LMFDB - Embedded newform 2151.1.f.b.1672.1

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			1672.1
Root			$-0.719340 - 0.694658i$ of defining polynomial
Character	$\chi$	$=$	2151.1672
Dual form			2151.1.f.b.238.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.990268 + 1.71519i) q^{2} +(0.669131 - 0.743145i) q^{3} +(-1.46126 - 2.53098i) q^{4} +(-0.848048 - 1.46886i) q^{5} +(0.612019 + 1.88360i) q^{6} +3.80763 q^{8} +(-0.104528 - 0.994522i) q^{9} +O(q^{10})$	\(q+(-0.990268 + 1.71519i) q^{2} +(0.669131 - 0.743145i) q^{3} +(-1.46126 - 2.53098i) q^{4} +(-0.848048 - 1.46886i) q^{5} +(0.612019 + 1.88360i) q^{6} +3.80763 q^{8} +(-0.104528 - 0.994522i) q^{9} +3.35918 q^{10} +(0.997564 - 1.72783i) q^{11} +(-2.85866 - 0.607627i) q^{12} +(-1.65903 - 0.352638i) q^{15} +(-2.30931 + 3.99984i) q^{16} +0.347296 q^{17} +(1.80931 + 0.805557i) q^{18} +(-2.47844 + 4.29278i) q^{20} +(1.97571 + 3.42203i) q^{22} +(2.54780 - 2.82962i) q^{24} +(-0.938371 + 1.62531i) q^{25} +(-0.809017 - 0.587785i) q^{27} +(0.241922 - 0.419021i) q^{29} +(2.24773 - 2.49636i) q^{30} +(0.615661 + 1.06636i) q^{31} +(-2.66986 - 4.62433i) q^{32} +(-0.616528 - 1.89748i) q^{33} +(-0.343916 + 0.595681i) q^{34} +(-2.36437 + 1.71782i) q^{36} +(-3.22905 - 5.59288i) q^{40} -5.83081 q^{44} +(-1.37217 + 0.996940i) q^{45} +(1.42723 + 4.39257i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(-1.85848 - 3.21898i) q^{50} +(0.232387 - 0.258091i) q^{51} +(1.80931 - 0.805557i) q^{54} -3.38393 q^{55} +(0.479135 + 0.829886i) q^{58} +(1.53176 + 4.71427i) q^{60} +(-0.173648 + 0.300767i) q^{61} -2.43868 q^{62} +5.95688 q^{64} +(3.86508 + 0.821547i) q^{66} +(0.939693 + 1.62760i) q^{67} +(-0.507491 - 0.879000i) q^{68} -1.61803 q^{71} +(-0.398005 - 3.78677i) q^{72} +(0.579945 + 1.78489i) q^{75} +7.83362 q^{80} +(-0.978148 + 0.207912i) q^{81} +(0.719340 - 1.24593i) q^{83} +(-0.294524 - 0.510131i) q^{85} +(-0.149516 - 0.460163i) q^{87} +(3.79835 - 6.57894i) q^{88} +(-0.351130 - 3.34078i) q^{90} +(1.20442 + 0.256006i) q^{93} +(-5.22303 - 1.11019i) q^{96} +1.98054 q^{98} +(-1.82264 - 0.811492i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times$.

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

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 $

$2$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.374607	+	0.927184i	$0.622222\pi$
$2$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.615661	+	0.788011i	$0.711111\pi$
$3$	0.669131	−	0.743145i	0.669131	−	0.743145i
$3$	0.669131	−	0.743145i	0.669131	−	0.743145i
$4$	−1.46126	−	2.53098i	−1.46126	−	2.53098i
$5$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	−0.882948	−	0.469472i	$-0.844444\pi$
$5$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	0.0348995	−	0.999391i	$-0.488889\pi$
$6$	0.612019	+	1.88360i	0.612019	+	1.88360i
$7$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	3.80763			3.80763
$9$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$10$	3.35918			3.35918
$11$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.438371	−	0.898794i	$-0.355556\pi$
$11$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.559193	−	0.829038i	$-0.311111\pi$
$12$	−2.85866	−	0.607627i	−2.85866	−	0.607627i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$	−1.65903	−	0.352638i	−1.65903	−	0.352638i
$16$	−2.30931	+	3.99984i	−2.30931	+	3.99984i
$17$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$17$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$18$	1.80931	+	0.805557i	1.80931	+	0.805557i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−2.47844	+	4.29278i	−2.47844	+	4.29278i
$21$		0			0
$22$	1.97571	+	3.42203i	1.97571	+	3.42203i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	2.54780	−	2.82962i	2.54780	−	2.82962i
$25$	−0.938371	+	1.62531i	−0.938371	+	1.62531i
$26$		0			0
$27$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$28$		0			0
$29$	0.241922	−	0.419021i	0.241922	−	0.419021i	−0.719340	−	0.694658i	$-0.755556\pi$
$29$	0.241922	−	0.419021i	0.241922	−	0.419021i	0.961262	+	0.275637i	$0.0888889\pi$
$30$	2.24773	−	2.49636i	2.24773	−	2.49636i
$31$	0.615661	+	1.06636i	0.615661	+	1.06636i	0.990268	+	0.139173i	$0.0444444\pi$
$31$	0.615661	+	1.06636i	0.615661	+	1.06636i	−0.374607	+	0.927184i	$0.622222\pi$
$32$	−2.66986	−	4.62433i	−2.66986	−	4.62433i
$33$	−0.616528	−	1.89748i	−0.616528	−	1.89748i
$34$	−0.343916	+	0.595681i	−0.343916	+	0.595681i
$35$		0			0
$36$	−2.36437	+	1.71782i	−2.36437	+	1.71782i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	−3.22905	−	5.59288i	−3.22905	−	5.59288i
$41$		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	$0.666667\pi$
$44$	−5.83081			−5.83081
$45$	−1.37217	+	0.996940i	−1.37217	+	0.996940i
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	1.42723	+	4.39257i	1.42723	+	4.39257i
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$	−1.85848	−	3.21898i	−1.85848	−	3.21898i
$51$	0.232387	−	0.258091i	0.232387	−	0.258091i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	1.80931	−	0.805557i	1.80931	−	0.805557i
$55$	−3.38393			−3.38393
$56$		0			0
$57$		0			0
$58$	0.479135	+	0.829886i	0.479135	+	0.829886i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$	1.53176	+	4.71427i	1.53176	+	4.71427i
$61$	−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	$0.222222\pi$
$62$	−2.43868			−2.43868
$63$		0			0
$64$	5.95688			5.95688
$65$		0			0
$66$	3.86508	+	0.821547i	3.86508	+	0.821547i
$67$	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	$0.444444\pi$
$68$	−0.507491	−	0.879000i	−0.507491	−	0.879000i
$69$		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	$0.800000\pi$
$72$	−0.398005	−	3.78677i	−0.398005	−	3.78677i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	0.579945	+	1.78489i	0.579945	+	1.78489i
$76$		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	$0.666667\pi$
$80$	7.83362			7.83362
$81$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$82$		0			0
$83$	0.719340	−	1.24593i	0.719340	−	1.24593i	−0.241922	−	0.970296i	$-0.577778\pi$
$83$	0.719340	−	1.24593i	0.719340	−	1.24593i	0.961262	−	0.275637i	$-0.0888889\pi$
$84$		0			0
$85$	−0.294524	−	0.510131i	−0.294524	−	0.510131i
$86$		0			0
$87$	−0.149516	−	0.460163i	−0.149516	−	0.460163i
$88$	3.79835	−	6.57894i	3.79835	−	6.57894i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−0.351130	−	3.34078i	−0.351130	−	3.34078i
$91$		0			0
$92$		0			0
$93$	1.20442	+	0.256006i	1.20442	+	0.256006i
$94$		0			0
$95$		0			0
$96$	−5.22303	−	1.11019i	−5.22303	−	1.11019i
$97$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$97$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$98$	1.98054			1.98054
$99$	−1.82264	−	0.811492i	−1.82264	−	0.811492i
$100$	5.48482			5.48482
$101$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$101$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$102$	0.212552	+	0.654168i	0.212552	+	0.654168i
$103$		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			$\pi$
$108$	−0.305487	+	2.90651i	−0.305487	+	2.90651i
$109$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$109$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$110$	3.35100	−	5.80410i	3.35100	−	5.80410i
$111$		0			0
$112$		0			0
$113$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.669131	+	0.743145i	$0.266667\pi$
$113$	0.978148	+	1.69420i	0.978148	+	1.69420i	0.309017	+	0.951057i	$0.400000\pi$
$114$		0			0
$115$		0			0
$116$	−1.41404			−1.41404
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−6.31698	−	1.34271i	−6.31698	−	1.34271i
$121$	−1.49027	−	2.58122i	−1.49027	−	2.58122i
$122$	−0.343916	−	0.595681i	−0.343916	−	0.595681i
$123$		0			0
$124$	1.79929	−	3.11645i	1.79929	−	3.11645i
$125$	1.48704			1.48704
$126$		0			0
$127$	1.11839			1.11839			0.559193	−	0.829038i	$-0.311111\pi$
$127$	1.11839			1.11839			0.559193	+	0.829038i	$0.311111\pi$
$128$	−3.22905	+	5.59288i	−3.22905	+	5.59288i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	−3.90157	+	4.33314i	−3.90157	+	4.33314i
$133$		0			0
$134$	−3.72219			−3.72219
$135$	−0.177290	+	1.68680i	−0.177290	+	1.68680i
$136$	1.32237			1.32237
$137$		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
$142$	1.60229	−	2.77524i	1.60229	−	2.77524i
$143$		0			0
$144$	4.21932	+	1.87856i	4.21932	+	1.87856i
$145$	−0.820646			−0.820646
$146$		0			0
$147$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	−3.63573	−	0.772799i	−3.63573	−	0.772799i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$	−0.0363024	−	0.345394i	−0.0363024	−	0.345394i
$154$		0			0
$155$	1.04422	−	1.80864i	1.04422	−	1.80864i
$156$		0			0
$157$	0.241922	+	0.419021i	0.241922	+	0.419021i	0.961262	−	0.275637i	$-0.0888889\pi$
$157$	0.241922	+	0.419021i	0.241922	+	0.419021i	−0.719340	+	0.694658i	$0.755556\pi$
$158$		0			0
$159$		0			0
$160$	−4.52834	+	7.84331i	−4.52834	+	7.84331i
$161$		0			0
$162$	0.612019	−	1.88360i	0.612019	−	1.88360i
$163$	0.0697990			0.0697990			0.0348995	−	0.999391i	$-0.488889\pi$
$163$	0.0697990			0.0697990			0.0348995	+	0.999391i	$0.488889\pi$
$164$		0			0
$165$	−2.26429	+	2.51475i	−2.26429	+	2.51475i
$166$	1.42468	+	2.46762i	1.42468	+	2.46762i
$167$		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
$170$	1.16663			1.16663
$171$		0			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$	0.937330	+	0.199236i	0.937330	+	0.199236i
$175$		0			0
$176$	4.60737	+	7.98020i	4.60737	+	7.98020i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	4.52834	+	2.01615i	4.52834	+	2.01615i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	0.107320	+	0.330298i	0.107320	+	0.330298i
$184$		0			0
$185$		0			0
$186$	−1.63180	+	1.81229i	−1.63180	+	1.81229i
$187$	0.346450	−	0.600070i	0.346450	−	0.600070i
$188$		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	$0.666667\pi$
$192$	3.98593	−	4.42683i	3.98593	−	4.42683i
$193$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	0.848048	−	0.529919i	$-0.177778\pi$
$193$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	−0.882948	+	0.469472i	$0.844444\pi$
$194$		0			0
$195$		0			0
$196$	−1.46126	+	2.53098i	−1.46126	+	2.53098i
$197$	0.876742			0.876742			0.438371	−	0.898794i	$-0.355556\pi$
$197$	0.876742			0.876742			0.438371	+	0.898794i	$0.355556\pi$
$198$	3.19677	−	2.32259i	3.19677	−	2.32259i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−3.57297	+	6.18856i	−3.57297	+	6.18856i
$201$	1.83832	+	0.390746i	1.83832	+	0.390746i
$202$	−0.612019	−	1.06005i	−0.612019	−	1.06005i
$203$		0			0
$204$	−0.992802	−	0.211027i	−0.992802	−	0.211027i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.719340	+	1.24593i	0.719340	+	1.24593i	0.961262	+	0.275637i	$0.0888889\pi$
$211$	0.719340	+	1.24593i	0.719340	+	1.24593i	−0.241922	+	0.970296i	$0.577778\pi$
$212$		0			0
$213$	−1.08268	+	1.20243i	−1.08268	+	1.20243i
$214$		0			0
$215$		0			0
$216$	−3.08043	−	2.23807i	−3.08043	−	2.23807i
$217$		0			0
$218$	0.207022	−	0.358573i	0.207022	−	0.358573i
$219$		0			0
$220$	4.94481	+	8.56466i	4.94481	+	8.56466i
$221$		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
$225$	1.71449	+	0.763340i	1.71449	+	0.763340i
$226$	−3.87451			−3.87451
$227$		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
$232$	0.921148	−	1.59548i	0.921148	−	1.59548i
$233$		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
$240$	5.24172	−	5.82152i	5.24172	−	5.82152i
$241$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$241$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$242$	5.90306			5.90306
$243$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$244$	1.01498			1.01498
$245$	−0.848048	+	1.46886i	−0.848048	+	1.46886i
$246$		0			0
$247$		0			0
$248$	2.34421	+	4.06029i	2.34421	+	4.06029i
$249$	−0.444576	−	1.36827i	−0.444576	−	1.36827i
$250$	−1.47257	+	2.55056i	−1.47257	+	2.55056i
$251$	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
$254$	−1.10750	+	1.91825i	−1.10750	+	1.91825i
$255$	−0.576176	−	0.122470i	−0.576176	−	0.122470i
$256$	−3.41681	−	5.91809i	−3.41681	−	5.91809i
$257$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	−0.997564	−	0.0697565i	$-0.977778\pi$
$257$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	0.438371	−	0.898794i	$-0.355556\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.442013	−	0.196797i	−0.442013	−	0.196797i
$262$		0			0
$263$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.241922	+	0.970296i	$0.577778\pi$
$263$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.719340	+	0.694658i	$0.755556\pi$
$264$	−2.34751	−	7.22489i	−2.34751	−	7.22489i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	2.74627	−	4.75669i	2.74627	−	4.75669i
$269$	−1.23132			−1.23132			−0.615661	−	0.788011i	$-0.711111\pi$
$269$	−1.23132			−1.23132			−0.615661	+	0.788011i	$0.711111\pi$
$270$	−2.71763	−	1.97448i	−2.71763	−	1.97448i
$271$	−1.99513			−1.99513			−0.997564	−	0.0697565i	$-0.977778\pi$
$271$	−1.99513			−1.99513			−0.997564	+	0.0697565i	$0.977778\pi$
$272$	−0.802015	+	1.38913i	−0.802015	+	1.38913i
$273$		0			0
$274$		0			0
$275$	1.87217	+	3.24269i	1.87217	+	3.24269i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$	0.996161	−	0.723753i	0.996161	−	0.723753i
$280$		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
$283$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.615661	−	0.788011i	$-0.711111\pi$
$283$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.374607	−	0.927184i	$-0.622222\pi$
$284$	2.36437	+	4.09521i	2.36437	+	4.09521i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−4.31992	+	3.13861i	−4.31992	+	3.13861i
$289$	−0.879385			−0.879385
$290$	0.812659	−	1.40757i	0.812659	−	1.40757i
$291$		0			0
$292$		0			0
$293$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$293$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$294$	1.32524	−	1.47183i	1.32524	−	1.47183i
$295$		0			0
$296$		0			0
$297$	−1.82264	+	0.811492i	−1.82264	+	0.811492i
$298$		0			0
$299$		0			0
$300$	3.67006	−	4.07602i	3.67006	−	4.07602i
$301$		0			0
$302$		0			0
$303$	0.190983	+	0.587785i	0.190983	+	0.587785i
$304$		0			0
$305$	0.589048			0.589048
$306$	0.628367	+	0.279767i	0.628367	+	0.279767i
$307$	1.69610			1.69610			0.848048	−	0.529919i	$-0.177778\pi$
$307$	1.69610			1.69610			0.848048	+	0.529919i	$0.177778\pi$
$308$		0			0
$309$		0			0
$310$	2.06812	+	3.58208i	2.06812	+	3.58208i
$311$	−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	$0.666667\pi$
$314$	−0.958270			−0.958270
$315$		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
$319$	−0.482665	−	0.836001i	−0.482665	−	0.836001i
$320$	−5.05172	−	8.74984i	−5.05172	−	8.74984i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.95555	+	2.17186i	1.95555	+	2.17186i
$325$		0			0
$326$	−0.0691197	+	0.119719i	−0.0691197	+	0.119719i
$327$	−0.139886	+	0.155360i	−0.139886	+	0.155360i
$328$		0			0
$329$		0			0
$330$	−2.07103	−	6.37398i	−2.07103	−	6.37398i
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	−4.20457			−4.20457
$333$		0			0
$334$		0			0
$335$	1.59381	−	2.76056i	1.59381	−	2.76056i
$336$		0			0
$337$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	−0.882948	−	0.469472i	$-0.844444\pi$
$337$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	0.0348995	−	0.999391i	$-0.488889\pi$
$338$	−0.990268	−	1.71519i	−0.990268	−	1.71519i
$339$	1.91355	+	0.406737i	1.91355	+	0.406737i
$340$	−0.860753	+	1.49087i	−0.860753	+	1.49087i
$341$	2.45665			2.45665
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.615661	+	1.06636i	0.615661	+	1.06636i	0.990268	+	0.139173i	$0.0444444\pi$
$347$	0.615661	+	1.06636i	0.615661	+	1.06636i	−0.374607	+	0.927184i	$0.622222\pi$
$348$	−0.946181	+	1.05084i	−0.946181	+	1.05084i
$349$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	0.438371	+	0.898794i	$0.355556\pi$
$349$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	−0.997564	+	0.0697565i	$0.977778\pi$
$350$		0			0
$351$		0			0
$352$	−10.6534			−10.6534
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$	1.37217	+	2.37667i	1.37217	+	2.37667i
$356$		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	$0.400000\pi$
$360$	−5.22471	+	3.79598i	−5.22471	+	3.79598i
$361$	1.00000			1.00000
$362$		0			0
$363$	−2.91540	−	0.619688i	−2.91540	−	0.619688i
$364$		0			0
$365$		0			0
$366$	−0.672802	−	0.143009i	−0.672802	−	0.143009i
$367$	0.719340	−	1.24593i	0.719340	−	1.24593i	−0.241922	−	0.970296i	$-0.577778\pi$
$367$	0.719340	−	1.24593i	0.719340	−	1.24593i	0.961262	−	0.275637i	$-0.0888889\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.11202	−	3.42244i	−1.11202	−	3.42244i
$373$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$373$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$374$	0.686157	+	1.18846i	0.686157	+	1.18846i
$375$	0.995023	−	1.10509i	0.995023	−	1.10509i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	0.748346	−	0.831123i	0.748346	−	0.831123i
$382$		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	$0.666667\pi$
$384$	1.99566	+	6.14202i	1.99566	+	6.14202i
$385$		0			0
$386$	0.138239			0.138239
$387$		0			0
$388$		0			0
$389$	0.374607	−	0.648838i	0.374607	−	0.648838i	−0.615661	−	0.788011i	$-0.711111\pi$
$389$	0.374607	−	0.648838i	0.374607	−	0.648838i	0.990268	+	0.139173i	$0.0444444\pi$
$390$		0			0
$391$		0			0
$392$	−1.90381	−	3.29750i	−1.90381	−	3.29750i
$393$		0			0
$394$	−0.868210	+	1.50378i	−0.868210	+	1.50378i
$395$		0			0
$396$	0.609485	+	5.79887i	0.609485	+	5.79887i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−4.33398	−	7.50667i	−4.33398	−	7.50667i
$401$	0.719340	+	1.24593i	0.719340	+	1.24593i	0.961262	+	0.275637i	$0.0888889\pi$
$401$	0.719340	+	1.24593i	0.719340	+	1.24593i	−0.241922	+	0.970296i	$0.577778\pi$
$402$	−2.49063	+	2.76613i	−2.49063	+	2.76613i
$403$		0			0
$404$	1.80622			1.80622
$405$	1.13491	+	1.26045i	1.13491	+	1.26045i
$406$		0			0
$407$		0			0
$408$	0.884842	−	0.982716i	0.884842	−	0.982716i
$409$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.848048	+	0.529919i	$0.177778\pi$
$409$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.0348995	+	0.999391i	$0.488889\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−2.44014			−2.44014
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.719340	−	0.694658i	$-0.755556\pi$
$419$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.241922	−	0.970296i	$-0.577778\pi$
$420$		0			0
$421$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$421$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$422$	−2.84936			−2.84936
$423$		0			0
$424$		0			0
$425$	−0.325893	+	0.564463i	−0.325893	+	0.564463i
$426$	−0.990268	−	3.04773i	−0.990268	−	3.04773i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.0697990			0.0697990			0.0348995	−	0.999391i	$-0.488889\pi$
$431$	0.0697990			0.0697990			0.0348995	+	0.999391i	$0.488889\pi$
$432$	4.21932	−	1.87856i	4.21932	−	1.87856i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−0.549119	+	0.609859i	−0.549119	+	0.609859i
$436$	0.305487	+	0.529119i	0.305487	+	0.529119i
$437$		0			0
$438$		0			0
$439$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$439$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$440$	−12.8847			−12.8847
$441$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$442$		0			0
$443$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.0348995	−	0.999391i	$-0.488889\pi$
$443$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.848048	−	0.529919i	$-0.177778\pi$
$444$		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			$\pi$
$450$	−3.00708	+	2.18477i	−3.00708	+	2.18477i
$451$		0			0
$452$	2.85866	−	4.95134i	2.85866	−	4.95134i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$457$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$458$		0			0
$459$	−0.280969	−	0.204136i	−0.280969	−	0.204136i
$460$		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	$0.333333\pi$
$464$	1.11735	+	1.93530i	1.11735	+	1.93530i
$465$	−0.645364	−	1.98623i	−0.645364	−	1.98623i
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.473271	+	0.100597i	0.473271	+	0.100597i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.98054			1.98054
$479$	0.374607	−	0.648838i	0.374607	−	0.648838i	−0.615661	−	0.788011i	$-0.711111\pi$
$479$	0.374607	−	0.648838i	0.374607	−	0.648838i	0.990268	+	0.139173i	$0.0444444\pi$
$480$	2.79867	+	8.61341i	2.79867	+	8.61341i
$481$		0			0
$482$	1.93726	+	3.35543i	1.93726	+	3.35543i
$483$		0			0
$484$	−4.35534	+	7.54368i	−4.35534	+	7.54368i
$485$		0			0
$486$	−0.990268	−	1.71519i	−0.990268	−	1.71519i
$487$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$487$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$488$	−0.661187	+	1.14521i	−0.661187	+	1.14521i
$489$	0.0467046	−	0.0518708i	0.0467046	−	0.0518708i
$490$	−1.67959	−	2.90914i	−1.67959	−	2.90914i
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$	0.0840186	−	0.145524i	0.0840186	−	0.145524i
$494$		0			0
$495$	0.353717	+	3.36539i	0.353717	+	3.36539i
$496$	−5.68701			−5.68701
$497$		0			0
$498$	2.78709	+	0.592415i	2.78709	+	0.592415i
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	−2.17295	−	3.76367i	−2.17295	−	3.76367i
$501$		0			0
$502$	−1.51718	+	2.62783i	−1.51718	+	2.62783i
$503$	−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
$505$	1.04825			1.04825
$506$		0			0
$507$	0.309017	+	0.951057i	0.309017	+	0.951057i
$508$	−1.63425	−	2.83061i	−1.63425	−	2.83061i
$509$	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	$0.444444\pi$
$510$	0.780628	−	0.866976i	0.780628	−	0.866976i
$511$		0			0
$512$	7.07614			7.07614
$513$		0			0
$514$	2.21500			2.21500
$515$		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			$\pi$
$522$	0.775257	−	0.563257i	0.775257	−	0.563257i
$523$	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
$526$	−1.90381	−	3.29750i	−1.90381	−	3.29750i
$527$	0.213817	+	0.370342i	0.213817	+	0.370342i
$528$	9.01337	+	1.91585i	9.01337	+	1.91585i
$529$	−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
$536$	3.57800	+	6.19728i	3.57800	+	6.19728i
$537$		0			0
$538$	1.21934	−	2.11196i	1.21934	−	2.11196i
$539$	−1.99513			−1.99513
$540$	4.52834	−	2.01615i	4.52834	−	2.01615i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	1.97571	−	3.42203i	1.97571	−	3.42203i
$543$		0			0
$544$	−0.927232	−	1.60601i	−0.927232	−	1.60601i
$545$	0.177290	+	0.307076i	0.177290	+	0.307076i
$546$		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
$549$	0.317271	+	0.141258i	0.317271	+	0.141258i
$550$	−7.41580			−7.41580
$551$		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			$\pi$
$558$	0.254911	+	2.42532i	0.254911	+	2.42532i
$559$		0			0
$560$		0			0
$561$	−0.214118	−	0.658988i	−0.214118	−	0.658988i
$562$		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
$565$	1.65903	−	2.87353i	1.65903	−	2.87353i
$566$	3.92252			3.92252
$567$		0			0
$568$	−6.16087			−6.16087
$569$	0.615661	−	1.06636i	0.615661	−	1.06636i	−0.374607	−	0.927184i	$-0.622222\pi$
$569$	0.615661	−	1.06636i	0.615661	−	1.06636i	0.990268	−	0.139173i	$-0.0444444\pi$
$570$		0			0
$571$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$571$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.622664	−	5.92425i	−0.622664	−	5.92425i
$577$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$577$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$578$	0.870827	−	1.50832i	0.870827	−	1.50832i
$579$	−0.0682737	−	0.0145120i	−0.0682737	−	0.0145120i
$580$	1.19918	+	2.07704i	1.19918	+	2.07704i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−3.20457			−3.20457
$587$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	−0.882948	−	0.469472i	$-0.844444\pi$
$587$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	0.848048	+	0.529919i	$0.177778\pi$
$588$	0.903109	+	2.77948i	0.903109	+	2.77948i
$589$		0			0
$590$		0			0
$591$	0.586655	−	0.651546i	0.586655	−	0.651546i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	0.413036	−	3.92978i	0.413036	−	3.92978i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	−0.978148	−	0.207912i	$-0.933333\pi$
$599$	−0.669131	−	1.15897i	−0.669131	−	1.15897i	0.309017	−	0.951057i	$-0.400000\pi$
$600$	2.20822	+	6.79619i	2.20822	+	6.79619i
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$	1.52045	−	1.10467i	1.52045	−	1.10467i
$604$		0			0
$605$	−2.52764	+	4.37800i	−2.52764	+	4.37800i
$606$	−1.19729	−	0.254492i	−1.19729	−	0.254492i
$607$		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
$610$	−0.583315	+	1.01033i	−0.583315	+	1.01033i
$611$		0			0
$612$	−0.821137	+	0.596591i	−0.821137	+	0.596591i
$613$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$613$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$614$	−1.67959	+	2.90914i	−1.67959	+	2.90914i
$615$		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	$0.666667\pi$
$620$	−6.10352			−6.10352
$621$		0			0
$622$	3.03436			3.03436
$623$		0			0
$624$		0			0
$625$	−0.322710	−	0.558950i	−0.322710	−	0.558950i
$626$		0			0
$627$		0			0
$628$	0.707022	−	1.22460i	0.707022	−	1.22460i
$629$		0			0
$630$		0			0
$631$	1.69610			1.69610			0.848048	−	0.529919i	$-0.177778\pi$
$631$	1.69610			1.69610			0.848048	+	0.529919i	$0.177778\pi$
$632$		0			0
$633$	1.40724	+	0.299118i	1.40724	+	0.299118i
$634$		0			0
$635$	−0.948445	−	1.64275i	−0.948445	−	1.64275i
$636$		0			0
$637$		0			0
$638$	1.91187			1.91187
$639$	0.169131	+	1.60917i	0.169131	+	1.60917i
$640$	10.9536			10.9536
$641$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	−0.997564	−	0.0697565i	$-0.977778\pi$
$641$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	0.559193	+	0.829038i	$0.311111\pi$
$642$		0			0
$643$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	−0.997564	−	0.0697565i	$-0.977778\pi$
$643$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	0.438371	−	0.898794i	$-0.355556\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.749213			−0.749213			−0.374607	−	0.927184i	$-0.622222\pi$
$647$	−0.749213			−0.749213			−0.374607	+	0.927184i	$0.622222\pi$
$648$	−3.72442	+	0.791650i	−3.72442	+	0.791650i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.101995	−	0.176660i	−0.101995	−	0.176660i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$	−0.127947	−	0.393780i	−0.127947	−	0.393780i
$655$		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	$0.666667\pi$
$660$	9.67350	+	2.05617i	9.67350	+	2.05617i
$661$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.559193	+	0.829038i	$0.311111\pi$
$661$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.438371	+	0.898794i	$0.355556\pi$
$662$		0			0
$663$		0			0
$664$	2.73898	−	4.74405i	2.73898	−	4.74405i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	3.15660	+	5.46739i	3.15660	+	5.46739i
$671$	0.346450	+	0.600070i	0.346450	+	0.600070i
$672$		0			0
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$	3.35918			3.35918
$675$	1.71449	−	0.763340i	1.71449	−	0.763340i
$676$	2.92252			2.92252
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$	−2.59256	+	2.87932i	−2.59256	+	2.87932i
$679$		0			0
$680$	−1.12144	−	1.94239i	−1.12144	−	1.94239i
$681$		0			0
$682$	−2.43274	+	4.21363i	−2.43274	+	4.21363i
$683$		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
$694$	−2.43868			−2.43868
$695$		0			0
$696$	−0.569301	−	1.75213i	−0.569301	−	1.75213i
$697$		0			0
$698$	−1.10750	−	1.91825i	−1.10750	−	1.91825i
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	5.94237	−	10.2925i	5.94237	−	10.2925i
$705$		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	$0.666667\pi$
$710$	−5.43527			−5.43527
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−0.978148	−	0.207912i	−0.978148	−	0.207912i
$718$	−0.612019	+	1.06005i	−0.612019	+	1.06005i
$719$	−1.76590			−1.76590			−0.882948	−	0.469472i	$-0.844444\pi$
$719$	−1.76590			−1.76590			−0.882948	+	0.469472i	$0.844444\pi$
$720$	−0.818837	−	7.79071i	−0.818837	−	7.79071i
$721$		0			0
$722$	−0.990268	+	1.71519i	−0.990268	+	1.71519i
$723$	−0.604528	−	1.86055i	−0.604528	−	1.86055i
$724$		0			0
$725$	0.454025	+	0.786394i	0.454025	+	0.786394i
$726$	3.94992	−	4.38683i	3.94992	−	4.38683i
$727$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$727$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$728$		0			0
$729$	0.309017	+	0.951057i	0.309017	+	0.951057i
$730$		0			0
$731$		0			0
$732$	0.679155	−	0.754278i	0.679155	−	0.754278i
$733$	0.615661	+	1.06636i	0.615661	+	1.06636i	0.990268	+	0.139173i	$0.0444444\pi$
$733$	0.615661	+	1.06636i	0.615661	+	1.06636i	−0.374607	+	0.927184i	$0.622222\pi$
$734$	1.42468	+	2.46762i	1.42468	+	2.46762i
$735$	0.524123	+	1.61308i	0.524123	+	1.61308i
$736$		0			0
$737$	3.74961			3.74961
$738$		0			0
$739$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$739$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$740$		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	$0.333333\pi$
$744$	4.58597	+	0.974777i	4.58597	+	0.974777i
$745$		0			0
$746$	−3.20457			−3.20457
$747$	−1.31430	−	0.585164i	−1.31430	−	0.585164i
$748$	−2.02502			−2.02502
$749$		0			0
$750$	0.910097	+	2.80099i	0.910097	+	2.80099i
$751$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$751$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$752$		0			0
$753$	1.02517	−	1.13856i	1.02517	−	1.13856i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.98054			1.98054			0.990268	−	0.139173i	$-0.0444444\pi$
$757$	1.98054			1.98054			0.990268	+	0.139173i	$0.0444444\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$761$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$762$	0.684474	+	2.10659i	0.684474	+	2.10659i
$763$		0			0
$764$		0			0
$765$	−0.476550	+	0.346234i	−0.476550	+	0.346234i
$766$	−1.98054			−1.98054
$767$		0			0
$768$	−6.68429	−	1.42079i	−6.68429	−	1.42079i
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$	−1.09395	−	0.232525i	−1.09395	−	0.232525i
$772$	−0.101995	+	0.176660i	−0.101995	+	0.176660i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−2.31088			−2.31088
$776$		0			0
$777$		0			0
$778$	0.741922	+	1.28505i	0.741922	+	1.28505i
$779$		0			0
$780$		0			0
$781$	−1.61409	+	2.79569i	−1.61409	+	2.79569i
$782$		0			0
$783$	−0.442013	+	0.196797i	−0.442013	+	0.196797i
$784$	4.61862			4.61862
$785$	0.410323	−	0.710700i	0.410323	−	0.710700i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	−1.28115	−	2.21902i	−1.28115	−	2.21902i
$789$	0.594092	+	1.82843i	0.594092	+	1.82843i
$790$		0			0
$791$		0			0
$792$	−6.93993	−	3.08986i	−6.93993	−	3.08986i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.374607	+	0.648838i	0.374607	+	0.648838i	0.990268	−	0.139173i	$-0.0444444\pi$
$797$	0.374607	+	0.648838i	0.374607	+	0.648838i	−0.615661	+	0.788011i	$0.711111\pi$
$798$		0			0
$799$		0			0
$800$	10.0213			10.0213
$801$		0			0
$802$	−2.84936			−2.84936
$803$		0			0
$804$	−1.69729	−	5.22372i	−1.69729	−	5.22372i
$805$		0			0
$806$		0			0
$807$	−0.823916	+	0.915051i	−0.823916	+	0.915051i
$808$	−1.17662	+	2.03797i	−1.17662	+	2.03797i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−3.28577	+	0.698413i	−3.28577	+	0.698413i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−1.33500	+	1.48267i	−1.33500	+	1.48267i
$814$		0			0
$815$	−0.0591929	−	0.102525i	−0.0591929	−	0.102525i
$816$	0.495672	+	1.52552i	0.495672	+	1.52552i
$817$		0			0
$818$	−3.49742			−3.49742
$819$		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
$825$	3.66252	+	0.778492i	3.66252	+	0.778492i
$826$		0			0
$827$	−0.483844			−0.483844			−0.241922	−	0.970296i	$-0.577778\pi$
$827$	−0.483844			−0.483844			−0.241922	+	0.970296i	$0.577778\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$	2.41639	−	4.18531i	2.41639	−	4.18531i
$831$		0			0
$832$		0			0
$833$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.128708	−	1.22458i	0.128708	−	1.22458i
$838$	3.80763			3.80763
$839$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	−0.997564	−	0.0697565i	$-0.977778\pi$
$839$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	0.559193	+	0.829038i	$0.311111\pi$
$840$		0			0
$841$	0.382948	+	0.663285i	0.382948	+	0.663285i
$842$	1.93726	+	3.35543i	1.93726	+	3.35543i
$843$		0			0
$844$	2.10229	−	3.64127i	2.10229	−	3.64127i
$845$	1.69610			1.69610
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.93726	−	0.411777i	−1.93726	−	0.411777i
$850$	−0.645443	−	1.11794i	−0.645443	−	1.11794i
$851$		0			0
$852$	4.62541	+	0.983161i	4.62541	+	0.983161i
$853$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.0348995	−	0.999391i	$-0.488889\pi$
$853$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.848048	−	0.529919i	$-0.177778\pi$
$854$		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
$859$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	0.559193	−	0.829038i	$-0.311111\pi$
$859$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	−0.997564	+	0.0697565i	$0.977778\pi$
$860$		0			0
$861$		0			0
$862$	−0.0691197	+	0.119719i	−0.0691197	+	0.119719i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.558152	+	5.31046i	−0.558152	+	5.31046i
$865$		0			0
$866$		0			0
$867$	−0.588424	+	0.653511i	−0.588424	+	0.653511i
$868$		0			0
$869$		0			0
$870$	−0.502251	−	1.54577i	−0.502251	−	1.54577i
$871$		0			0
$872$	−0.796011			−0.796011
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.719340	+	1.24593i	0.719340	+	1.24593i	0.961262	+	0.275637i	$0.0888889\pi$
$877$	0.719340	+	1.24593i	0.719340	+	1.24593i	−0.241922	+	0.970296i	$0.577778\pi$
$878$	−1.32524	−	2.29538i	−1.32524	−	2.29538i
$879$	1.58268	+	0.336408i	1.58268	+	0.336408i
$880$	7.81454	−	13.5352i	7.81454	−	13.5352i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.207022	−	1.96969i	−0.207022	−	1.96969i
$883$	−0.749213			−0.749213			−0.374607	−	0.927184i	$-0.622222\pi$
$883$	−0.749213			−0.749213			−0.374607	+	0.927184i	$0.622222\pi$
$884$		0			0
$885$		0			0
$886$	1.74871	+	3.02885i	1.74871	+	3.02885i
$887$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$887$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.616528	+	1.89748i	−0.616528	+	1.89748i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	0.595768			0.595768
$900$	−0.573320	−	5.45478i	−0.573320	−	5.45478i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	3.72442	+	6.45089i	3.72442	+	6.45089i
$905$		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
$909$	0.564602	+	0.251377i	0.564602	+	0.251377i
$910$		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
$913$	−1.43518	−	2.48580i	−1.43518	−	2.48580i
$914$	1.60229	+	2.77524i	1.60229	+	2.77524i
$915$	0.394150	−	0.437748i	0.394150	−	0.437748i
$916$		0			0
$917$		0			0
$918$	0.628367	−	0.279767i	0.628367	−	0.279767i
$919$	0.0697990			0.0697990			0.0348995	−	0.999391i	$-0.488889\pi$
$919$	0.0697990			0.0697990			0.0348995	+	0.999391i	$0.488889\pi$
$920$		0			0
$921$	1.13491	−	1.26045i	1.13491	−	1.26045i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−2.58359			−2.58359
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$	4.04585	+	0.859972i	4.04585	+	0.859972i
$931$		0			0
$932$		0			0
$933$	−1.49861	−	0.318539i	−1.49861	−	0.318539i
$934$		0			0
$935$	−1.17523			−1.17523
$936$		0			0
$937$	−0.749213			−0.749213			−0.374607	−	0.927184i	$-0.622222\pi$
$937$	−0.749213			−0.749213			−0.374607	+	0.927184i	$0.622222\pi$
$938$		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	$0.333333\pi$
$942$	−0.641208	+	0.712133i	−0.641208	+	0.712133i
$943$		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
$956$	−1.46126	+	2.53098i	−1.46126	+	2.53098i
$957$	−0.944236	−	0.200703i	−0.944236	−	0.200703i
$958$	0.741922	+	1.28505i	0.741922	+	1.28505i
$959$		0			0
$960$	−9.88266	−	2.10062i	−9.88266	−	2.10062i
$961$	−0.258078	+	0.447004i	−0.258078	+	0.447004i
$962$		0			0
$963$		0			0
$964$	−5.71732			−5.71732
$965$	−0.0591929	+	0.102525i	−0.0591929	+	0.102525i
$966$		0			0
$967$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$967$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$968$	−5.67439	−	9.82832i	−5.67439	−	9.82832i
$969$		0			0
$970$		0			0
$971$	−1.76590			−1.76590			−0.882948	−	0.469472i	$-0.844444\pi$
$971$	−1.76590			−1.76590			−0.882948	+	0.469472i	$0.844444\pi$
$972$	2.92252			2.92252
$973$		0			0
$974$	−1.51718	+	2.62783i	−1.51718	+	2.62783i
$975$		0			0
$976$	−0.802015	−	1.38913i	−0.802015	−	1.38913i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$	0.0427183	+	0.131474i	0.0427183	+	0.131474i
$979$		0			0
$980$	4.95688			4.95688
$981$	0.0218524	+	0.207912i	0.0218524	+	0.207912i
$982$		0			0
$983$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	0.438371	+	0.898794i	$0.355556\pi$
$983$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	−0.997564	+	0.0697565i	$0.977778\pi$
$984$		0			0
$985$	−0.743520	−	1.28781i	−0.743520	−	1.28781i
$986$	0.166402	+	0.288216i	0.166402	+	0.288216i
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−6.12258	−	2.72595i	−6.12258	−	2.72595i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	3.28746	−	5.69404i	3.28746	−	5.69404i
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−2.81341	+	3.12461i	−2.81341	+	3.12461i
$997$		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.b.1672.1	yes	24
9.4	even	3	inner	2151.1.f.b.238.1	✓	24
239.238	odd	2	CM	2151.1.f.b.1672.1	yes	24
2151.238	odd	6	inner	2151.1.f.b.238.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.1.f.b.238.1	✓	24	9.4	even	3	inner
2151.1.f.b.238.1	✓	24	2151.238	odd	6	inner
2151.1.f.b.1672.1	yes	24	1.1	even	1	trivial
2151.1.f.b.1672.1	yes	24	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)

\(f(q)\)	\(=\)	\(q+(-0.990268 + 1.71519i) q^{2} +(0.669131 - 0.743145i) q^{3} +(-1.46126 - 2.53098i) q^{4} +(-0.848048 - 1.46886i) q^{5} +(0.612019 + 1.88360i) q^{6} +3.80763 q^{8} +(-0.104528 - 0.994522i) q^{9} +O(q^{10})\)	\(q+(-0.990268 + 1.71519i) q^{2} +(0.669131 - 0.743145i) q^{3} +(-1.46126 - 2.53098i) q^{4} +(-0.848048 - 1.46886i) q^{5} +(0.612019 + 1.88360i) q^{6} +3.80763 q^{8} +(-0.104528 - 0.994522i) q^{9} +3.35918 q^{10} +(0.997564 - 1.72783i) q^{11} +(-2.85866 - 0.607627i) q^{12} +(-1.65903 - 0.352638i) q^{15} +(-2.30931 + 3.99984i) q^{16} +0.347296 q^{17} +(1.80931 + 0.805557i) q^{18} +(-2.47844 + 4.29278i) q^{20} +(1.97571 + 3.42203i) q^{22} +(2.54780 - 2.82962i) q^{24} +(-0.938371 + 1.62531i) q^{25} +(-0.809017 - 0.587785i) q^{27} +(0.241922 - 0.419021i) q^{29} +(2.24773 - 2.49636i) q^{30} +(0.615661 + 1.06636i) q^{31} +(-2.66986 - 4.62433i) q^{32} +(-0.616528 - 1.89748i) q^{33} +(-0.343916 + 0.595681i) q^{34} +(-2.36437 + 1.71782i) q^{36} +(-3.22905 - 5.59288i) q^{40} -5.83081 q^{44} +(-1.37217 + 0.996940i) q^{45} +(1.42723 + 4.39257i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(-1.85848 - 3.21898i) q^{50} +(0.232387 - 0.258091i) q^{51} +(1.80931 - 0.805557i) q^{54} -3.38393 q^{55} +(0.479135 + 0.829886i) q^{58} +(1.53176 + 4.71427i) q^{60} +(-0.173648 + 0.300767i) q^{61} -2.43868 q^{62} +5.95688 q^{64} +(3.86508 + 0.821547i) q^{66} +(0.939693 + 1.62760i) q^{67} +(-0.507491 - 0.879000i) q^{68} -1.61803 q^{71} +(-0.398005 - 3.78677i) q^{72} +(0.579945 + 1.78489i) q^{75} +7.83362 q^{80} +(-0.978148 + 0.207912i) q^{81} +(0.719340 - 1.24593i) q^{83} +(-0.294524 - 0.510131i) q^{85} +(-0.149516 - 0.460163i) q^{87} +(3.79835 - 6.57894i) q^{88} +(-0.351130 - 3.34078i) q^{90} +(1.20442 + 0.256006i) q^{93} +(-5.22303 - 1.11019i) q^{96} +1.98054 q^{98} +(-1.82264 - 0.811492i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 3 q^{3} - 12 q^{4} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) 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

Introduction

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