LMFDB - Embedded newform 855.1.z.d.664.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,1,Mod(94,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("855.94");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	855.z (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:	$0.426700585801$
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			664.3
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	855.664
Dual form			855.1.z.d.94.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.258819 - 0.448288i) q^{2} +(0.258819 + 0.965926i) q^{3} +(0.366025 + 0.633975i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 + 0.133975i) q^{6} +0.896575 q^{8} +(-0.866025 + 0.500000i) q^{9} +O(q^{10})$	\(q+(0.258819 - 0.448288i) q^{2} +(0.258819 + 0.965926i) q^{3} +(0.366025 + 0.633975i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 + 0.133975i) q^{6} +0.896575 q^{8} +(-0.866025 + 0.500000i) q^{9} +0.517638 q^{10} +(0.866025 - 1.50000i) q^{11} +(-0.517638 + 0.517638i) q^{12} +(-0.965926 - 1.67303i) q^{13} +(-0.707107 + 0.707107i) q^{15} +(-0.133975 + 0.232051i) q^{16} +0.517638i q^{18} -1.00000 q^{19} +(-0.366025 + 0.633975i) q^{20} +(-0.448288 - 0.776457i) q^{22} +(0.232051 + 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{26} +(-0.707107 - 0.707107i) q^{27} +(0.133975 + 0.500000i) q^{30} +(0.517638 + 0.896575i) q^{32} +(1.67303 + 0.448288i) q^{33} +(-0.633975 - 0.366025i) q^{36} -1.93185 q^{37} +(-0.258819 + 0.448288i) q^{38} +(1.36603 - 1.36603i) q^{39} +(0.448288 + 0.776457i) q^{40} +1.26795 q^{44} +(-0.866025 - 0.500000i) q^{45} +(-0.258819 - 0.0693504i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(0.258819 + 0.448288i) q^{50} +(0.707107 - 1.22474i) q^{52} +0.517638 q^{53} +(-0.500000 + 0.133975i) q^{54} +1.73205 q^{55} +(-0.258819 - 0.965926i) q^{57} +(-0.707107 - 0.189469i) q^{60} +0.267949 q^{64} +(0.965926 - 1.67303i) q^{65} +(0.633975 - 0.633975i) q^{66} +(0.707107 + 1.22474i) q^{67} +(-0.776457 + 0.448288i) q^{72} +(-0.500000 + 0.866025i) q^{74} +(-0.965926 - 0.258819i) q^{75} +(-0.366025 - 0.633975i) q^{76} +(-0.258819 - 0.965926i) q^{78} -0.267949 q^{80} +(0.500000 - 0.866025i) q^{81} +(0.776457 - 1.34486i) q^{88} +(-0.448288 + 0.258819i) q^{90} +(-0.500000 - 0.866025i) q^{95} +(-0.732051 + 0.732051i) q^{96} +(-0.707107 + 1.22474i) q^{97} -0.517638 q^{98} +1.73205i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{4} + 4 q^{5} + 4 q^{6}+O(q^{10}) $ 8 * q - 4 * q^4 + 4 * q^5 + 4 * q^6	$ 8 q - 4 q^{4} + 4 q^{5} + 4 q^{6} - 8 q^{16} - 8 q^{19} + 4 q^{20} - 12 q^{24} - 4 q^{25} - 8 q^{26} + 8 q^{30} - 12 q^{36} + 4 q^{39} + 24 q^{44} - 4 q^{49} - 4 q^{54} + 16 q^{64} + 12 q^{66} - 4 q^{74} + 4 q^{76} - 16 q^{80} + 4 q^{81} - 4 q^{95} + 8 q^{96}+O(q^{100}) $ 8 * q - 4 * q^4 + 4 * q^5 + 4 * q^6 - 8 * q^16 - 8 * q^19 + 4 * q^20 - 12 * q^24 - 4 * q^25 - 8 * q^26 + 8 * q^30 - 12 * q^36 + 4 * q^39 + 24 * q^44 - 4 * q^49 - 4 * q^54 + 16 * q^64 + 12 * q^66 - 4 * q^74 + 4 * q^76 - 16 * q^80 + 4 * q^81 - 4 * q^95 + 8 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$.

$n$	$172$	$191$	$496$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$

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 $

$2$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$3$	0.258819	+	0.965926i	0.258819	+	0.965926i
$3$	0.258819	+	0.965926i	0.258819	+	0.965926i
$4$	0.366025	+	0.633975i	0.366025	+	0.633975i
$5$	0.500000	+	0.866025i	0.500000	+	0.866025i
$5$	0.500000	+	0.866025i	0.500000	+	0.866025i
$6$	0.500000	+	0.133975i	0.500000	+	0.133975i
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	0.896575			0.896575
$9$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$10$	0.517638			0.517638
$11$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$11$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$12$	−0.517638	+	0.517638i	−0.517638	+	0.517638i
$13$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$13$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$14$		0			0
$15$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$16$	−0.133975	+	0.232051i	−0.133975	+	0.232051i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$			0.517638i			0.517638i
$19$	−1.00000			−1.00000
$19$	−1.00000			−1.00000
$20$	−0.366025	+	0.633975i	−0.366025	+	0.633975i
$21$		0			0
$22$	−0.448288	−	0.776457i	−0.448288	−	0.776457i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	0.232051	+	0.866025i	0.232051	+	0.866025i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$	−1.00000			−1.00000
$27$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$	0.133975	+	0.500000i	0.133975	+	0.500000i
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	0.517638	+	0.896575i	0.517638	+	0.896575i
$33$	1.67303	+	0.448288i	1.67303	+	0.448288i
$34$		0			0
$35$		0			0
$36$	−0.633975	−	0.366025i	−0.633975	−	0.366025i
$37$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$37$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$38$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$39$	1.36603	−	1.36603i	1.36603	−	1.36603i
$40$	0.448288	+	0.776457i	0.448288	+	0.776457i
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$	1.26795			1.26795
$45$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	−0.258819	−	0.0693504i	−0.258819	−	0.0693504i
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$	0.258819	+	0.448288i	0.258819	+	0.448288i
$51$		0			0
$52$	0.707107	−	1.22474i	0.707107	−	1.22474i
$53$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$53$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$54$	−0.500000	+	0.133975i	−0.500000	+	0.133975i
$55$	1.73205			1.73205
$56$		0			0
$57$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$	−0.707107	−	0.189469i	−0.707107	−	0.189469i
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$		0			0
$64$	0.267949			0.267949
$65$	0.965926	−	1.67303i	0.965926	−	1.67303i
$66$	0.633975	−	0.633975i	0.633975	−	0.633975i
$67$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$67$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−0.776457	+	0.448288i	−0.776457	+	0.448288i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$75$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$76$	−0.366025	−	0.633975i	−0.366025	−	0.633975i
$77$		0			0
$78$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$	−0.267949			−0.267949
$81$	0.500000	−	0.866025i	0.500000	−	0.866025i
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.776457	−	1.34486i	0.776457	−	1.34486i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$96$	−0.732051	+	0.732051i	−0.732051	+	0.732051i
$97$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$97$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$98$	−0.517638			−0.517638
$99$			1.73205i			1.73205i
$100$	−0.732051			−0.732051
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$103$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$104$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$105$		0			0
$106$	0.133975	−	0.232051i	0.133975	−	0.232051i
$107$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$107$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$108$	0.189469	−	0.707107i	0.189469	−	0.707107i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	0.448288	−	0.776457i	0.448288	−	0.776457i
$111$	−0.500000	−	1.86603i	−0.500000	−	1.86603i
$112$		0			0
$113$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$113$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$114$	−0.500000	−	0.133975i	−0.500000	−	0.133975i
$115$		0			0
$116$		0			0
$117$	1.67303	+	0.965926i	1.67303	+	0.965926i
$118$		0			0
$119$		0			0
$120$	−0.633975	+	0.633975i	−0.633975	+	0.633975i
$121$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−1.00000
$126$		0			0
$127$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$127$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$128$	−0.448288	+	0.776457i	−0.448288	+	0.776457i
$129$		0			0
$130$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$131$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$131$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$132$	0.328169	+	1.22474i	0.328169	+	1.22474i
$133$		0			0
$134$	0.732051			0.732051
$135$	0.258819	−	0.965926i	0.258819	−	0.965926i
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$139$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−3.34607			−3.34607
$144$		−	0.267949i		−	0.267949i
$145$		0			0
$146$		0			0
$147$	0.707107	−	0.707107i	0.707107	−	0.707107i
$148$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$149$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$149$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$150$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	−0.896575			−0.896575
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.36603	+	0.366025i	1.36603	+	0.366025i
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$	0.133975	+	0.500000i	0.133975	+	0.500000i
$160$	−0.517638	+	0.896575i	−0.517638	+	0.896575i
$161$		0			0
$162$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	0.448288	+	1.67303i	0.448288	+	1.67303i
$166$		0			0
$167$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$167$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$168$		0			0
$169$	−1.36603	+	2.36603i	−1.36603	+	2.36603i
$170$		0			0
$171$	0.866025	−	0.500000i	0.866025	−	0.500000i
$172$		0			0
$173$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$173$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$	0.232051	+	0.401924i	0.232051	+	0.401924i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		−	0.732051i		−	0.732051i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−0.517638			−0.517638
$191$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$191$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$192$	0.0693504	+	0.258819i	0.0693504	+	0.258819i
$193$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$193$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$194$	0.366025	+	0.633975i	0.366025	+	0.633975i
$195$	1.86603	+	0.500000i	1.86603	+	0.500000i
$196$	0.366025	−	0.633975i	0.366025	−	0.633975i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	0.776457	+	0.448288i	0.776457	+	0.448288i
$199$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$199$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$200$	−0.448288	+	0.776457i	−0.448288	+	0.776457i
$201$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	0.267949			0.267949
$207$		0			0
$208$	0.517638			0.517638
$209$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$	0.189469	+	0.328169i	0.189469	+	0.328169i
$213$		0			0
$214$	0.500000	−	0.866025i	0.500000	−	0.866025i
$215$		0			0
$216$	−0.633975	−	0.633975i	−0.633975	−	0.633975i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	0.633975	+	1.09808i	0.633975	+	1.09808i
$221$		0			0
$222$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$223$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$223$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$		−	1.00000i		−	1.00000i
$226$	0.732051			0.732051
$227$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$227$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$228$	0.517638	−	0.517638i	0.517638	−	0.517638i
$229$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$229$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	0.866025	−	0.500000i	0.866025	−	0.500000i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$239$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$240$	−0.0693504	−	0.258819i	−0.0693504	−	0.258819i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−1.03528			−1.03528
$243$	0.965926	+	0.258819i	0.965926	+	0.258819i
$244$		0			0
$245$	0.500000	−	0.866025i	0.500000	−	0.866025i
$246$		0			0
$247$	0.965926	+	1.67303i	0.965926	+	1.67303i
$248$		0			0
$249$		0			0
$250$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$251$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$251$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$252$		0			0
$253$		0			0
$254$	0.366025	−	0.633975i	0.366025	−	0.633975i
$255$		0			0
$256$	0.366025	+	0.633975i	0.366025	+	0.633975i
$257$	−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
$259$		0			0
$260$	1.41421			1.41421
$261$		0			0
$262$	0.517638			0.517638
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$	1.50000	+	0.401924i	1.50000	+	0.401924i
$265$	0.258819	+	0.448288i	0.258819	+	0.448288i
$266$		0			0
$267$		0			0
$268$	−0.517638	+	0.896575i	−0.517638	+	0.896575i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	−0.366025	−	0.366025i	−0.366025	−	0.366025i
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.866025	+	1.50000i	0.866025	+	1.50000i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$	0.707107	−	0.707107i	0.707107	−	0.707107i
$286$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$287$		0			0
$288$	−0.896575	−	0.517638i	−0.896575	−	0.517638i
$289$	1.00000			1.00000
$290$		0			0
$291$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$292$		0			0
$293$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$293$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$294$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$295$		0			0
$296$	−1.73205			−1.73205
$297$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$298$	−0.896575			−0.896575
$299$		0			0
$300$	−0.189469	−	0.707107i	−0.189469	−	0.707107i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.133975	−	0.232051i	0.133975	−	0.232051i
$305$		0			0
$306$		0			0
$307$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$307$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$308$		0			0
$309$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$	1.22474	−	1.22474i	1.22474	−	1.22474i
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$317$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$318$	0.258819	+	0.0693504i	0.258819	+	0.0693504i
$319$		0			0
$320$	0.133975	+	0.232051i	0.133975	+	0.232051i
$321$	0.500000	+	1.86603i	0.500000	+	1.86603i
$322$		0			0
$323$		0			0
$324$	0.732051			0.732051
$325$	1.93185			1.93185
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	0.866025	+	0.232051i	0.866025	+	0.232051i
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	1.67303	−	0.965926i	1.67303	−	0.965926i
$334$	−0.267949			−0.267949
$335$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$336$		0			0
$337$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$337$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$338$	0.707107	+	1.22474i	0.707107	+	1.22474i
$339$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$340$		0			0
$341$		0			0
$342$		−	0.517638i		−	0.517638i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.500000	+	0.866025i	0.500000	+	0.866025i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$349$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$350$		0			0
$351$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$352$	1.79315			1.79315
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$359$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$360$	−0.776457	−	0.448288i	−0.776457	−	0.448288i
$361$	1.00000			1.00000
$362$		0			0
$363$	1.41421	−	1.41421i	1.41421	−	1.41421i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$	−1.00000			−1.00000
$371$		0			0
$372$		0			0
$373$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$373$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$374$		0			0
$375$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	0.366025	−	0.633975i	0.366025	−	0.633975i
$381$	0.366025	+	1.36603i	0.366025	+	1.36603i
$382$	0.258819	+	0.448288i	0.258819	+	0.448288i
$383$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$383$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$384$	−0.866025	−	0.232051i	−0.866025	−	0.232051i
$385$		0			0
$386$	−0.267949			−0.267949
$387$		0			0
$388$	−1.03528			−1.03528
$389$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$389$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$390$	0.707107	−	0.707107i	0.707107	−	0.707107i
$391$		0			0
$392$	−0.448288	−	0.776457i	−0.448288	−	0.776457i
$393$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$394$		0			0
$395$		0			0
$396$	−1.09808	+	0.633975i	−1.09808	+	0.633975i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	0.258819	−	0.448288i	0.258819	−	0.448288i
$399$		0			0
$400$	−0.133975	−	0.232051i	−0.133975	−	0.232051i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$	0.189469	+	0.707107i	0.189469	+	0.707107i
$403$		0			0
$404$		0			0
$405$	1.00000			1.00000
$406$		0			0
$407$	−1.67303	+	2.89778i	−1.67303	+	2.89778i
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		0			0
$412$	−0.189469	+	0.328169i	−0.189469	+	0.328169i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	1.00000	−	1.73205i	1.00000	−	1.73205i
$417$		0			0
$418$	0.448288	+	0.776457i	0.448288	+	0.776457i
$419$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$	0.464102			0.464102
$425$		0			0
$426$		0			0
$427$		0			0
$428$	0.707107	+	1.22474i	0.707107	+	1.22474i
$429$	−0.866025	−	3.23205i	−0.866025	−	3.23205i
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	0.258819	−	0.0693504i	0.258819	−	0.0693504i
$433$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$433$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$	1.55291			1.55291
$441$	0.866025	+	0.500000i	0.866025	+	0.500000i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	1.00000	−	1.00000i	1.00000	−	1.00000i
$445$		0			0
$446$	0.133975	+	0.232051i	0.133975	+	0.232051i
$447$	1.22474	−	1.22474i	1.22474	−	1.22474i
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$451$		0			0
$452$	−0.517638	+	0.896575i	−0.517638	+	0.896575i
$453$		0			0
$454$	−0.366025	−	0.633975i	−0.366025	−	0.633975i
$455$		0			0
$456$	−0.232051	−	0.866025i	−0.232051	−	0.866025i
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	−0.896575			−0.896575
$459$		0			0
$460$		0			0
$461$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$461$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$			1.41421i			1.41421i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.500000	−	0.866025i	0.500000	−	0.866025i
$476$		0			0
$477$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$478$	0.517638			0.517638
$479$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$479$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$480$	−1.00000	−	0.267949i	−1.00000	−	0.267949i
$481$	1.86603	+	3.23205i	1.86603	+	3.23205i
$482$		0			0
$483$		0			0
$484$	0.732051	−	1.26795i	0.732051	−	1.26795i
$485$	−1.41421			−1.41421
$486$	0.366025	−	0.366025i	0.366025	−	0.366025i
$487$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$487$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$488$		0			0
$489$		0			0
$490$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$491$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$491$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$	1.00000			1.00000
$495$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$499$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$500$	−0.366025	−	0.633975i	−0.366025	−	0.633975i
$501$	0.366025	−	0.366025i	0.366025	−	0.366025i
$502$	0.258819	−	0.448288i	0.258819	−	0.448288i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−2.63896	−	0.707107i	−2.63896	−	0.707107i
$508$	0.517638	+	0.896575i	0.517638	+	0.896575i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	−0.517638			−0.517638
$513$	0.707107	+	0.707107i	0.707107	+	0.707107i
$514$	−0.732051			−0.732051
$515$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$520$	0.866025	−	1.50000i	0.866025	−	1.50000i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$523$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$524$	−0.366025	+	0.633975i	−0.366025	+	0.633975i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−0.328169	+	0.328169i	−0.328169	+	0.328169i
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$	0.267949			0.267949
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.965926	+	1.67303i	0.965926	+	1.67303i
$536$	0.633975	+	1.09808i	0.633975	+	1.09808i
$537$		0			0
$538$		0			0
$539$	−1.73205			−1.73205
$540$	0.707107	−	0.189469i	0.707107	−	0.189469i
$541$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$541$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$547$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$548$		0			0
$549$		0			0
$550$	0.896575			0.896575
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	1.36603	−	1.36603i	1.36603	−	1.36603i
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$563$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$564$		0			0
$565$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$571$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$571$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$572$	−1.22474	−	2.12132i	−1.22474	−	2.12132i
$573$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$574$		0			0
$575$		0			0
$576$	−0.232051	+	0.133975i	−0.232051	+	0.133975i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.258819	−	0.448288i	0.258819	−	0.448288i
$579$	0.366025	−	0.366025i	0.366025	−	0.366025i
$580$		0			0
$581$		0			0
$582$	−0.517638	+	0.517638i	−0.517638	+	0.517638i
$583$	0.448288	−	0.776457i	0.448288	−	0.776457i
$584$		0			0
$585$			1.93185i			1.93185i
$586$	−0.732051			−0.732051
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$	0.707107	+	0.189469i	0.707107	+	0.189469i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.258819	−	0.448288i	0.258819	−	0.448288i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−0.232051	+	0.866025i	−0.232051	+	0.866025i
$595$		0			0
$596$	0.633975	−	1.09808i	0.633975	−	1.09808i
$597$	0.258819	+	0.965926i	0.258819	+	0.965926i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	−0.866025	−	0.232051i	−0.866025	−	0.232051i
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$604$		0			0
$605$	1.00000	−	1.73205i	1.00000	−	1.73205i
$606$		0			0
$607$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$607$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$608$	−0.517638	−	0.896575i	−0.517638	−	0.896575i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−0.133975	+	0.232051i	−0.133975	+	0.232051i
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$	0.0693504	+	0.258819i	0.0693504	+	0.258819i
$619$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$619$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	0.133975	+	0.500000i	0.133975	+	0.500000i
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$631$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$632$		0			0
$633$		0			0
$634$	−0.366025	−	0.633975i	−0.366025	−	0.633975i
$635$	0.707107	+	1.22474i	0.707107	+	1.22474i
$636$	−0.267949	+	0.267949i	−0.267949	+	0.267949i
$637$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$638$		0			0
$639$		0			0
$640$	−0.896575			−0.896575
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	0.965926	+	0.258819i	0.965926	+	0.258819i
$643$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	0.448288	−	0.776457i	0.448288	−	0.776457i
$649$		0			0
$650$	0.500000	−	0.866025i	0.500000	−	0.866025i
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$	−0.896575	+	0.896575i	−0.896575	+	0.896575i
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		−	1.00000i		−	1.00000i
$667$		0			0
$668$	0.189469	−	0.328169i	0.189469	−	0.328169i
$669$	−0.500000	−	0.133975i	−0.500000	−	0.133975i
$670$	0.366025	+	0.633975i	0.366025	+	0.633975i
$671$		0			0
$672$		0			0
$673$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$673$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$674$	0.732051			0.732051
$675$	0.965926	−	0.258819i	0.965926	−	0.258819i
$676$	−2.00000			−2.00000
$677$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$677$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$678$	0.189469	+	0.707107i	0.189469	+	0.707107i
$679$		0			0
$680$		0			0
$681$	1.36603	+	0.366025i	1.36603	+	0.366025i
$682$		0			0
$683$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$683$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$684$	0.633975	+	0.366025i	0.633975	+	0.366025i
$685$		0			0
$686$		0			0
$687$	1.22474	−	1.22474i	1.22474	−	1.22474i
$688$		0			0
$689$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$690$		0			0
$691$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$691$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$692$	−1.41421			−1.41421
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	0.258819	+	0.448288i	0.258819	+	0.448288i
$699$		0			0
$700$		0			0
$701$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$701$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$702$	0.707107	+	0.707107i	0.707107	+	0.707107i
$703$	1.93185			1.93185
$704$	0.232051	−	0.401924i	0.232051	−	0.401924i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$709$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−1.67303	−	2.89778i	−1.67303	−	2.89778i
$716$		0			0
$717$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$718$	0.448288	−	0.776457i	0.448288	−	0.776457i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$	0.232051	−	0.133975i	0.232051	−	0.133975i
$721$		0			0
$722$	0.258819	−	0.448288i	0.258819	−	0.448288i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−0.267949	−	1.00000i	−0.267949	−	1.00000i
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$	0.965926	+	0.258819i	0.965926	+	0.258819i
$736$		0			0
$737$	2.44949			2.44949
$738$		0			0
$739$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$739$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$740$	0.707107	−	1.22474i	0.707107	−	1.22474i
$741$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$742$		0			0
$743$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$743$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$744$		0			0
$745$	0.866025	−	1.50000i	0.866025	−	1.50000i
$746$	−0.267949			−0.267949
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−0.500000	−	0.133975i	−0.500000	−	0.133975i
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$	0.258819	+	0.965926i	0.258819	+	0.965926i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$	−0.448288	−	0.776457i	−0.448288	−	0.776457i
$761$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$761$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$762$	0.707107	+	0.189469i	0.707107	+	0.189469i
$763$		0			0
$764$	−0.732051			−0.732051
$765$		0			0
$766$	−1.00000			−1.00000
$767$		0			0
$768$	−0.517638	+	0.517638i	−0.517638	+	0.517638i
$769$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$769$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$770$		0			0
$771$	1.00000	−	1.00000i	1.00000	−	1.00000i
$772$	0.189469	−	0.328169i	0.189469	−	0.328169i
$773$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$773$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$774$		0			0
$775$		0			0
$776$	−0.633975	+	1.09808i	−0.633975	+	1.09808i
$777$		0			0
$778$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$779$		0			0
$780$	0.366025	+	1.36603i	0.366025	+	1.36603i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.267949			0.267949
$785$		0			0
$786$	0.133975	+	0.500000i	0.133975	+	0.500000i
$787$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$787$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$			1.55291i			1.55291i
$793$		0			0
$794$		0			0
$795$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$796$	0.366025	+	0.633975i	0.366025	+	0.633975i
$797$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$797$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$798$		0			0
$799$		0			0
$800$	−1.03528			−1.03528
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−1.00000	−	0.267949i	−1.00000	−	0.267949i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$809$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$810$	0.258819	−	0.448288i	0.258819	−	0.448288i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$	0.866025	+	1.50000i	0.866025	+	1.50000i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$821$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$	0.232051	+	0.401924i	0.232051	+	0.401924i
$825$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$826$		0			0
$827$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$827$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$833$		0			0
$834$		0			0
$835$	0.258819	−	0.448288i	0.258819	−	0.448288i
$836$	−1.26795			−1.26795
$837$		0			0
$838$	−1.03528			−1.03528
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.73205			−2.73205
$846$		0			0
$847$		0			0
$848$	−0.0693504	+	0.120118i	−0.0693504	+	0.120118i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$	0.866025	+	0.500000i	0.866025	+	0.500000i
$856$	1.73205			1.73205
$857$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$857$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$858$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$859$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$859$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$863$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$864$	0.267949	−	1.00000i	0.267949	−	1.00000i
$865$	−1.93185			−1.93185
$866$	−0.366025	+	0.633975i	−0.366025	+	0.633975i
$867$	0.258819	+	0.965926i	0.258819	+	0.965926i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	1.36603	−	2.36603i	1.36603	−	2.36603i
$872$		0			0
$873$		−	1.41421i		−	1.41421i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$877$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$878$		0			0
$879$	1.00000	−	1.00000i	1.00000	−	1.00000i
$880$	−0.232051	+	0.401924i	−0.232051	+	0.401924i
$881$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$881$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$882$	0.448288	−	0.258819i	0.448288	−	0.258819i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$887$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$888$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$889$		0			0
$890$		0			0
$891$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$892$	−0.378937			−0.378937
$893$		0			0
$894$	−0.232051	−	0.866025i	−0.232051	−	0.866025i
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.633975	−	0.366025i	0.633975	−	0.366025i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.633975	+	1.09808i	0.633975	+	1.09808i
$905$		0			0
$906$		0			0
$907$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$907$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$908$	1.03528			1.03528
$909$		0			0
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$	0.258819	+	0.0693504i	0.258819	+	0.0693504i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.633975	−	1.09808i	0.633975	−	1.09808i
$917$		0			0
$918$		0			0
$919$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$919$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$	−0.133975	−	0.500000i	−0.133975	−	0.500000i
$922$	0.517638	+	0.896575i	0.517638	+	0.896575i
$923$		0			0
$924$		0			0
$925$	0.965926	−	1.67303i	0.965926	−	1.67303i
$926$		0			0
$927$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$928$		0			0
$929$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$929$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$930$		0			0
$931$	0.500000	+	0.866025i	0.500000	+	0.866025i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	1.50000	+	0.866025i	1.50000	+	0.866025i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		0			0
$950$	−0.258819	−	0.448288i	−0.258819	−	0.448288i
$951$	1.36603	+	0.366025i	1.36603	+	0.366025i
$952$		0			0
$953$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$953$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$954$			0.267949i			0.267949i
$955$	−1.00000			−1.00000
$956$	−0.366025	+	0.633975i	−0.366025	+	0.633975i
$957$		0			0
$958$	0.448288	+	0.776457i	0.448288	+	0.776457i
$959$		0			0
$960$	−0.189469	+	0.189469i	−0.189469	+	0.189469i
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$	1.93185			1.93185
$963$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$964$		0			0
$965$	0.258819	−	0.448288i	0.258819	−	0.448288i
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$	−0.896575	−	1.55291i	−0.896575	−	1.55291i
$969$		0			0
$970$	−0.366025	+	0.633975i	−0.366025	+	0.633975i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	0.189469	+	0.707107i	0.189469	+	0.707107i
$973$		0			0
$974$	−0.366025	+	0.633975i	−0.366025	+	0.633975i
$975$	0.500000	+	1.86603i	0.500000	+	1.86603i
$976$		0			0
$977$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$977$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$978$		0			0
$979$		0			0
$980$	0.732051			0.732051
$981$		0			0
$982$	−0.517638			−0.517638
$983$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$983$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$989$		0			0
$990$			0.896575i			0.896575i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.500000	+	0.866025i	0.500000	+	0.866025i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	0.896575			0.896575
$999$	1.36603	+	1.36603i	1.36603	+	1.36603i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.1.z.d.664.3	yes	8
3.2	odd	2		2565.1.z.d.1234.2		8
5.4	even	2	inner	855.1.z.d.664.2	yes	8
9.4	even	3	inner	855.1.z.d.94.3	yes	8
9.5	odd	6		2565.1.z.d.2089.2		8
15.14	odd	2		2565.1.z.d.1234.3		8
19.18	odd	2	inner	855.1.z.d.664.2	yes	8
45.4	even	6	inner	855.1.z.d.94.2	✓	8
45.14	odd	6		2565.1.z.d.2089.3		8
57.56	even	2		2565.1.z.d.1234.3		8
95.94	odd	2	CM	855.1.z.d.664.3	yes	8
171.94	odd	6	inner	855.1.z.d.94.2	✓	8
171.113	even	6		2565.1.z.d.2089.3		8
285.284	even	2		2565.1.z.d.1234.2		8
855.94	odd	6	inner	855.1.z.d.94.3	yes	8
855.284	even	6		2565.1.z.d.2089.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
855.1.z.d.94.2	✓	8	45.4	even	6	inner
855.1.z.d.94.2	✓	8	171.94	odd	6	inner
855.1.z.d.94.3	yes	8	9.4	even	3	inner
855.1.z.d.94.3	yes	8	855.94	odd	6	inner
855.1.z.d.664.2	yes	8	5.4	even	2	inner
855.1.z.d.664.2	yes	8	19.18	odd	2	inner
855.1.z.d.664.3	yes	8	1.1	even	1	trivial
855.1.z.d.664.3	yes	8	95.94	odd	2	CM
2565.1.z.d.1234.2		8	3.2	odd	2
2565.1.z.d.1234.2		8	285.284	even	2
2565.1.z.d.1234.3		8	15.14	odd	2
2565.1.z.d.1234.3		8	57.56	even	2
2565.1.z.d.2089.2		8	9.5	odd	6
2565.1.z.d.2089.2		8	855.284	even	6
2565.1.z.d.2089.3		8	45.14	odd	6
2565.1.z.d.2089.3		8	171.113	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:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	855.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.258819 - 0.448288i) q^{2} +(0.258819 + 0.965926i) q^{3} +(0.366025 + 0.633975i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 + 0.133975i) q^{6} +0.896575 q^{8} +(-0.866025 + 0.500000i) q^{9} +O(q^{10})\)	\(q+(0.258819 - 0.448288i) q^{2} +(0.258819 + 0.965926i) q^{3} +(0.366025 + 0.633975i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 + 0.133975i) q^{6} +0.896575 q^{8} +(-0.866025 + 0.500000i) q^{9} +0.517638 q^{10} +(0.866025 - 1.50000i) q^{11} +(-0.517638 + 0.517638i) q^{12} +(-0.965926 - 1.67303i) q^{13} +(-0.707107 + 0.707107i) q^{15} +(-0.133975 + 0.232051i) q^{16} +0.517638i q^{18} -1.00000 q^{19} +(-0.366025 + 0.633975i) q^{20} +(-0.448288 - 0.776457i) q^{22} +(0.232051 + 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} -1.00000 q^{26} +(-0.707107 - 0.707107i) q^{27} +(0.133975 + 0.500000i) q^{30} +(0.517638 + 0.896575i) q^{32} +(1.67303 + 0.448288i) q^{33} +(-0.633975 - 0.366025i) q^{36} -1.93185 q^{37} +(-0.258819 + 0.448288i) q^{38} +(1.36603 - 1.36603i) q^{39} +(0.448288 + 0.776457i) q^{40} +1.26795 q^{44} +(-0.866025 - 0.500000i) q^{45} +(-0.258819 - 0.0693504i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(0.258819 + 0.448288i) q^{50} +(0.707107 - 1.22474i) q^{52} +0.517638 q^{53} +(-0.500000 + 0.133975i) q^{54} +1.73205 q^{55} +(-0.258819 - 0.965926i) q^{57} +(-0.707107 - 0.189469i) q^{60} +0.267949 q^{64} +(0.965926 - 1.67303i) q^{65} +(0.633975 - 0.633975i) q^{66} +(0.707107 + 1.22474i) q^{67} +(-0.776457 + 0.448288i) q^{72} +(-0.500000 + 0.866025i) q^{74} +(-0.965926 - 0.258819i) q^{75} +(-0.366025 - 0.633975i) q^{76} +(-0.258819 - 0.965926i) q^{78} -0.267949 q^{80} +(0.500000 - 0.866025i) q^{81} +(0.776457 - 1.34486i) q^{88} +(-0.448288 + 0.258819i) q^{90} +(-0.500000 - 0.866025i) q^{95} +(-0.732051 + 0.732051i) q^{96} +(-0.707107 + 1.22474i) q^{97} -0.517638 q^{98} +1.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4} + 4 q^{5} + 4 q^{6}+O(q^{10}) \) 8 * q - 4 * q^4 + 4 * q^5 + 4 * q^6	\( 8 q - 4 q^{4} + 4 q^{5} + 4 q^{6} - 8 q^{16} - 8 q^{19} + 4 q^{20} - 12 q^{24} - 4 q^{25} - 8 q^{26} + 8 q^{30} - 12 q^{36} + 4 q^{39} + 24 q^{44} - 4 q^{49} - 4 q^{54} + 16 q^{64} + 12 q^{66} - 4 q^{74} + 4 q^{76} - 16 q^{80} + 4 q^{81} - 4 q^{95} + 8 q^{96}+O(q^{100}) \) 8 * q - 4 * q^4 + 4 * q^5 + 4 * q^6 - 8 * q^16 - 8 * q^19 + 4 * q^20 - 12 * q^24 - 4 * q^25 - 8 * q^26 + 8 * q^30 - 12 * q^36 + 4 * q^39 + 24 * q^44 - 4 * q^49 - 4 * q^54 + 16 * q^64 + 12 * q^66 - 4 * q^74 + 4 * q^76 - 16 * q^80 + 4 * q^81 - 4 * q^95 + 8 * q^96

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