LMFDB - Embedded newform 3864.1.bx.f.275.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3864,1,Mod(275,3864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3864, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 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("3864.275");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3864.bx (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.92838720881$
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 6.0.16826668992.2

Embedding invariants

Embedding label			275.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	3864.275
Dual form			3864.1.bx.f.3035.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +1.00000 q^{6} +(-0.866025 + 0.500000i) q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +1.00000 q^{6} +(-0.866025 + 0.500000i) q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.866025 - 1.50000i) q^{11} +(0.500000 - 0.866025i) q^{12} +1.00000i q^{14} +(-0.500000 + 0.866025i) q^{16} +(-0.866025 - 1.50000i) q^{17} +(0.500000 + 0.866025i) q^{18} +(-0.866025 - 0.500000i) q^{21} -1.73205 q^{22} +(-0.500000 + 0.866025i) q^{23} +(-0.500000 - 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +(0.866025 + 0.500000i) q^{28} -1.00000 q^{29} +(0.500000 + 0.866025i) q^{32} +(0.866025 - 1.50000i) q^{33} -1.73205 q^{34} +1.00000 q^{36} +(-0.866025 + 0.500000i) q^{42} +(-0.866025 + 1.50000i) q^{44} +(0.500000 + 0.866025i) q^{46} +(0.500000 - 0.866025i) q^{47} -1.00000 q^{48} +(0.500000 - 0.866025i) q^{49} -1.00000 q^{50} +(0.866025 - 1.50000i) q^{51} +(-0.500000 + 0.866025i) q^{54} +(0.866025 - 0.500000i) q^{56} +(-0.500000 + 0.866025i) q^{58} -1.00000i q^{63} +1.00000 q^{64} +(-0.866025 - 1.50000i) q^{66} +(-0.866025 + 1.50000i) q^{68} -1.00000 q^{69} +1.00000 q^{71} +(0.500000 - 0.866025i) q^{72} +(-0.500000 - 0.866025i) q^{73} +(0.500000 - 0.866025i) q^{75} +(1.50000 + 0.866025i) q^{77} +(-0.866025 + 1.50000i) q^{79} +(-0.500000 - 0.866025i) q^{81} +1.00000i q^{84} +(-0.500000 - 0.866025i) q^{87} +(0.866025 + 1.50000i) q^{88} +1.00000 q^{92} +(-0.500000 - 0.866025i) q^{94} +(-0.500000 + 0.866025i) q^{96} +(-0.500000 - 0.866025i) q^{98} +1.73205 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 2 q^{16} + 2 q^{18} - 2 q^{23} - 2 q^{24} - 2 q^{25} - 4 q^{27} - 4 q^{29} + 2 q^{32} + 4 q^{36} + 2 q^{46} + 2 q^{47} - 4 q^{48} + 2 q^{49} - 4 q^{50} - 2 q^{54} - 2 q^{58} + 4 q^{64} - 4 q^{69} + 4 q^{71} + 2 q^{72} - 2 q^{73} + 2 q^{75} + 6 q^{77} - 2 q^{81} - 2 q^{87} + 4 q^{92} - 2 q^{94} - 2 q^{96} - 2 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 - 2 * q^16 + 2 * q^18 - 2 * q^23 - 2 * q^24 - 2 * q^25 - 4 * q^27 - 4 * q^29 + 2 * q^32 + 4 * q^36 + 2 * q^46 + 2 * q^47 - 4 * q^48 + 2 * q^49 - 4 * q^50 - 2 * q^54 - 2 * q^58 + 4 * q^64 - 4 * q^69 + 4 * q^71 + 2 * q^72 - 2 * q^73 + 2 * q^75 + 6 * q^77 - 2 * q^81 - 2 * q^87 + 4 * q^92 - 2 * q^94 - 2 * q^96 - 2 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3864\mathbb{Z}\right)^\times$.

$n$	$967$	$1289$	$1933$	$2761$	$2857$
$\chi(n)$	$-1$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$	$-1$

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 $

$2$	0.500000	−	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			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$	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			0
$11$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$11$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$12$	0.500000	−	0.866025i	0.500000	−	0.866025i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$			1.00000i			1.00000i
$15$		0			0
$16$	−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		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$22$	−1.73205			−1.73205
$23$	−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
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$		0			0
$27$	−1.00000			−1.00000
$28$	0.866025	+	0.500000i	0.866025	+	0.500000i
$29$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$29$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		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
$33$	0.866025	−	1.50000i	0.866025	−	1.50000i
$34$	−1.73205			−1.73205
$35$		0			0
$36$	1.00000			1.00000
$37$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$37$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$45$		0			0
$46$	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
$50$	−1.00000			−1.00000
$51$	0.866025	−	1.50000i	0.866025	−	1.50000i
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$55$		0			0
$56$	0.866025	−	0.500000i	0.866025	−	0.500000i
$57$		0			0
$58$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$61$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$62$		0			0
$63$		−	1.00000i		−	1.00000i
$64$	1.00000			1.00000
$65$		0			0
$66$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$69$	−1.00000			−1.00000
$70$		0			0
$71$	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	0.500000	−	0.866025i	$-0.333333\pi$
$73$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$74$		0			0
$75$	0.500000	−	0.866025i	0.500000	−	0.866025i
$76$		0			0
$77$	1.50000	+	0.866025i	1.50000	+	0.866025i
$78$		0			0
$79$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$79$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$			1.00000i			1.00000i
$85$		0			0
$86$		0			0
$87$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$88$	0.866025	+	1.50000i	0.866025	+	1.50000i
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$		0			0
$92$	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			$\pi$
$98$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$99$	1.73205			1.73205
$100$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$101$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$101$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$102$	−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	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$	0.500000	+	0.866025i	0.500000	+	0.866025i
$109$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$109$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$110$		0			0
$111$		0			0
$112$		−	1.00000i		−	1.00000i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$	0.500000	+	0.866025i	0.500000	+	0.866025i
$117$		0			0
$118$		0			0
$119$	1.50000	+	0.866025i	1.50000	+	0.866025i
$120$		0			0
$121$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$127$		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			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	−1.73205			−1.73205
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.866025	+	1.50000i	0.866025	+	1.50000i
$137$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$137$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$138$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$139$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$139$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$	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
$147$	1.00000			1.00000
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$151$		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
$154$	1.50000	−	0.866025i	1.50000	−	0.866025i
$155$		0			0
$156$		0			0
$157$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$157$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$158$	0.866025	+	1.50000i	0.866025	+	1.50000i
$159$		0			0
$160$		0			0
$161$		−	1.00000i		−	1.00000i
$162$	−1.00000			−1.00000
$163$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$163$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.00000			−2.00000			−1.00000			$\pi$
$167$	−2.00000			−2.00000			−1.00000			$\pi$
$168$	0.866025	+	0.500000i	0.866025	+	0.500000i
$169$	1.00000			1.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$173$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$174$	−1.00000			−1.00000
$175$	0.866025	+	0.500000i	0.866025	+	0.500000i
$176$	1.73205			1.73205
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$181$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$182$		0			0
$183$		0			0
$184$	0.500000	−	0.866025i	0.500000	−	0.866025i
$185$		0			0
$186$		0			0
$187$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$188$	−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
$193$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$193$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$	−1.00000			−1.00000
$197$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$197$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$198$	0.866025	−	1.50000i	0.866025	−	1.50000i
$199$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$199$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$200$	0.500000	+	0.866025i	0.500000	+	0.866025i
$201$		0			0
$202$	−1.00000			−1.00000
$203$	0.866025	−	0.500000i	0.866025	−	0.500000i
$204$	−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			0
$209$		0			0
$210$		0			0
$211$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$211$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$	0.500000	+	0.866025i	0.500000	+	0.866025i
$214$		0			0
$215$		0			0
$216$	1.00000			1.00000
$217$		0			0
$218$	−1.73205			−1.73205
$219$	0.500000	−	0.866025i	0.500000	−	0.866025i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$225$	1.00000			1.00000
$226$		0			0
$227$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$227$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$228$		0			0
$229$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$229$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$230$		0			0
$231$			1.73205i			1.73205i
$232$	1.00000			1.00000
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.73205			−1.73205
$238$	1.50000	−	0.866025i	1.50000	−	0.866025i
$239$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$239$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	1.00000	+	1.73205i	1.00000	+	1.73205i
$243$	0.500000	−	0.866025i	0.500000	−	0.866025i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$251$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$252$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$253$	1.73205			1.73205
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.500000	−	0.866025i	0.500000	−	0.866025i
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$269$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$	1.73205			1.73205
$273$		0			0
$274$	1.73205			1.73205
$275$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$276$	0.500000	+	0.866025i	0.500000	+	0.866025i
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$279$		0			0
$280$		0			0
$281$	−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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$	−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
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	0.500000	−	0.866025i	0.500000	−	0.866025i
$295$		0			0
$296$		0			0
$297$	0.866025	+	1.50000i	0.866025	+	1.50000i
$298$		0			0
$299$		0			0
$300$	−1.00000			−1.00000
$301$		0			0
$302$		0			0
$303$	0.500000	−	0.866025i	0.500000	−	0.866025i
$304$		0			0
$305$		0			0
$306$	0.866025	−	1.50000i	0.866025	−	1.50000i
$307$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$307$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$308$		−	1.73205i		−	1.73205i
$309$	−1.73205			−1.73205
$310$		0			0
$311$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$311$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	−1.73205			−1.73205
$315$		0			0
$316$	1.73205			1.73205
$317$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$317$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$	0.866025	+	1.50000i	0.866025	+	1.50000i
$320$		0			0
$321$		0			0
$322$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$323$		0			0
$324$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$325$		0			0
$326$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$327$	0.866025	−	1.50000i	0.866025	−	1.50000i
$328$		0			0
$329$			1.00000i			1.00000i
$330$		0			0
$331$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$331$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$332$		0			0
$333$		0			0
$334$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$335$		0			0
$336$	0.866025	−	0.500000i	0.866025	−	0.500000i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	0.500000	−	0.866025i	0.500000	−	0.866025i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$			1.00000i			1.00000i
$344$		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	$0.333333\pi$
$348$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	0.866025	−	0.500000i	0.866025	−	0.500000i
$351$		0			0
$352$	0.866025	−	1.50000i	0.866025	−	1.50000i
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$			1.73205i			1.73205i
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$	0.866025	−	1.50000i	0.866025	−	1.50000i
$363$	−2.00000			−2.00000
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$373$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$374$	1.50000	+	2.59808i	1.50000	+	2.59808i
$375$		0			0
$376$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$377$		0			0
$378$		−	1.00000i		−	1.00000i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		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
$386$	2.00000			2.00000
$387$		0			0
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$	1.73205			1.73205
$392$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$393$		0			0
$394$	0.500000	−	0.866025i	0.500000	−	0.866025i
$395$		0			0
$396$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$	1.73205			1.73205
$399$		0			0
$400$	1.00000			1.00000
$401$	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	$-0.166667\pi$
$402$		0			0
$403$		0			0
$404$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$405$		0			0
$406$		−	1.00000i		−	1.00000i
$407$		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
$413$		0			0
$414$	−1.00000			−1.00000
$415$		0			0
$416$		0			0
$417$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$418$		0			0
$419$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$419$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$420$		0			0
$421$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$421$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$422$	0.500000	−	0.866025i	0.500000	−	0.866025i
$423$	0.500000	+	0.866025i	0.500000	+	0.866025i
$424$		0			0
$425$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$426$	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
$436$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$437$		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
$441$	0.500000	+	0.866025i	0.500000	+	0.866025i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	0.500000	−	0.866025i	0.500000	−	0.866025i
$451$		0			0
$452$		0			0
$453$		0			0
$454$	1.73205			1.73205
$455$		0			0
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$459$	0.866025	+	1.50000i	0.866025	+	1.50000i
$460$		0			0
$461$	−2.00000			−2.00000			−1.00000			$\pi$
$461$	−2.00000			−2.00000			−1.00000			$\pi$
$462$	1.50000	+	0.866025i	1.50000	+	0.866025i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	0.500000	−	0.866025i	0.500000	−	0.866025i
$465$		0			0
$466$		0			0
$467$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$467$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.866025	−	1.50000i	0.866025	−	1.50000i
$472$		0			0
$473$		0			0
$474$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$475$		0			0
$476$		−	1.73205i		−	1.73205i
$477$		0			0
$478$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	0.866025	−	0.500000i	0.866025	−	0.500000i
$484$	2.00000			2.00000
$485$		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
$489$	1.00000			1.00000
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$	0.866025	+	1.50000i	0.866025	+	1.50000i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$498$		0			0
$499$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$499$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$500$		0			0
$501$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$502$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$			1.00000i			1.00000i
$505$		0			0
$506$	0.866025	−	1.50000i	0.866025	−	1.50000i
$507$	0.500000	+	0.866025i	0.500000	+	0.866025i
$508$		0			0
$509$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$509$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$510$		0			0
$511$	0.866025	+	0.500000i	0.866025	+	0.500000i
$512$	−1.00000			−1.00000
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	−1.73205			−1.73205
$518$		0			0
$519$	1.00000			1.00000
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$			1.00000i			1.00000i
$526$		0			0
$527$		0			0
$528$	0.866025	+	1.50000i	0.866025	+	1.50000i
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.00000			−1.00000
$539$	−1.73205			−1.73205
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$	0.866025	+	1.50000i	0.866025	+	1.50000i
$544$	0.866025	−	1.50000i	0.866025	−	1.50000i
$545$		0			0
$546$		0			0
$547$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$547$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$548$	0.866025	−	1.50000i	0.866025	−	1.50000i
$549$		0			0
$550$	0.866025	+	1.50000i	0.866025	+	1.50000i
$551$		0			0
$552$	1.00000			1.00000
$553$		−	1.73205i		−	1.73205i
$554$		0			0
$555$		0			0
$556$	0.500000	+	0.866025i	0.500000	+	0.866025i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−3.00000			−3.00000
$562$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$563$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$563$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$564$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$565$		0			0
$566$		0			0
$567$	0.866025	+	0.500000i	0.866025	+	0.500000i
$568$	−1.00000			−1.00000
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.00000			1.00000
$576$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$577$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$577$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$578$	1.00000	+	1.73205i	1.00000	+	1.73205i
$579$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.500000	+	0.866025i	0.500000	+	0.866025i
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$589$		0			0
$590$		0			0
$591$	0.500000	+	0.866025i	0.500000	+	0.866025i
$592$		0			0
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$	1.73205			1.73205
$595$		0			0
$596$		0			0
$597$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$598$		0			0
$599$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$599$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$600$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$601$	2.00000			2.00000			1.00000			$0$
$601$	2.00000			2.00000			1.00000			$0$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$	0.866025	+	0.500000i	0.866025	+	0.500000i
$610$		0			0
$611$		0			0
$612$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$613$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$613$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$614$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$615$		0			0
$616$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$617$	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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$	0.500000	−	0.866025i	0.500000	−	0.866025i
$622$	−1.00000			−1.00000
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$629$		0			0
$630$		0			0
$631$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$631$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$632$	0.866025	−	1.50000i	0.866025	−	1.50000i
$633$	0.500000	+	0.866025i	0.500000	+	0.866025i
$634$	1.00000	+	1.73205i	1.00000	+	1.73205i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	1.73205			1.73205
$639$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$640$		0			0
$641$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$641$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$645$		0			0
$646$		0			0
$647$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$647$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$648$	0.500000	+	0.866025i	0.500000	+	0.866025i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.00000			−1.00000
$653$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$653$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$654$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$655$		0			0
$656$		0			0
$657$	1.00000			1.00000
$658$	0.866025	+	0.500000i	0.866025	+	0.500000i
$659$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$659$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$661$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$662$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	0.500000	−	0.866025i	0.500000	−	0.866025i
$668$	1.00000	+	1.73205i	1.00000	+	1.73205i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		−	1.00000i		−	1.00000i
$673$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$673$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$674$		0			0
$675$	0.500000	+	0.866025i	0.500000	+	0.866025i
$676$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$		0			0
$685$		0			0
$686$	0.866025	+	0.500000i	0.866025	+	0.500000i
$687$	1.73205			1.73205
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$691$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$692$	−1.00000			−1.00000
$693$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$694$		0			0
$695$		0			0
$696$	0.500000	+	0.866025i	0.500000	+	0.866025i
$697$		0			0
$698$		0			0
$699$		0			0
$700$		−	1.00000i		−	1.00000i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$705$		0			0
$706$		0			0
$707$	0.866025	+	0.500000i	0.866025	+	0.500000i
$708$		0			0
$709$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$709$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$710$		0			0
$711$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$712$		0			0
$713$		0			0
$714$	1.50000	+	0.866025i	1.50000	+	0.866025i
$715$		0			0
$716$		0			0
$717$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$718$		0			0
$719$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$719$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$720$		0			0
$721$		−	1.73205i		−	1.73205i
$722$	−1.00000			−1.00000
$723$		0			0
$724$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$725$	0.500000	+	0.866025i	0.500000	+	0.866025i
$726$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$727$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$727$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$733$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$		0			0
$736$	−1.00000			−1.00000
$737$		0			0
$738$		0			0
$739$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$739$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$747$		0			0
$748$	3.00000			3.00000
$749$		0			0
$750$		0			0
$751$		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
$753$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$754$		0			0
$755$		0			0
$756$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$	0.866025	+	1.50000i	0.866025	+	1.50000i
$760$		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
$763$	1.50000	+	0.866025i	1.50000	+	0.866025i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	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
$772$	1.00000	−	1.73205i	1.00000	−	1.73205i
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$782$	0.866025	−	1.50000i	0.866025	−	1.50000i
$783$	1.00000			1.00000
$784$	0.500000	+	0.866025i	0.500000	+	0.866025i
$785$		0			0
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−1.73205			−1.73205
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.866025	−	1.50000i	0.866025	−	1.50000i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$	−1.73205			−1.73205
$800$	0.500000	−	0.866025i	0.500000	−	0.866025i
$801$		0			0
$802$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$803$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	0.500000	−	0.866025i	0.500000	−	0.866025i
$808$	0.500000	+	0.866025i	0.500000	+	0.866025i
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$811$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$812$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$813$		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			0
$820$		0			0
$821$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$821$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$822$	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
$825$	−1.73205			−1.73205
$826$		0			0
$827$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$827$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$828$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.73205			−1.73205
$834$	−1.00000			−1.00000
$835$		0			0
$836$		0			0
$837$		0			0
$838$	0.866025	−	1.50000i	0.866025	−	1.50000i
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$		0			0
$842$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$843$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$844$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$845$		0			0
$846$	1.00000			1.00000
$847$		−	2.00000i		−	2.00000i
$848$		0			0
$849$		0			0
$850$	0.866025	+	1.50000i	0.866025	+	1.50000i
$851$		0			0
$852$	0.500000	−	0.866025i	0.500000	−	0.866025i
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		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
$859$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$859$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$863$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$864$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$865$		0			0
$866$		0			0
$867$	−2.00000			−2.00000
$868$		0			0
$869$	3.00000			3.00000
$870$		0			0
$871$		0			0
$872$	0.866025	+	1.50000i	0.866025	+	1.50000i
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−1.00000			−1.00000
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$881$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$882$	1.00000			1.00000
$883$	−2.00000			−2.00000			−1.00000			$\pi$
$883$	−2.00000			−2.00000			−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$887$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$			1.00000i			1.00000i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$	0.866025	−	1.50000i	0.866025	−	1.50000i
$909$	1.00000			1.00000
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.73205			−1.73205
$917$		0			0
$918$	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	$-0.166667\pi$
$920$		0			0
$921$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$922$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$923$		0			0
$924$	1.50000	−	0.866025i	1.50000	−	0.866025i
$925$		0			0
$926$		0			0
$927$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$928$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$929$		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
$934$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$935$		0			0
$936$		0			0
$937$		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	$0.333333\pi$
$942$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$943$		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	$0.666667\pi$
$948$	0.866025	+	1.50000i	0.866025	+	1.50000i
$949$		0			0
$950$		0			0
$951$	−2.00000			−2.00000
$952$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$953$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$953$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$954$		0			0
$955$		0			0
$956$	0.500000	+	0.866025i	0.500000	+	0.866025i
$957$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$958$		0			0
$959$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$960$		0			0
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		−	1.00000i		−	1.00000i
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$	1.00000	−	1.73205i	1.00000	−	1.73205i
$969$		0			0
$970$		0			0
$971$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$971$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$972$	−1.00000			−1.00000
$973$	0.866025	−	0.500000i	0.866025	−	0.500000i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$	0.500000	−	0.866025i	0.500000	−	0.866025i
$979$		0			0
$980$		0			0
$981$	1.73205			1.73205
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$	1.73205			1.73205
$987$	−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
$993$	2.00000			2.00000
$994$			1.00000i			1.00000i
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3864.1.bx.f.275.1	yes	4
3.2	odd	2		3864.1.bx.e.275.1	✓	4
7.4	even	3	inner	3864.1.bx.f.3035.1	yes	4
8.3	odd	2		3864.1.bx.e.275.2	yes	4
21.11	odd	6		3864.1.bx.e.3035.1	yes	4
23.22	odd	2	inner	3864.1.bx.f.275.2	yes	4
24.11	even	2	inner	3864.1.bx.f.275.2	yes	4
56.11	odd	6		3864.1.bx.e.3035.2	yes	4
69.68	even	2		3864.1.bx.e.275.2	yes	4
161.137	odd	6	inner	3864.1.bx.f.3035.2	yes	4
168.11	even	6	inner	3864.1.bx.f.3035.2	yes	4
184.91	even	2		3864.1.bx.e.275.1	✓	4
483.137	even	6		3864.1.bx.e.3035.2	yes	4
552.275	odd	2	CM	3864.1.bx.f.275.1	yes	4
1288.459	even	6		3864.1.bx.e.3035.1	yes	4
3864.3035	odd	6	inner	3864.1.bx.f.3035.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.1.bx.e.275.1	✓	4	3.2	odd	2
3864.1.bx.e.275.1	✓	4	184.91	even	2
3864.1.bx.e.275.2	yes	4	8.3	odd	2
3864.1.bx.e.275.2	yes	4	69.68	even	2
3864.1.bx.e.3035.1	yes	4	21.11	odd	6
3864.1.bx.e.3035.1	yes	4	1288.459	even	6
3864.1.bx.e.3035.2	yes	4	56.11	odd	6
3864.1.bx.e.3035.2	yes	4	483.137	even	6
3864.1.bx.f.275.1	yes	4	1.1	even	1	trivial
3864.1.bx.f.275.1	yes	4	552.275	odd	2	CM
3864.1.bx.f.275.2	yes	4	23.22	odd	2	inner
3864.1.bx.f.275.2	yes	4	24.11	even	2	inner
3864.1.bx.f.3035.1	yes	4	7.4	even	3	inner
3864.1.bx.f.3035.1	yes	4	3864.3035	odd	6	inner
3864.1.bx.f.3035.2	yes	4	161.137	odd	6	inner
3864.1.bx.f.3035.2	yes	4	168.11	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3864 = 2^{3} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3864.bx (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +1.00000 q^{6} +(-0.866025 + 0.500000i) q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +1.00000 q^{6} +(-0.866025 + 0.500000i) q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.866025 - 1.50000i) q^{11} +(0.500000 - 0.866025i) q^{12} +1.00000i q^{14} +(-0.500000 + 0.866025i) q^{16} +(-0.866025 - 1.50000i) q^{17} +(0.500000 + 0.866025i) q^{18} +(-0.866025 - 0.500000i) q^{21} -1.73205 q^{22} +(-0.500000 + 0.866025i) q^{23} +(-0.500000 - 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +(0.866025 + 0.500000i) q^{28} -1.00000 q^{29} +(0.500000 + 0.866025i) q^{32} +(0.866025 - 1.50000i) q^{33} -1.73205 q^{34} +1.00000 q^{36} +(-0.866025 + 0.500000i) q^{42} +(-0.866025 + 1.50000i) q^{44} +(0.500000 + 0.866025i) q^{46} +(0.500000 - 0.866025i) q^{47} -1.00000 q^{48} +(0.500000 - 0.866025i) q^{49} -1.00000 q^{50} +(0.866025 - 1.50000i) q^{51} +(-0.500000 + 0.866025i) q^{54} +(0.866025 - 0.500000i) q^{56} +(-0.500000 + 0.866025i) q^{58} -1.00000i q^{63} +1.00000 q^{64} +(-0.866025 - 1.50000i) q^{66} +(-0.866025 + 1.50000i) q^{68} -1.00000 q^{69} +1.00000 q^{71} +(0.500000 - 0.866025i) q^{72} +(-0.500000 - 0.866025i) q^{73} +(0.500000 - 0.866025i) q^{75} +(1.50000 + 0.866025i) q^{77} +(-0.866025 + 1.50000i) q^{79} +(-0.500000 - 0.866025i) q^{81} +1.00000i q^{84} +(-0.500000 - 0.866025i) q^{87} +(0.866025 + 1.50000i) q^{88} +1.00000 q^{92} +(-0.500000 - 0.866025i) q^{94} +(-0.500000 + 0.866025i) q^{96} +(-0.500000 - 0.866025i) q^{98} +1.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{6} - 4 q^{8} - 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 4 * q^6 - 4 * q^8 - 2 * q^9	\( 4 q + 2 q^{2} + 2 q^{3} - 2 q^{4} + 4 q^{6} - 4 q^{8} - 2 q^{9} + 2 q^{12} - 2 q^{16} + 2 q^{18} - 2 q^{23} - 2 q^{24} - 2 q^{25} - 4 q^{27} - 4 q^{29} + 2 q^{32} + 4 q^{36} + 2 q^{46} + 2 q^{47} - 4 q^{48} + 2 q^{49} - 4 q^{50} - 2 q^{54} - 2 q^{58} + 4 q^{64} - 4 q^{69} + 4 q^{71} + 2 q^{72} - 2 q^{73} + 2 q^{75} + 6 q^{77} - 2 q^{81} - 2 q^{87} + 4 q^{92} - 2 q^{94} - 2 q^{96} - 2 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 2 * q^3 - 2 * q^4 + 4 * q^6 - 4 * q^8 - 2 * q^9 + 2 * q^12 - 2 * q^16 + 2 * q^18 - 2 * q^23 - 2 * q^24 - 2 * q^25 - 4 * q^27 - 4 * q^29 + 2 * q^32 + 4 * q^36 + 2 * q^46 + 2 * q^47 - 4 * q^48 + 2 * q^49 - 4 * q^50 - 2 * q^54 - 2 * q^58 + 4 * q^64 - 4 * q^69 + 4 * q^71 + 2 * q^72 - 2 * q^73 + 2 * q^75 + 6 * q^77 - 2 * q^81 - 2 * q^87 + 4 * q^92 - 2 * q^94 - 2 * q^96 - 2 * q^98

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