LMFDB - Embedded newform 3192.1.fm.c.1133.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3192,1,Mod(293,3192)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3192, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("3192.293");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3192.fm (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:	$1.59301552032$
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:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.209656254528.1

Embedding invariants

Embedding label			1133.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3192.1133
Dual form			3192.1.fm.c.293.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(1.50000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{6} -1.00000 q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(1.50000 + 0.866025i) q^{10} +1.00000 q^{12} +(-1.50000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-1.50000 - 0.866025i) q^{15} +(-0.500000 - 0.866025i) q^{16} -1.00000 q^{18} +(-0.500000 - 0.866025i) q^{19} +1.73205i q^{20} +(0.500000 + 0.866025i) q^{21} +(-1.50000 - 0.866025i) q^{23} +(0.500000 + 0.866025i) q^{24} +(1.00000 - 1.73205i) q^{25} -1.73205i q^{26} +1.00000 q^{27} +(0.500000 - 0.866025i) q^{28} -1.73205i q^{30} +(0.500000 - 0.866025i) q^{32} +(-1.50000 + 0.866025i) q^{35} +(-0.500000 - 0.866025i) q^{36} +(0.500000 - 0.866025i) q^{38} +1.73205i q^{39} +(-1.50000 + 0.866025i) q^{40} +(-0.500000 + 0.866025i) q^{42} +1.73205i q^{45} -1.73205i q^{46} +(-0.500000 + 0.866025i) q^{48} +1.00000 q^{49} +2.00000 q^{50} +(1.50000 - 0.866025i) q^{52} +(0.500000 + 0.866025i) q^{54} +1.00000 q^{56} +(-0.500000 + 0.866025i) q^{57} +(-0.500000 - 0.866025i) q^{59} +(1.50000 - 0.866025i) q^{60} +(0.500000 - 0.866025i) q^{61} +(0.500000 - 0.866025i) q^{63} +1.00000 q^{64} -3.00000 q^{65} +1.73205i q^{69} +(-1.50000 - 0.866025i) q^{70} +(0.500000 + 0.866025i) q^{71} +(0.500000 - 0.866025i) q^{72} -2.00000 q^{75} +1.00000 q^{76} +(-1.50000 + 0.866025i) q^{78} +(-1.50000 - 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +1.73205i q^{83} -1.00000 q^{84} +(-1.50000 + 0.866025i) q^{90} +(1.50000 + 0.866025i) q^{91} +(1.50000 - 0.866025i) q^{92} +(-1.50000 - 0.866025i) q^{95} -1.00000 q^{96} +(0.500000 + 0.866025i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{2} - q^{3} - q^{4} + 3 q^{5} + q^{6} - 2 q^{7} - 2 q^{8} - q^{9} + 3 q^{10} + 2 q^{12} - 3 q^{13} - q^{14} - 3 q^{15} - q^{16} - 2 q^{18} - q^{19} + q^{21} - 3 q^{23} + q^{24} + 2 q^{25}+ \cdots + q^{98}+O(q^{100})$ 2 * q + q^2 - q^3 - q^4 + 3 * q^5 + q^6 - 2 * q^7 - 2 * q^8 - q^9 + 3 * q^10 + 2 * q^12 - 3 * q^13 - q^14 - 3 * q^15 - q^16 - 2 * q^18 - q^19 + q^21 - 3 * q^23 + q^24 + 2 * q^25 + 2 * q^27 + q^28 + q^32 - 3 * q^35 - q^36 + q^38 - 3 * q^40 - q^42 - q^48 + 2 * q^49 + 4 * q^50 + 3 * q^52 + q^54 + 2 * q^56 - q^57 - q^59 + 3 * q^60 + q^61 + q^63 + 2 * q^64 - 6 * q^65 - 3 * q^70 + q^71 + q^72 - 4 * q^75 + 2 * q^76 - 3 * q^78 - 3 * q^80 - q^81 - 2 * q^84 - 3 * q^90 + 3 * q^91 + 3 * q^92 - 3 * q^95 - 2 * q^96 + q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3192\mathbb{Z}\right)^\times$ .

$n$	$799$	$913$	$1009$	$1597$	$2129$
$\chi(n)$	$1$	$-1$	$e\left(\frac{5}{6}\right)$	$-1$	$-1$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0.500000	+	0.866025i	0.500000	+	0.866025i
$2$	0.500000	+	0.866025i	0.500000	+	0.866025i
$3$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$3$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$4$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$5$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$5$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$6$	0.500000	−	0.866025i	0.500000	−	0.866025i
$7$	−1.00000			−1.00000
$7$	−1.00000			−1.00000
$8$	−1.00000			−1.00000
$9$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$10$	1.50000	+	0.866025i	1.50000	+	0.866025i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	1.00000			1.00000
$13$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$13$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$14$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$15$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$	−1.00000			−1.00000
$19$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$19$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$20$			1.73205i			1.73205i
$21$	0.500000	+	0.866025i	0.500000	+	0.866025i
$22$		0			0
$23$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$23$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$24$	0.500000	+	0.866025i	0.500000	+	0.866025i
$25$	1.00000	−	1.73205i	1.00000	−	1.73205i
$26$		−	1.73205i		−	1.73205i
$27$	1.00000			1.00000
$28$	0.500000	−	0.866025i	0.500000	−	0.866025i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		−	1.73205i		−	1.73205i
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	0.500000	−	0.866025i	0.500000	−	0.866025i
$33$		0			0
$34$		0			0
$35$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$36$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	0.500000	−	0.866025i	0.500000	−	0.866025i
$39$			1.73205i			1.73205i
$40$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$		0			0
$45$			1.73205i			1.73205i
$46$		−	1.73205i		−	1.73205i
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$49$	1.00000			1.00000
$50$	2.00000			2.00000
$51$		0			0
$52$	1.50000	−	0.866025i	1.50000	−	0.866025i
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$	0.500000	+	0.866025i	0.500000	+	0.866025i
$55$		0			0
$56$	1.00000			1.00000
$57$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$58$		0			0
$59$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$59$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$60$	1.50000	−	0.866025i	1.50000	−	0.866025i
$61$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$61$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$62$		0			0
$63$	0.500000	−	0.866025i	0.500000	−	0.866025i
$64$	1.00000			1.00000
$65$	−3.00000			−3.00000
$66$		0			0
$67$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$67$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$68$		0			0
$69$			1.73205i			1.73205i
$70$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$71$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$71$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$72$	0.500000	−	0.866025i	0.500000	−	0.866025i
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$	−2.00000			−2.00000
$76$	1.00000			1.00000
$77$		0			0
$78$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$83$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$84$	−1.00000			−1.00000
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$91$	1.50000	+	0.866025i	1.50000	+	0.866025i
$92$	1.50000	−	0.866025i	1.50000	−	0.866025i
$93$		0			0
$94$		0			0
$95$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$96$	−1.00000			−1.00000
$97$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$97$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$98$	0.500000	+	0.866025i	0.500000	+	0.866025i
$99$		0			0
$100$	1.00000	+	1.73205i	1.00000	+	1.73205i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	1.50000	+	0.866025i	1.50000	+	0.866025i
$105$	1.50000	+	0.866025i	1.50000	+	0.866025i
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$109$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$109$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$110$		0			0
$111$		0			0
$112$	0.500000	+	0.866025i	0.500000	+	0.866025i
$113$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$113$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$114$	−1.00000			−1.00000
$115$	−3.00000			−3.00000
$116$		0			0
$117$	1.50000	−	0.866025i	1.50000	−	0.866025i
$118$	0.500000	−	0.866025i	0.500000	−	0.866025i
$119$		0			0
$120$	1.50000	+	0.866025i	1.50000	+	0.866025i
$121$	−1.00000			−1.00000
$122$	1.00000			1.00000
$123$		0			0
$124$		0			0
$125$		−	1.73205i		−	1.73205i
$126$	1.00000			1.00000
$127$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$127$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$128$	0.500000	+	0.866025i	0.500000	+	0.866025i
$129$		0			0
$130$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$131$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$131$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$132$		0			0
$133$	0.500000	+	0.866025i	0.500000	+	0.866025i
$134$		0			0
$135$	1.50000	−	0.866025i	1.50000	−	0.866025i
$136$		0			0
$137$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$137$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$138$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$139$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$139$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$140$		−	1.73205i		−	1.73205i
$141$		0			0
$142$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$143$		0			0
$144$	1.00000			1.00000
$145$		0			0
$146$		0			0
$147$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$151$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$151$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$152$	0.500000	+	0.866025i	0.500000	+	0.866025i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$157$	−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			0
$159$		0			0
$160$		−	1.73205i		−	1.73205i
$161$	1.50000	+	0.866025i	1.50000	+	0.866025i
$162$	0.500000	−	0.866025i	0.500000	−	0.866025i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$169$	1.00000	+	1.73205i	1.00000	+	1.73205i
$170$		0			0
$171$	1.00000			1.00000
$172$		0			0
$173$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$173$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$174$		0			0
$175$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$176$		0			0
$177$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$181$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$181$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$182$			1.73205i			1.73205i
$183$	−1.00000			−1.00000
$184$	1.50000	+	0.866025i	1.50000	+	0.866025i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.00000			−1.00000
$190$		−	1.73205i		−	1.73205i
$191$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$191$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$192$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$193$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$193$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$194$		0			0
$195$	1.50000	+	2.59808i	1.50000	+	2.59808i
$196$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	1.50000	−	0.866025i	1.50000	−	0.866025i
$208$			1.73205i			1.73205i
$209$		0			0
$210$			1.73205i			1.73205i
$211$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$211$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$212$		0			0
$213$	0.500000	−	0.866025i	0.500000	−	0.866025i
$214$		0			0
$215$		0			0
$216$	−1.00000			−1.00000
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$223$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$224$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$225$	1.00000	+	1.73205i	1.00000	+	1.73205i
$226$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$227$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$227$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$228$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$229$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$229$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$230$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$231$		0			0
$232$		0			0
$233$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$233$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$234$	1.50000	+	0.866025i	1.50000	+	0.866025i
$235$		0			0
$236$	1.00000			1.00000
$237$		0			0
$238$		0			0
$239$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$239$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$240$			1.73205i			1.73205i
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$243$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$244$	0.500000	+	0.866025i	0.500000	+	0.866025i
$245$	1.50000	−	0.866025i	1.50000	−	0.866025i
$246$		0			0
$247$			1.73205i			1.73205i
$248$		0			0
$249$	1.50000	−	0.866025i	1.50000	−	0.866025i
$250$	1.50000	−	0.866025i	1.50000	−	0.866025i
$251$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$251$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$252$	0.500000	+	0.866025i	0.500000	+	0.866025i
$253$		0			0
$254$		−	1.73205i		−	1.73205i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$	1.50000	−	2.59808i	1.50000	−	2.59808i
$261$		0			0
$262$	1.50000	+	0.866025i	1.50000	+	0.866025i
$263$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$263$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$264$		0			0
$265$		0			0
$266$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$267$		0			0
$268$		0			0
$269$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$269$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$270$	1.50000	+	0.866025i	1.50000	+	0.866025i
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		−	1.73205i		−	1.73205i
$274$			1.73205i			1.73205i
$275$		0			0
$276$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	2.00000			2.00000
$279$		0			0
$280$	1.50000	−	0.866025i	1.50000	−	0.866025i
$281$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$281$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$283$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$284$	−1.00000			−1.00000
$285$			1.73205i			1.73205i
$286$		0			0
$287$		0			0
$288$	0.500000	+	0.866025i	0.500000	+	0.866025i
$289$	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	$0.666667\pi$
$294$	0.500000	−	0.866025i	0.500000	−	0.866025i
$295$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.50000	+	2.59808i	1.50000	+	2.59808i
$300$	1.00000	−	1.73205i	1.00000	−	1.73205i
$301$		0			0
$302$	1.50000	−	0.866025i	1.50000	−	0.866025i
$303$		0			0
$304$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$305$		−	1.73205i		−	1.73205i
$306$		0			0
$307$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$307$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		−	1.73205i		−	1.73205i
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	0.500000	−	0.866025i	0.500000	−	0.866025i
$315$		−	1.73205i		−	1.73205i
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$	1.50000	−	0.866025i	1.50000	−	0.866025i
$321$		0			0
$322$			1.73205i			1.73205i
$323$		0			0
$324$	1.00000			1.00000
$325$	−3.00000	+	1.73205i	−3.00000	+	1.73205i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	0.500000	−	0.866025i	0.500000	−	0.866025i
$337$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$337$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$338$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$339$	0.500000	+	0.866025i	0.500000	+	0.866025i
$340$		0			0
$341$		0			0
$342$	0.500000	+	0.866025i	0.500000	+	0.866025i
$343$	−1.00000			−1.00000
$344$		0			0
$345$	1.50000	+	2.59808i	1.50000	+	2.59808i
$346$	0.500000	−	0.866025i	0.500000	−	0.866025i
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	−2.00000			−2.00000			−1.00000			$\pi$
$349$	−2.00000			−2.00000			−1.00000			$\pi$
$350$	−2.00000			−2.00000
$351$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$	−1.00000			−1.00000
$355$	1.50000	+	0.866025i	1.50000	+	0.866025i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		−	1.73205i		−	1.73205i
$361$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$362$			1.73205i			1.73205i
$363$	0.500000	+	0.866025i	0.500000	+	0.866025i
$364$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$365$		0			0
$366$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$			1.73205i			1.73205i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$376$		0			0
$377$		0			0
$378$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	1.50000	−	0.866025i	1.50000	−	0.866025i
$381$			1.73205i			1.73205i
$382$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$	0.500000	−	0.866025i	0.500000	−	0.866025i
$385$		0			0
$386$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$387$		0			0
$388$		0			0
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$391$		0			0
$392$	−1.00000			−1.00000
$393$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$398$		0			0
$399$	0.500000	−	0.866025i	0.500000	−	0.866025i
$400$	−2.00000			−2.00000
$401$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$401$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$409$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$410$		0			0
$411$		−	1.73205i		−	1.73205i
$412$		0			0
$413$	0.500000	+	0.866025i	0.500000	+	0.866025i
$414$	1.50000	+	0.866025i	1.50000	+	0.866025i
$415$	1.50000	+	2.59808i	1.50000	+	2.59808i
$416$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$417$	−2.00000			−2.00000
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$421$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$421$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	1.00000			1.00000
$427$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$431$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$432$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$433$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$433$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			1.73205i			1.73205i
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$442$		0			0
$443$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$443$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.00000			−1.00000
$449$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$449$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$450$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$451$		0			0
$452$	0.500000	−	0.866025i	0.500000	−	0.866025i
$453$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$454$	0.500000	+	0.866025i	0.500000	+	0.866025i
$455$	3.00000			3.00000
$456$	0.500000	−	0.866025i	0.500000	−	0.866025i
$457$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$457$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$458$	0.500000	+	0.866025i	0.500000	+	0.866025i
$459$		0			0
$460$	1.50000	−	2.59808i	1.50000	−	2.59808i
$461$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$461$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$462$		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
$466$	1.50000	+	0.866025i	1.50000	+	0.866025i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$			1.73205i			1.73205i
$469$		0			0
$470$		0			0
$471$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$472$	0.500000	+	0.866025i	0.500000	+	0.866025i
$473$		0			0
$474$		0			0
$475$	−2.00000			−2.00000
$476$		0			0
$477$		0			0
$478$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$481$		0			0
$482$		0			0
$483$		−	1.73205i		−	1.73205i
$484$	0.500000	−	0.866025i	0.500000	−	0.866025i
$485$		0			0
$486$	−1.00000			−1.00000
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$489$		0			0
$490$	1.50000	+	0.866025i	1.50000	+	0.866025i
$491$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$491$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$492$		0			0
$493$		0			0
$494$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$495$		0			0
$496$		0			0
$497$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$498$	1.50000	+	0.866025i	1.50000	+	0.866025i
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	1.50000	+	0.866025i	1.50000	+	0.866025i
$501$		0			0
$502$			1.73205i			1.73205i
$503$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$503$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$504$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$505$		0			0
$506$		0			0
$507$	1.00000	−	1.73205i	1.00000	−	1.73205i
$508$	1.50000	−	0.866025i	1.50000	−	0.866025i
$509$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$509$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$520$	3.00000			3.00000
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$			1.73205i			1.73205i
$525$	2.00000			2.00000
$526$	1.50000	+	0.866025i	1.50000	+	0.866025i
$527$		0			0
$528$		0			0
$529$	1.00000	+	1.73205i	1.00000	+	1.73205i
$530$		0			0
$531$	1.00000			1.00000
$532$	−1.00000			−1.00000
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	1.00000	−	1.73205i	1.00000	−	1.73205i
$539$		0			0
$540$			1.73205i			1.73205i
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		−	1.73205i		−	1.73205i
$544$		0			0
$545$		0			0
$546$	1.50000	−	0.866025i	1.50000	−	0.866025i
$547$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$547$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$548$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$549$	0.500000	+	0.866025i	0.500000	+	0.866025i
$550$		0			0
$551$		0			0
$552$		−	1.73205i		−	1.73205i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	1.00000	+	1.73205i	1.00000	+	1.73205i
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$		0			0
$560$	1.50000	+	0.866025i	1.50000	+	0.866025i
$561$		0			0
$562$	−2.00000			−2.00000
$563$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$563$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$566$	0.500000	−	0.866025i	0.500000	−	0.866025i
$567$	0.500000	+	0.866025i	0.500000	+	0.866025i
$568$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$569$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$569$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$570$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$	1.50000	−	0.866025i	1.50000	−	0.866025i
$574$		0			0
$575$	−3.00000	+	1.73205i	−3.00000	+	1.73205i
$576$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	1.00000			1.00000
$579$	1.50000	+	0.866025i	1.50000	+	0.866025i
$580$		0			0
$581$		−	1.73205i		−	1.73205i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	1.50000	−	2.59808i	1.50000	−	2.59808i
$586$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$	1.00000			1.00000
$589$		0			0
$590$		−	1.73205i		−	1.73205i
$591$		0			0
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$599$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$599$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$600$	2.00000			2.00000
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$		0			0
$604$	1.50000	+	0.866025i	1.50000	+	0.866025i
$605$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	−1.00000			−1.00000
$609$		0			0
$610$	1.50000	−	0.866025i	1.50000	−	0.866025i
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$615$		0			0
$616$		0			0
$617$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$617$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$618$		0			0
$619$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$619$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$622$		0			0
$623$		0			0
$624$	1.50000	−	0.866025i	1.50000	−	0.866025i
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	1.00000			1.00000
$629$		0			0
$630$	1.50000	−	0.866025i	1.50000	−	0.866025i
$631$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$631$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.00000			−3.00000
$636$		0			0
$637$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$638$		0			0
$639$	−1.00000			−1.00000
$640$	1.50000	+	0.866025i	1.50000	+	0.866025i
$641$	−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
$643$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$643$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$644$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	0.500000	+	0.866025i	0.500000	+	0.866025i
$649$		0			0
$650$	−3.00000	−	1.73205i	−3.00000	−	1.73205i
$651$		0			0
$652$		0			0
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$	1.50000	−	2.59808i	1.50000	−	2.59808i
$656$		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
$661$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$661$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		−	1.73205i		−	1.73205i
$665$	1.50000	+	0.866025i	1.50000	+	0.866025i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	1.00000			1.00000
$673$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$673$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$674$	1.50000	+	0.866025i	1.50000	+	0.866025i
$675$	1.00000	−	1.73205i	1.00000	−	1.73205i
$676$	−2.00000			−2.00000
$677$	−2.00000			−2.00000			−1.00000			$\pi$
$677$	−2.00000			−2.00000			−1.00000			$\pi$
$678$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$679$		0			0
$680$		0			0
$681$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$685$	3.00000			3.00000
$686$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$687$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$688$		0			0
$689$		0			0
$690$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$691$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$691$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$692$	1.00000			1.00000
$693$		0			0
$694$		0			0
$695$		−	3.46410i		−	3.46410i
$696$		0			0
$697$		0			0
$698$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$699$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$700$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$701$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$701$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$702$		−	1.73205i		−	1.73205i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$			1.73205i			1.73205i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1.50000	−	0.866025i	1.50000	−	0.866025i
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$	1.50000	−	0.866025i	1.50000	−	0.866025i
$721$		0			0
$722$	−1.00000			−1.00000
$723$		0			0
$724$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$725$		0			0
$726$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$	0.500000	−	0.866025i	0.500000	−	0.866025i
$733$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$733$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$736$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$737$		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
$741$	1.50000	−	0.866025i	1.50000	−	0.866025i
$742$		0			0
$743$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$743$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$748$		0			0
$749$		0			0
$750$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$		−	1.73205i		−	1.73205i
$754$		0			0
$755$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$756$	0.500000	−	0.866025i	0.500000	−	0.866025i
$757$		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
$760$	1.50000	+	0.866025i	1.50000	+	0.866025i
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$763$		0			0
$764$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$765$		0			0
$766$		0			0
$767$			1.73205i			1.73205i
$768$	1.00000			1.00000
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$		−	1.73205i		−	1.73205i
$773$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$773$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−3.00000			−3.00000
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$785$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$786$		−	1.73205i		−	1.73205i
$787$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$787$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$788$		0			0
$789$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$790$		0			0
$791$	1.00000			1.00000
$792$		0			0
$793$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$794$	1.00000	−	1.73205i	1.00000	−	1.73205i
$795$		0			0
$796$		0			0
$797$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$797$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$798$	1.00000			1.00000
$799$		0			0
$800$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$801$		0			0
$802$	0.500000	−	0.866025i	0.500000	−	0.866025i
$803$		0			0
$804$		0			0
$805$	3.00000			3.00000
$806$		0			0
$807$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$808$		0			0
$809$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$809$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$810$		−	1.73205i		−	1.73205i
$811$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$811$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$	1.50000	−	0.866025i	1.50000	−	0.866025i
$823$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$823$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$824$		0			0
$825$		0			0
$826$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$827$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$827$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$828$			1.73205i			1.73205i
$829$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$829$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$830$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$831$		0			0
$832$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$833$		0			0
$834$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$		0			0
$843$	2.00000			2.00000
$844$		0			0
$845$	3.00000	+	1.73205i	3.00000	+	1.73205i
$846$		0			0
$847$	1.00000			1.00000
$848$		0			0
$849$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$850$		0			0
$851$		0			0
$852$	0.500000	+	0.866025i	0.500000	+	0.866025i
$853$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$853$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$854$	−1.00000			−1.00000
$855$	1.50000	−	0.866025i	1.50000	−	0.866025i
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$859$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$860$		0			0
$861$		0			0
$862$	2.00000			2.00000
$863$	2.00000			2.00000			1.00000			$0$
$863$	2.00000			2.00000			1.00000			$0$
$864$	0.500000	−	0.866025i	0.500000	−	0.866025i
$865$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$866$		0			0
$867$	−1.00000			−1.00000
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$875$			1.73205i			1.73205i
$876$		0			0
$877$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$877$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$878$		0			0
$879$	0.500000	+	0.866025i	0.500000	+	0.866025i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	−1.00000			−1.00000
$883$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$883$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$884$		0			0
$885$			1.73205i			1.73205i
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	1.50000	+	0.866025i	1.50000	+	0.866025i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$897$	1.50000	−	2.59808i	1.50000	−	2.59808i
$898$	0.500000	+	0.866025i	0.500000	+	0.866025i
$899$		0			0
$900$	−2.00000			−2.00000
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.00000			1.00000
$905$	3.00000			3.00000
$906$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$907$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$907$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$908$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$909$		0			0
$910$	1.50000	+	2.59808i	1.50000	+	2.59808i
$911$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$911$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$912$	1.00000			1.00000
$913$		0			0
$914$	0.500000	+	0.866025i	0.500000	+	0.866025i
$915$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$916$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$917$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$918$		0			0
$919$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$919$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$920$	3.00000			3.00000
$921$	1.50000	+	0.866025i	1.50000	+	0.866025i
$922$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$923$		−	1.73205i		−	1.73205i
$924$		0			0
$925$		0			0
$926$	0.500000	+	0.866025i	0.500000	+	0.866025i
$927$		0			0
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$932$			1.73205i			1.73205i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$937$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$937$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$941$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$942$	−1.00000			−1.00000
$943$		0			0
$944$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$945$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$		0			0
$950$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$951$		0			0
$952$		0			0
$953$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$953$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$954$		0			0
$955$	1.50000	+	2.59808i	1.50000	+	2.59808i
$956$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$957$		0			0
$958$		0			0
$959$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$960$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$961$	−1.00000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$966$	1.50000	−	0.866025i	1.50000	−	0.866025i
$967$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$967$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$968$	1.00000			1.00000
$969$		0			0
$970$		0			0
$971$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$971$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$972$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$973$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$974$		0			0
$975$	3.00000	+	1.73205i	3.00000	+	1.73205i
$976$	−1.00000			−1.00000
$977$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$977$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$			1.73205i			1.73205i
$981$		0			0
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$989$		0			0
$990$		0			0
$991$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$991$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$	0.500000	−	0.866025i	0.500000	−	0.866025i
$995$		0			0
$996$			1.73205i			1.73205i
$997$	−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

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3192.1.fm.c.1133.1	yes	2
3.2	odd	2		3192.1.fm.b.1133.1	yes	2
7.6	odd	2		3192.1.fm.d.1133.1	yes	2
8.5	even	2		3192.1.fm.d.1133.1	yes	2
19.8	odd	6		3192.1.fm.b.293.1	yes	2
21.20	even	2		3192.1.fm.a.1133.1	yes	2
24.5	odd	2		3192.1.fm.a.1133.1	yes	2
56.13	odd	2	CM	3192.1.fm.c.1133.1	yes	2
57.8	even	6	inner	3192.1.fm.c.293.1	yes	2
133.27	even	6		3192.1.fm.a.293.1	✓	2
152.141	odd	6		3192.1.fm.a.293.1	✓	2
168.125	even	2		3192.1.fm.b.1133.1	yes	2
399.293	odd	6		3192.1.fm.d.293.1	yes	2
456.293	even	6		3192.1.fm.d.293.1	yes	2
1064.293	even	6		3192.1.fm.b.293.1	yes	2
3192.293	odd	6	inner	3192.1.fm.c.293.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.1.fm.a.293.1	✓	2	133.27	even	6
3192.1.fm.a.293.1	✓	2	152.141	odd	6
3192.1.fm.a.1133.1	yes	2	21.20	even	2
3192.1.fm.a.1133.1	yes	2	24.5	odd	2
3192.1.fm.b.293.1	yes	2	19.8	odd	6
3192.1.fm.b.293.1	yes	2	1064.293	even	6
3192.1.fm.b.1133.1	yes	2	3.2	odd	2
3192.1.fm.b.1133.1	yes	2	168.125	even	2
3192.1.fm.c.293.1	yes	2	57.8	even	6	inner
3192.1.fm.c.293.1	yes	2	3192.293	odd	6	inner
3192.1.fm.c.1133.1	yes	2	1.1	even	1	trivial
3192.1.fm.c.1133.1	yes	2	56.13	odd	2	CM
3192.1.fm.d.293.1	yes	2	399.293	odd	6
3192.1.fm.d.293.1	yes	2	456.293	even	6
3192.1.fm.d.1133.1	yes	2	7.6	odd	2
3192.1.fm.d.1133.1	yes	2	8.5	even	2

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

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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	3192.1.fm.c.1133.1

Level	$3192$
Weight	$1$
Character	3192.1133
Analytic conductor	$1.593$
Analytic rank	$0$
Dimension	$2$
Projective image	$D_{6}$
CM discriminant	-56
Inner twists	$4$