LMFDB - Embedded newform 2151.1.f.b.238.3

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:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{45})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1 $ x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{45}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{45} - \cdots)$

Embedding invariants

Embedding label			238.3
Root			$-0.997564 - 0.0697565i$ of defining polynomial
Character	$\chi$	$=$	2151.238
Dual form			2151.1.f.b.1672.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.848048 - 1.46886i) q^{2} +(-0.978148 - 0.207912i) q^{3} +(-0.938371 + 1.62531i) q^{4} +(0.615661 - 1.06636i) q^{5} +(0.524123 + 1.61308i) q^{6} +1.48704 q^{8} +(0.913545 + 0.406737i) q^{9} +O(q^{10})$	\(q+(-0.848048 - 1.46886i) q^{2} +(-0.978148 - 0.207912i) q^{3} +(-0.938371 + 1.62531i) q^{4} +(0.615661 - 1.06636i) q^{5} +(0.524123 + 1.61308i) q^{6} +1.48704 q^{8} +(0.913545 + 0.406737i) q^{9} -2.08844 q^{10} +(-0.961262 - 1.66495i) q^{11} +(1.25579 - 1.39469i) q^{12} +(-0.823916 + 0.915051i) q^{15} +(-0.322710 - 0.558950i) q^{16} +1.53209 q^{17} +(-0.177290 - 1.68680i) q^{18} +(1.15544 + 2.00128i) q^{20} +(-1.63039 + 2.82392i) q^{22} +(-1.45454 - 0.309173i) q^{24} +(-0.258078 - 0.447004i) q^{25} +(-0.809017 - 0.587785i) q^{27} +(-0.559193 - 0.968551i) q^{29} +(2.04280 + 0.434212i) q^{30} +(0.882948 - 1.52931i) q^{31} +(0.196173 - 0.339782i) q^{32} +(0.594092 + 1.82843i) q^{33} +(-1.29929 - 2.25043i) q^{34} +(-1.51832 + 1.10312i) q^{36} +(0.915513 - 1.58571i) q^{40} +3.60808 q^{44} +(0.996161 - 0.723753i) q^{45} +(0.199446 + 0.613830i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.437725 + 0.758162i) q^{50} +(-1.49861 - 0.318539i) q^{51} +(-0.177290 + 1.68680i) q^{54} -2.36725 q^{55} +(-0.948445 + 1.64275i) q^{58} +(-0.714100 - 2.19777i) q^{60} +(-0.766044 - 1.32683i) q^{61} -2.99513 q^{62} -1.31088 q^{64} +(2.18189 - 2.42324i) q^{66} +(-0.173648 + 0.300767i) q^{67} +(-1.43767 + 2.49011i) q^{68} -1.61803 q^{71} +(1.35848 + 0.604833i) q^{72} +(0.159501 + 0.490894i) q^{75} -0.794720 q^{80} +(0.669131 + 0.743145i) q^{81} +(0.997564 + 1.72783i) q^{83} +(0.943248 - 1.63375i) q^{85} +(0.345600 + 1.06365i) q^{87} +(-1.42943 - 2.47585i) q^{88} +(-1.90789 - 0.849446i) q^{90} +(-1.18161 + 1.31232i) q^{93} +(-0.262531 + 0.291570i) q^{96} +1.69610 q^{98} +(-0.200958 - 1.91199i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 3 q^{3} - 12 q^{4} + 6 q^{8} + 3 q^{9}+O(q^{10}) $ 24 * q + 3 * q^3 - 12 * q^4 + 6 * q^8 + 3 * q^9	$ 24 q + 3 q^{3} - 12 q^{4} + 6 q^{8} + 3 q^{9} + 6 q^{10} + 3 q^{12} - 12 q^{16} - 3 q^{20} - 3 q^{22} - 3 q^{24} - 12 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} - 12 q^{49} + 6 q^{50} - 12 q^{55} - 3 q^{58} - 9 q^{60} - 24 q^{62} + 30 q^{64} + 27 q^{66} - 3 q^{68} - 12 q^{71} - 18 q^{72} - 6 q^{75} + 54 q^{80} + 3 q^{81} - 3 q^{85} - 3 q^{88} - 18 q^{90} - 3 q^{96}+O(q^{100}) $ 24 * q + 3 * q^3 - 12 * q^4 + 6 * q^8 + 3 * q^9 + 6 * q^10 + 3 * q^12 - 12 * q^16 - 3 * q^20 - 3 * q^22 - 3 * q^24 - 12 * 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 - 12 * q^49 + 6 * q^50 - 12 * q^55 - 3 * q^58 - 9 * q^60 - 24 * q^62 + 30 * q^64 + 27 * q^66 - 3 * q^68 - 12 * q^71 - 18 * q^72 - 6 * q^75 + 54 * q^80 + 3 * q^81 - 3 * q^85 - 3 * q^88 - 18 * q^90 - 3 * q^96

Display 100 coefficients 10 coefficients

Character values

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

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

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 $

$2$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	−0.882948	−	0.469472i	$-0.844444\pi$
$2$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	0.0348995	−	0.999391i	$-0.488889\pi$
$3$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$3$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$4$	−0.938371	+	1.62531i	−0.938371	+	1.62531i
$5$	0.615661	−	1.06636i	0.615661	−	1.06636i	−0.374607	−	0.927184i	$-0.622222\pi$
$5$	0.615661	−	1.06636i	0.615661	−	1.06636i	0.990268	−	0.139173i	$-0.0444444\pi$
$6$	0.524123	+	1.61308i	0.524123	+	1.61308i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	1.48704			1.48704
$9$	0.913545	+	0.406737i	0.913545	+	0.406737i
$10$	−2.08844			−2.08844
$11$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.719340	−	0.694658i	$-0.755556\pi$
$11$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.241922	−	0.970296i	$-0.577778\pi$
$12$	1.25579	−	1.39469i	1.25579	−	1.39469i
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$	−0.823916	+	0.915051i	−0.823916	+	0.915051i
$16$	−0.322710	−	0.558950i	−0.322710	−	0.558950i
$17$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$17$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$18$	−0.177290	−	1.68680i	−0.177290	−	1.68680i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	1.15544	+	2.00128i	1.15544	+	2.00128i
$21$		0			0
$22$	−1.63039	+	2.82392i	−1.63039	+	2.82392i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	−1.45454	−	0.309173i	−1.45454	−	0.309173i
$25$	−0.258078	−	0.447004i	−0.258078	−	0.447004i
$26$		0			0
$27$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$28$		0			0
$29$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	−0.997564	−	0.0697565i	$-0.977778\pi$
$29$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	0.438371	−	0.898794i	$-0.355556\pi$
$30$	2.04280	+	0.434212i	2.04280	+	0.434212i
$31$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.0348995	−	0.999391i	$-0.488889\pi$
$31$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.848048	−	0.529919i	$-0.177778\pi$
$32$	0.196173	−	0.339782i	0.196173	−	0.339782i
$33$	0.594092	+	1.82843i	0.594092	+	1.82843i
$34$	−1.29929	−	2.25043i	−1.29929	−	2.25043i
$35$		0			0
$36$	−1.51832	+	1.10312i	−1.51832	+	1.10312i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	0.915513	−	1.58571i	0.915513	−	1.58571i
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$	3.60808			3.60808
$45$	0.996161	−	0.723753i	0.996161	−	0.723753i
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	0.199446	+	0.613830i	0.199446	+	0.613830i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	−0.437725	+	0.758162i	−0.437725	+	0.758162i
$51$	−1.49861	−	0.318539i	−1.49861	−	0.318539i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−0.177290	+	1.68680i	−0.177290	+	1.68680i
$55$	−2.36725			−2.36725
$56$		0			0
$57$		0			0
$58$	−0.948445	+	1.64275i	−0.948445	+	1.64275i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	−0.714100	−	2.19777i	−0.714100	−	2.19777i
$61$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$61$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$62$	−2.99513			−2.99513
$63$		0			0
$64$	−1.31088			−1.31088
$65$		0			0
$66$	2.18189	−	2.42324i	2.18189	−	2.42324i
$67$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$67$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$68$	−1.43767	+	2.49011i	−1.43767	+	2.49011i
$69$		0			0
$70$		0			0
$71$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$71$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$72$	1.35848	+	0.604833i	1.35848	+	0.604833i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	0.159501	+	0.490894i	0.159501	+	0.490894i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$	−0.794720			−0.794720
$81$	0.669131	+	0.743145i	0.669131	+	0.743145i
$82$		0			0
$83$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.559193	+	0.829038i	$0.311111\pi$
$83$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.438371	+	0.898794i	$0.355556\pi$
$84$		0			0
$85$	0.943248	−	1.63375i	0.943248	−	1.63375i
$86$		0			0
$87$	0.345600	+	1.06365i	0.345600	+	1.06365i
$88$	−1.42943	−	2.47585i	−1.42943	−	2.47585i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−1.90789	−	0.849446i	−1.90789	−	0.849446i
$91$		0			0
$92$		0			0
$93$	−1.18161	+	1.31232i	−1.18161	+	1.31232i
$94$		0			0
$95$		0			0
$96$	−0.262531	+	0.291570i	−0.262531	+	0.291570i
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$	1.69610			1.69610
$99$	−0.200958	−	1.91199i	−0.200958	−	1.91199i
$100$	0.968692			0.968692
$101$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$101$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$102$	0.803002	+	2.47139i	0.803002	+	2.47139i
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	1.71449	−	0.763340i	1.71449	−	0.763340i
$109$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$109$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$110$	2.00754	+	3.47716i	2.00754	+	3.47716i
$111$		0			0
$112$		0			0
$113$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$113$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$114$		0			0
$115$		0			0
$116$	2.09892			2.09892
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−1.22520	+	1.36072i	−1.22520	+	1.36072i
$121$	−1.34805	+	2.33489i	−1.34805	+	2.33489i
$122$	−1.29929	+	2.25043i	−1.29929	+	2.25043i
$123$		0			0
$124$	1.65707	+	2.87012i	1.65707	+	2.87012i
$125$	0.595768			0.595768
$126$		0			0
$127$	−1.43868			−1.43868			−0.719340	−	0.694658i	$-0.755556\pi$
$127$	−1.43868			−1.43868			−0.719340	+	0.694658i	$0.755556\pi$
$128$	0.915513	+	1.58571i	0.915513	+	1.58571i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	−3.52924	−	0.750162i	−3.52924	−	0.750162i
$133$		0			0
$134$	0.589048			0.589048
$135$	−1.12487	+	0.500824i	−1.12487	+	0.500824i
$136$	2.27828			2.27828
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$		0			0
$142$	1.37217	+	2.37667i	1.37217	+	2.37667i
$143$		0			0
$144$	−0.0674647	−	0.641884i	−0.0674647	−	0.641884i
$145$	−1.37709			−1.37709
$146$		0			0
$147$	0.669131	−	0.743145i	0.669131	−	0.743145i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	0.585791	−	0.650587i	0.585791	−	0.650587i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$	1.39963	+	0.623157i	1.39963	+	0.623157i
$154$		0			0
$155$	−1.08719	−	1.88307i	−1.08719	−	1.88307i
$156$		0			0
$157$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	0.438371	+	0.898794i	$0.355556\pi$
$157$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	−0.997564	+	0.0697565i	$0.977778\pi$
$158$		0			0
$159$		0			0
$160$	−0.241552	−	0.418381i	−0.241552	−	0.418381i
$161$		0			0
$162$	0.524123	−	1.61308i	0.524123	−	1.61308i
$163$	1.98054			1.98054			0.990268	−	0.139173i	$-0.0444444\pi$
$163$	1.98054			1.98054			0.990268	+	0.139173i	$0.0444444\pi$
$164$		0			0
$165$	2.31552	+	0.492178i	2.31552	+	0.492178i
$166$	1.69196	−	2.93057i	1.69196	−	2.93057i
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$	−3.19968			−3.19968
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$	1.26927	−	1.40966i	1.26927	−	1.40966i
$175$		0			0
$176$	−0.620417	+	1.07459i	−0.620417	+	1.07459i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	0.241552	+	2.29822i	0.241552	+	2.29822i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	0.473442	+	1.45710i	0.473442	+	1.45710i
$184$		0			0
$185$		0			0
$186$	2.92968	+	0.622722i	2.92968	+	0.622722i
$187$	−1.47274	−	2.55086i	−1.47274	−	2.55086i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	1.28223	+	0.272546i	1.28223	+	0.272546i
$193$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.374607	+	0.927184i	$0.622222\pi$
$193$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.615661	+	0.788011i	$0.711111\pi$
$194$		0			0
$195$		0			0
$196$	−0.938371	−	1.62531i	−0.938371	−	1.62531i
$197$	−0.483844			−0.483844			−0.241922	−	0.970296i	$-0.577778\pi$
$197$	−0.483844			−0.483844			−0.241922	+	0.970296i	$0.577778\pi$
$198$	−2.63803	+	1.91664i	−2.63803	+	1.91664i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.383772	−	0.664713i	−0.383772	−	0.664713i
$201$	0.232387	−	0.258091i	0.232387	−	0.258091i
$202$	−0.524123	+	0.907807i	−0.524123	+	0.907807i
$203$		0			0
$204$	1.92398	−	2.13679i	1.92398	−	2.13679i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.438371	−	0.898794i	$-0.355556\pi$
$211$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.559193	−	0.829038i	$-0.311111\pi$
$212$		0			0
$213$	1.58268	+	0.336408i	1.58268	+	0.336408i
$214$		0			0
$215$		0			0
$216$	−1.20304	−	0.874060i	−1.20304	−	0.874060i
$217$		0			0
$218$	−1.54946	−	2.68375i	−1.54946	−	2.68375i
$219$		0			0
$220$	2.22136	−	3.84750i	2.22136	−	3.84750i
$221$		0			0
$222$		0			0
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$		0			0
$225$	−0.0539530	−	0.513329i	−0.0539530	−	0.513329i
$226$	2.26982			2.26982
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$	−0.831542	−	1.44027i	−0.831542	−	1.44027i
$233$		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
$240$	0.777353	+	0.165232i	0.777353	+	0.165232i
$241$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$241$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$242$	4.57284			4.57284
$243$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$244$	2.87534			2.87534
$245$	0.615661	+	1.06636i	0.615661	+	1.06636i
$246$		0			0
$247$		0			0
$248$	1.31298	−	2.27414i	1.31298	−	2.27414i
$249$	−0.616528	−	1.89748i	−0.616528	−	1.89748i
$250$	−0.505240	−	0.875101i	−0.505240	−	0.875101i
$251$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$251$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$252$		0			0
$253$		0			0
$254$	1.22007	+	2.11322i	1.22007	+	2.11322i
$255$	−1.26231	+	1.40194i	−1.26231	+	1.40194i
$256$	0.897360	−	1.55427i	0.897360	−	1.55427i
$257$	0.719340	−	1.24593i	0.719340	−	1.24593i	−0.241922	−	0.970296i	$-0.577778\pi$
$257$	0.719340	−	1.24593i	0.719340	−	1.24593i	0.961262	−	0.275637i	$-0.0888889\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.116903	−	1.11226i	−0.116903	−	1.11226i
$262$		0			0
$263$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	0.559193	−	0.829038i	$-0.311111\pi$
$263$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	−0.997564	+	0.0697565i	$0.977778\pi$
$264$	0.883439	+	2.71894i	0.883439	+	2.71894i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.325893	−	0.564463i	−0.325893	−	0.564463i
$269$	−1.76590			−1.76590			−0.882948	−	0.469472i	$-0.844444\pi$
$269$	−1.76590			−1.76590			−0.882948	+	0.469472i	$0.844444\pi$
$270$	1.68959	+	1.22756i	1.68959	+	1.22756i
$271$	1.92252			1.92252			0.961262	−	0.275637i	$-0.0888889\pi$
$271$	1.92252			1.92252			0.961262	+	0.275637i	$0.0888889\pi$
$272$	−0.494420	−	0.856360i	−0.494420	−	0.856360i
$273$		0			0
$274$		0			0
$275$	−0.496161	+	0.859376i	−0.496161	+	0.859376i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$	1.42864	−	1.03797i	1.42864	−	1.03797i
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	0.0348995	+	0.999391i	$0.488889\pi$
$283$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	−0.882948	+	0.469472i	$0.844444\pi$
$284$	1.51832	−	2.62980i	1.51832	−	2.62980i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.317415	−	0.230615i	0.317415	−	0.230615i
$289$	1.34730			1.34730
$290$	1.16784	+	2.02276i	1.16784	+	2.02276i
$291$		0			0
$292$		0			0
$293$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$293$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$294$	−1.65903	−	0.352638i	−1.65903	−	0.352638i
$295$		0			0
$296$		0			0
$297$	−0.200958	+	1.91199i	−0.200958	+	1.91199i
$298$		0			0
$299$		0			0
$300$	−0.947524	−	0.201402i	−0.947524	−	0.201402i
$301$		0			0
$302$		0			0
$303$	0.190983	+	0.587785i	0.190983	+	0.587785i
$304$		0			0
$305$	−1.88650			−1.88650
$306$	−0.271625	−	2.58433i	−0.271625	−	2.58433i
$307$	−1.23132			−1.23132			−0.615661	−	0.788011i	$-0.711111\pi$
$307$	−1.23132			−1.23132			−0.615661	+	0.788011i	$0.711111\pi$
$308$		0			0
$309$		0			0
$310$	−1.84398	+	3.19388i	−1.84398	+	3.19388i
$311$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$311$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	1.89689			1.89689
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$	−1.07506	+	1.86206i	−1.07506	+	1.86206i
$320$	−0.807056	+	1.39786i	−0.807056	+	1.39786i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.83573	+	0.390197i	−1.83573	+	0.390197i
$325$		0			0
$326$	−1.67959	−	2.90914i	−1.67959	−	2.90914i
$327$	−1.78716	−	0.379874i	−1.78716	−	0.379874i
$328$		0			0
$329$		0			0
$330$	−1.24073	−	3.81857i	−1.24073	−	3.81857i
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	−3.74434			−3.74434
$333$		0			0
$334$		0			0
$335$	0.213817	+	0.370342i	0.213817	+	0.370342i
$336$		0			0
$337$	0.615661	−	1.06636i	0.615661	−	1.06636i	−0.374607	−	0.927184i	$-0.622222\pi$
$337$	0.615661	−	1.06636i	0.615661	−	1.06636i	0.990268	−	0.139173i	$-0.0444444\pi$
$338$	−0.848048	+	1.46886i	−0.848048	+	1.46886i
$339$	0.895472	−	0.994522i	0.895472	−	0.994522i
$340$	1.77023	+	3.06613i	1.77023	+	3.06613i
$341$	−3.39497			−3.39497
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.0348995	−	0.999391i	$-0.488889\pi$
$347$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.848048	−	0.529919i	$-0.177778\pi$
$348$	−2.05306	−	0.436390i	−2.05306	−	0.436390i
$349$	0.719340	+	1.24593i	0.719340	+	1.24593i	0.961262	+	0.275637i	$0.0888889\pi$
$349$	0.719340	+	1.24593i	0.719340	+	1.24593i	−0.241922	+	0.970296i	$0.577778\pi$
$350$		0			0
$351$		0			0
$352$	−0.754294			−0.754294
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$	−0.996161	+	1.72540i	−0.996161	+	1.72540i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$359$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$360$	1.48133	−	1.07625i	1.48133	−	1.07625i
$361$	1.00000			1.00000
$362$		0			0
$363$	1.80404	−	2.00359i	1.80404	−	2.00359i
$364$		0			0
$365$		0			0
$366$	1.73878	−	1.93111i	1.73878	−	1.93111i
$367$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.559193	+	0.829038i	$0.311111\pi$
$367$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.438371	+	0.898794i	$0.355556\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.02412	−	3.15193i	−1.02412	−	3.15193i
$373$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$373$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$374$	−2.49791	+	4.32650i	−2.49791	+	4.32650i
$375$	−0.582749	−	0.123867i	−0.582749	−	0.123867i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	1.40724	+	0.299118i	1.40724	+	0.299118i
$382$		0			0
$383$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$383$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$384$	−0.565818	−	1.74141i	−0.565818	−	1.74141i
$385$		0			0
$386$	3.35918			3.35918
$387$		0			0
$388$		0			0
$389$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	0.848048	−	0.529919i	$-0.177778\pi$
$389$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	−0.882948	+	0.469472i	$0.844444\pi$
$390$		0			0
$391$		0			0
$392$	−0.743520	+	1.28781i	−0.743520	+	1.28781i
$393$		0			0
$394$	0.410323	+	0.710700i	0.410323	+	0.710700i
$395$		0			0
$396$	3.29615	+	1.46754i	3.29615	+	1.46754i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−0.166569	+	0.288505i	−0.166569	+	0.288505i
$401$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.438371	−	0.898794i	$-0.355556\pi$
$401$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.559193	−	0.829038i	$-0.311111\pi$
$402$	−0.576176	−	0.122470i	−0.576176	−	0.122470i
$403$		0			0
$404$	1.15989			1.15989
$405$	1.20442	−	0.256006i	1.20442	−	0.256006i
$406$		0			0
$407$		0			0
$408$	−2.22849	−	0.473680i	−2.22849	−	0.473680i
$409$	0.374607	−	0.648838i	0.374607	−	0.648838i	−0.615661	−	0.788011i	$-0.711111\pi$
$409$	0.374607	−	0.648838i	0.374607	−	0.648838i	0.990268	+	0.139173i	$0.0444444\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	2.45665			2.45665
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	−0.997564	−	0.0697565i	$-0.977778\pi$
$419$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	0.559193	+	0.829038i	$0.311111\pi$
$420$		0			0
$421$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$421$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$422$	−3.38393			−3.38393
$423$		0			0
$424$		0			0
$425$	−0.395399	−	0.684850i	−0.395399	−	0.684850i
$426$	−0.848048	−	2.61002i	−0.848048	−	2.61002i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.98054			1.98054			0.990268	−	0.139173i	$-0.0444444\pi$
$431$	1.98054			1.98054			0.990268	+	0.139173i	$0.0444444\pi$
$432$	−0.0674647	+	0.641884i	−0.0674647	+	0.641884i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	1.34700	+	0.286314i	1.34700	+	0.286314i
$436$	−1.71449	+	2.96958i	−1.71449	+	2.96958i
$437$		0			0
$438$		0			0
$439$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$439$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$440$	−3.52019			−3.52019
$441$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$442$		0			0
$443$	0.374607	+	0.648838i	0.374607	+	0.648838i	0.990268	−	0.139173i	$-0.0444444\pi$
$443$	0.374607	+	0.648838i	0.374607	+	0.648838i	−0.615661	+	0.788011i	$0.711111\pi$
$444$		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			$\pi$
$450$	−0.708254	+	0.514577i	−0.708254	+	0.514577i
$451$		0			0
$452$	−1.25579	−	2.17508i	−1.25579	−	2.17508i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$457$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$458$		0			0
$459$	−1.23949	−	0.900539i	−1.23949	−	0.900539i
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$	−0.360914	+	0.625121i	−0.360914	+	0.625121i
$465$	0.671923	+	2.06797i	0.671923	+	2.06797i
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.748346	−	0.831123i	0.748346	−	0.831123i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.69610			1.69610
$479$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	0.848048	−	0.529919i	$-0.177778\pi$
$479$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	−0.882948	+	0.469472i	$0.844444\pi$
$480$	0.149288	+	0.459460i	0.149288	+	0.459460i
$481$		0			0
$482$	−1.13491	+	1.96572i	−1.13491	+	1.96572i
$483$		0			0
$484$	−2.52994	−	4.38198i	−2.52994	−	4.38198i
$485$		0			0
$486$	−0.848048	+	1.46886i	−0.848048	+	1.46886i
$487$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$487$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$488$	−1.13914	−	1.97305i	−1.13914	−	1.97305i
$489$	−1.93726	−	0.411777i	−1.93726	−	0.411777i
$490$	1.04422	−	1.80864i	1.04422	−	1.80864i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$	−0.856733	−	1.48391i	−0.856733	−	1.48391i
$494$		0			0
$495$	−2.16259	−	0.962846i	−2.16259	−	0.962846i
$496$	−1.13974			−1.13974
$497$		0			0
$498$	−2.26429	+	2.51475i	−2.26429	+	2.51475i
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	−0.559051	+	0.968306i	−0.559051	+	0.968306i
$501$		0			0
$502$	1.59381	+	2.76056i	1.59381	+	2.76056i
$503$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$503$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$504$		0			0
$505$	−0.760999			−0.760999
$506$		0			0
$507$	0.309017	+	0.951057i	0.309017	+	0.951057i
$508$	1.35002	−	2.33830i	1.35002	−	2.33830i
$509$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$509$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$510$	3.12976	+	0.665251i	3.12976	+	0.665251i
$511$		0			0
$512$	−1.21299			−1.21299
$513$		0			0
$514$	−2.44014			−2.44014
$515$		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			$\pi$
$522$	−1.53462	+	1.11496i	−1.53462	+	1.11496i
$523$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$523$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$524$		0			0
$525$		0			0
$526$	−0.743520	+	1.28781i	−0.743520	+	1.28781i
$527$	1.35275	−	2.34304i	1.35275	−	2.34304i
$528$	0.830280	−	0.922119i	0.830280	−	0.922119i
$529$	−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
$536$	−0.258222	+	0.447253i	−0.258222	+	0.447253i
$537$		0			0
$538$	1.49756	+	2.59386i	1.49756	+	2.59386i
$539$	1.92252			1.92252
$540$	0.241552	−	2.29822i	0.241552	−	2.29822i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	−1.63039	−	2.82392i	−1.63039	−	2.82392i
$543$		0			0
$544$	0.300554	−	0.520576i	0.300554	−	0.520576i
$545$	1.12487	−	1.94833i	1.12487	−	1.94833i
$546$		0			0
$547$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$547$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$548$		0			0
$549$	−0.160147	−	1.52370i	−0.160147	−	1.52370i
$550$	1.68307			1.68307
$551$		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			$\pi$
$558$	−2.73619	−	1.21823i	−2.73619	−	1.21823i
$559$		0			0
$560$		0			0
$561$	0.910202	+	2.80131i	0.910202	+	2.80131i
$562$		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
$565$	0.823916	+	1.42706i	0.823916	+	1.42706i
$566$	2.87674			2.87674
$567$		0			0
$568$	−2.40608			−2.40608
$569$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.848048	+	0.529919i	$0.177778\pi$
$569$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.0348995	+	0.999391i	$0.488889\pi$
$570$		0			0
$571$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$571$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1.19754	−	0.533181i	−1.19754	−	0.533181i
$577$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$577$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$578$	−1.14257	−	1.97899i	−1.14257	−	1.97899i
$579$	1.32524	−	1.47183i	1.32524	−	1.47183i
$580$	1.29223	−	2.23820i	1.29223	−	2.23820i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−2.74434			−2.74434
$587$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.615661	−	0.788011i	$-0.711111\pi$
$587$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.374607	−	0.927184i	$-0.622222\pi$
$588$	0.579945	+	1.78489i	0.579945	+	1.78489i
$589$		0			0
$590$		0			0
$591$	0.473271	+	0.100597i	0.473271	+	0.100597i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	2.97887	−	1.32628i	2.97887	−	1.32628i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$599$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$600$	0.237184	+	0.729978i	0.237184	+	0.729978i
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$	−0.280969	+	0.204136i	−0.280969	+	0.204136i
$604$		0			0
$605$	1.65988	+	2.87500i	1.65988	+	2.87500i
$606$	0.701413	−	0.778998i	0.701413	−	0.778998i
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$	1.59984	+	2.77100i	1.59984	+	2.77100i
$611$		0			0
$612$	−2.32620	+	1.69008i	−2.32620	+	1.69008i
$613$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$613$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$614$	1.04422	+	1.80864i	1.04422	+	1.80864i
$615$		0			0
$616$		0			0
$617$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$	4.08076			4.08076
$621$		0			0
$622$	−3.18762			−3.18762
$623$		0			0
$624$		0			0
$625$	0.624869	−	1.08231i	0.624869	−	1.08231i
$626$		0			0
$627$		0			0
$628$	−1.04946	−	1.81772i	−1.04946	−	1.81772i
$629$		0			0
$630$		0			0
$631$	−1.23132			−1.23132			−0.615661	−	0.788011i	$-0.711111\pi$
$631$	−1.23132			−1.23132			−0.615661	+	0.788011i	$0.711111\pi$
$632$		0			0
$633$	−1.33500	+	1.48267i	−1.33500	+	1.48267i
$634$		0			0
$635$	−0.885740	+	1.53415i	−0.885740	+	1.53415i
$636$		0			0
$637$		0			0
$638$	3.64682			3.64682
$639$	−1.47815	−	0.658114i	−1.47815	−	0.658114i
$640$	2.25458			2.25458
$641$	0.241922	+	0.419021i	0.241922	+	0.419021i	0.961262	−	0.275637i	$-0.0888889\pi$
$641$	0.241922	+	0.419021i	0.241922	+	0.419021i	−0.719340	+	0.694658i	$0.755556\pi$
$642$		0			0
$643$	0.719340	−	1.24593i	0.719340	−	1.24593i	−0.241922	−	0.970296i	$-0.577778\pi$
$643$	0.719340	−	1.24593i	0.719340	−	1.24593i	0.961262	−	0.275637i	$-0.0888889\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.0697990			0.0697990			0.0348995	−	0.999391i	$-0.488889\pi$
$647$	0.0697990			0.0697990			0.0348995	+	0.999391i	$0.488889\pi$
$648$	0.995023	+	1.10509i	0.995023	+	1.10509i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.85848	+	3.21898i	−1.85848	+	3.21898i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	0.957620	+	2.94725i	0.957620	+	2.94725i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$	−2.97276	+	3.30158i	−2.97276	+	3.30158i
$661$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.241922	+	0.970296i	$0.577778\pi$
$661$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.719340	+	0.694658i	$0.755556\pi$
$662$		0			0
$663$		0			0
$664$	1.48342	+	2.56935i	1.48342	+	2.56935i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	0.362654	−	0.628135i	0.362654	−	0.628135i
$671$	−1.47274	+	2.55086i	−1.47274	+	2.55086i
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$	−2.08844			−2.08844
$675$	−0.0539530	+	0.513329i	−0.0539530	+	0.513329i
$676$	1.87674			1.87674
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$	−2.22022	−	0.471922i	−2.22022	−	0.471922i
$679$		0			0
$680$	1.40265	−	2.42946i	1.40265	−	2.42946i
$681$		0			0
$682$	2.87910	+	4.98675i	2.87910	+	4.98675i
$683$		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
$691$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$691$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$692$		0			0
$693$		0			0
$694$	−2.99513			−2.99513
$695$		0			0
$696$	0.513921	+	1.58169i	0.513921	+	1.58169i
$697$		0			0
$698$	1.22007	−	2.11322i	1.22007	−	2.11322i
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	1.26009	+	2.18255i	1.26009	+	2.18255i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$	3.37917			3.37917
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	0.669131	−	0.743145i	0.669131	−	0.743145i
$718$	−0.524123	−	0.907807i	−0.524123	−	0.907807i
$719$	−0.749213			−0.749213			−0.374607	−	0.927184i	$-0.622222\pi$
$719$	−0.749213			−0.749213			−0.374607	+	0.927184i	$0.622222\pi$
$720$	−0.726013	−	0.323242i	−0.726013	−	0.323242i
$721$		0			0
$722$	−0.848048	−	1.46886i	−0.848048	−	1.46886i
$723$	0.413545	+	1.27276i	0.413545	+	1.27276i
$724$		0			0
$725$	−0.288631	+	0.499923i	−0.288631	+	0.499923i
$726$	−4.47291	−	0.950747i	−4.47291	−	0.950747i
$727$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$727$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$728$		0			0
$729$	0.309017	+	0.951057i	0.309017	+	0.951057i
$730$		0			0
$731$		0			0
$732$	−2.81250	−	0.597816i	−2.81250	−	0.597816i
$733$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.0348995	−	0.999391i	$-0.488889\pi$
$733$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.848048	−	0.529919i	$-0.177778\pi$
$734$	1.69196	−	2.93057i	1.69196	−	2.93057i
$735$	−0.380500	−	1.17106i	−0.380500	−	1.17106i
$736$		0			0
$737$	0.667685			0.667685
$738$		0			0
$739$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$739$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$	−1.75711	+	1.95147i	−1.75711	+	1.95147i
$745$		0			0
$746$	−2.74434			−2.74434
$747$	0.208548	+	1.98420i	0.208548	+	1.98420i
$748$	5.52790			5.52790
$749$		0			0
$750$	0.312255	+	0.961023i	0.312255	+	0.961023i
$751$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$751$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$752$		0			0
$753$	1.83832	+	0.390746i	1.83832	+	0.390746i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.69610			1.69610			0.848048	−	0.529919i	$-0.177778\pi$
$757$	1.69610			1.69610			0.848048	+	0.529919i	$0.177778\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$761$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$762$	−0.754044	−	2.32071i	−0.754044	−	2.32071i
$763$		0			0
$764$		0			0
$765$	1.52621	−	1.10885i	1.52621	−	1.10885i
$766$	−1.69610			−1.69610
$767$		0			0
$768$	−1.20090	+	1.33374i	−1.20090	+	1.33374i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$	−0.962665	+	1.06915i	−0.962665	+	1.06915i
$772$	−1.85848	−	3.21898i	−1.85848	−	3.21898i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−0.911478			−0.911478
$776$		0			0
$777$		0			0
$778$	−0.0591929	+	0.102525i	−0.0591929	+	0.102525i
$779$		0			0
$780$		0			0
$781$	1.55535	+	2.69395i	1.55535	+	2.69395i
$782$		0			0
$783$	−0.116903	+	1.11226i	−0.116903	+	1.11226i
$784$	0.645419			0.645419
$785$	0.688547	+	1.19260i	0.688547	+	1.19260i
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	0.454025	−	0.786394i	0.454025	−	0.786394i
$789$	0.270928	+	0.833831i	0.270928	+	0.833831i
$790$		0			0
$791$		0			0
$792$	−0.298833	−	2.84321i	−0.298833	−	2.84321i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	−0.882948	−	0.469472i	$-0.844444\pi$
$797$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	0.848048	+	0.529919i	$0.177778\pi$
$798$		0			0
$799$		0			0
$800$	−0.202512			−0.202512
$801$		0			0
$802$	−3.38393			−3.38393
$803$		0			0
$804$	0.201413	+	0.619885i	0.201413	+	0.619885i
$805$		0			0
$806$		0			0
$807$	1.72731	+	0.367150i	1.72731	+	0.367150i
$808$	−0.459520	−	0.795913i	−0.459520	−	0.795913i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−1.39744	−	1.55201i	−1.39744	−	1.55201i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−1.88051	−	0.399715i	−1.88051	−	0.399715i
$814$		0			0
$815$	1.21934	−	2.11196i	1.21934	−	2.11196i
$816$	0.305568	+	0.940443i	0.305568	+	0.940443i
$817$		0			0
$818$	−1.27074			−1.27074
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$	0.663993	−	0.737439i	0.663993	−	0.737439i
$826$		0			0
$827$	1.11839			1.11839			0.559193	−	0.829038i	$-0.311111\pi$
$827$	1.11839			1.11839			0.559193	+	0.829038i	$0.311111\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$	−2.08335	−	3.60848i	−2.08335	−	3.60848i
$831$		0			0
$832$		0			0
$833$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.61323	+	0.718254i	−1.61323	+	0.718254i
$838$	1.48704			1.48704
$839$	0.241922	+	0.419021i	0.241922	+	0.419021i	0.961262	−	0.275637i	$-0.0888889\pi$
$839$	0.241922	+	0.419021i	0.241922	+	0.419021i	−0.719340	+	0.694658i	$0.755556\pi$
$840$		0			0
$841$	−0.125393	+	0.217188i	−0.125393	+	0.217188i
$842$	−1.13491	+	1.96572i	−1.13491	+	1.96572i
$843$		0			0
$844$	1.87217	+	3.24269i	1.87217	+	3.24269i
$845$	−1.23132			−1.23132
$846$		0			0
$847$		0			0
$848$		0			0
$849$	1.13491	−	1.26045i	1.13491	−	1.26045i
$850$	−0.670634	+	1.16157i	−0.670634	+	1.16157i
$851$		0			0
$852$	−2.03190	+	2.25666i	−2.03190	+	2.25666i
$853$	0.374607	+	0.648838i	0.374607	+	0.648838i	0.990268	−	0.139173i	$-0.0444444\pi$
$853$	0.374607	+	0.648838i	0.374607	+	0.648838i	−0.615661	+	0.788011i	$0.711111\pi$
$854$		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
$859$	0.241922	−	0.419021i	0.241922	−	0.419021i	−0.719340	−	0.694658i	$-0.755556\pi$
$859$	0.241922	−	0.419021i	0.241922	−	0.419021i	0.961262	+	0.275637i	$0.0888889\pi$
$860$		0			0
$861$		0			0
$862$	−1.67959	−	2.90914i	−1.67959	−	2.90914i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.358426	+	0.159581i	−0.358426	+	0.159581i
$865$		0			0
$866$		0			0
$867$	−1.31785	−	0.280119i	−1.31785	−	0.280119i
$868$		0			0
$869$		0			0
$870$	−0.721766	−	2.22137i	−0.721766	−	2.22137i
$871$		0			0
$872$	2.71696			2.71696
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.438371	−	0.898794i	$-0.355556\pi$
$877$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.559193	−	0.829038i	$-0.311111\pi$
$878$	1.65903	−	2.87353i	1.65903	−	2.87353i
$879$	−1.08268	+	1.20243i	−1.08268	+	1.20243i
$880$	0.763934	+	1.32317i	0.763934	+	1.32317i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	1.54946	+	0.689864i	1.54946	+	0.689864i
$883$	0.0697990			0.0697990			0.0348995	−	0.999391i	$-0.488889\pi$
$883$	0.0697990			0.0697990			0.0348995	+	0.999391i	$0.488889\pi$
$884$		0			0
$885$		0			0
$886$	0.635369	−	1.10049i	0.635369	−	1.10049i
$887$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$887$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.594092	−	1.82843i	0.594092	−	1.82843i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.97495			−1.97495
$900$	0.884944	+	0.394003i	0.884944	+	0.394003i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.995023	+	1.72343i	−0.995023	+	1.72343i
$905$		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			0
$909$	−0.0646021	−	0.614648i	−0.0646021	−	0.614648i
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$		0			0
$913$	1.91784	−	3.32180i	1.91784	−	3.32180i
$914$	1.37217	−	2.37667i	1.37217	−	2.37667i
$915$	1.84527	+	0.392225i	1.84527	+	0.392225i
$916$		0			0
$917$		0			0
$918$	−0.271625	+	2.58433i	−0.271625	+	2.58433i
$919$	1.98054			1.98054			0.990268	−	0.139173i	$-0.0444444\pi$
$919$	1.98054			1.98054			0.990268	+	0.139173i	$0.0444444\pi$
$920$		0			0
$921$	1.20442	+	0.256006i	1.20442	+	0.256006i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−0.438794			−0.438794
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$	2.46773	−	2.74070i	2.46773	−	2.74070i
$931$		0			0
$932$		0			0
$933$	−1.25755	+	1.39666i	−1.25755	+	1.39666i
$934$		0			0
$935$	−3.62683			−3.62683
$936$		0			0
$937$	0.0697990			0.0697990			0.0348995	−	0.999391i	$-0.488889\pi$
$937$	0.0697990			0.0697990			0.0348995	+	0.999391i	$0.488889\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$	−1.85544	−	0.394386i	−1.85544	−	0.394386i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		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
$956$	−0.938371	−	1.62531i	−0.938371	−	1.62531i
$957$	1.43871	−	1.59785i	1.43871	−	1.59785i
$958$	−0.0591929	+	0.102525i	−0.0591929	+	0.102525i
$959$		0			0
$960$	1.08005	−	1.19952i	1.08005	−	1.19952i
$961$	−1.05919	−	1.83458i	−1.05919	−	1.83458i
$962$		0			0
$963$		0			0
$964$	2.51157			2.51157
$965$	1.21934	+	2.11196i	1.21934	+	2.11196i
$966$		0			0
$967$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$967$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$968$	−2.00460	+	3.47207i	−2.00460	+	3.47207i
$969$		0			0
$970$		0			0
$971$	−0.749213			−0.749213			−0.374607	−	0.927184i	$-0.622222\pi$
$971$	−0.749213			−0.749213			−0.374607	+	0.927184i	$0.622222\pi$
$972$	1.87674			1.87674
$973$		0			0
$974$	1.59381	+	2.76056i	1.59381	+	2.76056i
$975$		0			0
$976$	−0.494420	+	0.856360i	−0.494420	+	0.856360i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$	1.03804	+	3.19477i	1.03804	+	3.19477i
$979$		0			0
$980$	−2.31088			−2.31088
$981$	1.66913	+	0.743145i	1.66913	+	0.743145i
$982$		0			0
$983$	0.719340	+	1.24593i	0.719340	+	1.24593i	0.961262	+	0.275637i	$0.0888889\pi$
$983$	0.719340	+	1.24593i	0.719340	+	1.24593i	−0.241922	+	0.970296i	$0.577778\pi$
$984$		0			0
$985$	−0.297884	+	0.515950i	−0.297884	+	0.515950i
$986$	−1.45310	+	2.51685i	−1.45310	+	2.51685i
$987$		0			0
$988$		0			0
$989$		0			0
$990$	0.419690	+	3.99308i	0.419690	+	3.99308i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−0.346421	−	0.600019i	−0.346421	−	0.600019i
$993$		0			0
$994$		0			0
$995$		0			0
$996$	3.66252	+	0.778492i	3.66252	+	0.778492i
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.1.f.b.238.3	✓	24
9.7	even	3	inner	2151.1.f.b.1672.3	yes	24
239.238	odd	2	CM	2151.1.f.b.238.3	✓	24
2151.1672	odd	6	inner	2151.1.f.b.1672.3	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.1.f.b.238.3	✓	24	1.1	even	1	trivial
2151.1.f.b.238.3	✓	24	239.238	odd	2	CM
2151.1.f.b.1672.3	yes	24	9.7	even	3	inner
2151.1.f.b.1672.3	yes	24	2151.1672	odd	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2151.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.848048 - 1.46886i) q^{2} +(-0.978148 - 0.207912i) q^{3} +(-0.938371 + 1.62531i) q^{4} +(0.615661 - 1.06636i) q^{5} +(0.524123 + 1.61308i) q^{6} +1.48704 q^{8} +(0.913545 + 0.406737i) q^{9} +O(q^{10})\)	\(q+(-0.848048 - 1.46886i) q^{2} +(-0.978148 - 0.207912i) q^{3} +(-0.938371 + 1.62531i) q^{4} +(0.615661 - 1.06636i) q^{5} +(0.524123 + 1.61308i) q^{6} +1.48704 q^{8} +(0.913545 + 0.406737i) q^{9} -2.08844 q^{10} +(-0.961262 - 1.66495i) q^{11} +(1.25579 - 1.39469i) q^{12} +(-0.823916 + 0.915051i) q^{15} +(-0.322710 - 0.558950i) q^{16} +1.53209 q^{17} +(-0.177290 - 1.68680i) q^{18} +(1.15544 + 2.00128i) q^{20} +(-1.63039 + 2.82392i) q^{22} +(-1.45454 - 0.309173i) q^{24} +(-0.258078 - 0.447004i) q^{25} +(-0.809017 - 0.587785i) q^{27} +(-0.559193 - 0.968551i) q^{29} +(2.04280 + 0.434212i) q^{30} +(0.882948 - 1.52931i) q^{31} +(0.196173 - 0.339782i) q^{32} +(0.594092 + 1.82843i) q^{33} +(-1.29929 - 2.25043i) q^{34} +(-1.51832 + 1.10312i) q^{36} +(0.915513 - 1.58571i) q^{40} +3.60808 q^{44} +(0.996161 - 0.723753i) q^{45} +(0.199446 + 0.613830i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.437725 + 0.758162i) q^{50} +(-1.49861 - 0.318539i) q^{51} +(-0.177290 + 1.68680i) q^{54} -2.36725 q^{55} +(-0.948445 + 1.64275i) q^{58} +(-0.714100 - 2.19777i) q^{60} +(-0.766044 - 1.32683i) q^{61} -2.99513 q^{62} -1.31088 q^{64} +(2.18189 - 2.42324i) q^{66} +(-0.173648 + 0.300767i) q^{67} +(-1.43767 + 2.49011i) q^{68} -1.61803 q^{71} +(1.35848 + 0.604833i) q^{72} +(0.159501 + 0.490894i) q^{75} -0.794720 q^{80} +(0.669131 + 0.743145i) q^{81} +(0.997564 + 1.72783i) q^{83} +(0.943248 - 1.63375i) q^{85} +(0.345600 + 1.06365i) q^{87} +(-1.42943 - 2.47585i) q^{88} +(-1.90789 - 0.849446i) q^{90} +(-1.18161 + 1.31232i) q^{93} +(-0.262531 + 0.291570i) q^{96} +1.69610 q^{98} +(-0.200958 - 1.91199i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 3 q^{3} - 12 q^{4} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) 24 * q + 3 * q^3 - 12 * q^4 + 6 * q^8 + 3 * q^9	\( 24 q + 3 q^{3} - 12 q^{4} + 6 q^{8} + 3 q^{9} + 6 q^{10} + 3 q^{12} - 12 q^{16} - 3 q^{20} - 3 q^{22} - 3 q^{24} - 12 q^{25} - 6 q^{27} - 3 q^{30} - 3 q^{32} + 6 q^{34} - 6 q^{36} - 3 q^{40} + 6 q^{44} - 6 q^{48} - 12 q^{49} + 6 q^{50} - 12 q^{55} - 3 q^{58} - 9 q^{60} - 24 q^{62} + 30 q^{64} + 27 q^{66} - 3 q^{68} - 12 q^{71} - 18 q^{72} - 6 q^{75} + 54 q^{80} + 3 q^{81} - 3 q^{85} - 3 q^{88} - 18 q^{90} - 3 q^{96}+O(q^{100}) \) 24 * q + 3 * q^3 - 12 * q^4 + 6 * q^8 + 3 * q^9 + 6 * q^10 + 3 * q^12 - 12 * q^16 - 3 * q^20 - 3 * q^22 - 3 * q^24 - 12 * 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 - 12 * q^49 + 6 * q^50 - 12 * q^55 - 3 * q^58 - 9 * q^60 - 24 * q^62 + 30 * q^64 + 27 * q^66 - 3 * q^68 - 12 * q^71 - 18 * q^72 - 6 * q^75 + 54 * q^80 + 3 * q^81 - 3 * q^85 - 3 * q^88 - 18 * q^90 - 3 * q^96

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