LMFDB - Embedded newform 1932.1.w.e.275.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1932,1,Mod(275,1932)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1932, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1932.275");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1932.w (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:	$0.964193604407$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.2194782912.1

Embedding invariants

Embedding label			275.2
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1932.275
Dual form			1932.1.w.e.1103.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.866025 - 1.50000i) q^{5} -1.00000 q^{6} +(-0.866025 - 0.500000i) q^{7} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.866025 - 1.50000i) q^{5} -1.00000 q^{6} +(-0.866025 - 0.500000i) q^{7} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(0.866025 + 1.50000i) q^{10} +(0.500000 - 0.866025i) q^{12} +1.00000 q^{13} +(0.866025 - 0.500000i) q^{14} +1.73205 q^{15} +(-0.500000 + 0.866025i) q^{16} +(-0.866025 - 1.50000i) q^{17} +(-0.500000 - 0.866025i) q^{18} -1.73205 q^{20} -1.00000i q^{21} +(0.500000 - 0.866025i) q^{23} +(0.500000 + 0.866025i) q^{24} +(-1.00000 - 1.73205i) q^{25} +(-0.500000 + 0.866025i) q^{26} -1.00000 q^{27} +1.00000i q^{28} +(-0.866025 + 1.50000i) q^{30} +(-0.500000 - 0.866025i) q^{32} +1.73205 q^{34} +(-1.50000 + 0.866025i) q^{35} +1.00000 q^{36} +(0.500000 + 0.866025i) q^{39} +(0.866025 - 1.50000i) q^{40} +(0.866025 + 0.500000i) q^{42} +(0.866025 + 1.50000i) q^{45} +(0.500000 + 0.866025i) q^{46} +(0.500000 - 0.866025i) q^{47} -1.00000 q^{48} +(0.500000 + 0.866025i) q^{49} +2.00000 q^{50} +(0.866025 - 1.50000i) q^{51} +(-0.500000 - 0.866025i) q^{52} +(0.866025 + 1.50000i) q^{53} +(0.500000 - 0.866025i) q^{54} +(-0.866025 - 0.500000i) q^{56} +(-1.00000 - 1.73205i) q^{59} +(-0.866025 - 1.50000i) q^{60} +(0.866025 - 0.500000i) q^{63} +1.00000 q^{64} +(0.866025 - 1.50000i) q^{65} +(0.866025 + 1.50000i) q^{67} +(-0.866025 + 1.50000i) q^{68} +1.00000 q^{69} -1.73205i q^{70} +1.00000 q^{71} +(-0.500000 + 0.866025i) q^{72} +(0.500000 + 0.866025i) q^{73} +(1.00000 - 1.73205i) q^{75} -1.00000 q^{78} +(0.866025 + 1.50000i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.866025 + 0.500000i) q^{84} -3.00000 q^{85} -1.73205 q^{90} +(-0.866025 - 0.500000i) q^{91} -1.00000 q^{92} +(0.500000 + 0.866025i) q^{94} +(0.500000 - 0.866025i) q^{96} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 4 q^{6} + 4 q^{8} - 2 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 - 4 * q^6 + 4 * q^8 - 2 * q^9	$ 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 4 q^{6} + 4 q^{8} - 2 q^{9} + 2 q^{12} + 4 q^{13} - 2 q^{16} - 2 q^{18} + 2 q^{23} + 2 q^{24} - 4 q^{25} - 2 q^{26} - 4 q^{27} - 2 q^{32} - 6 q^{35} + 4 q^{36} + 2 q^{39} + 2 q^{46} + 2 q^{47} - 4 q^{48} + 2 q^{49} + 8 q^{50} - 2 q^{52} + 2 q^{54} - 4 q^{59} + 4 q^{64} + 4 q^{69} + 4 q^{71} - 2 q^{72} + 2 q^{73} + 4 q^{75} - 4 q^{78} - 2 q^{81} - 12 q^{85} - 4 q^{92} + 2 q^{94} + 2 q^{96} - 4 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 - 4 * q^6 + 4 * q^8 - 2 * q^9 + 2 * q^12 + 4 * q^13 - 2 * q^16 - 2 * q^18 + 2 * q^23 + 2 * q^24 - 4 * q^25 - 2 * q^26 - 4 * q^27 - 2 * q^32 - 6 * q^35 + 4 * q^36 + 2 * q^39 + 2 * q^46 + 2 * q^47 - 4 * q^48 + 2 * q^49 + 8 * q^50 - 2 * q^52 + 2 * q^54 - 4 * q^59 + 4 * q^64 + 4 * q^69 + 4 * q^71 - 2 * q^72 + 2 * q^73 + 4 * q^75 - 4 * q^78 - 2 * q^81 - 12 * q^85 - 4 * q^92 + 2 * q^94 + 2 * q^96 - 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1932\mathbb{Z}\right)^\times$.

$n$	$829$	$925$	$967$	$1289$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$2$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$3$	0.500000	+	0.866025i	0.500000	+	0.866025i
$3$	0.500000	+	0.866025i	0.500000	+	0.866025i
$4$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$5$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$6$	−1.00000			−1.00000
$7$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$7$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$8$	1.00000			1.00000
$9$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$10$	0.866025	+	1.50000i	0.866025	+	1.50000i
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	0.500000	−	0.866025i	0.500000	−	0.866025i
$13$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$13$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$14$	0.866025	−	0.500000i	0.866025	−	0.500000i
$15$	1.73205			1.73205
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$17$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$18$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$	−1.73205			−1.73205
$21$		−	1.00000i		−	1.00000i
$22$		0			0
$23$	0.500000	−	0.866025i	0.500000	−	0.866025i
$23$	0.500000	−	0.866025i	0.500000	−	0.866025i
$24$	0.500000	+	0.866025i	0.500000	+	0.866025i
$25$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$26$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$27$	−1.00000			−1.00000
$28$			1.00000i			1.00000i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$33$		0			0
$34$	1.73205			1.73205
$35$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$36$	1.00000			1.00000
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$	0.500000	+	0.866025i	0.500000	+	0.866025i
$40$	0.866025	−	1.50000i	0.866025	−	1.50000i
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	0.866025	+	0.500000i	0.866025	+	0.500000i
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	0.866025	+	1.50000i	0.866025	+	1.50000i
$46$	0.500000	+	0.866025i	0.500000	+	0.866025i
$47$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$47$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$48$	−1.00000			−1.00000
$49$	0.500000	+	0.866025i	0.500000	+	0.866025i
$50$	2.00000			2.00000
$51$	0.866025	−	1.50000i	0.866025	−	1.50000i
$52$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$53$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$53$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$54$	0.500000	−	0.866025i	0.500000	−	0.866025i
$55$		0			0
$56$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$57$		0			0
$58$		0			0
$59$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$59$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$60$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$	0.866025	−	0.500000i	0.866025	−	0.500000i
$64$	1.00000			1.00000
$65$	0.866025	−	1.50000i	0.866025	−	1.50000i
$66$		0			0
$67$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$67$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$68$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$69$	1.00000			1.00000
$70$		−	1.73205i		−	1.73205i
$71$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$71$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$72$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$73$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$73$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$	1.00000	−	1.73205i	1.00000	−	1.73205i
$76$		0			0
$77$		0			0
$78$	−1.00000			−1.00000
$79$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$79$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$80$	0.866025	+	1.50000i	0.866025	+	1.50000i
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$85$	−3.00000			−3.00000
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$	−1.73205			−1.73205
$91$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$92$	−1.00000			−1.00000
$93$		0			0
$94$	0.500000	+	0.866025i	0.500000	+	0.866025i
$95$		0			0
$96$	0.500000	−	0.866025i	0.500000	−	0.866025i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	−1.00000			−1.00000
$99$		0			0
$100$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$	0.866025	+	1.50000i	0.866025	+	1.50000i
$103$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$103$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$104$	1.00000			1.00000
$105$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$106$	−1.73205			−1.73205
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	0.500000	+	0.866025i	0.500000	+	0.866025i
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$	0.866025	−	0.500000i	0.866025	−	0.500000i
$113$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$113$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$114$		0			0
$115$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$116$		0			0
$117$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$118$	2.00000			2.00000
$119$			1.73205i			1.73205i
$120$	1.73205			1.73205
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.73205			−1.73205
$126$			1.00000i			1.00000i
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$129$		0			0
$130$	0.866025	+	1.50000i	0.866025	+	1.50000i
$131$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$131$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$	−1.73205			−1.73205
$135$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$136$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$137$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$137$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$138$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$	1.50000	+	0.866025i	1.50000	+	0.866025i
$141$	1.00000			1.00000
$142$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$143$		0			0
$144$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$145$		0			0
$146$	−1.00000			−1.00000
$147$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$148$		0			0
$149$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$149$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$150$	1.00000	+	1.73205i	1.00000	+	1.73205i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$	1.73205			1.73205
$154$		0			0
$155$		0			0
$156$	0.500000	−	0.866025i	0.500000	−	0.866025i
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$160$	−1.73205			−1.73205
$161$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$162$	1.00000			1.00000
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$167$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$168$		−	1.00000i		−	1.00000i
$169$		0			0
$170$	1.50000	−	2.59808i	1.50000	−	2.59808i
$171$		0			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$			2.00000i			2.00000i
$176$		0			0
$177$	1.00000	−	1.73205i	1.00000	−	1.73205i
$178$		0			0
$179$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$179$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$180$	0.866025	−	1.50000i	0.866025	−	1.50000i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$	0.866025	−	0.500000i	0.866025	−	0.500000i
$183$		0			0
$184$	0.500000	−	0.866025i	0.500000	−	0.866025i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−1.00000			−1.00000
$189$	0.866025	+	0.500000i	0.866025	+	0.500000i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	0.500000	+	0.866025i	0.500000	+	0.866025i
$193$	−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
$195$	1.73205			1.73205
$196$	0.500000	−	0.866025i	0.500000	−	0.866025i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$201$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$202$		0			0
$203$		0			0
$204$	−1.73205			−1.73205
$205$		0			0
$206$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$207$	0.500000	+	0.866025i	0.500000	+	0.866025i
$208$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$209$		0			0
$210$	1.50000	−	0.866025i	1.50000	−	0.866025i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$	0.866025	−	1.50000i	0.866025	−	1.50000i
$213$	0.500000	+	0.866025i	0.500000	+	0.866025i
$214$		0			0
$215$		0			0
$216$	−1.00000			−1.00000
$217$		0			0
$218$		0			0
$219$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$220$		0			0
$221$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$			1.00000i			1.00000i
$225$	2.00000			2.00000
$226$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$	1.73205			1.73205
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$235$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$236$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$237$		0			0
$238$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$239$	−2.00000			−2.00000			−1.00000			$\pi$
$239$	−2.00000			−2.00000			−1.00000			$\pi$
$240$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$243$	0.500000	−	0.866025i	0.500000	−	0.866025i
$244$		0			0
$245$	1.73205			1.73205
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.866025	−	1.50000i	0.866025	−	1.50000i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$253$		0			0
$254$		0			0
$255$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$	−1.73205			−1.73205
$261$		0			0
$262$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$	3.00000			3.00000
$266$		0			0
$267$		0			0
$268$	0.866025	−	1.50000i	0.866025	−	1.50000i
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$	1.73205			1.73205
$273$		−	1.00000i		−	1.00000i
$274$	1.73205			1.73205
$275$		0			0
$276$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$281$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$281$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$282$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$283$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$283$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$284$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.00000			1.00000
$289$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$290$		0			0
$291$		0			0
$292$	0.500000	−	0.866025i	0.500000	−	0.866025i
$293$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$293$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$294$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$295$	−3.46410			−3.46410
$296$		0			0
$297$		0			0
$298$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$299$	0.500000	−	0.866025i	0.500000	−	0.866025i
$300$	−2.00000			−2.00000
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	−1.73205			−1.73205
$310$		0			0
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$312$	0.500000	+	0.866025i	0.500000	+	0.866025i
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		−	1.73205i		−	1.73205i
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$319$		0			0
$320$	0.866025	−	1.50000i	0.866025	−	1.50000i
$321$		0			0
$322$		−	1.00000i		−	1.00000i
$323$		0			0
$324$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$325$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$335$	3.00000			3.00000
$336$	0.866025	+	0.500000i	0.866025	+	0.500000i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	0.866025	+	1.50000i	0.866025	+	1.50000i
$340$	1.50000	+	2.59808i	1.50000	+	2.59808i
$341$		0			0
$342$		0			0
$343$		−	1.00000i		−	1.00000i
$344$		0			0
$345$	0.866025	−	1.50000i	0.866025	−	1.50000i
$346$		0			0
$347$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$347$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$349$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$350$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$351$	−1.00000			−1.00000
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$	1.00000	+	1.73205i	1.00000	+	1.73205i
$355$	0.866025	−	1.50000i	0.866025	−	1.50000i
$356$		0			0
$357$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$358$	−1.00000			−1.00000
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$	0.866025	+	1.50000i	0.866025	+	1.50000i
$361$	0.500000	+	0.866025i	0.500000	+	0.866025i
$362$		0			0
$363$	−1.00000			−1.00000
$364$			1.00000i			1.00000i
$365$	1.73205			1.73205
$366$		0			0
$367$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$367$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$368$	0.500000	+	0.866025i	0.500000	+	0.866025i
$369$		0			0
$370$		0			0
$371$		−	1.73205i		−	1.73205i
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$376$	0.500000	−	0.866025i	0.500000	−	0.866025i
$377$		0			0
$378$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$379$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$379$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$	−1.00000			−1.00000
$385$		0			0
$386$	1.00000			1.00000
$387$		0			0
$388$		0			0
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$391$	−1.73205			−1.73205
$392$	0.500000	+	0.866025i	0.500000	+	0.866025i
$393$	−1.00000			−1.00000
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$397$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$398$		0			0
$399$		0			0
$400$	2.00000			2.00000
$401$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$401$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$402$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$403$		0			0
$404$		0			0
$405$	−1.73205			−1.73205
$406$		0			0
$407$		0			0
$408$	0.866025	−	1.50000i	0.866025	−	1.50000i
$409$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$409$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$410$		0			0
$411$	0.866025	−	1.50000i	0.866025	−	1.50000i
$412$	1.73205			1.73205
$413$			2.00000i			2.00000i
$414$	−1.00000			−1.00000
$415$		0			0
$416$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$			1.73205i			1.73205i
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$	0.500000	+	0.866025i	0.500000	+	0.866025i
$424$	0.866025	+	1.50000i	0.866025	+	1.50000i
$425$	−1.73205	+	3.00000i	−1.73205	+	3.00000i
$426$	−1.00000			−1.00000
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	0.500000	−	0.866025i	0.500000	−	0.866025i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−1.00000			−1.00000
$442$	1.73205			1.73205
$443$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$443$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−1.73205			−1.73205
$448$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$451$		0			0
$452$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$453$		0			0
$454$		0			0
$455$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$	0.866025	+	1.50000i	0.866025	+	1.50000i
$460$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$	1.00000			1.00000
$469$		−	1.73205i		−	1.73205i
$470$	1.73205			1.73205
$471$		0			0
$472$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$473$		0			0
$474$		0			0
$475$		0			0
$476$	1.50000	−	0.866025i	1.50000	−	0.866025i
$477$	−1.73205			−1.73205
$478$	1.00000	−	1.73205i	1.00000	−	1.73205i
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$481$		0			0
$482$		0			0
$483$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$484$	1.00000			1.00000
$485$		0			0
$486$	0.500000	+	0.866025i	0.500000	+	0.866025i
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$		0			0
$489$		0			0
$490$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$491$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$491$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	0.866025	+	1.50000i	0.866025	+	1.50000i
$501$	0.500000	+	0.866025i	0.500000	+	0.866025i
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	0.866025	−	0.500000i	0.866025	−	0.500000i
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$	3.00000			3.00000
$511$		−	1.00000i		−	1.00000i
$512$	1.00000			1.00000
$513$		0			0
$514$		0			0
$515$	1.50000	+	2.59808i	1.50000	+	2.59808i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.866025	−	1.50000i	0.866025	−	1.50000i
$521$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$521$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$522$		0			0
$523$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$523$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$524$	1.00000			1.00000
$525$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$531$	2.00000			2.00000
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.866025	+	1.50000i	0.866025	+	1.50000i
$537$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$538$		0			0
$539$		0			0
$540$	1.73205			1.73205
$541$	−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
$544$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$545$		0			0
$546$	0.866025	+	0.500000i	0.866025	+	0.500000i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$549$		0			0
$550$		0			0
$551$		0			0
$552$	1.00000			1.00000
$553$		0			0
$554$	−1.00000			−1.00000
$555$		0			0
$556$		0			0
$557$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$557$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		0			0
$560$		−	1.73205i		−	1.73205i
$561$		0			0
$562$	0.866025	−	1.50000i	0.866025	−	1.50000i
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$565$	1.50000	−	2.59808i	1.50000	−	2.59808i
$566$	1.73205			1.73205
$567$			1.00000i			1.00000i
$568$	1.00000			1.00000
$569$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$569$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$	−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	$-0.5\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.00000			−2.00000
$576$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$577$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$577$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$578$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$579$	0.500000	−	0.866025i	0.500000	−	0.866025i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.500000	+	0.866025i	0.500000	+	0.866025i
$585$	0.866025	+	1.50000i	0.866025	+	1.50000i
$586$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$587$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$587$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$588$	1.00000			1.00000
$589$		0			0
$590$	1.73205	−	3.00000i	1.73205	−	3.00000i
$591$		0			0
$592$		0			0
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$	2.59808	+	1.50000i	2.59808	+	1.50000i
$596$	1.73205			1.73205
$597$		0			0
$598$	0.500000	+	0.866025i	0.500000	+	0.866025i
$599$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$599$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$600$	1.00000	−	1.73205i	1.00000	−	1.73205i
$601$	−2.00000			−2.00000			−1.00000			$\pi$
$601$	−2.00000			−2.00000			−1.00000			$\pi$
$602$		0			0
$603$	−1.73205			−1.73205
$604$		0			0
$605$	0.866025	+	1.50000i	0.866025	+	1.50000i
$606$		0			0
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.500000	−	0.866025i	0.500000	−	0.866025i
$612$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$617$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$618$	0.866025	−	1.50000i	0.866025	−	1.50000i
$619$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$619$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$620$		0			0
$621$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$622$	−1.00000			−1.00000
$623$		0			0
$624$	−1.00000			−1.00000
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	1.50000	+	0.866025i	1.50000	+	0.866025i
$631$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$631$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	1.73205			1.73205
$637$	0.500000	+	0.866025i	0.500000	+	0.866025i
$638$		0			0
$639$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$640$	0.866025	+	1.50000i	0.866025	+	1.50000i
$641$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$641$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$	0.866025	+	0.500000i	0.866025	+	0.500000i
$645$		0			0
$646$		0			0
$647$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$648$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$649$		0			0
$650$	2.00000			2.00000
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$	0.866025	+	1.50000i	0.866025	+	1.50000i
$656$		0			0
$657$	−1.00000			−1.00000
$658$		−	1.00000i		−	1.00000i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$	0.866025	−	1.50000i	0.866025	−	1.50000i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$669$		0			0
$670$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$671$		0			0
$672$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$673$	−2.00000			−2.00000			−1.00000			$\pi$
$673$	−2.00000			−2.00000			−1.00000			$\pi$
$674$		0			0
$675$	1.00000	+	1.73205i	1.00000	+	1.73205i
$676$		0			0
$677$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$677$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$678$	−1.73205			−1.73205
$679$		0			0
$680$	−3.00000			−3.00000
$681$		0			0
$682$		0			0
$683$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$683$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$684$		0			0
$685$	−3.00000			−3.00000
$686$	0.866025	+	0.500000i	0.866025	+	0.500000i
$687$		0			0
$688$		0			0
$689$	0.866025	+	1.50000i	0.866025	+	1.50000i
$690$	0.866025	+	1.50000i	0.866025	+	1.50000i
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$	−1.00000			−1.00000
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$699$		0			0
$700$	1.73205	−	1.00000i	1.73205	−	1.00000i
$701$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$701$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$702$	0.500000	−	0.866025i	0.500000	−	0.866025i
$703$		0			0
$704$		0			0
$705$	0.866025	−	1.50000i	0.866025	−	1.50000i
$706$		0			0
$707$		0			0
$708$	−2.00000			−2.00000
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$	0.866025	+	1.50000i	0.866025	+	1.50000i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		−	1.73205i		−	1.73205i
$715$		0			0
$716$	0.500000	−	0.866025i	0.500000	−	0.866025i
$717$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$718$		0			0
$719$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$719$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$720$	−1.73205			−1.73205
$721$	1.50000	−	0.866025i	1.50000	−	0.866025i
$722$	−1.00000			−1.00000
$723$		0			0
$724$		0			0
$725$		0			0
$726$	0.500000	−	0.866025i	0.500000	−	0.866025i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$729$	1.00000			1.00000
$730$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$	−1.73205			−1.73205
$735$	0.866025	+	1.50000i	0.866025	+	1.50000i
$736$	−1.00000			−1.00000
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$	1.50000	+	0.866025i	1.50000	+	0.866025i
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	1.50000	+	2.59808i	1.50000	+	2.59808i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$	1.73205			1.73205
$751$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$751$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$752$	0.500000	+	0.866025i	0.500000	+	0.866025i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		−	1.00000i		−	1.00000i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.50000	−	2.59808i	1.50000	−	2.59808i
$766$		0			0
$767$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$768$	0.500000	−	0.866025i	0.500000	−	0.866025i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$773$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$773$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$781$		0			0
$782$	0.866025	−	1.50000i	0.866025	−	1.50000i
$783$		0			0
$784$	−1.00000			−1.00000
$785$		0			0
$786$	0.500000	−	0.866025i	0.500000	−	0.866025i
$787$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$787$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$792$		0			0
$793$		0			0
$794$	0.500000	+	0.866025i	0.500000	+	0.866025i
$795$	1.50000	+	2.59808i	1.50000	+	2.59808i
$796$		0			0
$797$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$797$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$798$		0			0
$799$	−1.73205			−1.73205
$800$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$801$		0			0
$802$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$803$		0			0
$804$	1.73205			1.73205
$805$			1.73205i			1.73205i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$	0.866025	−	1.50000i	0.866025	−	1.50000i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	0.866025	+	1.50000i	0.866025	+	1.50000i
$817$		0			0
$818$	1.00000			1.00000
$819$	0.866025	−	0.500000i	0.866025	−	0.500000i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	0.866025	+	1.50000i	0.866025	+	1.50000i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$825$		0			0
$826$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	0.500000	−	0.866025i	0.500000	−	0.866025i
$829$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$829$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$830$		0			0
$831$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$832$	1.00000			1.00000
$833$	0.866025	−	1.50000i	0.866025	−	1.50000i
$834$		0			0
$835$	0.866025	−	1.50000i	0.866025	−	1.50000i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$841$	1.00000			1.00000
$842$		0			0
$843$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$844$		0			0
$845$		0			0
$846$	−1.00000			−1.00000
$847$	0.866025	−	0.500000i	0.866025	−	0.500000i
$848$	−1.73205			−1.73205
$849$	0.866025	−	1.50000i	0.866025	−	1.50000i
$850$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$851$		0			0
$852$	0.500000	−	0.866025i	0.500000	−	0.866025i
$853$	2.00000			2.00000			1.00000			$0$
$853$	2.00000			2.00000			1.00000			$0$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$863$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$864$	0.500000	+	0.866025i	0.500000	+	0.866025i
$865$		0			0
$866$		0			0
$867$	−2.00000			−2.00000
$868$		0			0
$869$		0			0
$870$		0			0
$871$	0.866025	+	1.50000i	0.866025	+	1.50000i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.50000	+	0.866025i	1.50000	+	0.866025i
$876$	1.00000			1.00000
$877$	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			0
$879$	0.866025	+	1.50000i	0.866025	+	1.50000i
$880$		0			0
$881$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$881$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$882$	0.500000	−	0.866025i	0.500000	−	0.866025i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$885$	−1.73205	−	3.00000i	−1.73205	−	3.00000i
$886$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$887$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$887$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	0.866025	−	1.50000i	0.866025	−	1.50000i
$895$	1.73205			1.73205
$896$	0.866025	−	0.500000i	0.866025	−	0.500000i
$897$	1.00000			1.00000
$898$		0			0
$899$		0			0
$900$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$901$	1.50000	−	2.59808i	1.50000	−	2.59808i
$902$		0			0
$903$		0			0
$904$	1.73205			1.73205
$905$		0			0
$906$		0			0
$907$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$907$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$908$		0			0
$909$		0			0
$910$		−	1.73205i		−	1.73205i
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.866025	−	0.500000i	0.866025	−	0.500000i
$918$	−1.73205			−1.73205
$919$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$919$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$920$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$921$		0			0
$922$		0			0
$923$	1.00000			1.00000
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$934$		0			0
$935$		0			0
$936$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	1.50000	+	0.866025i	1.50000	+	0.866025i
$939$		0			0
$940$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$941$		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
$944$	2.00000			2.00000
$945$	1.50000	−	0.866025i	1.50000	−	0.866025i
$946$		0			0
$947$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$947$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$948$		0			0
$949$	0.500000	+	0.866025i	0.500000	+	0.866025i
$950$		0			0
$951$		0			0
$952$			1.73205i			1.73205i
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$	0.866025	−	1.50000i	0.866025	−	1.50000i
$955$		0			0
$956$	1.00000	+	1.73205i	1.00000	+	1.73205i
$957$		0			0
$958$		0			0
$959$			1.73205i			1.73205i
$960$	1.73205			1.73205
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.73205			−1.73205
$966$	0.866025	−	0.500000i	0.866025	−	0.500000i
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	−1.00000			−1.00000
$973$		0			0
$974$		0			0
$975$	1.00000	−	1.73205i	1.00000	−	1.73205i
$976$		0			0
$977$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$977$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$981$		0			0
$982$	0.500000	−	0.866025i	0.500000	−	0.866025i
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		0			0
$994$	0.866025	−	0.500000i	0.866025	−	0.500000i
$995$		0			0
$996$		0			0
$997$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$997$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1932.1.w.e.275.2	yes	4
3.2	odd	2		1932.1.w.f.275.1	yes	4
4.3	odd	2		1932.1.w.f.275.2	yes	4
7.4	even	3	inner	1932.1.w.e.1103.2	yes	4
12.11	even	2	inner	1932.1.w.e.275.1	✓	4
21.11	odd	6		1932.1.w.f.1103.1	yes	4
23.22	odd	2	inner	1932.1.w.e.275.1	✓	4
28.11	odd	6		1932.1.w.f.1103.2	yes	4
69.68	even	2		1932.1.w.f.275.2	yes	4
84.11	even	6	inner	1932.1.w.e.1103.1	yes	4
92.91	even	2		1932.1.w.f.275.1	yes	4
161.137	odd	6	inner	1932.1.w.e.1103.1	yes	4
276.275	odd	2	CM	1932.1.w.e.275.2	yes	4
483.137	even	6		1932.1.w.f.1103.2	yes	4
644.459	even	6		1932.1.w.f.1103.1	yes	4
1932.1103	odd	6	inner	1932.1.w.e.1103.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1932.1.w.e.275.1	✓	4	12.11	even	2	inner
1932.1.w.e.275.1	✓	4	23.22	odd	2	inner
1932.1.w.e.275.2	yes	4	1.1	even	1	trivial
1932.1.w.e.275.2	yes	4	276.275	odd	2	CM
1932.1.w.e.1103.1	yes	4	84.11	even	6	inner
1932.1.w.e.1103.1	yes	4	161.137	odd	6	inner
1932.1.w.e.1103.2	yes	4	7.4	even	3	inner
1932.1.w.e.1103.2	yes	4	1932.1103	odd	6	inner
1932.1.w.f.275.1	yes	4	3.2	odd	2
1932.1.w.f.275.1	yes	4	92.91	even	2
1932.1.w.f.275.2	yes	4	4.3	odd	2
1932.1.w.f.275.2	yes	4	69.68	even	2
1932.1.w.f.1103.1	yes	4	21.11	odd	6
1932.1.w.f.1103.1	yes	4	644.459	even	6
1932.1.w.f.1103.2	yes	4	28.11	odd	6
1932.1.w.f.1103.2	yes	4	483.137	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:	\( N \)	\(=\)	\( 1932 = 2^{2} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1932.w (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.866025 - 1.50000i) q^{5} -1.00000 q^{6} +(-0.866025 - 0.500000i) q^{7} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.866025 - 1.50000i) q^{5} -1.00000 q^{6} +(-0.866025 - 0.500000i) q^{7} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(0.866025 + 1.50000i) q^{10} +(0.500000 - 0.866025i) q^{12} +1.00000 q^{13} +(0.866025 - 0.500000i) q^{14} +1.73205 q^{15} +(-0.500000 + 0.866025i) q^{16} +(-0.866025 - 1.50000i) q^{17} +(-0.500000 - 0.866025i) q^{18} -1.73205 q^{20} -1.00000i q^{21} +(0.500000 - 0.866025i) q^{23} +(0.500000 + 0.866025i) q^{24} +(-1.00000 - 1.73205i) q^{25} +(-0.500000 + 0.866025i) q^{26} -1.00000 q^{27} +1.00000i q^{28} +(-0.866025 + 1.50000i) q^{30} +(-0.500000 - 0.866025i) q^{32} +1.73205 q^{34} +(-1.50000 + 0.866025i) q^{35} +1.00000 q^{36} +(0.500000 + 0.866025i) q^{39} +(0.866025 - 1.50000i) q^{40} +(0.866025 + 0.500000i) q^{42} +(0.866025 + 1.50000i) q^{45} +(0.500000 + 0.866025i) q^{46} +(0.500000 - 0.866025i) q^{47} -1.00000 q^{48} +(0.500000 + 0.866025i) q^{49} +2.00000 q^{50} +(0.866025 - 1.50000i) q^{51} +(-0.500000 - 0.866025i) q^{52} +(0.866025 + 1.50000i) q^{53} +(0.500000 - 0.866025i) q^{54} +(-0.866025 - 0.500000i) q^{56} +(-1.00000 - 1.73205i) q^{59} +(-0.866025 - 1.50000i) q^{60} +(0.866025 - 0.500000i) q^{63} +1.00000 q^{64} +(0.866025 - 1.50000i) q^{65} +(0.866025 + 1.50000i) q^{67} +(-0.866025 + 1.50000i) q^{68} +1.00000 q^{69} -1.73205i q^{70} +1.00000 q^{71} +(-0.500000 + 0.866025i) q^{72} +(0.500000 + 0.866025i) q^{73} +(1.00000 - 1.73205i) q^{75} -1.00000 q^{78} +(0.866025 + 1.50000i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.866025 + 0.500000i) q^{84} -3.00000 q^{85} -1.73205 q^{90} +(-0.866025 - 0.500000i) q^{91} -1.00000 q^{92} +(0.500000 + 0.866025i) q^{94} +(0.500000 - 0.866025i) q^{96} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 4 q^{6} + 4 q^{8} - 2 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 - 4 * q^6 + 4 * q^8 - 2 * q^9	\( 4 q - 2 q^{2} + 2 q^{3} - 2 q^{4} - 4 q^{6} + 4 q^{8} - 2 q^{9} + 2 q^{12} + 4 q^{13} - 2 q^{16} - 2 q^{18} + 2 q^{23} + 2 q^{24} - 4 q^{25} - 2 q^{26} - 4 q^{27} - 2 q^{32} - 6 q^{35} + 4 q^{36} + 2 q^{39} + 2 q^{46} + 2 q^{47} - 4 q^{48} + 2 q^{49} + 8 q^{50} - 2 q^{52} + 2 q^{54} - 4 q^{59} + 4 q^{64} + 4 q^{69} + 4 q^{71} - 2 q^{72} + 2 q^{73} + 4 q^{75} - 4 q^{78} - 2 q^{81} - 12 q^{85} - 4 q^{92} + 2 q^{94} + 2 q^{96} - 4 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 2 * q^3 - 2 * q^4 - 4 * q^6 + 4 * q^8 - 2 * q^9 + 2 * q^12 + 4 * q^13 - 2 * q^16 - 2 * q^18 + 2 * q^23 + 2 * q^24 - 4 * q^25 - 2 * q^26 - 4 * q^27 - 2 * q^32 - 6 * q^35 + 4 * q^36 + 2 * q^39 + 2 * q^46 + 2 * q^47 - 4 * q^48 + 2 * q^49 + 8 * q^50 - 2 * q^52 + 2 * q^54 - 4 * q^59 + 4 * q^64 + 4 * q^69 + 4 * q^71 - 2 * q^72 + 2 * q^73 + 4 * q^75 - 4 * q^78 - 2 * q^81 - 12 * q^85 - 4 * q^92 + 2 * q^94 + 2 * q^96 - 4 * q^98

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