LMFDB - Embedded newform 2151.1.f.a.1672.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,1,Mod(238,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("2151.238");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2151.f (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:	$1.07348884217$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.1733990286981681.1

Embedding invariants

Embedding label			1672.2
Root			$-0.766044 + 0.642788i$ of defining polynomial
Character	$\chi$	$=$	2151.1672
Dual form			2151.1.f.a.238.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.173648 + 0.300767i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(0.439693 + 0.761570i) q^{4} +(-0.766044 - 1.32683i) q^{5} +0.347296 q^{6} -0.652704 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.173648 + 0.300767i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(0.439693 + 0.761570i) q^{4} +(-0.766044 - 1.32683i) q^{5} +0.347296 q^{6} -0.652704 q^{8} +(-0.500000 + 0.866025i) q^{9} +0.532089 q^{10} +(-0.766044 + 1.32683i) q^{11} +(0.439693 - 0.761570i) q^{12} +(-0.766044 + 1.32683i) q^{15} +(-0.326352 + 0.565258i) q^{16} +0.347296 q^{17} +(-0.173648 - 0.300767i) q^{18} +(0.673648 - 1.16679i) q^{20} +(-0.266044 - 0.460802i) q^{22} +(0.326352 + 0.565258i) q^{24} +(-0.673648 + 1.16679i) q^{25} +1.00000 q^{27} +(-0.766044 + 1.32683i) q^{29} +(-0.266044 - 0.460802i) q^{30} +(0.939693 + 1.62760i) q^{31} +(-0.439693 - 0.761570i) q^{32} +1.53209 q^{33} +(-0.0603074 + 0.104455i) q^{34} -0.879385 q^{36} +(0.500000 + 0.866025i) q^{40} -1.34730 q^{44} +1.53209 q^{45} +0.652704 q^{48} +(-0.500000 - 0.866025i) q^{49} +(-0.233956 - 0.405223i) q^{50} +(-0.173648 - 0.300767i) q^{51} +(-0.173648 + 0.300767i) q^{54} +2.34730 q^{55} +(-0.266044 - 0.460802i) q^{58} -1.34730 q^{60} +(-0.173648 + 0.300767i) q^{61} -0.652704 q^{62} -0.347296 q^{64} +(-0.266044 + 0.460802i) q^{66} +(0.939693 + 1.62760i) q^{67} +(0.152704 + 0.264490i) q^{68} +2.00000 q^{71} +(0.326352 - 0.565258i) q^{72} +1.34730 q^{75} +1.00000 q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.173648 + 0.300767i) q^{83} +(-0.266044 - 0.460802i) q^{85} +1.53209 q^{87} +(0.500000 - 0.866025i) q^{88} +(-0.266044 + 0.460802i) q^{90} +(0.939693 - 1.62760i) q^{93} +(-0.439693 + 0.761570i) q^{96} +0.347296 q^{98} +(-0.766044 - 1.32683i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 3 q^{4} - 6 q^{8} - 3 q^{9}+O(q^{10}) $ 6 * q - 3 * q^3 - 3 * q^4 - 6 * q^8 - 3 * q^9	$ 6 q - 3 q^{3} - 3 q^{4} - 6 q^{8} - 3 q^{9} - 6 q^{10} - 3 q^{12} - 3 q^{16} + 3 q^{20} + 3 q^{22} + 3 q^{24} - 3 q^{25} + 6 q^{27} + 3 q^{30} + 3 q^{32} - 6 q^{34} + 6 q^{36} + 3 q^{40} - 6 q^{44} + 6 q^{48} - 3 q^{49} - 6 q^{50} + 12 q^{55} + 3 q^{58} - 6 q^{60} - 6 q^{62} + 3 q^{66} + 3 q^{68} + 12 q^{71} + 3 q^{72} + 6 q^{75} + 6 q^{80} - 3 q^{81} + 3 q^{85} + 3 q^{88} + 3 q^{90} + 3 q^{96}+O(q^{100}) $ 6 * q - 3 * q^3 - 3 * q^4 - 6 * q^8 - 3 * q^9 - 6 * q^10 - 3 * q^12 - 3 * q^16 + 3 * q^20 + 3 * q^22 + 3 * q^24 - 3 * q^25 + 6 * q^27 + 3 * q^30 + 3 * q^32 - 6 * q^34 + 6 * q^36 + 3 * q^40 - 6 * q^44 + 6 * q^48 - 3 * q^49 - 6 * q^50 + 12 * q^55 + 3 * q^58 - 6 * q^60 - 6 * q^62 + 3 * q^66 + 3 * q^68 + 12 * q^71 + 3 * q^72 + 6 * q^75 + 6 * q^80 - 3 * q^81 + 3 * q^85 + 3 * q^88 + 3 * q^90 + 3 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times$.

$n$	$479$	$1441$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$

Coefficient data

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

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

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

$2$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$2$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$3$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$3$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$4$	0.439693	+	0.761570i	0.439693	+	0.761570i
$5$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$5$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$6$	0.347296			0.347296
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	−0.652704			−0.652704
$9$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$10$	0.532089			0.532089
$11$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$11$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$12$	0.439693	−	0.761570i	0.439693	−	0.761570i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$16$	−0.326352	+	0.565258i	−0.326352	+	0.565258i
$17$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$17$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$18$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	0.673648	−	1.16679i	0.673648	−	1.16679i
$21$		0			0
$22$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	0.326352	+	0.565258i	0.326352	+	0.565258i
$25$	−0.673648	+	1.16679i	−0.673648	+	1.16679i
$26$		0			0
$27$	1.00000			1.00000
$28$		0			0
$29$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$29$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$30$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$31$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$31$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$32$	−0.439693	−	0.761570i	−0.439693	−	0.761570i
$33$	1.53209			1.53209
$34$	−0.0603074	+	0.104455i	−0.0603074	+	0.104455i
$35$		0			0
$36$	−0.879385			−0.879385
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	0.500000	+	0.866025i	0.500000	+	0.866025i
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$	−1.34730			−1.34730
$45$	1.53209			1.53209
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	0.652704			0.652704
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$	−0.233956	−	0.405223i	−0.233956	−	0.405223i
$51$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$55$	2.34730			2.34730
$56$		0			0
$57$		0			0
$58$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$	−1.34730			−1.34730
$61$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$61$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$62$	−0.652704			−0.652704
$63$		0			0
$64$	−0.347296			−0.347296
$65$		0			0
$66$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$67$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$67$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$68$	0.152704	+	0.264490i	0.152704	+	0.264490i
$69$		0			0
$70$		0			0
$71$	2.00000			2.00000			1.00000			$0$
$71$	2.00000			2.00000			1.00000			$0$
$72$	0.326352	−	0.565258i	0.326352	−	0.565258i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	1.34730			1.34730
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$	1.00000			1.00000
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$83$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$84$		0			0
$85$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$86$		0			0
$87$	1.53209			1.53209
$88$	0.500000	−	0.866025i	0.500000	−	0.866025i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$91$		0			0
$92$		0			0
$93$	0.939693	−	1.62760i	0.939693	−	1.62760i
$94$		0			0
$95$		0			0
$96$	−0.439693	+	0.761570i	−0.439693	+	0.761570i
$97$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$97$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$98$	0.347296			0.347296
$99$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$100$	−1.18479			−1.18479
$101$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$101$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$102$	0.120615			0.120615
$103$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$103$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	0.439693	+	0.761570i	0.439693	+	0.761570i
$109$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$109$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$110$	−0.407604	+	0.705990i	−0.407604	+	0.705990i
$111$		0			0
$112$		0			0
$113$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$113$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$		0			0
$116$	−1.34730			−1.34730
$117$		0			0
$118$		0			0
$119$		0			0
$120$	0.500000	−	0.866025i	0.500000	−	0.866025i
$121$	−0.673648	−	1.16679i	−0.673648	−	1.16679i
$122$	−0.0603074	−	0.104455i	−0.0603074	−	0.104455i
$123$		0			0
$124$	−0.826352	+	1.43128i	−0.826352	+	1.43128i
$125$	0.532089			0.532089
$126$		0			0
$127$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$127$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$128$	0.500000	−	0.866025i	0.500000	−	0.866025i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	0.673648	+	1.16679i	0.673648	+	1.16679i
$133$		0			0
$134$	−0.652704			−0.652704
$135$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$136$	−0.226682			−0.226682
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$		0			0
$142$	−0.347296	+	0.601535i	−0.347296	+	0.601535i
$143$		0			0
$144$	−0.326352	−	0.565258i	−0.326352	−	0.565258i
$145$	2.34730			2.34730
$146$		0			0
$147$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	−0.233956	+	0.405223i	−0.233956	+	0.405223i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$154$		0			0
$155$	1.43969	−	2.49362i	1.43969	−	2.49362i
$156$		0			0
$157$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$157$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$158$		0			0
$159$		0			0
$160$	−0.673648	+	1.16679i	−0.673648	+	1.16679i
$161$		0			0
$162$	0.347296			0.347296
$163$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$163$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$164$		0			0
$165$	−1.17365	−	2.03282i	−1.17365	−	2.03282i
$166$	−0.0603074	−	0.104455i	−0.0603074	−	0.104455i
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$	0.184793			0.184793
$171$		0			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$175$		0			0
$176$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	0.673648	+	1.16679i	0.673648	+	1.16679i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	0.347296			0.347296
$184$		0			0
$185$		0			0
$186$	0.326352	+	0.565258i	0.326352	+	0.565258i
$187$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	0.173648	+	0.300767i	0.173648	+	0.300767i
$193$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$193$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$194$		0			0
$195$		0			0
$196$	0.439693	−	0.761570i	0.439693	−	0.761570i
$197$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$197$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$198$	0.532089			0.532089
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	0.439693	−	0.761570i	0.439693	−	0.761570i
$201$	0.939693	−	1.62760i	0.939693	−	1.62760i
$202$	−0.347296	−	0.601535i	−0.347296	−	0.601535i
$203$		0			0
$204$	0.152704	−	0.264490i	0.152704	−	0.264490i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$211$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$212$		0			0
$213$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$214$		0			0
$215$		0			0
$216$	−0.652704			−0.652704
$217$		0			0
$218$	0.173648	−	0.300767i	0.173648	−	0.300767i
$219$		0			0
$220$	1.03209	+	1.78763i	1.03209	+	1.78763i
$221$		0			0
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$	−0.673648	−	1.16679i	−0.673648	−	1.16679i
$226$	−0.347296			−0.347296
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$	0.500000	−	0.866025i	0.500000	−	0.866025i
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$239$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$240$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$242$	0.467911			0.467911
$243$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$244$	−0.305407			−0.305407
$245$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$246$		0			0
$247$		0			0
$248$	−0.613341	−	1.06234i	−0.613341	−	1.06234i
$249$	0.347296			0.347296
$250$	−0.0923963	+	0.160035i	−0.0923963	+	0.160035i
$251$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$251$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$252$		0			0
$253$		0			0
$254$	0.326352	−	0.565258i	0.326352	−	0.565258i
$255$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$256$		0			0
$257$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$257$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$262$		0			0
$263$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$263$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$264$	−1.00000			−1.00000
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.826352	+	1.43128i	−0.826352	+	1.43128i
$269$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$269$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$270$	0.532089			0.532089
$271$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$271$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$272$	−0.113341	+	0.196312i	−0.113341	+	0.196312i
$273$		0			0
$274$		0			0
$275$	−1.03209	−	1.78763i	−1.03209	−	1.78763i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$	−1.87939			−1.87939
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$283$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$284$	0.879385	+	1.52314i	0.879385	+	1.52314i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.879385			0.879385
$289$	−0.879385			−0.879385
$290$	−0.407604	+	0.705990i	−0.407604	+	0.705990i
$291$		0			0
$292$		0			0
$293$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$293$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$294$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$295$		0			0
$296$		0			0
$297$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$298$		0			0
$299$		0			0
$300$	0.592396	+	1.02606i	0.592396	+	1.02606i
$301$		0			0
$302$		0			0
$303$	2.00000			2.00000
$304$		0			0
$305$	0.532089			0.532089
$306$	−0.0603074	−	0.104455i	−0.0603074	−	0.104455i
$307$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$307$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$308$		0			0
$309$		0			0
$310$	0.500000	+	0.866025i	0.500000	+	0.866025i
$311$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$311$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	0.532089			0.532089
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$	−1.17365	−	2.03282i	−1.17365	−	2.03282i
$320$	0.266044	+	0.460802i	0.266044	+	0.460802i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.439693	−	0.761570i	0.439693	−	0.761570i
$325$		0			0
$326$	0.326352	−	0.565258i	0.326352	−	0.565258i
$327$	0.500000	+	0.866025i	0.500000	+	0.866025i
$328$		0			0
$329$		0			0
$330$	0.815207			0.815207
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	−0.305407			−0.305407
$333$		0			0
$334$		0			0
$335$	1.43969	−	2.49362i	1.43969	−	2.49362i
$336$		0			0
$337$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$337$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$338$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$339$	0.500000	−	0.866025i	0.500000	−	0.866025i
$340$	0.233956	−	0.405223i	0.233956	−	0.405223i
$341$	−2.87939			−2.87939
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$347$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$348$	0.673648	+	1.16679i	0.673648	+	1.16679i
$349$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$349$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$350$		0			0
$351$		0			0
$352$	1.34730			1.34730
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$	−1.53209	−	2.65366i	−1.53209	−	2.65366i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	2.00000			2.00000			1.00000			$0$
$359$	2.00000			2.00000			1.00000			$0$
$360$	−1.00000			−1.00000
$361$	1.00000			1.00000
$362$		0			0
$363$	−0.673648	+	1.16679i	−0.673648	+	1.16679i
$364$		0			0
$365$		0			0
$366$	−0.0603074	+	0.104455i	−0.0603074	+	0.104455i
$367$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$367$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	1.65270			1.65270
$373$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$373$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$374$	−0.0923963	−	0.160035i	−0.0923963	−	0.160035i
$375$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	0.939693	+	1.62760i	0.939693	+	1.62760i
$382$		0			0
$383$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$383$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$384$	−1.00000			−1.00000
$385$		0			0
$386$	−0.652704			−0.652704
$387$		0			0
$388$		0			0
$389$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$389$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$390$		0			0
$391$		0			0
$392$	0.326352	+	0.565258i	0.326352	+	0.565258i
$393$		0			0
$394$	−0.0603074	+	0.104455i	−0.0603074	+	0.104455i
$395$		0			0
$396$	0.673648	−	1.16679i	0.673648	−	1.16679i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−0.439693	−	0.761570i	−0.439693	−	0.761570i
$401$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$401$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$402$	0.326352	+	0.565258i	0.326352	+	0.565258i
$403$		0			0
$404$	−1.75877			−1.75877
$405$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$406$		0			0
$407$		0			0
$408$	0.113341	+	0.196312i	0.113341	+	0.196312i
$409$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$409$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	0.532089			0.532089
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$419$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$420$		0			0
$421$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$422$	0.120615			0.120615
$423$		0			0
$424$		0			0
$425$	−0.233956	+	0.405223i	−0.233956	+	0.405223i
$426$	0.694593			0.694593
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$431$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$432$	−0.326352	+	0.565258i	−0.326352	+	0.565258i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−1.17365	−	2.03282i	−1.17365	−	2.03282i
$436$	−0.439693	−	0.761570i	−0.439693	−	0.761570i
$437$		0			0
$438$		0			0
$439$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$439$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$440$	−1.53209			−1.53209
$441$	1.00000			1.00000
$442$		0			0
$443$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$443$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	0.467911			0.467911
$451$		0			0
$452$	−0.439693	+	0.761570i	−0.439693	+	0.761570i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$457$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$	0.347296			0.347296
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$465$	−2.87939			−2.87939
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	0.347296			0.347296
$479$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$479$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$480$	1.34730			1.34730
$481$		0			0
$482$	0.173648	+	0.300767i	0.173648	+	0.300767i
$483$		0			0
$484$	0.592396	−	1.02606i	0.592396	−	1.02606i
$485$		0			0
$486$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$487$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$487$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$488$	0.113341	−	0.196312i	0.113341	−	0.196312i
$489$	0.939693	+	1.62760i	0.939693	+	1.62760i
$490$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$494$		0			0
$495$	−1.17365	+	2.03282i	−1.17365	+	2.03282i
$496$	−1.22668			−1.22668
$497$		0			0
$498$	−0.0603074	+	0.104455i	−0.0603074	+	0.104455i
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	0.233956	+	0.405223i	0.233956	+	0.405223i
$501$		0			0
$502$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$503$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$503$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$504$		0			0
$505$	3.06418			3.06418
$506$		0			0
$507$	1.00000			1.00000
$508$	−0.826352	−	1.43128i	−0.826352	−	1.43128i
$509$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$509$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$510$	−0.0923963	−	0.160035i	−0.0923963	−	0.160035i
$511$		0			0
$512$	1.00000			1.00000
$513$		0			0
$514$	−0.652704			−0.652704
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	0.532089			0.532089
$523$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$523$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$524$		0			0
$525$		0			0
$526$	0.326352	+	0.565258i	0.326352	+	0.565258i
$527$	0.326352	+	0.565258i	0.326352	+	0.565258i
$528$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$529$	−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
$536$	−0.613341	−	1.06234i	−0.613341	−	1.06234i
$537$		0			0
$538$	0.326352	−	0.565258i	0.326352	−	0.565258i
$539$	1.53209			1.53209
$540$	0.673648	−	1.16679i	0.673648	−	1.16679i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$543$		0			0
$544$	−0.152704	−	0.264490i	−0.152704	−	0.264490i
$545$	0.766044	+	1.32683i	0.766044	+	1.32683i
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$		0			0
$549$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$550$	0.716881			0.716881
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$	0.326352	−	0.565258i	0.326352	−	0.565258i
$559$		0			0
$560$		0			0
$561$	0.532089			0.532089
$562$		0			0
$563$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$563$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$564$		0			0
$565$	0.766044	−	1.32683i	0.766044	−	1.32683i
$566$	0.120615			0.120615
$567$		0			0
$568$	−1.30541			−1.30541
$569$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$569$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$570$		0			0
$571$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$571$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.173648	−	0.300767i	0.173648	−	0.300767i
$577$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$577$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$578$	0.152704	−	0.264490i	0.152704	−	0.264490i
$579$	0.939693	−	1.62760i	0.939693	−	1.62760i
$580$	1.03209	+	1.78763i	1.03209	+	1.78763i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.694593			0.694593
$587$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$587$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$588$	−0.879385			−0.879385
$589$		0			0
$590$		0			0
$591$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		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	$0.666667\pi$
$600$	−0.879385			−0.879385
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$	−1.87939			−1.87939
$604$		0			0
$605$	−1.03209	+	1.78763i	−1.03209	+	1.78763i
$606$	−0.347296	+	0.601535i	−0.347296	+	0.601535i
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$	−0.0923963	+	0.160035i	−0.0923963	+	0.160035i
$611$		0			0
$612$	−0.305407			−0.305407
$613$	2.00000			2.00000			1.00000			$0$
$613$	2.00000			2.00000			1.00000			$0$
$614$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$	2.53209			2.53209
$621$		0			0
$622$	0.532089			0.532089
$623$		0			0
$624$		0			0
$625$	0.266044	+	0.460802i	0.266044	+	0.460802i
$626$		0			0
$627$		0			0
$628$	0.673648	−	1.16679i	0.673648	−	1.16679i
$629$		0			0
$630$		0			0
$631$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$631$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$632$		0			0
$633$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$634$		0			0
$635$	1.43969	+	2.49362i	1.43969	+	2.49362i
$636$		0			0
$637$		0			0
$638$	0.815207			0.815207
$639$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$640$	−1.53209			−1.53209
$641$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$641$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$642$		0			0
$643$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$643$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$647$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$648$	0.326352	+	0.565258i	0.326352	+	0.565258i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.826352	−	1.43128i	−0.826352	−	1.43128i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$	−0.347296			−0.347296
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$	1.03209	−	1.78763i	1.03209	−	1.78763i
$661$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$661$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$662$		0			0
$663$		0			0
$664$	0.113341	−	0.196312i	0.113341	−	0.196312i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	0.500000	+	0.866025i	0.500000	+	0.866025i
$671$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$672$		0			0
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$	0.532089			0.532089
$675$	−0.673648	+	1.16679i	−0.673648	+	1.16679i
$676$	−0.879385			−0.879385
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$	0.173648	+	0.300767i	0.173648	+	0.300767i
$679$		0			0
$680$	0.173648	+	0.300767i	0.173648	+	0.300767i
$681$		0			0
$682$	0.500000	−	0.866025i	0.500000	−	0.866025i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$691$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$692$		0			0
$693$		0			0
$694$	−0.652704			−0.652704
$695$		0			0
$696$	−1.00000			−1.00000
$697$		0			0
$698$	0.326352	+	0.565258i	0.326352	+	0.565258i
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	0.266044	−	0.460802i	0.266044	−	0.460802i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$	1.06418			1.06418
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$718$	−0.347296	+	0.601535i	−0.347296	+	0.601535i
$719$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$719$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$720$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$721$		0			0
$722$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$723$	−1.00000			−1.00000
$724$		0			0
$725$	−1.03209	−	1.78763i	−1.03209	−	1.78763i
$726$	−0.233956	−	0.405223i	−0.233956	−	0.405223i
$727$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$727$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$	0.152704	+	0.264490i	0.152704	+	0.264490i
$733$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$733$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$734$	−0.0603074	−	0.104455i	−0.0603074	−	0.104455i
$735$	1.53209			1.53209
$736$		0			0
$737$	−2.87939			−2.87939
$738$		0			0
$739$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$	−0.613341	+	1.06234i	−0.613341	+	1.06234i
$745$		0			0
$746$	0.694593			0.694593
$747$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$748$	−0.467911			−0.467911
$749$		0			0
$750$	0.184793			0.184793
$751$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$751$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$757$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$761$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$762$	−0.652704			−0.652704
$763$		0			0
$764$		0			0
$765$	0.532089			0.532089
$766$	−0.347296			−0.347296
$767$		0			0
$768$		0			0
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$	0.939693	−	1.62760i	0.939693	−	1.62760i
$772$	−0.826352	+	1.43128i	−0.826352	+	1.43128i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−2.53209			−2.53209
$776$		0			0
$777$		0			0
$778$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$779$		0			0
$780$		0			0
$781$	−1.53209	+	2.65366i	−1.53209	+	2.65366i
$782$		0			0
$783$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$784$	0.652704			0.652704
$785$	−1.17365	+	2.03282i	−1.17365	+	2.03282i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	0.152704	+	0.264490i	0.152704	+	0.264490i
$789$	−1.87939			−1.87939
$790$		0			0
$791$		0			0
$792$	0.500000	+	0.866025i	0.500000	+	0.866025i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$797$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$798$		0			0
$799$		0			0
$800$	1.18479			1.18479
$801$		0			0
$802$	0.120615			0.120615
$803$		0			0
$804$	1.65270			1.65270
$805$		0			0
$806$		0			0
$807$	0.939693	+	1.62760i	0.939693	+	1.62760i
$808$	0.652704	−	1.13052i	0.652704	−	1.13052i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$814$		0			0
$815$	1.43969	+	2.49362i	1.43969	+	2.49362i
$816$	0.226682			0.226682
$817$		0			0
$818$	0.120615			0.120615
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$	−1.03209	+	1.78763i	−1.03209	+	1.78763i
$826$		0			0
$827$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$827$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$	−0.0923963	+	0.160035i	−0.0923963	+	0.160035i
$831$		0			0
$832$		0			0
$833$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.939693	+	1.62760i	0.939693	+	1.62760i
$838$	−0.652704			−0.652704
$839$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$839$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$840$		0			0
$841$	−0.673648	−	1.16679i	−0.673648	−	1.16679i
$842$	0.173648	+	0.300767i	0.173648	+	0.300767i
$843$		0			0
$844$	0.152704	−	0.264490i	0.152704	−	0.264490i
$845$	1.53209			1.53209
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$850$	−0.0812519	−	0.140732i	−0.0812519	−	0.140732i
$851$		0			0
$852$	0.879385	−	1.52314i	0.879385	−	1.52314i
$853$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$853$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$859$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$860$		0			0
$861$		0			0
$862$	0.326352	−	0.565258i	0.326352	−	0.565258i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.439693	−	0.761570i	−0.439693	−	0.761570i
$865$		0			0
$866$		0			0
$867$	0.439693	+	0.761570i	0.439693	+	0.761570i
$868$		0			0
$869$		0			0
$870$	0.815207			0.815207
$871$		0			0
$872$	0.652704			0.652704
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$877$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$878$	0.173648	+	0.300767i	0.173648	+	0.300767i
$879$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$880$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$883$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$883$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$884$		0			0
$885$		0			0
$886$	−0.0603074	−	0.104455i	−0.0603074	−	0.104455i
$887$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$887$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.53209			1.53209
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.87939			−2.87939
$900$	0.592396	−	1.02606i	0.592396	−	1.02606i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.326352	−	0.565258i	−0.326352	−	0.565258i
$905$		0			0
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$		0			0
$913$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$914$	−0.347296	−	0.601535i	−0.347296	−	0.601535i
$915$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$916$		0			0
$917$		0			0
$918$	−0.0603074	+	0.104455i	−0.0603074	+	0.104455i
$919$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$919$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$920$		0			0
$921$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	1.34730			1.34730
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$	0.500000	−	0.866025i	0.500000	−	0.866025i
$931$		0			0
$932$		0			0
$933$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$934$		0			0
$935$	0.815207			0.815207
$936$		0			0
$937$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$937$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$938$		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	$0.333333\pi$
$942$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	0.439693	−	0.761570i	0.439693	−	0.761570i
$957$	−1.17365	+	2.03282i	−1.17365	+	2.03282i
$958$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$959$		0			0
$960$	0.266044	−	0.460802i	0.266044	−	0.460802i
$961$	−1.26604	+	2.19285i	−1.26604	+	2.19285i
$962$		0			0
$963$		0			0
$964$	0.879385			0.879385
$965$	1.43969	−	2.49362i	1.43969	−	2.49362i
$966$		0			0
$967$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$967$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$968$	0.439693	+	0.761570i	0.439693	+	0.761570i
$969$		0			0
$970$		0			0
$971$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$971$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$972$	−0.879385			−0.879385
$973$		0			0
$974$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$975$		0			0
$976$	−0.113341	−	0.196312i	−0.113341	−	0.196312i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$	−0.652704			−0.652704
$979$		0			0
$980$	−1.34730			−1.34730
$981$	0.500000	−	0.866025i	0.500000	−	0.866025i
$982$		0			0
$983$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$983$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$984$		0			0
$985$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$986$	−0.0923963	−	0.160035i	−0.0923963	−	0.160035i
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−0.407604	−	0.705990i	−0.407604	−	0.705990i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	0.826352	−	1.43128i	0.826352	−	1.43128i
$993$		0			0
$994$		0			0
$995$		0			0
$996$	0.152704	+	0.264490i	0.152704	+	0.264490i
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.1.f.a.1672.2	yes	6
9.4	even	3	inner	2151.1.f.a.238.2	✓	6
239.238	odd	2	CM	2151.1.f.a.1672.2	yes	6
2151.238	odd	6	inner	2151.1.f.a.238.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.1.f.a.238.2	✓	6	9.4	even	3	inner
2151.1.f.a.238.2	✓	6	2151.238	odd	6	inner
2151.1.f.a.1672.2	yes	6	1.1	even	1	trivial
2151.1.f.a.1672.2	yes	6	239.238	odd	2	CM

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2151.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.173648 + 0.300767i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(0.439693 + 0.761570i) q^{4} +(-0.766044 - 1.32683i) q^{5} +0.347296 q^{6} -0.652704 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.173648 + 0.300767i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(0.439693 + 0.761570i) q^{4} +(-0.766044 - 1.32683i) q^{5} +0.347296 q^{6} -0.652704 q^{8} +(-0.500000 + 0.866025i) q^{9} +0.532089 q^{10} +(-0.766044 + 1.32683i) q^{11} +(0.439693 - 0.761570i) q^{12} +(-0.766044 + 1.32683i) q^{15} +(-0.326352 + 0.565258i) q^{16} +0.347296 q^{17} +(-0.173648 - 0.300767i) q^{18} +(0.673648 - 1.16679i) q^{20} +(-0.266044 - 0.460802i) q^{22} +(0.326352 + 0.565258i) q^{24} +(-0.673648 + 1.16679i) q^{25} +1.00000 q^{27} +(-0.766044 + 1.32683i) q^{29} +(-0.266044 - 0.460802i) q^{30} +(0.939693 + 1.62760i) q^{31} +(-0.439693 - 0.761570i) q^{32} +1.53209 q^{33} +(-0.0603074 + 0.104455i) q^{34} -0.879385 q^{36} +(0.500000 + 0.866025i) q^{40} -1.34730 q^{44} +1.53209 q^{45} +0.652704 q^{48} +(-0.500000 - 0.866025i) q^{49} +(-0.233956 - 0.405223i) q^{50} +(-0.173648 - 0.300767i) q^{51} +(-0.173648 + 0.300767i) q^{54} +2.34730 q^{55} +(-0.266044 - 0.460802i) q^{58} -1.34730 q^{60} +(-0.173648 + 0.300767i) q^{61} -0.652704 q^{62} -0.347296 q^{64} +(-0.266044 + 0.460802i) q^{66} +(0.939693 + 1.62760i) q^{67} +(0.152704 + 0.264490i) q^{68} +2.00000 q^{71} +(0.326352 - 0.565258i) q^{72} +1.34730 q^{75} +1.00000 q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.173648 + 0.300767i) q^{83} +(-0.266044 - 0.460802i) q^{85} +1.53209 q^{87} +(0.500000 - 0.866025i) q^{88} +(-0.266044 + 0.460802i) q^{90} +(0.939693 - 1.62760i) q^{93} +(-0.439693 + 0.761570i) q^{96} +0.347296 q^{98} +(-0.766044 - 1.32683i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 3 q^{4} - 6 q^{8} - 3 q^{9}+O(q^{10}) \) 6 * q - 3 * q^3 - 3 * q^4 - 6 * q^8 - 3 * q^9	\( 6 q - 3 q^{3} - 3 q^{4} - 6 q^{8} - 3 q^{9} - 6 q^{10} - 3 q^{12} - 3 q^{16} + 3 q^{20} + 3 q^{22} + 3 q^{24} - 3 q^{25} + 6 q^{27} + 3 q^{30} + 3 q^{32} - 6 q^{34} + 6 q^{36} + 3 q^{40} - 6 q^{44} + 6 q^{48} - 3 q^{49} - 6 q^{50} + 12 q^{55} + 3 q^{58} - 6 q^{60} - 6 q^{62} + 3 q^{66} + 3 q^{68} + 12 q^{71} + 3 q^{72} + 6 q^{75} + 6 q^{80} - 3 q^{81} + 3 q^{85} + 3 q^{88} + 3 q^{90} + 3 q^{96}+O(q^{100}) \) 6 * q - 3 * q^3 - 3 * q^4 - 6 * q^8 - 3 * q^9 - 6 * q^10 - 3 * q^12 - 3 * q^16 + 3 * q^20 + 3 * q^22 + 3 * q^24 - 3 * q^25 + 6 * q^27 + 3 * q^30 + 3 * q^32 - 6 * q^34 + 6 * q^36 + 3 * q^40 - 6 * q^44 + 6 * q^48 - 3 * q^49 - 6 * q^50 + 12 * q^55 + 3 * q^58 - 6 * q^60 - 6 * q^62 + 3 * q^66 + 3 * q^68 + 12 * q^71 + 3 * q^72 + 6 * q^75 + 6 * q^80 - 3 * q^81 + 3 * q^85 + 3 * q^88 + 3 * q^90 + 3 * q^96

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