LMFDB - Embedded newform 2151.1.f.b.238.11

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			238.11
Root			$0.438371 + 0.898794i$ of defining polynomial
Character	$\chi$	$=$	2151.238
Dual form			2151.1.f.b.1672.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.882948 + 1.52931i) q^{2} +(-0.978148 - 0.207912i) q^{3} +(-1.05919 + 1.83458i) q^{4} +(0.374607 - 0.648838i) q^{5} +(-0.545692 - 1.67947i) q^{6} -1.97495 q^{8} +(0.913545 + 0.406737i) q^{9} +O(q^{10})$	\(q+(0.882948 + 1.52931i) q^{2} +(-0.978148 - 0.207912i) q^{3} +(-1.05919 + 1.83458i) q^{4} +(0.374607 - 0.648838i) q^{5} +(-0.545692 - 1.67947i) q^{6} -1.97495 q^{8} +(0.913545 + 0.406737i) q^{9} +1.32303 q^{10} +(0.241922 + 0.419021i) q^{11} +(1.41748 - 1.57427i) q^{12} +(-0.501321 + 0.556774i) q^{15} +(-0.684586 - 1.18574i) q^{16} +0.347296 q^{17} +(0.184586 + 1.75622i) q^{18} +(0.793561 + 1.37449i) q^{20} +(-0.427209 + 0.739947i) q^{22} +(1.93179 + 0.410616i) q^{24} +(0.219340 + 0.379908i) q^{25} +(-0.809017 - 0.587785i) q^{27} +(0.997564 + 1.72783i) q^{29} +(-1.29412 - 0.275074i) q^{30} +(-0.0348995 + 0.0604477i) q^{31} +(0.221432 - 0.383531i) q^{32} +(-0.149516 - 0.460163i) q^{33} +(0.306644 + 0.531124i) q^{34} +(-1.71381 + 1.24516i) q^{36} +(-0.739830 + 1.28142i) q^{40} -1.02497 q^{44} +(0.606126 - 0.440376i) q^{45} +(0.423098 + 1.30216i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.387331 + 0.670877i) q^{50} +(-0.339707 - 0.0722070i) q^{51} +(0.184586 - 1.75622i) q^{54} +0.362502 q^{55} +(-1.76159 + 3.05117i) q^{58} +(-0.490448 - 1.50944i) q^{60} +(-0.173648 - 0.300767i) q^{61} -0.123258 q^{62} -0.587123 q^{64} +(0.571717 - 0.634956i) q^{66} +(0.939693 - 1.62760i) q^{67} +(-0.367854 + 0.637142i) q^{68} -1.61803 q^{71} +(-1.80421 - 0.803285i) q^{72} +(-0.135559 - 0.417209i) q^{75} -1.02580 q^{80} +(0.669131 + 0.743145i) q^{81} +(-0.438371 - 0.759281i) q^{83} +(0.130100 - 0.225339i) q^{85} +(-0.616528 - 1.89748i) q^{87} +(-0.477784 - 0.827546i) q^{88} +(1.20865 + 0.538126i) q^{90} +(0.0467046 - 0.0518708i) q^{93} +(-0.296333 + 0.329112i) q^{96} -1.76590 q^{98} +(0.0505754 + 0.481193i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.848048	+	0.529919i	$0.177778\pi$
$2$	0.882948	+	1.52931i	0.882948	+	1.52931i	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
$4$	−1.05919	+	1.83458i	−1.05919	+	1.83458i
$5$	0.374607	−	0.648838i	0.374607	−	0.648838i	−0.615661	−	0.788011i	$-0.711111\pi$
$5$	0.374607	−	0.648838i	0.374607	−	0.648838i	0.990268	+	0.139173i	$0.0444444\pi$
$6$	−0.545692	−	1.67947i	−0.545692	−	1.67947i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	−1.97495			−1.97495
$9$	0.913545	+	0.406737i	0.913545	+	0.406737i
$10$	1.32303			1.32303
$11$	0.241922	+	0.419021i	0.241922	+	0.419021i	0.961262	−	0.275637i	$-0.0888889\pi$
$11$	0.241922	+	0.419021i	0.241922	+	0.419021i	−0.719340	+	0.694658i	$0.755556\pi$
$12$	1.41748	−	1.57427i	1.41748	−	1.57427i
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$	−0.501321	+	0.556774i	−0.501321	+	0.556774i
$16$	−0.684586	−	1.18574i	−0.684586	−	1.18574i
$17$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$17$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$18$	0.184586	+	1.75622i	0.184586	+	1.75622i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	0.793561	+	1.37449i	0.793561	+	1.37449i
$21$		0			0
$22$	−0.427209	+	0.739947i	−0.427209	+	0.739947i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	1.93179	+	0.410616i	1.93179	+	0.410616i
$25$	0.219340	+	0.379908i	0.219340	+	0.379908i
$26$		0			0
$27$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$28$		0			0
$29$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.559193	+	0.829038i	$0.311111\pi$
$29$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.438371	+	0.898794i	$0.355556\pi$
$30$	−1.29412	−	0.275074i	−1.29412	−	0.275074i
$31$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	−0.882948	−	0.469472i	$-0.844444\pi$
$31$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	0.848048	+	0.529919i	$0.177778\pi$
$32$	0.221432	−	0.383531i	0.221432	−	0.383531i
$33$	−0.149516	−	0.460163i	−0.149516	−	0.460163i
$34$	0.306644	+	0.531124i	0.306644	+	0.531124i
$35$		0			0
$36$	−1.71381	+	1.24516i	−1.71381	+	1.24516i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	−0.739830	+	1.28142i	−0.739830	+	1.28142i
$41$		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	$0.333333\pi$
$44$	−1.02497			−1.02497
$45$	0.606126	−	0.440376i	0.606126	−	0.440376i
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	0.423098	+	1.30216i	0.423098	+	1.30216i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	−0.387331	+	0.670877i	−0.387331	+	0.670877i
$51$	−0.339707	−	0.0722070i	−0.339707	−	0.0722070i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	0.184586	−	1.75622i	0.184586	−	1.75622i
$55$	0.362502			0.362502
$56$		0			0
$57$		0			0
$58$	−1.76159	+	3.05117i	−1.76159	+	3.05117i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	−0.490448	−	1.50944i	−0.490448	−	1.50944i
$61$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$61$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$62$	−0.123258			−0.123258
$63$		0			0
$64$	−0.587123			−0.587123
$65$		0			0
$66$	0.571717	−	0.634956i	0.571717	−	0.634956i
$67$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$67$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$68$	−0.367854	+	0.637142i	−0.367854	+	0.637142i
$69$		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	$0.800000\pi$
$72$	−1.80421	−	0.803285i	−1.80421	−	0.803285i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−0.135559	−	0.417209i	−0.135559	−	0.417209i
$76$		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	$0.333333\pi$
$80$	−1.02580			−1.02580
$81$	0.669131	+	0.743145i	0.669131	+	0.743145i
$82$		0			0
$83$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	0.559193	−	0.829038i	$-0.311111\pi$
$83$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	−0.997564	+	0.0697565i	$0.977778\pi$
$84$		0			0
$85$	0.130100	−	0.225339i	0.130100	−	0.225339i
$86$		0			0
$87$	−0.616528	−	1.89748i	−0.616528	−	1.89748i
$88$	−0.477784	−	0.827546i	−0.477784	−	0.827546i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	1.20865	+	0.538126i	1.20865	+	0.538126i
$91$		0			0
$92$		0			0
$93$	0.0467046	−	0.0518708i	0.0467046	−	0.0518708i
$94$		0			0
$95$		0			0
$96$	−0.296333	+	0.329112i	−0.296333	+	0.329112i
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$	−1.76590			−1.76590
$99$	0.0505754	+	0.481193i	0.0505754	+	0.481193i
$100$	−0.929293			−0.929293
$101$	−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	$0.933333\pi$
$102$	−0.189517	−	0.583272i	−0.189517	−	0.583272i
$103$		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			$\pi$
$108$	1.93524	−	0.861625i	1.93524	−	0.861625i
$109$	1.82709			1.82709			0.913545	−	0.406737i	$-0.133333\pi$
$109$	1.82709			1.82709			0.913545	+	0.406737i	$0.133333\pi$
$110$	0.320070	+	0.554378i	0.320070	+	0.554378i
$111$		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
$116$	−4.22645			−4.22645
$117$		0			0
$118$		0			0
$119$		0			0
$120$	0.990086	−	1.09960i	0.990086	−	1.09960i
$121$	0.382948	−	0.663285i	0.382948	−	0.663285i
$122$	0.306644	−	0.531124i	0.306644	−	0.531124i
$123$		0			0
$124$	−0.0739306	−	0.128052i	−0.0739306	−	0.128052i
$125$	1.07788			1.07788
$126$		0			0
$127$	1.92252			1.92252			0.961262	−	0.275637i	$-0.0888889\pi$
$127$	1.92252			1.92252			0.961262	+	0.275637i	$0.0888889\pi$
$128$	−0.739830	−	1.28142i	−0.739830	−	1.28142i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	1.00257	+	0.213103i	1.00257	+	0.213103i
$133$		0			0
$134$	3.31880			3.31880
$135$	−0.684440	+	0.304732i	−0.684440	+	0.304732i
$136$	−0.685894			−0.685894
$137$		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
$142$	−1.42864	−	2.47448i	−1.42864	−	2.47448i
$143$		0			0
$144$	−0.143118	−	1.36167i	−0.143118	−	1.36167i
$145$	1.49478			1.49478
$146$		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	$0.666667\pi$
$150$	0.518350	−	0.575686i	0.518350	−	0.575686i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$	0.317271	+	0.141258i	0.317271	+	0.141258i
$154$		0			0
$155$	0.0261472	+	0.0452882i	0.0261472	+	0.0452882i
$156$		0			0
$157$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.438371	−	0.898794i	$-0.355556\pi$
$157$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.559193	−	0.829038i	$-0.311111\pi$
$158$		0			0
$159$		0			0
$160$	−0.165899	−	0.287346i	−0.165899	−	0.287346i
$161$		0			0
$162$	−0.545692	+	1.67947i	−0.545692	+	1.67947i
$163$	−1.23132			−1.23132			−0.615661	−	0.788011i	$-0.711111\pi$
$163$	−1.23132			−1.23132			−0.615661	+	0.788011i	$0.711111\pi$
$164$		0			0
$165$	−0.354581	−	0.0753684i	−0.354581	−	0.0753684i
$166$	0.774117	−	1.34081i	0.774117	−	1.34081i
$167$		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
$170$	0.459484			0.459484
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$	2.35747	−	2.61824i	2.35747	−	2.61824i
$175$		0			0
$176$	0.331233	−	0.573712i	0.331233	−	0.573712i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	0.165899	+	1.57843i	0.165899	+	1.57843i
$181$		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
$186$	0.120564	+	0.0256267i	0.120564	+	0.0256267i
$187$	0.0840186	+	0.145524i	0.0840186	+	0.145524i
$188$		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	$0.333333\pi$
$192$	0.574293	+	0.122070i	0.574293	+	0.122070i
$193$	0.615661	−	1.06636i	0.615661	−	1.06636i	−0.374607	−	0.927184i	$-0.622222\pi$
$193$	0.615661	−	1.06636i	0.615661	−	1.06636i	0.990268	−	0.139173i	$-0.0444444\pi$
$194$		0			0
$195$		0			0
$196$	−1.05919	−	1.83458i	−1.05919	−	1.83458i
$197$	−1.43868			−1.43868			−0.719340	−	0.694658i	$-0.755556\pi$
$197$	−1.43868			−1.43868			−0.719340	+	0.694658i	$0.755556\pi$
$198$	−0.691238	+	0.502214i	−0.691238	+	0.502214i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.433186	−	0.750299i	−0.433186	−	0.750299i
$201$	−1.25755	+	1.39666i	−1.25755	+	1.39666i
$202$	0.545692	−	0.945166i	0.545692	−	0.945166i
$203$		0			0
$204$	0.492285	−	0.546737i	0.492285	−	0.546737i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	−0.997564	−	0.0697565i	$-0.977778\pi$
$211$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	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
$216$	1.59777	+	1.16085i	1.59777	+	1.16085i
$217$		0			0
$218$	1.61323	+	2.79419i	1.61323	+	2.79419i
$219$		0			0
$220$	−0.383960	+	0.665038i	−0.383960	+	0.665038i
$221$		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
$225$	0.0458545	+	0.436276i	0.0458545	+	0.436276i
$226$	−2.36323			−2.36323
$227$		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
$232$	−1.97014	−	3.41238i	−1.97014	−	3.41238i
$233$		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
$240$	1.00339	+	0.213276i	1.00339	+	0.213276i
$241$	−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	$-0.400000\pi$
$242$	1.35249			1.35249
$243$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$244$	0.735708			0.735708
$245$	0.374607	+	0.648838i	0.374607	+	0.648838i
$246$		0			0
$247$		0			0
$248$	0.0689248	−	0.119381i	0.0689248	−	0.119381i
$249$	0.270928	+	0.833831i	0.270928	+	0.833831i
$250$	0.951710	+	1.64841i	0.951710	+	1.64841i
$251$	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
$254$	1.69749	+	2.94013i	1.69749	+	2.94013i
$255$	−0.174107	+	0.193366i	−0.174107	+	0.193366i
$256$	1.01290	−	1.75440i	1.01290	−	1.75440i
$257$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.241922	+	0.970296i	$0.577778\pi$
$257$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.719340	+	0.694658i	$0.755556\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.208548	+	1.98420i	0.208548	+	1.98420i
$262$		0			0
$263$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	−0.997564	−	0.0697565i	$-0.977778\pi$
$263$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	0.438371	−	0.898794i	$-0.355556\pi$
$264$	0.295287	+	0.908799i	0.295287	+	0.908799i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.99063	+	3.44787i	1.99063	+	3.44787i
$269$	0.0697990			0.0697990			0.0348995	−	0.999391i	$-0.488889\pi$
$269$	0.0697990			0.0697990			0.0348995	+	0.999391i	$0.488889\pi$
$270$	−1.07036	−	0.777659i	−1.07036	−	0.777659i
$271$	−0.483844			−0.483844			−0.241922	−	0.970296i	$-0.577778\pi$
$271$	−0.483844			−0.483844			−0.241922	+	0.970296i	$0.577778\pi$
$272$	−0.237754	−	0.411803i	−0.237754	−	0.411803i
$273$		0			0
$274$		0			0
$275$	−0.106126	+	0.183816i	−0.106126	+	0.183816i
$276$		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
$279$	−0.0564686	+	0.0410268i	−0.0564686	+	0.0410268i
$280$		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
$283$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.0348995	−	0.999391i	$-0.488889\pi$
$283$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.848048	−	0.529919i	$-0.177778\pi$
$284$	1.71381	−	2.96841i	1.71381	−	2.96841i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.358284	−	0.260308i	0.358284	−	0.260308i
$289$	−0.879385			−0.879385
$290$	1.31981	+	2.28598i	1.31981	+	2.28598i
$291$		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	$-0.133333\pi$
$294$	1.72731	+	0.367150i	1.72731	+	0.367150i
$295$		0			0
$296$		0			0
$297$	0.0505754	−	0.481193i	0.0505754	−	0.481193i
$298$		0			0
$299$		0			0
$300$	0.908985	+	0.193211i	0.908985	+	0.193211i
$301$		0			0
$302$		0			0
$303$	0.190983	+	0.587785i	0.190983	+	0.587785i
$304$		0			0
$305$	−0.260199			−0.260199
$306$	0.0641062	+	0.609929i	0.0641062	+	0.609929i
$307$	−0.749213			−0.749213			−0.374607	−	0.927184i	$-0.622222\pi$
$307$	−0.749213			−0.749213			−0.374607	+	0.927184i	$0.622222\pi$
$308$		0			0
$309$		0			0
$310$	−0.0461731	+	0.0799742i	−0.0461731	+	0.0799742i
$311$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$311$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	3.52319			3.52319
$315$		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
$319$	−0.482665	+	0.836001i	−0.482665	+	0.836001i
$320$	−0.219940	+	0.380947i	−0.219940	+	0.380947i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−2.07209	+	0.440437i	−2.07209	+	0.440437i
$325$		0			0
$326$	−1.08719	−	1.88307i	−1.08719	−	1.88307i
$327$	−1.78716	−	0.379874i	−1.78716	−	0.379874i
$328$		0			0
$329$		0			0
$330$	−0.197814	−	0.608810i	−0.197814	−	0.608810i
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	1.85728			1.85728
$333$		0			0
$334$		0			0
$335$	−0.704030	−	1.21942i	−0.704030	−	1.21942i
$336$		0			0
$337$	0.374607	−	0.648838i	0.374607	−	0.648838i	−0.615661	−	0.788011i	$-0.711111\pi$
$337$	0.374607	−	0.648838i	0.374607	−	0.648838i	0.990268	+	0.139173i	$0.0444444\pi$
$338$	0.882948	−	1.52931i	0.882948	−	1.52931i
$339$	0.895472	−	0.994522i	0.895472	−	0.994522i
$340$	0.275601	+	0.477355i	0.275601	+	0.477355i
$341$	−0.0337718			−0.0337718
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	−0.882948	−	0.469472i	$-0.844444\pi$
$347$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	0.848048	+	0.529919i	$0.177778\pi$
$348$	4.13409	+	0.878729i	4.13409	+	0.878729i
$349$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.719340	−	0.694658i	$-0.755556\pi$
$349$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.241922	−	0.970296i	$-0.577778\pi$
$350$		0			0
$351$		0			0
$352$	0.214277			0.214277
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$	−0.606126	+	1.04984i	−0.606126	+	1.04984i
$356$		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	$0.400000\pi$
$360$	−1.19707	+	0.869722i	−1.19707	+	0.869722i
$361$	1.00000			1.00000
$362$		0			0
$363$	−0.512484	+	0.569171i	−0.512484	+	0.569171i
$364$		0			0
$365$		0			0
$366$	−0.410370	+	0.455763i	−0.410370	+	0.455763i
$367$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	0.559193	−	0.829038i	$-0.311111\pi$
$367$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	−0.997564	+	0.0697565i	$0.977778\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.0456916	+	0.140624i	0.0456916	+	0.140624i
$373$	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	$-0.133333\pi$
$374$	−0.148368	+	0.256981i	−0.148368	+	0.256981i
$375$	−1.05432	−	0.224103i	−1.05432	−	0.224103i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−1.88051	−	0.399715i	−1.88051	−	0.399715i
$382$		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			$0$
$384$	0.457240	+	1.40724i	0.457240	+	1.40724i
$385$		0			0
$386$	2.17439			2.17439
$387$		0			0
$388$		0			0
$389$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	−0.882948	−	0.469472i	$-0.844444\pi$
$389$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	0.0348995	−	0.999391i	$-0.488889\pi$
$390$		0			0
$391$		0			0
$392$	0.987476	−	1.71036i	0.987476	−	1.71036i
$393$		0			0
$394$	−1.27028	−	2.20019i	−1.27028	−	2.20019i
$395$		0			0
$396$	−0.936355	−	0.416892i	−0.936355	−	0.416892i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	0.300314	−	0.520159i	0.300314	−	0.520159i
$401$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	−0.997564	−	0.0697565i	$-0.977778\pi$
$401$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	0.559193	+	0.829038i	$0.311111\pi$
$402$	−3.24627	−	0.690017i	−3.24627	−	0.690017i
$403$		0			0
$404$	1.30923			1.30923
$405$	0.732841	−	0.155770i	0.732841	−	0.155770i
$406$		0			0
$407$		0			0
$408$	0.670905	+	0.142605i	0.670905	+	0.142605i
$409$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.374607	+	0.927184i	$0.622222\pi$
$409$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.615661	+	0.788011i	$0.711111\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−0.656867			−0.656867
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	0.438371	+	0.898794i	$0.355556\pi$
$419$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	−0.997564	+	0.0697565i	$0.977778\pi$
$420$		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	$-0.400000\pi$
$422$	−1.54823			−1.54823
$423$		0			0
$424$		0			0
$425$	0.0761759	+	0.131941i	0.0761759	+	0.131941i
$426$	0.882948	+	2.71743i	0.882948	+	2.71743i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.23132			−1.23132			−0.615661	−	0.788011i	$-0.711111\pi$
$431$	−1.23132			−1.23132			−0.615661	+	0.788011i	$0.711111\pi$
$432$	−0.143118	+	1.36167i	−0.143118	+	1.36167i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−1.46211	−	0.310781i	−1.46211	−	0.310781i
$436$	−1.93524	+	3.35194i	−1.93524	+	3.35194i
$437$		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	$0.400000\pi$
$440$	−0.715924			−0.715924
$441$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$442$		0			0
$443$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.615661	−	0.788011i	$-0.711111\pi$
$443$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.374607	−	0.927184i	$-0.622222\pi$
$444$		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			$\pi$
$450$	−0.626715	+	0.455335i	−0.626715	+	0.455335i
$451$		0			0
$452$	−1.41748	−	2.45514i	−1.41748	−	2.45514i
$453$		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
$459$	−0.280969	−	0.204136i	−0.280969	−	0.204136i
$460$		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	$0.666667\pi$
$464$	1.36584	−	2.36570i	1.36584	−	2.36570i
$465$	−0.0161598	−	0.0497349i	−0.0161598	−	0.0497349i
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1.33500	+	1.48267i	−1.33500	+	1.48267i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−1.76590			−1.76590
$479$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	−0.882948	−	0.469472i	$-0.844444\pi$
$479$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	0.0348995	−	0.999391i	$-0.488889\pi$
$480$	0.102532	+	0.315560i	0.102532	+	0.315560i
$481$		0			0
$482$	1.18161	−	2.04662i	1.18161	−	2.04662i
$483$		0			0
$484$	0.811231	+	1.40509i	0.811231	+	1.40509i
$485$		0			0
$486$	0.882948	−	1.52931i	0.882948	−	1.52931i
$487$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$487$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$488$	0.342947	+	0.594001i	0.342947	+	0.594001i
$489$	1.20442	+	0.256006i	1.20442	+	0.256006i
$490$	−0.661516	+	1.14578i	−0.661516	+	1.14578i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$	0.346450	+	0.600070i	0.346450	+	0.600070i
$494$		0			0
$495$	0.331162	+	0.147443i	0.331162	+	0.147443i
$496$	0.0955669			0.0955669
$497$		0			0
$498$	−1.03597	+	1.15056i	−1.03597	+	1.15056i
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	−1.14168	+	1.97745i	−1.14168	+	1.97745i
$501$		0			0
$502$	1.35275	+	2.34304i	1.35275	+	2.34304i
$503$	−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
$505$	−0.463039			−0.463039
$506$		0			0
$507$	0.309017	+	0.951057i	0.309017	+	0.951057i
$508$	−2.03632	+	3.52702i	−2.03632	+	3.52702i
$509$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$509$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$510$	−0.449443	−	0.0955321i	−0.449443	−	0.0955321i
$511$		0			0
$512$	2.09769			2.09769
$513$		0			0
$514$	−3.39497			−3.39497
$515$		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			$\pi$
$522$	−2.85032	+	2.07088i	−2.85032	+	2.07088i
$523$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$523$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$524$		0			0
$525$		0			0
$526$	0.987476	−	1.71036i	0.987476	−	1.71036i
$527$	−0.0121205	+	0.0209933i	−0.0121205	+	0.0209933i
$528$	−0.443276	+	0.492308i	−0.443276	+	0.492308i
$529$	−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
$536$	−1.85585	+	3.21442i	−1.85585	+	3.21442i
$537$		0			0
$538$	0.0616289	+	0.106744i	0.0616289	+	0.106744i
$539$	−0.483844			−0.483844
$540$	0.165899	−	1.57843i	0.165899	−	1.57843i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	−0.427209	−	0.739947i	−0.427209	−	0.739947i
$543$		0			0
$544$	0.0769024	−	0.133199i	0.0769024	−	0.133199i
$545$	0.684440	−	1.18549i	0.684440	−	1.18549i
$546$		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
$549$	−0.0363024	−	0.345394i	−0.0363024	−	0.345394i
$550$	−0.374815			−0.374815
$551$		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			$\pi$
$558$	−0.112602	−	0.0501334i	−0.112602	−	0.0501334i
$559$		0			0
$560$		0			0
$561$	−0.0519263	−	0.159813i	−0.0519263	−	0.159813i
$562$		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
$565$	0.501321	+	0.868314i	0.501321	+	0.868314i
$566$	3.11839			3.11839
$567$		0			0
$568$	3.19554			3.19554
$569$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	0.848048	−	0.529919i	$-0.177778\pi$
$569$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	−0.882948	+	0.469472i	$0.844444\pi$
$570$		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
$576$	−0.536363	−	0.238804i	−0.536363	−	0.238804i
$577$	−0.209057			−0.209057			−0.104528	−	0.994522i	$-0.533333\pi$
$577$	−0.209057			−0.209057			−0.104528	+	0.994522i	$0.533333\pi$
$578$	−0.776451	−	1.34485i	−0.776451	−	1.34485i
$579$	−0.823916	+	0.915051i	−0.823916	+	0.915051i
$580$	−1.58326	+	2.74228i	−1.58326	+	2.74228i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	2.85728			2.85728
$587$	0.615661	+	1.06636i	0.615661	+	1.06636i	0.990268	+	0.139173i	$0.0444444\pi$
$587$	0.615661	+	1.06636i	0.615661	+	1.06636i	−0.374607	+	0.927184i	$0.622222\pi$
$588$	0.654617	+	2.01470i	0.654617	+	2.01470i
$589$		0			0
$590$		0			0
$591$	1.40724	+	0.299118i	1.40724	+	0.299118i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	0.780549	−	0.347523i	0.780549	−	0.347523i
$595$		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	$-0.266667\pi$
$600$	0.267723	+	0.823968i	0.267723	+	0.823968i
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$	1.52045	−	1.10467i	1.52045	−	1.10467i
$604$		0			0
$605$	−0.286909	−	0.496942i	−0.286909	−	0.496942i
$606$	−0.730278	+	0.811056i	−0.730278	+	0.811056i
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$	−0.229742	−	0.397925i	−0.229742	−	0.397925i
$611$		0			0
$612$	−0.595200	+	0.432438i	−0.595200	+	0.432438i
$613$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$613$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$614$	−0.661516	−	1.14578i	−0.661516	−	1.14578i
$615$		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	$0.333333\pi$
$620$	−0.110780			−0.110780
$621$		0			0
$622$	−2.70551			−2.70551
$623$		0			0
$624$		0			0
$625$	0.184440	−	0.319460i	0.184440	−	0.319460i
$626$		0			0
$627$		0			0
$628$	2.11323	+	3.66021i	2.11323	+	3.66021i
$629$		0			0
$630$		0			0
$631$	−0.749213			−0.749213			−0.374607	−	0.927184i	$-0.622222\pi$
$631$	−0.749213			−0.749213			−0.374607	+	0.927184i	$0.622222\pi$
$632$		0			0
$633$	0.586655	−	0.651546i	0.586655	−	0.651546i
$634$		0			0
$635$	0.720190	−	1.24741i	0.720190	−	1.24741i
$636$		0			0
$637$		0			0
$638$	−1.70467			−1.70467
$639$	−1.47815	−	0.658114i	−1.47815	−	0.658114i
$640$	−1.10858			−1.10858
$641$	0.719340	+	1.24593i	0.719340	+	1.24593i	0.961262	+	0.275637i	$0.0888889\pi$
$641$	0.719340	+	1.24593i	0.719340	+	1.24593i	−0.241922	+	0.970296i	$0.577778\pi$
$642$		0			0
$643$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.241922	+	0.970296i	$0.577778\pi$
$643$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.719340	+	0.694658i	$0.755556\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.69610			1.69610			0.848048	−	0.529919i	$-0.177778\pi$
$647$	1.69610			1.69610			0.848048	+	0.529919i	$0.177778\pi$
$648$	−1.32150	−	1.46768i	−1.32150	−	1.46768i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.30421	−	2.25896i	1.30421	−	2.25896i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	−0.997028	−	3.06854i	−0.997028	−	3.06854i
$655$		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	$0.333333\pi$
$660$	0.513838	−	0.570675i	0.513838	−	0.570675i
$661$	0.241922	−	0.419021i	0.241922	−	0.419021i	−0.719340	−	0.694658i	$-0.755556\pi$
$661$	0.241922	−	0.419021i	0.241922	−	0.419021i	0.961262	+	0.275637i	$0.0888889\pi$
$662$		0			0
$663$		0			0
$664$	0.865762	+	1.49954i	0.865762	+	1.49954i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	1.24324	−	2.15336i	1.24324	−	2.15336i
$671$	0.0840186	−	0.145524i	0.0840186	−	0.145524i
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$	1.32303			1.32303
$675$	0.0458545	−	0.436276i	0.0458545	−	0.436276i
$676$	2.11839			2.11839
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$	2.31159	+	0.491343i	2.31159	+	0.491343i
$679$		0			0
$680$	−0.256940	+	0.445034i	−0.256940	+	0.445034i
$681$		0			0
$682$	−0.0298187	−	0.0516476i	−0.0298187	−	0.0516476i
$683$		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	−0.939693	−	0.342020i	$-0.888889\pi$
$691$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$692$		0			0
$693$		0			0
$694$	−0.123258			−0.123258
$695$		0			0
$696$	1.21761	+	3.74743i	1.21761	+	3.74743i
$697$		0			0
$698$	1.69749	−	2.94013i	1.69749	−	2.94013i
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−0.142038	−	0.246017i	−0.142038	−	0.246017i
$705$		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	$0.333333\pi$
$710$	−2.14071			−2.14071
$711$		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
$718$	0.545692	+	0.945166i	0.545692	+	0.945166i
$719$	1.98054			1.98054			0.990268	−	0.139173i	$-0.0444444\pi$
$719$	1.98054			1.98054			0.990268	+	0.139173i	$0.0444444\pi$
$720$	−0.937117	−	0.417231i	−0.937117	−	0.417231i
$721$		0			0
$722$	0.882948	+	1.52931i	0.882948	+	1.52931i
$723$	0.413545	+	1.27276i	0.413545	+	1.27276i
$724$		0			0
$725$	−0.437611	+	0.757964i	−0.437611	+	0.757964i
$726$	−1.32294	−	0.281199i	−1.32294	−	0.281199i
$727$	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
$732$	−0.719631	−	0.152962i	−0.719631	−	0.152962i
$733$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	−0.882948	−	0.469472i	$-0.844444\pi$
$733$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	0.848048	+	0.529919i	$0.177778\pi$
$734$	0.774117	−	1.34081i	0.774117	−	1.34081i
$735$	−0.231520	−	0.712544i	−0.231520	−	0.712544i
$736$		0			0
$737$	0.909329			0.909329
$738$		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	$0.666667\pi$
$744$	−0.0922394	+	0.102442i	−0.0922394	+	0.102442i
$745$		0			0
$746$	2.85728			2.85728
$747$	−0.0916445	−	0.871939i	−0.0916445	−	0.871939i
$748$	−0.355968			−0.355968
$749$		0			0
$750$	−0.588189	−	1.81026i	−0.588189	−	1.81026i
$751$	−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
$753$	−1.49861	−	0.318539i	−1.49861	−	0.318539i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.76590			−1.76590			−0.882948	−	0.469472i	$-0.844444\pi$
$757$	−1.76590			−1.76590			−0.882948	+	0.469472i	$0.844444\pi$
$758$		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	$0.800000\pi$
$762$	−1.04910	−	3.22881i	−1.04910	−	3.22881i
$763$		0			0
$764$		0			0
$765$	0.210505	−	0.152941i	0.210505	−	0.152941i
$766$	1.76590			1.76590
$767$		0			0
$768$	−1.35553	+	1.50546i	−1.35553	+	1.50546i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$	1.28642	−	1.42871i	1.28642	−	1.42871i
$772$	1.30421	+	2.25896i	1.30421	+	2.25896i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−0.0306194			−0.0306194
$776$		0			0
$777$		0			0
$778$	1.49756	−	2.59386i	1.49756	−	2.59386i
$779$		0			0
$780$		0			0
$781$	−0.391438	−	0.677990i	−0.391438	−	0.677990i
$782$		0			0
$783$	0.208548	−	1.98420i	0.208548	−	1.98420i
$784$	1.36917			1.36917
$785$	−0.747388	−	1.29451i	−0.747388	−	1.29451i
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	1.52384	−	2.63937i	1.52384	−	2.63937i
$789$	0.345600	+	1.06365i	0.345600	+	1.06365i
$790$		0			0
$791$		0			0
$792$	−0.0998841	−	0.950334i	−0.0998841	−	0.950334i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	0.0348995	+	0.999391i	$0.488889\pi$
$797$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	−0.882948	+	0.469472i	$0.844444\pi$
$798$		0			0
$799$		0			0
$800$	0.194275			0.194275
$801$		0			0
$802$	−1.54823			−1.54823
$803$		0			0
$804$	−1.23028	−	3.78641i	−1.23028	−	3.78641i
$805$		0			0
$806$		0			0
$807$	−0.0682737	−	0.0145120i	−0.0682737	−	0.0145120i
$808$	0.610294	+	1.05706i	0.610294	+	1.05706i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	0.885281	+	0.983204i	0.885281	+	0.983204i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	0.473271	+	0.100597i	0.473271	+	0.100597i
$814$		0			0
$815$	−0.461262	+	0.798929i	−0.461262	+	0.798929i
$816$	0.146940	+	0.452236i	0.146940	+	0.452236i
$817$		0			0
$818$	−3.49742			−3.49742
$819$		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
$825$	0.142025	−	0.157734i	0.142025	−	0.157734i
$826$		0			0
$827$	−1.99513			−1.99513			−0.997564	−	0.0697565i	$-0.977778\pi$
$827$	−1.99513			−1.99513			−0.997564	+	0.0697565i	$0.977778\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$	−0.579979	−	1.00455i	−0.579979	−	1.00455i
$831$		0			0
$832$		0			0
$833$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.0637646	−	0.0283898i	0.0637646	−	0.0283898i
$838$	−1.97495			−1.97495
$839$	0.719340	+	1.24593i	0.719340	+	1.24593i	0.961262	+	0.275637i	$0.0888889\pi$
$839$	0.719340	+	1.24593i	0.719340	+	1.24593i	−0.241922	+	0.970296i	$0.577778\pi$
$840$		0			0
$841$	−1.49027	+	2.58122i	−1.49027	+	2.58122i
$842$	1.18161	−	2.04662i	1.18161	−	2.04662i
$843$		0			0
$844$	−0.928639	−	1.60845i	−0.928639	−	1.60845i
$845$	−0.749213			−0.749213
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.18161	+	1.31232i	−1.18161	+	1.31232i
$850$	−0.134519	+	0.232993i	−0.134519	+	0.232993i
$851$		0			0
$852$	−2.29353	+	2.54722i	−2.29353	+	2.54722i
$853$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.615661	−	0.788011i	$-0.711111\pi$
$853$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.374607	−	0.927184i	$-0.622222\pi$
$854$		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
$859$	0.719340	−	1.24593i	0.719340	−	1.24593i	−0.241922	−	0.970296i	$-0.577778\pi$
$859$	0.719340	−	1.24593i	0.719340	−	1.24593i	0.961262	−	0.275637i	$-0.0888889\pi$
$860$		0			0
$861$		0			0
$862$	−1.08719	−	1.88307i	−1.08719	−	1.88307i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.404576	+	0.180129i	−0.404576	+	0.180129i
$865$		0			0
$866$		0			0
$867$	0.860169	+	0.182834i	0.860169	+	0.182834i
$868$		0			0
$869$		0			0
$870$	−0.815687	−	2.51043i	−0.815687	−	2.51043i
$871$		0			0
$872$	−3.60842			−3.60842
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	−0.997564	−	0.0697565i	$-0.977778\pi$
$877$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	0.559193	+	0.829038i	$0.311111\pi$
$878$	−1.72731	+	2.99178i	−1.72731	+	2.99178i
$879$	−1.08268	+	1.20243i	−1.08268	+	1.20243i
$880$	−0.248164	−	0.429833i	−0.248164	−	0.429833i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.61323	−	0.718254i	−1.61323	−	0.718254i
$883$	1.69610			1.69610			0.848048	−	0.529919i	$-0.177778\pi$
$883$	1.69610			1.69610			0.848048	+	0.529919i	$0.177778\pi$
$884$		0			0
$885$		0			0
$886$	1.74871	−	3.02885i	1.74871	−	3.02885i
$887$	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
$891$	−0.149516	+	0.460163i	−0.149516	+	0.460163i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.139258			−0.139258
$900$	−0.848951	−	0.377977i	−0.848951	−	0.377977i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.32150	−	2.28891i	1.32150	−	2.28891i
$905$		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
$913$	0.212103	−	0.367373i	0.212103	−	0.367373i
$914$	−1.42864	+	2.47448i	−1.42864	+	2.47448i
$915$	0.254513	+	0.0540984i	0.254513	+	0.0540984i
$916$		0			0
$917$		0			0
$918$	0.0641062	−	0.609929i	0.0641062	−	0.609929i
$919$	−1.23132			−1.23132			−0.615661	−	0.788011i	$-0.711111\pi$
$919$	−1.23132			−1.23132			−0.615661	+	0.788011i	$0.711111\pi$
$920$		0			0
$921$	0.732841	+	0.155770i	0.732841	+	0.155770i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	0.883569			0.883569
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$	0.0617917	−	0.0686267i	0.0617917	−	0.0686267i
$931$		0			0
$932$		0			0
$933$	1.02517	−	1.13856i	1.02517	−	1.13856i
$934$		0			0
$935$	0.125896			0.125896
$936$		0			0
$937$	1.69610			1.69610			0.848048	−	0.529919i	$-0.177778\pi$
$937$	1.69610			1.69610			0.848048	+	0.529919i	$0.177778\pi$
$938$		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	$0.666667\pi$
$942$	−3.44620	−	0.732512i	−3.44620	−	0.732512i
$943$		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
$956$	−1.05919	−	1.83458i	−1.05919	−	1.83458i
$957$	0.645932	−	0.717380i	0.645932	−	0.717380i
$958$	1.49756	−	2.59386i	1.49756	−	2.59386i
$959$		0			0
$960$	0.294337	−	0.326895i	0.294337	−	0.326895i
$961$	0.497564	+	0.861806i	0.497564	+	0.861806i
$962$		0			0
$963$		0			0
$964$	2.83495			2.83495
$965$	−0.461262	−	0.798929i	−0.461262	−	0.798929i
$966$		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	$0.133333\pi$
$968$	−0.756303	+	1.30996i	−0.756303	+	1.30996i
$969$		0			0
$970$		0			0
$971$	1.98054			1.98054			0.990268	−	0.139173i	$-0.0444444\pi$
$971$	1.98054			1.98054			0.990268	+	0.139173i	$0.0444444\pi$
$972$	2.11839			2.11839
$973$		0			0
$974$	1.35275	+	2.34304i	1.35275	+	2.34304i
$975$		0			0
$976$	−0.237754	+	0.411803i	−0.237754	+	0.411803i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$	0.671923	+	2.06797i	0.671923	+	2.06797i
$979$		0			0
$980$	−1.58712			−1.58712
$981$	1.66913	+	0.743145i	1.66913	+	0.743145i
$982$		0			0
$983$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.719340	−	0.694658i	$-0.755556\pi$
$983$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.241922	−	0.970296i	$-0.577778\pi$
$984$		0			0
$985$	−0.538939	+	0.933469i	−0.538939	+	0.933469i
$986$	−0.611795	+	1.05966i	−0.611795	+	1.05966i
$987$		0			0
$988$		0			0
$989$		0			0
$990$	0.0669129	+	0.636634i	0.0669129	+	0.636634i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	0.0154557	+	0.0267701i	0.0154557	+	0.0267701i
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−1.81669	−	0.386150i	−1.81669	−	0.386150i
$997$		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.238.11	✓	24
9.7	even	3	inner	2151.1.f.b.1672.11	yes	24
239.238	odd	2	CM	2151.1.f.b.238.11	✓	24
2151.1672	odd	6	inner	2151.1.f.b.1672.11	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.1.f.b.238.11	✓	24	1.1	even	1	trivial
2151.1.f.b.238.11	✓	24	239.238	odd	2	CM
2151.1.f.b.1672.11	yes	24	9.7	even	3	inner
2151.1.f.b.1672.11	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)

\(f(q)\)	\(=\)	\(q+(0.882948 + 1.52931i) q^{2} +(-0.978148 - 0.207912i) q^{3} +(-1.05919 + 1.83458i) q^{4} +(0.374607 - 0.648838i) q^{5} +(-0.545692 - 1.67947i) q^{6} -1.97495 q^{8} +(0.913545 + 0.406737i) q^{9} +O(q^{10})\)	\(q+(0.882948 + 1.52931i) q^{2} +(-0.978148 - 0.207912i) q^{3} +(-1.05919 + 1.83458i) q^{4} +(0.374607 - 0.648838i) q^{5} +(-0.545692 - 1.67947i) q^{6} -1.97495 q^{8} +(0.913545 + 0.406737i) q^{9} +1.32303 q^{10} +(0.241922 + 0.419021i) q^{11} +(1.41748 - 1.57427i) q^{12} +(-0.501321 + 0.556774i) q^{15} +(-0.684586 - 1.18574i) q^{16} +0.347296 q^{17} +(0.184586 + 1.75622i) q^{18} +(0.793561 + 1.37449i) q^{20} +(-0.427209 + 0.739947i) q^{22} +(1.93179 + 0.410616i) q^{24} +(0.219340 + 0.379908i) q^{25} +(-0.809017 - 0.587785i) q^{27} +(0.997564 + 1.72783i) q^{29} +(-1.29412 - 0.275074i) q^{30} +(-0.0348995 + 0.0604477i) q^{31} +(0.221432 - 0.383531i) q^{32} +(-0.149516 - 0.460163i) q^{33} +(0.306644 + 0.531124i) q^{34} +(-1.71381 + 1.24516i) q^{36} +(-0.739830 + 1.28142i) q^{40} -1.02497 q^{44} +(0.606126 - 0.440376i) q^{45} +(0.423098 + 1.30216i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.387331 + 0.670877i) q^{50} +(-0.339707 - 0.0722070i) q^{51} +(0.184586 - 1.75622i) q^{54} +0.362502 q^{55} +(-1.76159 + 3.05117i) q^{58} +(-0.490448 - 1.50944i) q^{60} +(-0.173648 - 0.300767i) q^{61} -0.123258 q^{62} -0.587123 q^{64} +(0.571717 - 0.634956i) q^{66} +(0.939693 - 1.62760i) q^{67} +(-0.367854 + 0.637142i) q^{68} -1.61803 q^{71} +(-1.80421 - 0.803285i) q^{72} +(-0.135559 - 0.417209i) q^{75} -1.02580 q^{80} +(0.669131 + 0.743145i) q^{81} +(-0.438371 - 0.759281i) q^{83} +(0.130100 - 0.225339i) q^{85} +(-0.616528 - 1.89748i) q^{87} +(-0.477784 - 0.827546i) q^{88} +(1.20865 + 0.538126i) q^{90} +(0.0467046 - 0.0518708i) q^{93} +(-0.296333 + 0.329112i) q^{96} -1.76590 q^{98} +(0.0505754 + 0.481193i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 3 q^{3} - 12 q^{4} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) 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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