LMFDB - Embedded newform 684.1.q.a.163.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,1,Mod(163,684)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 4]))

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

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

chi := DirichletCharacter("684.163");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	684.q (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:	$0.341360468641$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.17328.1

Embedding invariants

Embedding label			163.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	684.163
Dual form			684.1.q.a.235.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.965926 + 0.258819i) q^{2} +(0.866025 - 0.500000i) q^{4} +(0.707107 - 1.22474i) q^{5} +1.00000i q^{7} +(-0.707107 + 0.707107i) q^{8} +(-0.366025 + 1.36603i) q^{10} +1.41421i q^{11} +(0.500000 + 0.866025i) q^{13} +(-0.258819 - 0.965926i) q^{14} +(0.500000 - 0.866025i) q^{16} -1.00000i q^{19} -1.41421i q^{20} +(-0.366025 - 1.36603i) q^{22} +(1.22474 - 0.707107i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-0.707107 - 0.707107i) q^{26} +(0.500000 + 0.866025i) q^{28} -1.00000i q^{31} +(-0.258819 + 0.965926i) q^{32} +(1.22474 + 0.707107i) q^{35} -1.00000 q^{37} +(0.258819 + 0.965926i) q^{38} +(0.366025 + 1.36603i) q^{40} +(-0.866025 - 0.500000i) q^{43} +(0.707107 + 1.22474i) q^{44} +(-1.00000 + 1.00000i) q^{46} +(0.707107 + 0.707107i) q^{50} +(0.866025 + 0.500000i) q^{52} +(0.707107 + 1.22474i) q^{53} +(1.73205 + 1.00000i) q^{55} +(-0.707107 - 0.707107i) q^{56} +(-1.22474 - 0.707107i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(0.258819 + 0.965926i) q^{62} -1.00000i q^{64} +1.41421 q^{65} +(-0.866025 + 0.500000i) q^{67} +(-1.36603 - 0.366025i) q^{70} +(-0.500000 + 0.866025i) q^{73} +(0.965926 - 0.258819i) q^{74} +(-0.500000 - 0.866025i) q^{76} -1.41421 q^{77} +(0.866025 + 0.500000i) q^{79} +(-0.707107 - 1.22474i) q^{80} +(0.965926 + 0.258819i) q^{86} +(-1.00000 - 1.00000i) q^{88} +(-0.707107 - 1.22474i) q^{89} +(-0.866025 + 0.500000i) q^{91} +(0.707107 - 1.22474i) q^{92} +(-1.22474 - 0.707107i) q^{95} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{10} + 4 q^{13} + 4 q^{16} + 4 q^{22} - 4 q^{25} + 4 q^{28} - 8 q^{37} - 4 q^{40} - 8 q^{46} - 4 q^{61} - 4 q^{70} - 4 q^{73} - 4 q^{76} - 8 q^{88}+O(q^{100}) $ 8 * q + 4 * q^10 + 4 * q^13 + 4 * q^16 + 4 * q^22 - 4 * q^25 + 4 * q^28 - 8 * q^37 - 4 * q^40 - 8 * q^46 - 4 * q^61 - 4 * q^70 - 4 * q^73 - 4 * q^76 - 8 * q^88

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$	$1$

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 $

$2$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$2$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$3$		0			0
$3$		0			0
$4$	0.866025	−	0.500000i	0.866025	−	0.500000i
$5$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$5$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$6$		0			0
$7$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$7$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$8$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$9$		0			0
$10$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$11$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$11$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$12$		0			0
$13$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$13$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$14$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$15$		0			0
$16$	0.500000	−	0.866025i	0.500000	−	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$		−	1.00000i		−	1.00000i
$19$		−	1.00000i		−	1.00000i
$20$		−	1.41421i		−	1.41421i
$21$		0			0
$22$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$23$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$23$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$27$		0			0
$28$	0.500000	+	0.866025i	0.500000	+	0.866025i
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$		0			0
$31$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$31$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$32$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$33$		0			0
$34$		0			0
$35$	1.22474	+	0.707107i	1.22474	+	0.707107i
$36$		0			0
$37$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$37$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$38$	0.258819	+	0.965926i	0.258819	+	0.965926i
$39$		0			0
$40$	0.366025	+	1.36603i	0.366025	+	1.36603i
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$43$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$44$	0.707107	+	1.22474i	0.707107	+	1.22474i
$45$		0			0
$46$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$		0			0
$50$	0.707107	+	0.707107i	0.707107	+	0.707107i
$51$		0			0
$52$	0.866025	+	0.500000i	0.866025	+	0.500000i
$53$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$53$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$54$		0			0
$55$	1.73205	+	1.00000i	1.73205	+	1.00000i
$56$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$57$		0			0
$58$		0			0
$59$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$59$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$60$		0			0
$61$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$61$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$62$	0.258819	+	0.965926i	0.258819	+	0.965926i
$63$		0			0
$64$		−	1.00000i		−	1.00000i
$65$	1.41421			1.41421
$66$		0			0
$67$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$67$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$68$		0			0
$69$		0			0
$70$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$71$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$71$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$72$		0			0
$73$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$73$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$74$	0.965926	−	0.258819i	0.965926	−	0.258819i
$75$		0			0
$76$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$77$	−1.41421			−1.41421
$78$		0			0
$79$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$79$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$80$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$81$		0			0
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$	0.965926	+	0.258819i	0.965926	+	0.258819i
$87$		0			0
$88$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$89$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$89$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$90$		0			0
$91$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$92$	0.707107	−	1.22474i	0.707107	−	1.22474i
$93$		0			0
$94$		0			0
$95$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$96$		0			0
$97$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$97$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$98$		0			0
$99$		0			0
$100$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$103$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$104$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$105$		0			0
$106$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$109$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$110$	−1.93185	−	0.517638i	−1.93185	−	0.517638i
$111$		0			0
$112$	0.866025	+	0.500000i	0.866025	+	0.500000i
$113$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$113$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		−	2.00000i		−	2.00000i
$116$		0			0
$117$		0			0
$118$	1.36603	+	0.366025i	1.36603	+	0.366025i
$119$		0			0
$120$		0			0
$121$	−1.00000			−1.00000
$122$	0.707107	+	0.707107i	0.707107	+	0.707107i
$123$		0			0
$124$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$125$		0			0
$126$		0			0
$127$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$127$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$128$	0.258819	+	0.965926i	0.258819	+	0.965926i
$129$		0			0
$130$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	1.00000			1.00000
$134$	0.707107	−	0.707107i	0.707107	−	0.707107i
$135$		0			0
$136$		0			0
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$139$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$140$	1.41421			1.41421
$141$		0			0
$142$		0			0
$143$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$144$		0			0
$145$		0			0
$146$	0.258819	−	0.965926i	0.258819	−	0.965926i
$147$		0			0
$148$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$149$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$149$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	0.707107	+	0.707107i	0.707107	+	0.707107i
$153$		0			0
$154$	1.36603	−	0.366025i	1.36603	−	0.366025i
$155$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$156$		0			0
$157$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$157$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$158$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$159$		0			0
$160$	1.00000	+	1.00000i	1.00000	+	1.00000i
$161$	0.707107	+	1.22474i	0.707107	+	1.22474i
$162$		0			0
$163$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$163$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$167$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$	−1.00000			−1.00000
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	0.866025	−	0.500000i	0.866025	−	0.500000i
$176$	1.22474	+	0.707107i	1.22474	+	0.707107i
$177$		0			0
$178$	1.00000	+	1.00000i	1.00000	+	1.00000i
$179$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$179$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$180$		0			0
$181$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$181$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$182$	0.707107	−	0.707107i	0.707107	−	0.707107i
$183$		0			0
$184$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$185$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	1.36603	+	0.366025i	1.36603	+	0.366025i
$191$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$191$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$192$		0			0
$193$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$193$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$197$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$199$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$200$	0.965926	+	0.258819i	0.965926	+	0.258819i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$207$		0			0
$208$	1.00000			1.00000
$209$	1.41421			1.41421
$210$		0			0
$211$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$211$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$212$	1.22474	+	0.707107i	1.22474	+	0.707107i
$213$		0			0
$214$		0			0
$215$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$216$		0			0
$217$	1.00000			1.00000
$218$		0			0
$219$		0			0
$220$	2.00000			2.00000
$221$		0			0
$222$		0			0
$223$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$223$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$224$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$225$		0			0
$226$	1.36603	−	0.366025i	1.36603	−	0.366025i
$227$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$227$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$228$		0			0
$229$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$229$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$230$	0.517638	+	1.93185i	0.517638	+	1.93185i
$231$		0			0
$232$		0			0
$233$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$233$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$234$		0			0
$235$		0			0
$236$	−1.41421			−1.41421
$237$		0			0
$238$		0			0
$239$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$239$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$240$		0			0
$241$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$241$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$242$	0.965926	−	0.258819i	0.965926	−	0.258819i
$243$		0			0
$244$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$245$		0			0
$246$		0			0
$247$	0.866025	−	0.500000i	0.866025	−	0.500000i
$248$	0.707107	+	0.707107i	0.707107	+	0.707107i
$249$		0			0
$250$		0			0
$251$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$		0			0
$253$	1.00000	+	1.73205i	1.00000	+	1.73205i
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$257$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$258$		0			0
$259$		−	1.00000i		−	1.00000i
$260$	1.22474	−	0.707107i	1.22474	−	0.707107i
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$	2.00000			2.00000
$266$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$267$		0			0
$268$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$269$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$269$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.22474	−	0.707107i	1.22474	−	0.707107i
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	0.707107	−	0.707107i	0.707107	−	0.707107i
$279$		0			0
$280$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$281$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$281$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$	1.00000	−	1.00000i	1.00000	−	1.00000i
$287$		0			0
$288$		0			0
$289$	0.500000	+	0.866025i	0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$			1.00000i			1.00000i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$296$	0.707107	−	0.707107i	0.707107	−	0.707107i
$297$		0			0
$298$	0.366025	−	1.36603i	0.366025	−	1.36603i
$299$	1.22474	+	0.707107i	1.22474	+	0.707107i
$300$		0			0
$301$	0.500000	−	0.866025i	0.500000	−	0.866025i
$302$		0			0
$303$		0			0
$304$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$305$	−1.41421			−1.41421
$306$		0			0
$307$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$307$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$308$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$309$		0			0
$310$	1.36603	+	0.366025i	1.36603	+	0.366025i
$311$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$311$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$312$		0			0
$313$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$313$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$314$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$315$		0			0
$316$	1.00000			1.00000
$317$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$317$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$318$		0			0
$319$		0			0
$320$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$321$		0			0
$322$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$323$		0			0
$324$		0			0
$325$	0.500000	−	0.866025i	0.500000	−	0.866025i
$326$	0.258819	+	0.965926i	0.258819	+	0.965926i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$331$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$332$		0			0
$333$		0			0
$334$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$335$			1.41421i			1.41421i
$336$		0			0
$337$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.41421			1.41421
$342$		0			0
$343$			1.00000i			1.00000i
$344$	0.965926	−	0.258819i	0.965926	−	0.258819i
$345$		0			0
$346$		0			0
$347$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$347$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$348$		0			0
$349$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$349$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$350$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$351$		0			0
$352$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$353$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$353$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$		0			0
$356$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$357$		0			0
$358$	0.366025	+	1.36603i	0.366025	+	1.36603i
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−1.00000			−1.00000
$362$		0			0
$363$		0			0
$364$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$365$	0.707107	+	1.22474i	0.707107	+	1.22474i
$366$		0			0
$367$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$367$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$368$		−	1.41421i		−	1.41421i
$369$		0			0
$370$	0.366025	−	1.36603i	0.366025	−	1.36603i
$371$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$379$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$380$	−1.41421			−1.41421
$381$		0			0
$382$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$383$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$383$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$384$		0			0
$385$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$386$	0.258819	−	0.965926i	0.258819	−	0.965926i
$387$		0			0
$388$		0			0
$389$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$389$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	1.36603	−	0.366025i	1.36603	−	0.366025i
$395$	1.22474	−	0.707107i	1.22474	−	0.707107i
$396$		0			0
$397$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$397$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$398$	0.707107	−	0.707107i	0.707107	−	0.707107i
$399$		0			0
$400$	−1.00000			−1.00000
$401$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$401$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$402$		0			0
$403$	0.866025	−	0.500000i	0.866025	−	0.500000i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	1.41421i		−	1.41421i
$408$		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			0
$412$	0.500000	+	0.866025i	0.500000	+	0.866025i
$413$	0.707107	−	1.22474i	0.707107	−	1.22474i
$414$		0			0
$415$		0			0
$416$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$417$		0			0
$418$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$419$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$419$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$421$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$422$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$423$		0			0
$424$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$425$		0			0
$426$		0			0
$427$	0.866025	−	0.500000i	0.866025	−	0.500000i
$428$		0			0
$429$		0			0
$430$	1.00000	−	1.00000i	1.00000	−	1.00000i
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$433$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$434$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$435$		0			0
$436$		0			0
$437$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$438$		0			0
$439$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$439$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$440$	−1.93185	+	0.517638i	−1.93185	+	0.517638i
$441$		0			0
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$	−2.00000			−2.00000
$446$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$447$		0			0
$448$	1.00000			1.00000
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$453$		0			0
$454$	0.366025	+	1.36603i	0.366025	+	1.36603i
$455$			1.41421i			1.41421i
$456$		0			0
$457$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$457$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$458$	0.965926	−	0.258819i	0.965926	−	0.258819i
$459$		0			0
$460$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$461$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$461$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$462$		0			0
$463$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$463$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$470$		0			0
$471$		0			0
$472$	1.36603	−	0.366025i	1.36603	−	0.366025i
$473$	0.707107	−	1.22474i	0.707107	−	1.22474i
$474$		0			0
$475$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$476$		0			0
$477$		0			0
$478$	0.366025	+	1.36603i	0.366025	+	1.36603i
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$482$	0.707107	+	0.707107i	0.707107	+	0.707107i
$483$		0			0
$484$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	0.965926	+	0.258819i	0.965926	+	0.258819i
$489$		0			0
$490$		0			0
$491$		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
$494$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$495$		0			0
$496$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$497$		0			0
$498$		0			0
$499$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$499$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$507$		0			0
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$512$	0.707107	+	0.707107i	0.707107	+	0.707107i
$513$		0			0
$514$	1.00000	+	1.00000i	1.00000	+	1.00000i
$515$	1.22474	+	0.707107i	1.22474	+	0.707107i
$516$		0			0
$517$		0			0
$518$	0.258819	+	0.965926i	0.258819	+	0.965926i
$519$		0			0
$520$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$521$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$521$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$522$		0			0
$523$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$523$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.500000	−	0.866025i	0.500000	−	0.866025i
$530$	−1.93185	+	0.517638i	−1.93185	+	0.517638i
$531$		0			0
$532$	0.866025	−	0.500000i	0.866025	−	0.500000i
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.258819	−	0.965926i	0.258819	−	0.965926i
$537$		0			0
$538$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$539$		0			0
$540$		0			0
$541$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$541$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$547$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$548$		0			0
$549$		0			0
$550$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$551$		0			0
$552$		0			0
$553$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$554$		0			0
$555$		0			0
$556$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		−	1.00000i		−	1.00000i
$560$	1.22474	−	0.707107i	1.22474	−	0.707107i
$561$		0			0
$562$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$571$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$572$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$573$		0			0
$574$		0			0
$575$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$576$		0			0
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$584$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$585$		0			0
$586$		0			0
$587$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$587$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$588$		0			0
$589$	−1.00000			−1.00000
$590$	1.41421	−	1.41421i	1.41421	−	1.41421i
$591$		0			0
$592$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$593$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$593$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$594$		0			0
$595$		0			0
$596$			1.41421i			1.41421i
$597$		0			0
$598$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$599$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$599$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$600$		0			0
$601$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$601$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$602$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$603$		0			0
$604$		0			0
$605$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$606$		0			0
$607$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$607$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$608$	0.965926	+	0.258819i	0.965926	+	0.258819i
$609$		0			0
$610$	1.36603	−	0.366025i	1.36603	−	0.366025i
$611$		0			0
$612$		0			0
$613$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$613$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$614$	1.93185	+	0.517638i	1.93185	+	0.517638i
$615$		0			0
$616$	1.00000	−	1.00000i	1.00000	−	1.00000i
$617$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$617$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$618$		0			0
$619$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$619$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$620$	−1.41421			−1.41421
$621$		0			0
$622$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$623$	1.22474	−	0.707107i	1.22474	−	0.707107i
$624$		0			0
$625$	0.500000	−	0.866025i	0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		−	1.00000i		−	1.00000i
$629$		0			0
$630$		0			0
$631$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$631$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$632$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$633$		0			0
$634$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	1.36603	+	0.366025i	1.36603	+	0.366025i
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$		0			0
$643$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$643$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$644$	1.22474	+	0.707107i	1.22474	+	0.707107i
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	1.00000	−	1.73205i	1.00000	−	1.73205i
$650$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$651$		0			0
$652$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$659$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$660$		0			0
$661$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$661$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$662$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$663$		0			0
$664$		0			0
$665$	0.707107	−	1.22474i	0.707107	−	1.22474i
$666$		0			0
$667$		0			0
$668$	0.707107	−	1.22474i	0.707107	−	1.22474i
$669$		0			0
$670$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$671$	1.22474	−	0.707107i	1.22474	−	0.707107i
$672$		0			0
$673$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$673$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$674$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$675$		0			0
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$683$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$683$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$		0			0
$686$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$687$		0			0
$688$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$689$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$	1.36603	+	0.366025i	1.36603	+	0.366025i
$695$			1.41421i			1.41421i
$696$		0			0
$697$		0			0
$698$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$699$		0			0
$700$	0.500000	−	0.866025i	0.500000	−	0.866025i
$701$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$701$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$702$		0			0
$703$			1.00000i			1.00000i
$704$	1.41421			1.41421
$705$		0			0
$706$	1.36603	−	0.366025i	1.36603	−	0.366025i
$707$		0			0
$708$		0			0
$709$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$709$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	1.36603	+	0.366025i	1.36603	+	0.366025i
$713$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$714$		0			0
$715$			2.00000i			2.00000i
$716$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$717$		0			0
$718$		0			0
$719$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$719$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$720$		0			0
$721$	−1.00000			−1.00000
$722$	0.965926	−	0.258819i	0.965926	−	0.258819i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$727$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$728$	0.258819	−	0.965926i	0.258819	−	0.965926i
$729$		0			0
$730$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$735$		0			0
$736$	0.366025	+	1.36603i	0.366025	+	1.36603i
$737$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$738$		0			0
$739$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$739$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$740$			1.41421i			1.41421i
$741$		0			0
$742$	1.00000	−	1.00000i	1.00000	−	1.00000i
$743$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$743$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$744$		0			0
$745$	1.00000	+	1.73205i	1.00000	+	1.73205i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$751$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$757$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$758$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$759$		0			0
$760$	1.36603	−	0.366025i	1.36603	−	0.366025i
$761$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$761$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$762$		0			0
$763$		0			0
$764$	0.707107	+	1.22474i	0.707107	+	1.22474i
$765$		0			0
$766$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$767$		−	1.41421i		−	1.41421i
$768$		0			0
$769$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$769$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$770$	0.517638	−	1.93185i	0.517638	−	1.93185i
$771$		0			0
$772$			1.00000i			1.00000i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$776$		0			0
$777$		0			0
$778$	1.00000	+	1.00000i	1.00000	+	1.00000i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$786$		0			0
$787$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$787$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$788$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$789$		0			0
$790$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$791$		−	1.41421i		−	1.41421i
$792$		0			0
$793$	0.500000	−	0.866025i	0.500000	−	0.866025i
$794$	0.258819	−	0.965926i	0.258819	−	0.965926i
$795$		0			0
$796$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	0.965926	−	0.258819i	0.965926	−	0.258819i
$801$		0			0
$802$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$803$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$804$		0			0
$805$	2.00000			2.00000
$806$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$807$		0			0
$808$		0			0
$809$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$809$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$810$		0			0
$811$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$811$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$812$		0			0
$813$		0			0
$814$	0.366025	+	1.36603i	0.366025	+	1.36603i
$815$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$816$		0			0
$817$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$825$		0			0
$826$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$827$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$827$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$828$		0			0
$829$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$829$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$	0.866025	−	0.500000i	0.866025	−	0.500000i
$833$		0			0
$834$		0			0
$835$		−	2.00000i		−	2.00000i
$836$	1.22474	−	0.707107i	1.22474	−	0.707107i
$837$		0			0
$838$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$839$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$839$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$840$		0			0
$841$	0.500000	−	0.866025i	0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	1.00000			1.00000
$845$		0			0
$846$		0			0
$847$		−	1.00000i		−	1.00000i
$848$	1.41421			1.41421
$849$		0			0
$850$		0			0
$851$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$852$		0			0
$853$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$853$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$854$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$859$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$860$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$	0.707107	+	0.707107i	0.707107	+	0.707107i
$867$		0			0
$868$	0.866025	−	0.500000i	0.866025	−	0.500000i
$869$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$870$		0			0
$871$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$872$		0			0
$873$		0			0
$874$	1.00000	+	1.00000i	1.00000	+	1.00000i
$875$		0			0
$876$		0			0
$877$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$877$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$878$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$879$		0			0
$880$	1.73205	−	1.00000i	1.73205	−	1.00000i
$881$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$881$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$882$		0			0
$883$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$883$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$887$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$888$		0			0
$889$		0			0
$890$	1.93185	−	0.517638i	1.93185	−	0.517638i
$891$		0			0
$892$	1.00000			1.00000
$893$		0			0
$894$		0			0
$895$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$896$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.00000	−	1.00000i	1.00000	−	1.00000i
$905$		0			0
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$909$		0			0
$910$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	0.965926	−	0.258819i	0.965926	−	0.258819i
$915$		0			0
$916$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$917$		0			0
$918$		0			0
$919$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$919$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$920$	1.41421	+	1.41421i	1.41421	+	1.41421i
$921$		0			0
$922$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$923$		0			0
$924$		0			0
$925$	0.500000	+	0.866025i	0.500000	+	0.866025i
$926$	0.258819	+	0.965926i	0.258819	+	0.965926i
$927$		0			0
$928$		0			0
$929$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$929$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$937$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$938$	0.707107	+	0.707107i	0.707107	+	0.707107i
$939$		0			0
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$945$		0			0
$946$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$	−1.00000			−1.00000
$950$	0.707107	−	0.707107i	0.707107	−	0.707107i
$951$		0			0
$952$		0			0
$953$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$953$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$954$		0			0
$955$	1.73205	+	1.00000i	1.73205	+	1.00000i
$956$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		0			0
$962$	0.707107	+	0.707107i	0.707107	+	0.707107i
$963$		0			0
$964$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$965$	0.707107	+	1.22474i	0.707107	+	1.22474i
$966$		0			0
$967$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$967$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$968$	0.707107	−	0.707107i	0.707107	−	0.707107i
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$974$		0			0
$975$		0			0
$976$	−1.00000			−1.00000
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$	1.73205	−	1.00000i	1.73205	−	1.00000i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$983$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$984$		0			0
$985$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$986$		0			0
$987$		0			0
$988$	0.500000	−	0.866025i	0.500000	−	0.866025i
$989$	−1.41421			−1.41421
$990$		0			0
$991$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$991$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$992$	0.965926	+	0.258819i	0.965926	+	0.258819i
$993$		0			0
$994$		0			0
$995$			1.41421i			1.41421i
$996$		0			0
$997$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$997$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$998$	0.965926	+	0.258819i	0.965926	+	0.258819i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.1.q.a.163.1	✓	8
3.2	odd	2	inner	684.1.q.a.163.4	yes	8
4.3	odd	2	inner	684.1.q.a.163.3	yes	8
12.11	even	2	inner	684.1.q.a.163.2	yes	8
19.7	even	3	inner	684.1.q.a.235.3	yes	8
57.26	odd	6	inner	684.1.q.a.235.2	yes	8
76.7	odd	6	inner	684.1.q.a.235.1	yes	8
228.83	even	6	inner	684.1.q.a.235.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.1.q.a.163.1	✓	8	1.1	even	1	trivial
684.1.q.a.163.2	yes	8	12.11	even	2	inner
684.1.q.a.163.3	yes	8	4.3	odd	2	inner
684.1.q.a.163.4	yes	8	3.2	odd	2	inner
684.1.q.a.235.1	yes	8	76.7	odd	6	inner
684.1.q.a.235.2	yes	8	57.26	odd	6	inner
684.1.q.a.235.3	yes	8	19.7	even	3	inner
684.1.q.a.235.4	yes	8	228.83	even	6	inner

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:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	684.q (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.965926 + 0.258819i) q^{2} +(0.866025 - 0.500000i) q^{4} +(0.707107 - 1.22474i) q^{5} +1.00000i q^{7} +(-0.707107 + 0.707107i) q^{8} +(-0.366025 + 1.36603i) q^{10} +1.41421i q^{11} +(0.500000 + 0.866025i) q^{13} +(-0.258819 - 0.965926i) q^{14} +(0.500000 - 0.866025i) q^{16} -1.00000i q^{19} -1.41421i q^{20} +(-0.366025 - 1.36603i) q^{22} +(1.22474 - 0.707107i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(-0.707107 - 0.707107i) q^{26} +(0.500000 + 0.866025i) q^{28} -1.00000i q^{31} +(-0.258819 + 0.965926i) q^{32} +(1.22474 + 0.707107i) q^{35} -1.00000 q^{37} +(0.258819 + 0.965926i) q^{38} +(0.366025 + 1.36603i) q^{40} +(-0.866025 - 0.500000i) q^{43} +(0.707107 + 1.22474i) q^{44} +(-1.00000 + 1.00000i) q^{46} +(0.707107 + 0.707107i) q^{50} +(0.866025 + 0.500000i) q^{52} +(0.707107 + 1.22474i) q^{53} +(1.73205 + 1.00000i) q^{55} +(-0.707107 - 0.707107i) q^{56} +(-1.22474 - 0.707107i) q^{59} +(-0.500000 - 0.866025i) q^{61} +(0.258819 + 0.965926i) q^{62} -1.00000i q^{64} +1.41421 q^{65} +(-0.866025 + 0.500000i) q^{67} +(-1.36603 - 0.366025i) q^{70} +(-0.500000 + 0.866025i) q^{73} +(0.965926 - 0.258819i) q^{74} +(-0.500000 - 0.866025i) q^{76} -1.41421 q^{77} +(0.866025 + 0.500000i) q^{79} +(-0.707107 - 1.22474i) q^{80} +(0.965926 + 0.258819i) q^{86} +(-1.00000 - 1.00000i) q^{88} +(-0.707107 - 1.22474i) q^{89} +(-0.866025 + 0.500000i) q^{91} +(0.707107 - 1.22474i) q^{92} +(-1.22474 - 0.707107i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{10} + 4 q^{13} + 4 q^{16} + 4 q^{22} - 4 q^{25} + 4 q^{28} - 8 q^{37} - 4 q^{40} - 8 q^{46} - 4 q^{61} - 4 q^{70} - 4 q^{73} - 4 q^{76} - 8 q^{88}+O(q^{100}) \) 8 * q + 4 * q^10 + 4 * q^13 + 4 * q^16 + 4 * q^22 - 4 * q^25 + 4 * q^28 - 8 * q^37 - 4 * q^40 - 8 * q^46 - 4 * q^61 - 4 * q^70 - 4 * q^73 - 4 * q^76 - 8 * q^88

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