LMFDB - Embedded newform 728.1.bl.a.69.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [728,1,Mod(69,728)]

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, 1]))

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

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

chi := DirichletCharacter("728.69");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 728 = 2^{3} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	728.bl (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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			69.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	728.69
Dual form			728.1.bl.a.517.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.866025 - 0.500000i) q^{2} +(-0.258819 + 0.448288i) q^{3} +(0.500000 + 0.866025i) q^{4} +1.93185i q^{5} +(0.448288 - 0.258819i) q^{6} +(0.866025 - 0.500000i) q^{7} -1.00000i q^{8} +(0.366025 + 0.633975i) q^{9} +(0.965926 - 1.67303i) q^{10} -0.517638 q^{12} +(-0.965926 - 0.258819i) q^{13} -1.00000 q^{14} +(-0.866025 - 0.500000i) q^{15} +(-0.500000 + 0.866025i) q^{16} -0.732051i q^{18} +(1.22474 - 0.707107i) q^{19} +(-1.67303 + 0.965926i) q^{20} +0.517638i q^{21} +(-0.500000 + 0.866025i) q^{23} +(0.448288 + 0.258819i) q^{24} -2.73205 q^{25} +(0.707107 + 0.707107i) q^{26} -0.896575 q^{27} +(0.866025 + 0.500000i) q^{28} +(0.500000 + 0.866025i) q^{30} +(0.866025 - 0.500000i) q^{32} +(0.965926 + 1.67303i) q^{35} +(-0.366025 + 0.633975i) q^{36} -1.41421 q^{38} +(0.366025 - 0.366025i) q^{39} +1.93185 q^{40} +(0.258819 - 0.448288i) q^{42} +(-1.22474 + 0.707107i) q^{45} +(0.866025 - 0.500000i) q^{46} +(-0.258819 - 0.448288i) q^{48} +(0.500000 - 0.866025i) q^{49} +(2.36603 + 1.36603i) q^{50} +(-0.258819 - 0.965926i) q^{52} +(0.776457 + 0.448288i) q^{54} +(-0.500000 - 0.866025i) q^{56} +0.732051i q^{57} +(1.67303 - 0.965926i) q^{59} -1.00000i q^{60} +(0.258819 + 0.448288i) q^{61} +(0.633975 + 0.366025i) q^{63} -1.00000 q^{64} +(0.500000 - 1.86603i) q^{65} +(-0.258819 - 0.448288i) q^{69} -1.93185i q^{70} +(-0.866025 + 0.500000i) q^{71} +(0.633975 - 0.366025i) q^{72} +(0.707107 - 1.22474i) q^{75} +(1.22474 + 0.707107i) q^{76} +(-0.500000 + 0.133975i) q^{78} +(-1.67303 - 0.965926i) q^{80} +(-0.133975 + 0.232051i) q^{81} -1.41421i q^{83} +(-0.448288 + 0.258819i) q^{84} +1.41421 q^{90} +(-0.965926 + 0.258819i) q^{91} -1.00000 q^{92} +(1.36603 + 2.36603i) q^{95} +0.517638i q^{96} +(-0.866025 + 0.500000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4} - 4 q^{9} - 8 q^{14} - 4 q^{16} - 4 q^{23} - 8 q^{25} + 4 q^{30} + 4 q^{36} - 4 q^{39} + 4 q^{49} + 12 q^{50} - 4 q^{56} + 12 q^{63} - 8 q^{64} + 4 q^{65} + 12 q^{72} - 4 q^{78} - 8 q^{81}+ \cdots + 4 q^{95}+O(q^{100}) $ 8 * q + 4 * q^4 - 4 * q^9 - 8 * q^14 - 4 * q^16 - 4 * q^23 - 8 * q^25 + 4 * q^30 + 4 * q^36 - 4 * q^39 + 4 * q^49 + 12 * q^50 - 4 * q^56 + 12 * q^63 - 8 * q^64 + 4 * q^65 + 12 * q^72 - 4 * q^78 - 8 * q^81 - 8 * q^92 + 4 * q^95

Character values

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

$n$	$183$	$365$	$521$	$561$
$\chi(n)$	$1$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$2$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$3$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$3$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$4$	0.500000	+	0.866025i	0.500000	+	0.866025i
$5$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$5$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$6$	0.448288	−	0.258819i	0.448288	−	0.258819i
$7$	0.866025	−	0.500000i	0.866025	−	0.500000i
$7$	0.866025	−	0.500000i	0.866025	−	0.500000i
$8$		−	1.00000i		−	1.00000i
$9$	0.366025	+	0.633975i	0.366025	+	0.633975i
$10$	0.965926	−	1.67303i	0.965926	−	1.67303i
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$	−0.517638			−0.517638
$13$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$13$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$14$	−1.00000			−1.00000
$15$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		−	0.732051i		−	0.732051i
$19$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$19$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$20$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$21$			0.517638i			0.517638i
$22$		0			0
$23$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$23$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$24$	0.448288	+	0.258819i	0.448288	+	0.258819i
$25$	−2.73205			−2.73205
$26$	0.707107	+	0.707107i	0.707107	+	0.707107i
$27$	−0.896575			−0.896575
$28$	0.866025	+	0.500000i	0.866025	+	0.500000i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$	0.500000	+	0.866025i	0.500000	+	0.866025i
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	0.866025	−	0.500000i	0.866025	−	0.500000i
$33$		0			0
$34$		0			0
$35$	0.965926	+	1.67303i	0.965926	+	1.67303i
$36$	−0.366025	+	0.633975i	−0.366025	+	0.633975i
$37$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$37$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$38$	−1.41421			−1.41421
$39$	0.366025	−	0.366025i	0.366025	−	0.366025i
$40$	1.93185			1.93185
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$	0.258819	−	0.448288i	0.258819	−	0.448288i
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$		0			0
$45$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$46$	0.866025	−	0.500000i	0.866025	−	0.500000i
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$49$	0.500000	−	0.866025i	0.500000	−	0.866025i
$50$	2.36603	+	1.36603i	2.36603	+	1.36603i
$51$		0			0
$52$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	0.776457	+	0.448288i	0.776457	+	0.448288i
$55$		0			0
$56$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$57$			0.732051i			0.732051i
$58$		0			0
$59$	1.67303	−	0.965926i	1.67303	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$59$	1.67303	−	0.965926i	1.67303	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$60$		−	1.00000i		−	1.00000i
$61$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$61$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$62$		0			0
$63$	0.633975	+	0.366025i	0.633975	+	0.366025i
$64$	−1.00000			−1.00000
$65$	0.500000	−	1.86603i	0.500000	−	1.86603i
$66$		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
$69$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$70$		−	1.93185i		−	1.93185i
$71$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$71$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$72$	0.633975	−	0.366025i	0.633975	−	0.366025i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	0.707107	−	1.22474i	0.707107	−	1.22474i
$76$	1.22474	+	0.707107i	1.22474	+	0.707107i
$77$		0			0
$78$	−0.500000	+	0.133975i	−0.500000	+	0.133975i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−1.67303	−	0.965926i	−1.67303	−	0.965926i
$81$	−0.133975	+	0.232051i	−0.133975	+	0.232051i
$82$		0			0
$83$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$83$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$84$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$	1.41421			1.41421
$91$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$92$	−1.00000			−1.00000
$93$		0			0
$94$		0			0
$95$	1.36603	+	2.36603i	1.36603	+	2.36603i
$96$			0.517638i			0.517638i
$97$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$97$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$98$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$99$		0			0
$100$	−1.36603	−	2.36603i	−1.36603	−	2.36603i
$101$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$101$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$105$	−1.00000			−1.00000
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	−0.448288	−	0.776457i	−0.448288	−	0.776457i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$			1.00000i			1.00000i
$113$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$113$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$114$	0.366025	−	0.633975i	0.366025	−	0.633975i
$115$	−1.67303	−	0.965926i	−1.67303	−	0.965926i
$116$		0			0
$117$	−0.189469	−	0.707107i	−0.189469	−	0.707107i
$118$	−1.93185			−1.93185
$119$		0			0
$120$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$121$	0.500000	+	0.866025i	0.500000	+	0.866025i
$122$		−	0.517638i		−	0.517638i
$123$		0			0
$124$		0			0
$125$		−	3.34607i		−	3.34607i
$126$	−0.366025	−	0.633975i	−0.366025	−	0.633975i
$127$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$127$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$128$	0.866025	+	0.500000i	0.866025	+	0.500000i
$129$		0			0
$130$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$131$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$131$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$132$		0			0
$133$	0.707107	−	1.22474i	0.707107	−	1.22474i
$134$		0			0
$135$		−	1.73205i		−	1.73205i
$136$		0			0
$137$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$137$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$138$			0.517638i			0.517638i
$139$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$139$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$140$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$141$		0			0
$142$	1.00000			1.00000
$143$		0			0
$144$	−0.732051			−0.732051
$145$		0			0
$146$		0			0
$147$	0.258819	+	0.448288i	0.258819	+	0.448288i
$148$		0			0
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$151$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$151$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$152$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.500000	+	0.133975i	0.500000	+	0.133975i
$157$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$157$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$	0.965926	+	1.67303i	0.965926	+	1.67303i
$161$			1.00000i			1.00000i
$162$	0.232051	−	0.133975i	0.232051	−	0.133975i
$163$		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
$166$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$	0.517638			0.517638
$169$	0.866025	+	0.500000i	0.866025	+	0.500000i
$170$		0			0
$171$	0.896575	+	0.517638i	0.896575	+	0.517638i
$172$		0			0
$173$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$173$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$174$		0			0
$175$	−2.36603	+	1.36603i	−2.36603	+	1.36603i
$176$		0			0
$177$			1.00000i			1.00000i
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$181$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$181$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$182$	0.965926	+	0.258819i	0.965926	+	0.258819i
$183$	−0.267949			−0.267949
$184$	0.866025	+	0.500000i	0.866025	+	0.500000i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.776457	+	0.448288i	−0.776457	+	0.448288i
$190$		−	2.73205i		−	2.73205i
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$	0.258819	−	0.448288i	0.258819	−	0.448288i
$193$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$193$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$194$		0			0
$195$	0.707107	+	0.707107i	0.707107	+	0.707107i
$196$	1.00000			1.00000
$197$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$197$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$			2.73205i			2.73205i
$201$		0			0
$202$	1.22474	−	0.707107i	1.22474	−	0.707107i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.732051			−0.732051
$208$	0.707107	−	0.707107i	0.707107	−	0.707107i
$209$		0			0
$210$	0.866025	+	0.500000i	0.866025	+	0.500000i
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$		0			0
$213$		−	0.517638i		−	0.517638i
$214$		0			0
$215$		0			0
$216$			0.896575i			0.896575i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$223$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$224$	0.500000	−	0.866025i	0.500000	−	0.866025i
$225$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$226$		0			0
$227$	−0.448288	+	0.258819i	−0.448288	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$227$	−0.448288	+	0.258819i	−0.448288	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$228$	−0.633975	+	0.366025i	−0.633975	+	0.366025i
$229$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$229$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$230$	0.965926	+	1.67303i	0.965926	+	1.67303i
$231$		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	$0.666667\pi$
$234$	−0.189469	+	0.707107i	−0.189469	+	0.707107i
$235$		0			0
$236$	1.67303	+	0.965926i	1.67303	+	0.965926i
$237$		0			0
$238$		0			0
$239$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$239$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$240$	0.866025	−	0.500000i	0.866025	−	0.500000i
$241$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$241$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$242$		−	1.00000i		−	1.00000i
$243$	−0.517638	−	0.896575i	−0.517638	−	0.896575i
$244$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$245$	1.67303	+	0.965926i	1.67303	+	0.965926i
$246$		0			0
$247$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$248$		0			0
$249$	0.633975	+	0.366025i	0.633975	+	0.366025i
$250$	−1.67303	+	2.89778i	−1.67303	+	2.89778i
$251$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$251$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$252$			0.732051i			0.732051i
$253$		0			0
$254$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$255$		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
$260$	1.86603	−	0.500000i	1.86603	−	0.500000i
$261$		0			0
$262$	0.448288	+	0.258819i	0.448288	+	0.258819i
$263$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$263$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$264$		0			0
$265$		0			0
$266$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$267$		0			0
$268$		0			0
$269$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$269$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$270$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$	0.133975	−	0.500000i	0.133975	−	0.500000i
$274$	−1.73205			−1.73205
$275$		0			0
$276$	0.258819	−	0.448288i	0.258819	−	0.448288i
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$			1.41421i			1.41421i
$279$		0			0
$280$	1.67303	−	0.965926i	1.67303	−	0.965926i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$283$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$284$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$285$	−1.41421			−1.41421
$286$		0			0
$287$		0			0
$288$	0.633975	+	0.366025i	0.633975	+	0.366025i
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$293$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$294$		−	0.517638i		−	0.517638i
$295$	1.86603	+	3.23205i	1.86603	+	3.23205i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.707107	−	0.707107i	0.707107	−	0.707107i
$300$	1.41421			1.41421
$301$		0			0
$302$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$303$	−0.366025	−	0.633975i	−0.366025	−	0.633975i
$304$			1.41421i			1.41421i
$305$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$306$		0			0
$307$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$307$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$	−0.366025	−	0.366025i	−0.366025	−	0.366025i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$315$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		−	1.93185i		−	1.93185i
$321$		0			0
$322$	0.500000	−	0.866025i	0.500000	−	0.866025i
$323$		0			0
$324$	−0.267949			−0.267949
$325$	2.63896	+	0.707107i	2.63896	+	0.707107i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$331$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$332$	1.22474	−	0.707107i	1.22474	−	0.707107i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−0.517638	−	0.896575i	−0.517638	−	0.896575i
$343$		−	1.00000i		−	1.00000i
$344$		0			0
$345$	0.866025	−	0.500000i	0.866025	−	0.500000i
$346$			0.517638i			0.517638i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	−1.67303	−	0.965926i	−1.67303	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$349$	−1.67303	−	0.965926i	−1.67303	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$350$	2.73205			2.73205
$351$	0.866025	+	0.232051i	0.866025	+	0.232051i
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$	0.500000	−	0.866025i	0.500000	−	0.866025i
$355$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$359$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$360$	0.707107	+	1.22474i	0.707107	+	1.22474i
$361$	0.500000	−	0.866025i	0.500000	−	0.866025i
$362$	−1.67303	−	0.965926i	−1.67303	−	0.965926i
$363$	−0.517638			−0.517638
$364$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$365$		0			0
$366$	0.232051	+	0.133975i	0.232051	+	0.133975i
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	−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	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	1.50000	+	0.866025i	1.50000	+	0.866025i
$376$		0			0
$377$		0			0
$378$	0.896575			0.896575
$379$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$379$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$380$	−1.36603	+	2.36603i	−1.36603	+	2.36603i
$381$	0.258819	+	0.448288i	0.258819	+	0.448288i
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$385$		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			$\pi$
$390$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$391$		0			0
$392$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$393$	0.133975	−	0.232051i	0.133975	−	0.232051i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.448288	+	0.258819i	−0.448288	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$397$	−0.448288	+	0.258819i	−0.448288	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$398$		0			0
$399$	0.366025	+	0.633975i	0.366025	+	0.633975i
$400$	1.36603	−	2.36603i	1.36603	−	2.36603i
$401$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$401$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$402$		0			0
$403$		0			0
$404$	−1.41421			−1.41421
$405$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$409$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$410$		0			0
$411$			0.896575i			0.896575i
$412$		0			0
$413$	0.965926	−	1.67303i	0.965926	−	1.67303i
$414$	0.633975	+	0.366025i	0.633975	+	0.366025i
$415$	2.73205			2.73205
$416$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$417$	0.732051			0.732051
$418$		0			0
$419$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$419$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$420$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$427$	0.448288	+	0.258819i	0.448288	+	0.258819i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$431$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$432$	0.448288	−	0.776457i	0.448288	−	0.776457i
$433$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$433$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			1.41421i			1.41421i
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	0.732051			0.732051
$442$		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
$448$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$449$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$449$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$450$			2.00000i			2.00000i
$451$		0			0
$452$		0			0
$453$	0.776457	+	0.448288i	0.776457	+	0.448288i
$454$	0.517638			0.517638
$455$	−0.500000	−	1.86603i	−0.500000	−	1.86603i
$456$	0.732051			0.732051
$457$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$457$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$458$	0.707107	−	1.22474i	0.707107	−	1.22474i
$459$		0			0
$460$		−	1.93185i		−	1.93185i
$461$	0.448288	−	0.258819i	0.448288	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$461$	0.448288	−	0.258819i	0.448288	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$462$		0			0
$463$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$463$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$464$		0			0
$465$		0			0
$466$	0.866025	+	0.500000i	0.866025	+	0.500000i
$467$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$467$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$468$	0.517638	−	0.517638i	0.517638	−	0.517638i
$469$		0			0
$470$		0			0
$471$	−0.366025	+	0.633975i	−0.366025	+	0.633975i
$472$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$473$		0			0
$474$		0			0
$475$	−3.34607	+	1.93185i	−3.34607	+	1.93185i
$476$		0			0
$477$		0			0
$478$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$	−1.00000			−1.00000
$481$		0			0
$482$		0			0
$483$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$484$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$485$		0			0
$486$			1.03528i			1.03528i
$487$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$487$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$488$	0.448288	−	0.258819i	0.448288	−	0.258819i
$489$		0			0
$490$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$	1.36603	+	0.366025i	1.36603	+	0.366025i
$495$		0			0
$496$		0			0
$497$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$498$	−0.366025	−	0.633975i	−0.366025	−	0.633975i
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$	2.89778	−	1.67303i	2.89778	−	1.67303i
$501$		0			0
$502$		−	1.93185i		−	1.93185i
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$	0.366025	−	0.633975i	0.366025	−	0.633975i
$505$	−2.36603	−	1.36603i	−2.36603	−	1.36603i
$506$		0			0
$507$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$508$	1.00000			1.00000
$509$	0.448288	+	0.258819i	0.448288	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$509$	0.448288	+	0.258819i	0.448288	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$510$		0			0
$511$		0			0
$512$			1.00000i			1.00000i
$513$	−1.09808	+	0.633975i	−1.09808	+	0.633975i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	0.267949			0.267949
$520$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$523$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$524$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$525$		−	1.41421i		−	1.41421i
$526$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$527$		0			0
$528$		0			0
$529$		0			0
$530$		0			0
$531$	1.22474	+	0.707107i	1.22474	+	0.707107i
$532$	1.41421			1.41421
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$			1.93185i			1.93185i
$539$		0			0
$540$	1.50000	−	0.866025i	1.50000	−	0.866025i
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$544$		0			0
$545$		0			0
$546$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	1.50000	+	0.866025i	1.50000	+	0.866025i
$549$	−0.189469	+	0.328169i	−0.189469	+	0.328169i
$550$		0			0
$551$		0			0
$552$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.707107	−	1.22474i	0.707107	−	1.22474i
$557$		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
$560$	−1.93185			−1.93185
$561$		0			0
$562$		0			0
$563$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$563$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$564$		0			0
$565$		0			0
$566$	1.67303	−	0.965926i	1.67303	−	0.965926i
$567$			0.267949i			0.267949i
$568$	0.500000	+	0.866025i	0.500000	+	0.866025i
$569$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$569$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$570$	1.22474	+	0.707107i	1.22474	+	0.707107i
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.36603	−	2.36603i	1.36603	−	2.36603i
$576$	−0.366025	−	0.633975i	−0.366025	−	0.633975i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	0.866025	−	0.500000i	0.866025	−	0.500000i
$579$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$580$		0			0
$581$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	1.36603	−	0.366025i	1.36603	−	0.366025i
$586$	−1.41421			−1.41421
$587$	−0.448288	−	0.258819i	−0.448288	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$587$	−0.448288	−	0.258819i	−0.448288	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$588$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$589$		0			0
$590$		−	3.73205i		−	3.73205i
$591$		0			0
$592$		0			0
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$599$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$599$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$600$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$	1.50000	−	0.866025i	1.50000	−	0.866025i
$605$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$606$			0.732051i			0.732051i
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$	0.707107	−	1.22474i	0.707107	−	1.22474i
$609$		0			0
$610$	1.00000			1.00000
$611$		0			0
$612$		0			0
$613$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$613$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$614$	0.965926	−	1.67303i	0.965926	−	1.67303i
$615$		0			0
$616$		0			0
$617$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$617$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$618$		0			0
$619$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$619$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$620$		0			0
$621$	0.448288	−	0.776457i	0.448288	−	0.776457i
$622$		0			0
$623$		0			0
$624$	0.133975	+	0.500000i	0.133975	+	0.500000i
$625$	3.73205			3.73205
$626$		0			0
$627$		0			0
$628$	0.707107	+	1.22474i	0.707107	+	1.22474i
$629$		0			0
$630$	1.22474	−	0.707107i	1.22474	−	0.707107i
$631$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$631$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.67303	+	0.965926i	1.67303	+	0.965926i
$636$		0			0
$637$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$638$		0			0
$639$	−0.633975	−	0.366025i	−0.633975	−	0.366025i
$640$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$641$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$641$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$642$		0			0
$643$	−1.67303	+	0.965926i	−1.67303	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$643$	−1.67303	+	0.965926i	−1.67303	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$644$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	0.232051	+	0.133975i	0.232051	+	0.133975i
$649$		0			0
$650$	−1.93185	−	1.93185i	−1.93185	−	1.93185i
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		−	1.00000i		−	1.00000i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	0.448288	+	0.258819i	0.448288	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$661$	0.448288	+	0.258819i	0.448288	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$662$		0			0
$663$		0			0
$664$	−1.41421			−1.41421
$665$	2.36603	+	1.36603i	2.36603	+	1.36603i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	0.258819	+	0.448288i	0.258819	+	0.448288i
$673$	−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	$0.666667\pi$
$674$		0			0
$675$	2.44949			2.44949
$676$			1.00000i			1.00000i
$677$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$677$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	0.267949i		−	0.267949i
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$			1.03528i			1.03528i
$685$	1.67303	+	2.89778i	1.67303	+	2.89778i
$686$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$687$	−0.633975	−	0.366025i	−0.633975	−	0.366025i
$688$		0			0
$689$		0			0
$690$	−1.00000			−1.00000
$691$	−1.67303	−	0.965926i	−1.67303	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$691$	−1.67303	−	0.965926i	−1.67303	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$692$	0.258819	−	0.448288i	0.258819	−	0.448288i
$693$		0			0
$694$		0			0
$695$	2.36603	−	1.36603i	2.36603	−	1.36603i
$696$		0			0
$697$		0			0
$698$	0.965926	+	1.67303i	0.965926	+	1.67303i
$699$	0.258819	−	0.448288i	0.258819	−	0.448288i
$700$	−2.36603	−	1.36603i	−2.36603	−	1.36603i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	−0.633975	−	0.633975i	−0.633975	−	0.633975i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$			1.41421i			1.41421i
$708$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$709$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$709$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$710$			1.93185i			1.93185i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	0.776457	+	0.448288i	0.776457	+	0.448288i
$718$	0.866025	−	1.50000i	0.866025	−	1.50000i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		−	1.41421i		−	1.41421i
$721$		0			0
$722$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$723$		0			0
$724$	0.965926	+	1.67303i	0.965926	+	1.67303i
$725$		0			0
$726$	0.448288	+	0.258819i	0.448288	+	0.258819i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$	0.258819	+	0.965926i	0.258819	+	0.965926i
$729$	0.267949			0.267949
$730$		0			0
$731$		0			0
$732$	−0.133975	−	0.232051i	−0.133975	−	0.232051i
$733$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$733$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$734$		0			0
$735$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$736$			1.00000i			1.00000i
$737$		0			0
$738$		0			0
$739$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$739$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$740$		0			0
$741$	0.189469	−	0.707107i	0.189469	−	0.707107i
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.896575	−	0.517638i	0.896575	−	0.517638i
$748$		0			0
$749$		0			0
$750$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$751$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$751$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$752$		0			0
$753$	−1.00000			−1.00000
$754$		0			0
$755$	3.34607			3.34607
$756$	−0.776457	−	0.448288i	−0.776457	−	0.448288i
$757$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$757$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$758$		0			0
$759$		0			0
$760$	2.36603	−	1.36603i	2.36603	−	1.36603i
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		−	0.517638i		−	0.517638i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1.86603	+	0.500000i	−1.86603	+	0.500000i
$768$	0.517638			0.517638
$769$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$769$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$770$		0			0
$771$		0			0
$772$			1.00000i			1.00000i
$773$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$773$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.500000	+	0.866025i	0.500000	+	0.866025i
$785$			2.73205i			2.73205i
$786$	−0.232051	+	0.133975i	−0.232051	+	0.133975i
$787$	0.448288	−	0.258819i	0.448288	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$787$	0.448288	−	0.258819i	0.448288	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$	0.448288	+	0.776457i	0.448288	+	0.776457i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$794$	0.517638			0.517638
$795$		0			0
$796$		0			0
$797$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$797$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$798$		−	0.732051i		−	0.732051i
$799$		0			0
$800$	−2.36603	+	1.36603i	−2.36603	+	1.36603i
$801$		0			0
$802$	1.00000	+	1.73205i	1.00000	+	1.73205i
$803$		0			0
$804$		0			0
$805$	−1.93185			−1.93185
$806$		0			0
$807$	1.00000			1.00000
$808$	1.22474	+	0.707107i	1.22474	+	0.707107i
$809$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$809$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$810$	0.258819	+	0.448288i	0.258819	+	0.448288i
$811$			0.517638i			0.517638i	0.965926	+	0.258819i	$0.0833333\pi$
$811$			0.517638i			0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−0.517638	−	0.517638i	−0.517638	−	0.517638i
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$	0.448288	−	0.776457i	0.448288	−	0.776457i
$823$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$823$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$824$		0			0
$825$		0			0
$826$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−0.366025	−	0.633975i	−0.366025	−	0.633975i
$829$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$829$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$830$	−2.36603	−	1.36603i	−2.36603	−	1.36603i
$831$		0			0
$832$	0.965926	+	0.258819i	0.965926	+	0.258819i
$833$		0			0
$834$	−0.633975	−	0.366025i	−0.633975	−	0.366025i
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$			1.00000i			1.00000i
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$846$		0			0
$847$	0.866025	+	0.500000i	0.866025	+	0.500000i
$848$		0			0
$849$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$850$		0			0
$851$		0			0
$852$	0.448288	−	0.258819i	0.448288	−	0.258819i
$853$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$853$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$854$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$855$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$859$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$860$		0			0
$861$		0			0
$862$	0.866025	+	1.50000i	0.866025	+	1.50000i
$863$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$863$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$864$	−0.776457	+	0.448288i	−0.776457	+	0.448288i
$865$	0.866025	−	0.500000i	0.866025	−	0.500000i
$866$		0			0
$867$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	0.707107	−	1.22474i	0.707107	−	1.22474i
$875$	−1.67303	−	2.89778i	−1.67303	−	2.89778i
$876$		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
$879$			0.732051i			0.732051i
$880$		0			0
$881$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$881$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$882$	−0.633975	−	0.366025i	−0.633975	−	0.366025i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$	−1.93185			−1.93185
$886$		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
$889$		−	1.00000i		−	1.00000i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	1.00000			1.00000
$897$	0.133975	+	0.500000i	0.133975	+	0.500000i
$898$	1.73205			1.73205
$899$		0			0
$900$	1.00000	−	1.73205i	1.00000	−	1.73205i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$			3.73205i			3.73205i
$906$	−0.448288	−	0.776457i	−0.448288	−	0.776457i
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$909$	−1.03528			−1.03528
$910$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$911$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$911$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$912$	−0.633975	−	0.366025i	−0.633975	−	0.366025i
$913$		0			0
$914$	0.500000	+	0.866025i	0.500000	+	0.866025i
$915$		−	0.517638i		−	0.517638i
$916$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$917$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$918$		0			0
$919$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$919$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$920$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$921$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$922$	−0.517638			−0.517638
$923$	0.965926	−	0.258819i	0.965926	−	0.258819i
$924$		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
$931$		−	1.41421i		−	1.41421i
$932$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$933$		0			0
$934$	1.67303	+	0.965926i	1.67303	+	0.965926i
$935$		0			0
$936$	−0.707107	+	0.189469i	−0.707107	+	0.189469i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$941$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$942$	0.633975	−	0.366025i	0.633975	−	0.366025i
$943$		0			0
$944$			1.93185i			1.93185i
$945$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$		0			0
$950$	3.86370			3.86370
$951$		0			0
$952$		0			0
$953$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$953$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	1.50000	−	0.866025i	1.50000	−	0.866025i
$957$		0			0
$958$		0			0
$959$	0.866025	−	1.50000i	0.866025	−	1.50000i
$960$	0.866025	+	0.500000i	0.866025	+	0.500000i
$961$	−1.00000			−1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$966$	0.258819	+	0.448288i	0.258819	+	0.448288i
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	0.866025	−	0.500000i	0.866025	−	0.500000i
$969$		0			0
$970$		0			0
$971$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$971$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$972$	0.517638	−	0.896575i	0.517638	−	0.896575i
$973$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$974$	1.73205			1.73205
$975$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$976$	−0.517638			−0.517638
$977$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$977$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$			1.93185i			1.93185i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$989$		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	$0.833333\pi$
$992$		0			0
$993$		0			0
$994$	0.866025	−	0.500000i	0.866025	−	0.500000i
$995$		0			0
$996$			0.732051i			0.732051i
$997$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$997$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	728.1.bl.a.69.1	✓	8
4.3	odd	2		2912.1.cb.a.433.3		8
7.6	odd	2	inner	728.1.bl.a.69.2	yes	8
8.3	odd	2		2912.1.cb.a.433.2		8
8.5	even	2	inner	728.1.bl.a.69.2	yes	8
13.10	even	6	inner	728.1.bl.a.517.1	yes	8
28.27	even	2		2912.1.cb.a.433.2		8
52.23	odd	6		2912.1.cb.a.881.3		8
56.13	odd	2	CM	728.1.bl.a.69.1	✓	8
56.27	even	2		2912.1.cb.a.433.3		8
91.62	odd	6	inner	728.1.bl.a.517.2	yes	8
104.75	odd	6		2912.1.cb.a.881.2		8
104.101	even	6	inner	728.1.bl.a.517.2	yes	8
364.335	even	6		2912.1.cb.a.881.2		8
728.517	odd	6	inner	728.1.bl.a.517.1	yes	8
728.699	even	6		2912.1.cb.a.881.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.1.bl.a.69.1	✓	8	1.1	even	1	trivial
728.1.bl.a.69.1	✓	8	56.13	odd	2	CM
728.1.bl.a.69.2	yes	8	7.6	odd	2	inner
728.1.bl.a.69.2	yes	8	8.5	even	2	inner
728.1.bl.a.517.1	yes	8	13.10	even	6	inner
728.1.bl.a.517.1	yes	8	728.517	odd	6	inner
728.1.bl.a.517.2	yes	8	91.62	odd	6	inner
728.1.bl.a.517.2	yes	8	104.101	even	6	inner
2912.1.cb.a.433.2		8	8.3	odd	2
2912.1.cb.a.433.2		8	28.27	even	2
2912.1.cb.a.433.3		8	4.3	odd	2
2912.1.cb.a.433.3		8	56.27	even	2
2912.1.cb.a.881.2		8	104.75	odd	6
2912.1.cb.a.881.2		8	364.335	even	6
2912.1.cb.a.881.3		8	52.23	odd	6
2912.1.cb.a.881.3		8	728.699	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	728.bl (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{2} +(-0.258819 + 0.448288i) q^{3} +(0.500000 + 0.866025i) q^{4} +1.93185i q^{5} +(0.448288 - 0.258819i) q^{6} +(0.866025 - 0.500000i) q^{7} -1.00000i q^{8} +(0.366025 + 0.633975i) q^{9} +(0.965926 - 1.67303i) q^{10} -0.517638 q^{12} +(-0.965926 - 0.258819i) q^{13} -1.00000 q^{14} +(-0.866025 - 0.500000i) q^{15} +(-0.500000 + 0.866025i) q^{16} -0.732051i q^{18} +(1.22474 - 0.707107i) q^{19} +(-1.67303 + 0.965926i) q^{20} +0.517638i q^{21} +(-0.500000 + 0.866025i) q^{23} +(0.448288 + 0.258819i) q^{24} -2.73205 q^{25} +(0.707107 + 0.707107i) q^{26} -0.896575 q^{27} +(0.866025 + 0.500000i) q^{28} +(0.500000 + 0.866025i) q^{30} +(0.866025 - 0.500000i) q^{32} +(0.965926 + 1.67303i) q^{35} +(-0.366025 + 0.633975i) q^{36} -1.41421 q^{38} +(0.366025 - 0.366025i) q^{39} +1.93185 q^{40} +(0.258819 - 0.448288i) q^{42} +(-1.22474 + 0.707107i) q^{45} +(0.866025 - 0.500000i) q^{46} +(-0.258819 - 0.448288i) q^{48} +(0.500000 - 0.866025i) q^{49} +(2.36603 + 1.36603i) q^{50} +(-0.258819 - 0.965926i) q^{52} +(0.776457 + 0.448288i) q^{54} +(-0.500000 - 0.866025i) q^{56} +0.732051i q^{57} +(1.67303 - 0.965926i) q^{59} -1.00000i q^{60} +(0.258819 + 0.448288i) q^{61} +(0.633975 + 0.366025i) q^{63} -1.00000 q^{64} +(0.500000 - 1.86603i) q^{65} +(-0.258819 - 0.448288i) q^{69} -1.93185i q^{70} +(-0.866025 + 0.500000i) q^{71} +(0.633975 - 0.366025i) q^{72} +(0.707107 - 1.22474i) q^{75} +(1.22474 + 0.707107i) q^{76} +(-0.500000 + 0.133975i) q^{78} +(-1.67303 - 0.965926i) q^{80} +(-0.133975 + 0.232051i) q^{81} -1.41421i q^{83} +(-0.448288 + 0.258819i) q^{84} +1.41421 q^{90} +(-0.965926 + 0.258819i) q^{91} -1.00000 q^{92} +(1.36603 + 2.36603i) q^{95} +0.517638i q^{96} +(-0.866025 + 0.500000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} - 4 q^{9} - 8 q^{14} - 4 q^{16} - 4 q^{23} - 8 q^{25} + 4 q^{30} + 4 q^{36} - 4 q^{39} + 4 q^{49} + 12 q^{50} - 4 q^{56} + 12 q^{63} - 8 q^{64} + 4 q^{65} + 12 q^{72} - 4 q^{78} - 8 q^{81}+ \cdots + 4 q^{95}+O(q^{100}) \) 8 * q + 4 * q^4 - 4 * q^9 - 8 * q^14 - 4 * q^16 - 4 * q^23 - 8 * q^25 + 4 * q^30 + 4 * q^36 - 4 * q^39 + 4 * q^49 + 12 * q^50 - 4 * q^56 + 12 * q^63 - 8 * q^64 + 4 * q^65 + 12 * q^72 - 4 * q^78 - 8 * q^81 - 8 * q^92 + 4 * q^95

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