - 1.50000i) q^{24} +(-0.500000 + 0.866025i) q^{25} +(0.500000 + 0.866025i) q^{26} +1.73205 q^{27} +(-0.866025 - 0.500000i) q^{28} +(0.500000 - 0.866025i) q^{32} +(1.50000 + 2.59808i) q^{33} -1.00000 q^{34} +2.00000 q^{36} +(-0.866025 + 1.50000i) q^{39} -1.73205i q^{42} +(0.866025 + 1.50000i) q^{44} +1.73205 q^{48} -1.00000 q^{49} -1.00000 q^{50} +(-0.866025 - 1.50000i) q^{51} +(-0.500000 + 0.866025i) q^{52} +(0.500000 - 0.866025i) q^{53} +(0.866025 + 1.50000i) q^{54} -1.00000i q^{56} +(1.73205 - 1.00000i) q^{63} +1.00000 q^{64} +(-1.50000 + 2.59808i) q^{66} +(-0.500000 - 0.866025i) q^{68} +1.73205 q^{71} +(1.00000 + 1.73205i) q^{72} +(-0.866025 - 1.50000i) q^{75} +(1.50000 + 0.866025i) q^{77} -1.73205 q^{78} +(0.866025 + 1.50000i) q^{79} +(-0.500000 + 0.866025i) q^{81} +(1.50000 - 0.866025i) q^{84} +(-0.866025 + 1.50000i) q^{88} +(-0.500000 - 0.866025i) q^{89} +1.00000i q^{91} +(0.866025 + 1.50000i) q^{96} +(-0.500000 - 0.866025i) q^{98} -3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 4 q^{9} + 4 q^{13} - 2 q^{16} - 2 q^{17} + 4 q^{18} - 6 q^{21} - 2 q^{25} + 2 q^{26} + 2 q^{32} + 6 q^{33} - 4 q^{34} + 8 q^{36} - 4 q^{49} - 4 q^{50} - 2 q^{52} + 2 q^{53}+ \cdots - 2 q^{98}+O(q^{100})$ 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^8 - 4 * q^9 + 4 * q^13 - 2 * q^16 - 2 * q^17 + 4 * q^18 - 6 * q^21 - 2 * q^25 + 2 * q^26 + 2 * q^32 + 6 * q^33 - 4 * q^34 + 8 * q^36 - 4 * q^49 - 4 * q^50 - 2 * q^52 + 2 * q^53 + 4 * q^64 - 6 * q^66 - 2 * q^68 + 4 * q^72 + 6 * q^77 - 2 * q^81 + 6 * q^84 - 2 * q^89 - 2 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/476\mathbb{Z}\right)^\times$ .

$n$	$239$	$309$	$409$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{2}{3}\right)$

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			1.00000i	$0.5\pi$
$3$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$4$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$	−1.73205			−1.73205
$7$			1.00000i			1.00000i
$7$			1.00000i			1.00000i
$8$	−1.00000			−1.00000
$9$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$10$		0			0
$11$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$11$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$12$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$13$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$13$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$14$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$18$	1.00000	−	1.73205i	1.00000	−	1.73205i
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$		0			0
$21$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$22$	1.73205			1.73205
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$	0.866025	−	1.50000i	0.866025	−	1.50000i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$	0.500000	+	0.866025i	0.500000	+	0.866025i
$27$	1.73205			1.73205
$28$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	0.500000	−	0.866025i	0.500000	−	0.866025i
$33$	1.50000	+	2.59808i	1.50000	+	2.59808i
$34$	−1.00000			−1.00000
$35$		0			0
$36$	2.00000			2.00000
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		−	1.73205i		−	1.73205i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.866025	+	1.50000i	0.866025	+	1.50000i
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	1.73205			1.73205
$49$	−1.00000			−1.00000
$50$	−1.00000			−1.00000
$51$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$52$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$53$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$53$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$54$	0.866025	+	1.50000i	0.866025	+	1.50000i
$55$		0			0
$56$		−	1.00000i		−	1.00000i
$57$		0			0
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$	1.73205	−	1.00000i	1.73205	−	1.00000i
$64$	1.00000			1.00000
$65$		0			0
$66$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$69$		0			0
$70$		0			0
$71$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$71$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$72$	1.00000	+	1.73205i	1.00000	+	1.73205i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$76$		0			0
$77$	1.50000	+	0.866025i	1.50000	+	0.866025i
$78$	−1.73205			−1.73205
$79$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$79$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$80$		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			$\pi$
$84$	1.50000	−	0.866025i	1.50000	−	0.866025i
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$89$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$89$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$90$		0			0
$91$			1.00000i			1.00000i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	0.866025	+	1.50000i	0.866025	+	1.50000i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$99$	−3.46410			−3.46410
$100$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$101$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$101$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$102$	0.866025	−	1.50000i	0.866025	−	1.50000i
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$	−1.00000			−1.00000
$105$		0			0
$106$	1.00000			1.00000
$107$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$107$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$108$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$	0.866025	−	0.500000i	0.866025	−	0.500000i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$118$		0			0
$119$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$120$		0			0
$121$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	1.73205	+	1.00000i	1.73205	+	1.00000i
$127$		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			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$	−3.00000			−3.00000
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.500000	−	0.866025i	0.500000	−	0.866025i
$137$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$137$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$138$		0			0
$139$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$139$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$140$		0			0
$141$		0			0
$142$	0.866025	+	1.50000i	0.866025	+	1.50000i
$143$	0.866025	−	1.50000i	0.866025	−	1.50000i
$144$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$145$		0			0
$146$		0			0
$147$	0.866025	−	1.50000i	0.866025	−	1.50000i
$148$		0			0
$149$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$149$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$150$	0.866025	−	1.50000i	0.866025	−	1.50000i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$	2.00000			2.00000
$154$			1.73205i			1.73205i
$155$		0			0
$156$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$157$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$157$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$158$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$159$	0.866025	+	1.50000i	0.866025	+	1.50000i
$160$		0			0
$161$		0			0
$162$	−1.00000			−1.00000
$163$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$163$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$167$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$168$	1.50000	+	0.866025i	1.50000	+	0.866025i
$169$		0			0
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$176$	−1.73205			−1.73205
$177$		0			0
$178$	0.500000	−	0.866025i	0.500000	−	0.866025i
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	0.866025	+	1.50000i	0.866025	+	1.50000i
$188$		0			0
$189$			1.73205i			1.73205i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		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			$\pi$
$198$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$199$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$199$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$200$	0.500000	−	0.866025i	0.500000	−	0.866025i
$201$		0			0
$202$	2.00000			2.00000
$203$		0			0
$204$	1.73205			1.73205
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$209$		0			0
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	0.500000	+	0.866025i	0.500000	+	0.866025i
$213$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$214$	0.866025	−	1.50000i	0.866025	−	1.50000i
$215$		0			0
$216$	−1.73205			−1.73205
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	0.866025	+	0.500000i	0.866025	+	0.500000i
$225$	2.00000			2.00000
$226$		0			0
$227$	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
$229$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$229$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$	−2.59808	+	1.50000i	−2.59808	+	1.50000i
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$	1.00000	−	1.73205i	1.00000	−	1.73205i
$235$		0			0
$236$		0			0
$237$	−3.00000			−3.00000
$238$		−	1.00000i		−	1.00000i
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	1.00000	−	1.73205i	1.00000	−	1.73205i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$			2.00000i			2.00000i
$253$		0			0
$254$		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
$260$		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	$0.666667\pi$
$264$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$265$		0			0
$266$		0			0
$267$	1.73205			1.73205
$268$		0			0
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$	1.00000			1.00000
$273$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$274$	−1.00000			−1.00000
$275$	0.866025	+	1.50000i	0.866025	+	1.50000i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$279$		0			0
$280$		0			0
$281$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$281$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$283$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$284$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$285$		0			0
$286$	1.73205			1.73205
$287$		0			0
$288$	−2.00000			−2.00000
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$293$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$294$	1.73205			1.73205
$295$		0			0
$296$		0			0
$297$	1.50000	−	2.59808i	1.50000	−	2.59808i
$298$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$299$		0			0
$300$	1.73205			1.73205
$301$		0			0
$302$		0			0
$303$	1.73205	+	3.00000i	1.73205	+	3.00000i
$304$		0			0
$305$		0			0
$306$	1.00000	+	1.73205i	1.00000	+	1.73205i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$309$		0			0
$310$		0			0
$311$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$311$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$312$	0.866025	−	1.50000i	0.866025	−	1.50000i
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	1.00000			1.00000
$315$		0			0
$316$	−1.73205			−1.73205
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$319$		0			0
$320$		0			0
$321$	3.00000			3.00000
$322$		0			0
$323$		0			0
$324$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$325$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$335$		0			0
$336$			1.73205i			1.73205i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	1.00000i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	2.00000			2.00000			1.00000			$0$
$349$	2.00000			2.00000			1.00000			$0$
$350$		−	1.00000i		−	1.00000i
$351$	1.73205			1.73205
$352$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$353$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$353$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$354$		0			0
$355$		0			0
$356$	1.00000			1.00000
$357$	1.50000	−	0.866025i	1.50000	−	0.866025i
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$362$		0			0
$363$	3.46410			3.46410
$364$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$365$		0			0
$366$		0			0
$367$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$367$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	0.866025	+	0.500000i	0.866025	+	0.500000i
$372$		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			$\pi$
$374$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$375$		0			0
$376$		0			0
$377$		0			0
$378$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$379$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$379$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$	−1.73205			−1.73205
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$389$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	1.00000			1.00000
$393$		0			0
$394$		0			0
$395$		0			0
$396$	1.73205	−	3.00000i	1.73205	−	3.00000i
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$	1.73205			1.73205
$399$		0			0
$400$	1.00000			1.00000
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$	1.00000	+	1.73205i	1.00000	+	1.73205i
$405$		0			0
$406$		0			0
$407$		0			0
$408$	0.866025	+	1.50000i	0.866025	+	1.50000i
$409$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$409$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$410$		0			0
$411$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.500000	−	0.866025i	0.500000	−	0.866025i
$417$	1.50000	−	2.59808i	1.50000	−	2.59808i
$418$		0			0
$419$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$419$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$420$		0			0
$421$	−2.00000			−2.00000			−1.00000			$\pi$
$421$	−2.00000			−2.00000			−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$425$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$426$	−3.00000			−3.00000
$427$		0			0
$428$	1.73205			1.73205
$429$	1.50000	+	2.59808i	1.50000	+	2.59808i
$430$		0			0
$431$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$431$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$432$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$433$	−2.00000			−2.00000			−1.00000			$\pi$
$433$	−2.00000			−2.00000			−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$439$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$440$		0			0
$441$	1.00000	+	1.73205i	1.00000	+	1.73205i
$442$	−1.00000			−1.00000
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−1.73205			−1.73205
$448$			1.00000i			1.00000i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	1.00000	+	1.73205i	1.00000	+	1.73205i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	1.73205			1.73205
$455$		0			0
$456$		0			0
$457$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$457$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$458$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$459$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$460$		0			0
$461$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$461$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$462$	−2.59808	−	1.50000i	−2.59808	−	1.50000i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$	2.00000			2.00000
$469$		0			0
$470$		0			0
$471$	0.866025	+	1.50000i	0.866025	+	1.50000i
$472$		0			0
$473$		0			0
$474$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$475$		0			0
$476$	0.866025	−	0.500000i	0.866025	−	0.500000i
$477$	−2.00000			−2.00000
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	2.00000			2.00000
$485$		0			0
$486$		0			0
$487$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$487$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$			1.73205i			1.73205i
$498$		0			0
$499$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$499$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$500$		0			0
$501$	1.50000	−	2.59808i	1.50000	−	2.59808i
$502$		0			0
$503$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$503$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$504$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$509$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$		0			0
$514$	0.500000	−	0.866025i	0.500000	−	0.866025i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$	1.50000	−	0.866025i	1.50000	−	0.866025i
$526$		0			0
$527$		0			0
$528$	1.50000	−	2.59808i	1.50000	−	2.59808i
$529$	0.500000	−	0.866025i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	0.866025	+	1.50000i	0.866025	+	1.50000i
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$540$		0			0
$541$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$541$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$542$		0			0
$543$		0			0
$544$	0.500000	+	0.866025i	0.500000	+	0.866025i
$545$		0			0
$546$		−	1.73205i		−	1.73205i
$547$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$547$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$548$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$549$		0			0
$550$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$551$		0			0
$552$		0			0
$553$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$554$		0			0
$555$		0			0
$556$	0.866025	−	1.50000i	0.866025	−	1.50000i
$557$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$557$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−3.00000			−3.00000
$562$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$	−1.73205			−1.73205
$567$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$568$	−1.73205			−1.73205
$569$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$569$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$571$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$572$	0.866025	+	1.50000i	0.866025	+	1.50000i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$577$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$577$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$578$	0.500000	−	0.866025i	0.500000	−	0.866025i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$584$		0			0
$585$		0			0
$586$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	0.866025	+	1.50000i	0.866025	+	1.50000i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$593$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$594$	3.00000			3.00000
$595$		0			0
$596$	−1.00000			−1.00000
$597$	1.50000	+	2.59808i	1.50000	+	2.59808i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	0.866025	+	1.50000i	0.866025	+	1.50000i
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−1.73205	+	3.00000i	−1.73205	+	3.00000i
$607$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$607$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$613$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$613$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$614$		0			0
$615$		0			0
$616$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$619$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$620$		0			0
$621$		0			0
$622$	−1.73205			−1.73205
$623$	0.866025	−	0.500000i	0.866025	−	0.500000i
$624$	1.73205			1.73205
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	0.500000	+	0.866025i	0.500000	+	0.866025i
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−1.73205			−1.73205
$637$	−1.00000			−1.00000
$638$		0			0
$639$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	1.50000	+	2.59808i	1.50000	+	2.59808i
$643$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$643$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$644$		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	$0.666667\pi$
$648$	0.500000	−	0.866025i	0.500000	−	0.866025i
$649$		0			0
$650$	−1.00000			−1.00000
$651$		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
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$661$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.866025	−	1.50000i	0.866025	−	1.50000i
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$676$		0			0
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	1.50000	+	2.59808i	1.50000	+	2.59808i
$682$		0			0
$683$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$683$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$684$		0			0
$685$		0			0
$686$	0.866025	−	0.500000i	0.866025	−	0.500000i
$687$	−3.46410			−3.46410
$688$		0			0
$689$	0.500000	−	0.866025i	0.500000	−	0.866025i
$690$		0			0
$691$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$691$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$692$		0			0
$693$		−	3.46410i		−	3.46410i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	1.00000	+	1.73205i	1.00000	+	1.73205i
$699$		0			0
$700$	0.866025	−	0.500000i	0.866025	−	0.500000i
$701$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$701$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$702$	0.866025	+	1.50000i	0.866025	+	1.50000i
$703$		0			0
$704$	0.866025	−	1.50000i	0.866025	−	1.50000i
$705$		0			0
$706$	1.00000			1.00000
$707$	1.73205	+	1.00000i	1.73205	+	1.00000i
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$	1.73205	−	3.00000i	1.73205	−	3.00000i
$712$	0.500000	+	0.866025i	0.500000	+	0.866025i
$713$		0			0
$714$	1.50000	+	0.866025i	1.50000	+	0.866025i
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$719$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	−1.00000			−1.00000
$723$		0			0
$724$		0			0
$725$		0			0
$726$	1.73205	+	3.00000i	1.73205	+	3.00000i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		−	1.00000i		−	1.00000i
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$733$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$734$	1.73205			1.73205
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$739$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$			1.00000i			1.00000i
$743$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$743$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$	0.500000	−	0.866025i	0.500000	−	0.866025i
$747$		0			0
$748$	−1.73205			−1.73205
$749$	1.50000	−	0.866025i	1.50000	−	0.866025i
$750$		0			0
$751$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$751$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$757$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$757$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$758$	0.866025	+	1.50000i	0.866025	+	1.50000i
$759$		0			0
$760$		0			0
$761$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$761$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$769$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$769$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$	1.73205			1.73205
$772$		0			0
$773$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$773$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−1.00000			−1.00000
$779$		0			0
$780$		0			0
$781$	1.50000	−	2.59808i	1.50000	−	2.59808i
$782$		0			0
$783$		0			0
$784$	0.500000	+	0.866025i	0.500000	+	0.866025i
$785$		0			0
$786$		0			0
$787$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$787$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	3.46410			3.46410
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.866025	+	1.50000i	0.866025	+	1.50000i
$797$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$797$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$	0.500000	+	0.866025i	0.500000	+	0.866025i
$801$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$811$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$817$		0			0
$818$	1.00000			1.00000
$819$	1.73205	−	1.00000i	1.73205	−	1.00000i
$820$		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
$823$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$823$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$	−3.00000			−3.00000
$826$		0			0
$827$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$827$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$828$		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
$831$		0			0
$832$	1.00000			1.00000
$833$	0.500000	−	0.866025i	0.500000	−	0.866025i
$834$	3.00000			3.00000
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$843$	0.866025	−	1.50000i	0.866025	−	1.50000i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1.73205	−	1.00000i	1.73205	−	1.00000i
$848$	−1.00000			−1.00000
$849$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$850$	0.500000	−	0.866025i	0.500000	−	0.866025i
$851$		0			0
$852$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	0.866025	+	1.50000i	0.866025	+	1.50000i
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$	−1.73205			−1.73205
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$	0.866025	−	1.50000i	0.866025	−	1.50000i
$865$		0			0
$866$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$867$	1.73205			1.73205
$868$		0			0
$869$	3.00000			3.00000
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$	0.866025	−	1.50000i	0.866025	−	1.50000i
$879$	0.866025	−	1.50000i	0.866025	−	1.50000i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$885$		0			0
$886$		0			0
$887$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$887$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.866025	+	1.50000i	0.866025	+	1.50000i
$892$		0			0
$893$		0			0
$894$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$895$		0			0
$896$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$901$	0.500000	+	0.866025i	0.500000	+	0.866025i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$907$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$908$	0.866025	+	1.50000i	0.866025	+	1.50000i
$909$	−4.00000			−4.00000
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$915$		0			0
$916$	−2.00000			−2.00000
$917$		0			0
$918$	−1.73205			−1.73205
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$		0			0
$922$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$923$	1.73205			1.73205
$924$		−	3.00000i		−	3.00000i
$925$		0			0
$926$		0			0
$927$		0			0
$928$		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
$933$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$934$		0			0
$935$		0			0
$936$	1.00000	+	1.73205i	1.00000	+	1.73205i
$937$	2.00000			2.00000			1.00000			$0$
$937$	2.00000			2.00000			1.00000			$0$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$947$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$948$	1.50000	−	2.59808i	1.50000	−	2.59808i
$949$		0			0
$950$		0			0
$951$		0			0
$952$	0.866025	+	0.500000i	0.866025	+	0.500000i
$953$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$953$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$954$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$960$		0			0
$961$	0.500000	+	0.866025i	0.500000	+	0.866025i
$962$		0			0
$963$	−1.73205	+	3.00000i	−1.73205	+	3.00000i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	1.00000	+	1.73205i	1.00000	+	1.73205i
$969$		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			0
$973$		−	1.73205i		−	1.73205i
$974$	1.73205			1.73205
$975$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$976$		0			0
$977$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$977$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$	−1.73205			−1.73205
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$983$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$991$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$992$		0			0
$993$		0			0
$994$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$995$		0			0
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	0.866025	−	1.50000i	0.866025	−	1.50000i
$999$		0			0

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	476.1.o.c.67.1	✓	4
4.3	odd	2	inner	476.1.o.c.67.2	yes	4
7.2	even	3	inner	476.1.o.c.135.1	yes	4
7.3	odd	6		3332.1.g.f.883.1		2
7.4	even	3		3332.1.g.g.883.2		2
7.5	odd	6		3332.1.o.f.2039.2		4
7.6	odd	2		3332.1.o.f.67.2		4
17.16	even	2	inner	476.1.o.c.67.2	yes	4
28.3	even	6		3332.1.g.f.883.2		2
28.11	odd	6		3332.1.g.g.883.1		2
28.19	even	6		3332.1.o.f.2039.1		4
28.23	odd	6	inner	476.1.o.c.135.2	yes	4
28.27	even	2		3332.1.o.f.67.1		4
68.67	odd	2	CM	476.1.o.c.67.1	✓	4
119.16	even	6	inner	476.1.o.c.135.2	yes	4
119.33	odd	6		3332.1.o.f.2039.1		4
119.67	even	6		3332.1.g.g.883.1		2
119.101	odd	6		3332.1.g.f.883.2		2
119.118	odd	2		3332.1.o.f.67.1		4
476.67	odd	6		3332.1.g.g.883.2		2
476.135	odd	6	inner	476.1.o.c.135.1	yes	4
476.271	even	6		3332.1.o.f.2039.2		4
476.339	even	6		3332.1.g.f.883.1		2
476.475	even	2		3332.1.o.f.67.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
476.1.o.c.67.1	✓	4	1.1	even	1	trivial
476.1.o.c.67.1	✓	4	68.67	odd	2	CM
476.1.o.c.67.2	yes	4	4.3	odd	2	inner
476.1.o.c.67.2	yes	4	17.16	even	2	inner
476.1.o.c.135.1	yes	4	7.2	even	3	inner
476.1.o.c.135.1	yes	4	476.135	odd	6	inner
476.1.o.c.135.2	yes	4	28.23	odd	6	inner
476.1.o.c.135.2	yes	4	119.16	even	6	inner
3332.1.g.f.883.1		2	7.3	odd	6
3332.1.g.f.883.1		2	476.339	even	6
3332.1.g.f.883.2		2	28.3	even	6
3332.1.g.f.883.2		2	119.101	odd	6
3332.1.g.g.883.1		2	28.11	odd	6
3332.1.g.g.883.1		2	119.67	even	6
3332.1.g.g.883.2		2	7.4	even	3
3332.1.g.g.883.2		2	476.67	odd	6
3332.1.o.f.67.1		4	28.27	even	2
3332.1.o.f.67.1		4	119.118	odd	2
3332.1.o.f.67.2		4	7.6	odd	2
3332.1.o.f.67.2		4	476.475	even	2
3332.1.o.f.2039.1		4	28.19	even	6
3332.1.o.f.2039.1		4	119.33	odd	6
3332.1.o.f.2039.2		4	7.5	odd	6
3332.1.o.f.2039.2		4	476.271	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

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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	476.1.o.c.67.1

Level	$476$
Weight	$1$
Character	476.67
Analytic conductor	$0.238$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{6}$
CM discriminant	-68
Inner twists	$8$