LMFDB - Embedded newform 2151.1.f.b.238.4

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			238.4
Root			$-0.615661 - 0.788011i$ of defining polynomial
Character	$\chi$	$=$	2151.238
Dual form			2151.1.f.b.1672.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.559193 - 0.968551i) q^{2} +(0.913545 - 0.406737i) q^{3} +(-0.125393 + 0.217188i) q^{4} +(0.719340 - 1.24593i) q^{5} +(-0.904793 - 0.657371i) q^{6} -0.837909 q^{8} +(0.669131 - 0.743145i) q^{9} +O(q^{10})$	\(q+(-0.559193 - 0.968551i) q^{2} +(0.913545 - 0.406737i) q^{3} +(-0.125393 + 0.217188i) q^{4} +(0.719340 - 1.24593i) q^{5} +(-0.904793 - 0.657371i) q^{6} -0.837909 q^{8} +(0.669131 - 0.743145i) q^{9} -1.60900 q^{10} +(0.882948 + 1.52931i) q^{11} +(-0.0262144 + 0.249413i) q^{12} +(0.150383 - 1.43080i) q^{15} +(0.593946 + 1.02875i) q^{16} -1.87939 q^{17} +(-1.09395 - 0.232525i) q^{18} +(0.180401 + 0.312464i) q^{20} +(0.987476 - 1.71036i) q^{22} +(-0.765468 + 0.340808i) q^{24} +(-0.534899 - 0.926473i) q^{25} +(0.309017 - 0.951057i) q^{27} +(-0.990268 - 1.71519i) q^{29} +(-1.46989 + 0.654439i) q^{30} +(0.997564 - 1.72783i) q^{31} +(0.245307 - 0.424883i) q^{32} +(1.42864 + 1.03797i) q^{33} +(1.05094 + 1.82028i) q^{34} +(0.0774974 + 0.238512i) q^{36} +(-0.602742 + 1.04398i) q^{40} -0.442863 q^{44} +(-0.444576 - 1.36827i) q^{45} +(0.961025 + 0.698226i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.598224 + 1.03615i) q^{50} +(-1.71690 + 0.764415i) q^{51} +(-1.09395 + 0.232525i) q^{54} +2.54056 q^{55} +(-1.10750 + 1.91825i) q^{58} +(0.291895 + 0.212074i) q^{60} +(0.939693 + 1.62760i) q^{61} -2.23132 q^{62} +0.639198 q^{64} +(0.206439 - 1.96413i) q^{66} +(-0.766044 + 1.32683i) q^{67} +(0.235663 - 0.408179i) q^{68} +0.618034 q^{71} +(-0.560671 + 0.622688i) q^{72} +(-0.865486 - 0.628812i) q^{75} +1.70900 q^{80} +(-0.104528 - 0.994522i) q^{81} +(0.615661 + 1.06636i) q^{83} +(-1.35192 + 2.34159i) q^{85} +(-1.60229 - 1.16413i) q^{87} +(-0.739830 - 1.28142i) q^{88} +(-1.07663 + 1.19572i) q^{90} +(0.208548 - 1.98420i) q^{93} +(0.0512830 - 0.487925i) q^{96} +1.11839 q^{98} +(1.72731 + 0.367150i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	−0.997564	−	0.0697565i	$-0.977778\pi$
$2$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	0.438371	−	0.898794i	$-0.355556\pi$
$3$	0.913545	−	0.406737i	0.913545	−	0.406737i
$3$	0.913545	−	0.406737i	0.913545	−	0.406737i
$4$	−0.125393	+	0.217188i	−0.125393	+	0.217188i
$5$	0.719340	−	1.24593i	0.719340	−	1.24593i	−0.241922	−	0.970296i	$-0.577778\pi$
$5$	0.719340	−	1.24593i	0.719340	−	1.24593i	0.961262	−	0.275637i	$-0.0888889\pi$
$6$	−0.904793	−	0.657371i	−0.904793	−	0.657371i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	−0.837909			−0.837909
$9$	0.669131	−	0.743145i	0.669131	−	0.743145i
$10$	−1.60900			−1.60900
$11$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.848048	+	0.529919i	$0.177778\pi$
$11$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.0348995	+	0.999391i	$0.488889\pi$
$12$	−0.0262144	+	0.249413i	−0.0262144	+	0.249413i
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$	0.150383	−	1.43080i	0.150383	−	1.43080i
$16$	0.593946	+	1.02875i	0.593946	+	1.02875i
$17$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$17$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$18$	−1.09395	−	0.232525i	−1.09395	−	0.232525i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	0.180401	+	0.312464i	0.180401	+	0.312464i
$21$		0			0
$22$	0.987476	−	1.71036i	0.987476	−	1.71036i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	−0.765468	+	0.340808i	−0.765468	+	0.340808i
$25$	−0.534899	−	0.926473i	−0.534899	−	0.926473i
$26$		0			0
$27$	0.309017	−	0.951057i	0.309017	−	0.951057i
$28$		0			0
$29$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.615661	−	0.788011i	$-0.711111\pi$
$29$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.374607	−	0.927184i	$-0.622222\pi$
$30$	−1.46989	+	0.654439i	−1.46989	+	0.654439i
$31$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.438371	−	0.898794i	$-0.355556\pi$
$31$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.559193	−	0.829038i	$-0.311111\pi$
$32$	0.245307	−	0.424883i	0.245307	−	0.424883i
$33$	1.42864	+	1.03797i	1.42864	+	1.03797i
$34$	1.05094	+	1.82028i	1.05094	+	1.82028i
$35$		0			0
$36$	0.0774974	+	0.238512i	0.0774974	+	0.238512i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	−0.602742	+	1.04398i	−0.602742	+	1.04398i
$41$		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	$0.333333\pi$
$44$	−0.442863			−0.442863
$45$	−0.444576	−	1.36827i	−0.444576	−	1.36827i
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	0.961025	+	0.698226i	0.961025	+	0.698226i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	−0.598224	+	1.03615i	−0.598224	+	1.03615i
$51$	−1.71690	+	0.764415i	−1.71690	+	0.764415i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−1.09395	+	0.232525i	−1.09395	+	0.232525i
$55$	2.54056			2.54056
$56$		0			0
$57$		0			0
$58$	−1.10750	+	1.91825i	−1.10750	+	1.91825i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	0.291895	+	0.212074i	0.291895	+	0.212074i
$61$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$61$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$62$	−2.23132			−2.23132
$63$		0			0
$64$	0.639198			0.639198
$65$		0			0
$66$	0.206439	−	1.96413i	0.206439	−	1.96413i
$67$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$67$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$68$	0.235663	−	0.408179i	0.235663	−	0.408179i
$69$		0			0
$70$		0			0
$71$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$71$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$72$	−0.560671	+	0.622688i	−0.560671	+	0.622688i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−0.865486	−	0.628812i	−0.865486	−	0.628812i
$76$		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	$0.333333\pi$
$80$	1.70900			1.70900
$81$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$82$		0			0
$83$	0.615661	+	1.06636i	0.615661	+	1.06636i	0.990268	+	0.139173i	$0.0444444\pi$
$83$	0.615661	+	1.06636i	0.615661	+	1.06636i	−0.374607	+	0.927184i	$0.622222\pi$
$84$		0			0
$85$	−1.35192	+	2.34159i	−1.35192	+	2.34159i
$86$		0			0
$87$	−1.60229	−	1.16413i	−1.60229	−	1.16413i
$88$	−0.739830	−	1.28142i	−0.739830	−	1.28142i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−1.07663	+	1.19572i	−1.07663	+	1.19572i
$91$		0			0
$92$		0			0
$93$	0.208548	−	1.98420i	0.208548	−	1.98420i
$94$		0			0
$95$		0			0
$96$	0.0512830	−	0.487925i	0.0512830	−	0.487925i
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$	1.11839			1.11839
$99$	1.72731	+	0.367150i	1.72731	+	0.367150i
$100$	0.268291			0.268291
$101$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$101$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$102$	1.70045	+	1.23545i	1.70045	+	1.23545i
$103$		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			$\pi$
$108$	0.167809	+	0.186371i	0.167809	+	0.186371i
$109$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$109$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$110$	−1.42066	−	2.46066i	−1.42066	−	2.46066i
$111$		0			0
$112$		0			0
$113$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$113$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$114$		0			0
$115$		0			0
$116$	0.496692			0.496692
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−0.126007	+	1.19888i	−0.126007	+	1.19888i
$121$	−1.05919	+	1.83458i	−1.05919	+	1.83458i
$122$	1.05094	−	1.82028i	1.05094	−	1.82028i
$123$		0			0
$124$	0.250176	+	0.433317i	0.250176	+	0.433317i
$125$	−0.100418			−0.100418
$126$		0			0
$127$	1.69610			1.69610			0.848048	−	0.529919i	$-0.177778\pi$
$127$	1.69610			1.69610			0.848048	+	0.529919i	$0.177778\pi$
$128$	−0.602742	−	1.04398i	−0.602742	−	1.04398i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	−0.404576	+	0.180129i	−0.404576	+	0.180129i
$133$		0			0
$134$	1.71347			1.71347
$135$	−0.962665	−	1.06915i	−0.962665	−	1.06915i
$136$	1.57475			1.57475
$137$		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
$142$	−0.345600	−	0.598597i	−0.345600	−	0.598597i
$143$		0			0
$144$	1.16193	+	0.246977i	1.16193	+	0.246977i
$145$	−2.84936			−2.84936
$146$		0			0
$147$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−0.125063	+	1.18989i	−0.125063	+	1.18989i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$	−1.25755	+	1.39666i	−1.25755	+	1.39666i
$154$		0			0
$155$	−1.43518	−	2.48580i	−1.43518	−	2.48580i
$156$		0			0
$157$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.374607	+	0.927184i	$0.622222\pi$
$157$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.615661	+	0.788011i	$0.711111\pi$
$158$		0			0
$159$		0			0
$160$	−0.352917	−	0.611271i	−0.352917	−	0.611271i
$161$		0			0
$162$	−0.904793	+	0.657371i	−0.904793	+	0.657371i
$163$	−0.483844			−0.483844			−0.241922	−	0.970296i	$-0.577778\pi$
$163$	−0.483844			−0.483844			−0.241922	+	0.970296i	$0.577778\pi$
$164$		0			0
$165$	2.32091	−	1.03334i	2.32091	−	1.03334i
$166$	0.688547	−	1.19260i	0.688547	−	1.19260i
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$	3.02393			3.02393
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$	−0.231531	+	2.20287i	−0.231531	+	2.20287i
$175$		0			0
$176$	−1.04885	+	1.81666i	−1.04885	+	1.81666i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	0.352917	+	0.0750149i	0.352917	+	0.0750149i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	1.52045	+	1.10467i	1.52045	+	1.10467i
$184$		0			0
$185$		0			0
$186$	−2.03841	+	0.907561i	−2.03841	+	0.907561i
$187$	−1.65940	−	2.87416i	−1.65940	−	2.87416i
$188$		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	$0.333333\pi$
$192$	0.583937	−	0.259985i	0.583937	−	0.259985i
$193$	0.241922	−	0.419021i	0.241922	−	0.419021i	−0.719340	−	0.694658i	$-0.755556\pi$
$193$	0.241922	−	0.419021i	0.241922	−	0.419021i	0.961262	+	0.275637i	$0.0888889\pi$
$194$		0			0
$195$		0			0
$196$	−0.125393	−	0.217188i	−0.125393	−	0.217188i
$197$	0.0697990			0.0697990			0.0348995	−	0.999391i	$-0.488889\pi$
$197$	0.0697990			0.0697990			0.0348995	+	0.999391i	$0.488889\pi$
$198$	−0.610294	−	1.87829i	−0.610294	−	1.87829i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	0.448197	+	0.776301i	0.448197	+	0.776301i
$201$	−0.160147	+	1.52370i	−0.160147	+	1.52370i
$202$	0.904793	−	1.56715i	0.904793	−	1.56715i
$203$		0			0
$204$	0.0492669	−	0.468743i	0.0492669	−	0.468743i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.615661	−	1.06636i	0.615661	−	1.06636i	−0.374607	−	0.927184i	$-0.622222\pi$
$211$	0.615661	−	1.06636i	0.615661	−	1.06636i	0.990268	−	0.139173i	$-0.0444444\pi$
$212$		0			0
$213$	0.564602	−	0.251377i	0.564602	−	0.251377i
$214$		0			0
$215$		0			0
$216$	−0.258928	+	0.796899i	−0.258928	+	0.796899i
$217$		0			0
$218$	−0.748346	−	1.29617i	−0.748346	−	1.29617i
$219$		0			0
$220$	−0.318569	+	0.551778i	−0.318569	+	0.551778i
$221$		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
$225$	−1.04642	−	0.222424i	−1.04642	−	0.222424i
$226$	−0.233806			−0.233806
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$	0.829755	+	1.43718i	0.829755	+	1.43718i
$233$		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
$240$	1.56125	−	0.695112i	1.56125	−	0.695112i
$241$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$241$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$242$	2.36917			2.36917
$243$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$244$	−0.471325			−0.471325
$245$	0.719340	+	1.24593i	0.719340	+	1.24593i
$246$		0			0
$247$		0			0
$248$	−0.835868	+	1.44777i	−0.835868	+	1.44777i
$249$	0.996161	+	0.723753i	0.996161	+	0.723753i
$250$	0.0561532	+	0.0972603i	0.0561532	+	0.0972603i
$251$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$251$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$252$		0			0
$253$		0			0
$254$	−0.948445	−	1.64275i	−0.948445	−	1.64275i
$255$	−0.282628	+	2.68902i	−0.282628	+	2.68902i
$256$	−0.354499	+	0.614010i	−0.354499	+	0.614010i
$257$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	0.0348995	+	0.999391i	$0.488889\pi$
$257$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	−0.882948	+	0.469472i	$0.844444\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−1.93726	−	0.411777i	−1.93726	−	0.411777i
$262$		0			0
$263$	0.374607	+	0.648838i	0.374607	+	0.648838i	0.990268	−	0.139173i	$-0.0444444\pi$
$263$	0.374607	+	0.648838i	0.374607	+	0.648838i	−0.615661	+	0.788011i	$0.711111\pi$
$264$	−1.19707	−	0.869722i	−1.19707	−	0.869722i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.192114	−	0.332751i	−0.192114	−	0.332751i
$269$	−1.99513			−1.99513			−0.997564	−	0.0697565i	$-0.977778\pi$
$269$	−1.99513			−1.99513			−0.997564	+	0.0697565i	$0.977778\pi$
$270$	−0.497208	+	1.53025i	−0.497208	+	1.53025i
$271$	−1.76590			−1.76590			−0.882948	−	0.469472i	$-0.844444\pi$
$271$	−1.76590			−1.76590			−0.882948	+	0.469472i	$0.844444\pi$
$272$	−1.11625	−	1.93341i	−1.11625	−	1.93341i
$273$		0			0
$274$		0			0
$275$	0.944576	−	1.63605i	0.944576	−	1.63605i
$276$		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
$279$	−0.616528	−	1.89748i	−0.616528	−	1.89748i
$280$		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
$283$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	0.438371	+	0.898794i	$0.355556\pi$
$283$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	−0.997564	+	0.0697565i	$0.977778\pi$
$284$	−0.0774974	+	0.134229i	−0.0774974	+	0.134229i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.151608	−	0.466601i	−0.151608	−	0.466601i
$289$	2.53209			2.53209
$290$	1.59334	+	2.75975i	1.59334	+	2.75975i
$291$		0			0
$292$		0			0
$293$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$293$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$294$	1.02170	−	0.454888i	1.02170	−	0.454888i
$295$		0			0
$296$		0			0
$297$	1.72731	−	0.367150i	1.72731	−	0.367150i
$298$		0			0
$299$		0			0
$300$	0.245096	−	0.109124i	0.245096	−	0.109124i
$301$		0			0
$302$		0			0
$303$	1.30902	+	0.951057i	1.30902	+	0.951057i
$304$		0			0
$305$	2.70383			2.70383
$306$	2.05595	+	0.437005i	2.05595	+	0.437005i
$307$	−1.43868			−1.43868			−0.719340	−	0.694658i	$-0.755556\pi$
$307$	−1.43868			−1.43868			−0.719340	+	0.694658i	$0.755556\pi$
$308$		0			0
$309$		0			0
$310$	−1.60508	+	2.78008i	−1.60508	+	2.78008i
$311$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$311$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	2.21500			2.21500
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$	1.74871	−	3.02885i	1.74871	−	3.02885i
$320$	0.459801	−	0.796398i	0.459801	−	0.796398i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.229105	+	0.102004i	0.229105	+	0.102004i
$325$		0			0
$326$	0.270562	+	0.468627i	0.270562	+	0.468627i
$327$	1.22256	−	0.544320i	1.22256	−	0.544320i
$328$		0			0
$329$		0			0
$330$	−2.29868	−	1.67009i	−2.29868	−	1.67009i
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	−0.308800			−0.308800
$333$		0			0
$334$		0			0
$335$	1.10209	+	1.90888i	1.10209	+	1.90888i
$336$		0			0
$337$	0.719340	−	1.24593i	0.719340	−	1.24593i	−0.241922	−	0.970296i	$-0.577778\pi$
$337$	0.719340	−	1.24593i	0.719340	−	1.24593i	0.961262	−	0.275637i	$-0.0888889\pi$
$338$	−0.559193	+	0.968551i	−0.559193	+	0.968551i
$339$	0.0218524	−	0.207912i	0.0218524	−	0.207912i
$340$	−0.339043	−	0.587239i	−0.339043	−	0.587239i
$341$	3.52319			3.52319
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.438371	−	0.898794i	$-0.355556\pi$
$347$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.559193	−	0.829038i	$-0.311111\pi$
$348$	0.453751	−	0.202023i	0.453751	−	0.202023i
$349$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	−0.882948	−	0.469472i	$-0.844444\pi$
$349$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	0.0348995	−	0.999391i	$-0.488889\pi$
$350$		0			0
$351$		0			0
$352$	0.866371			0.866371
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$	0.444576	−	0.770029i	0.444576	−	0.770029i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$359$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$360$	0.372515	+	1.14648i	0.372515	+	1.14648i
$361$	1.00000			1.00000
$362$		0			0
$363$	−0.221432	+	2.10678i	−0.221432	+	2.10678i
$364$		0			0
$365$		0			0
$366$	0.219706	−	2.09036i	0.219706	−	2.09036i
$367$	0.615661	+	1.06636i	0.615661	+	1.06636i	0.990268	+	0.139173i	$0.0444444\pi$
$367$	0.615661	+	1.06636i	0.615661	+	1.06636i	−0.374607	+	0.927184i	$0.622222\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.404793	+	0.294099i	0.404793	+	0.294099i
$373$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$373$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$374$	−1.85585	+	3.21442i	−1.85585	+	3.21442i
$375$	−0.0917368	+	0.0408438i	−0.0917368	+	0.0408438i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	1.54946	−	0.689864i	1.54946	−	0.689864i
$382$		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			$0$
$384$	−0.975256	−	0.708565i	−0.975256	−	0.708565i
$385$		0			0
$386$	−0.541124			−0.541124
$387$		0			0
$388$		0			0
$389$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	0.559193	−	0.829038i	$-0.311111\pi$
$389$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	−0.997564	+	0.0697565i	$0.977778\pi$
$390$		0			0
$391$		0			0
$392$	0.418955	−	0.725651i	0.418955	−	0.725651i
$393$		0			0
$394$	−0.0390311	−	0.0676039i	−0.0390311	−	0.0676039i
$395$		0			0
$396$	−0.296333	+	0.329112i	−0.296333	+	0.329112i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	0.635403	−	1.10055i	0.635403	−	1.10055i
$401$	0.615661	−	1.06636i	0.615661	−	1.06636i	−0.374607	−	0.927184i	$-0.622222\pi$
$401$	0.615661	−	1.06636i	0.615661	−	1.06636i	0.990268	−	0.139173i	$-0.0444444\pi$
$402$	1.56533	−	0.696930i	1.56533	−	0.696930i
$403$		0			0
$404$	−0.405782			−0.405782
$405$	−1.31430	−	0.585164i	−1.31430	−	0.585164i
$406$		0			0
$407$		0			0
$408$	1.43861	−	0.640510i	1.43861	−	0.640510i
$409$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.241922	+	0.970296i	$0.577778\pi$
$409$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.719340	+	0.694658i	$0.755556\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1.77148			1.77148
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.374607	−	0.648838i	0.374607	−	0.648838i	−0.615661	−	0.788011i	$-0.711111\pi$
$419$	0.374607	−	0.648838i	0.374607	−	0.648838i	0.990268	+	0.139173i	$0.0444444\pi$
$420$		0			0
$421$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$421$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$422$	−1.37709			−1.37709
$423$		0			0
$424$		0			0
$425$	1.00528	+	1.74120i	1.00528	+	1.74120i
$426$	−0.559193	−	0.406277i	−0.559193	−	0.406277i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.483844			−0.483844			−0.241922	−	0.970296i	$-0.577778\pi$
$431$	−0.483844			−0.483844			−0.241922	+	0.970296i	$0.577778\pi$
$432$	1.16193	−	0.246977i	1.16193	−	0.246977i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−2.60302	+	1.15894i	−2.60302	+	1.15894i
$436$	−0.167809	+	0.290654i	−0.167809	+	0.290654i
$437$		0			0
$438$		0			0
$439$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$439$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$440$	−2.12876			−2.12876
$441$	0.309017	+	0.951057i	0.309017	+	0.951057i
$442$		0			0
$443$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.719340	−	0.694658i	$-0.755556\pi$
$443$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.241922	−	0.970296i	$-0.577778\pi$
$444$		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			$\pi$
$450$	0.369723	+	1.13789i	0.369723	+	1.13789i
$451$		0			0
$452$	0.0262144	+	0.0454046i	0.0262144	+	0.0454046i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$457$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$458$		0			0
$459$	−0.580762	+	1.78740i	−0.580762	+	1.78740i
$460$		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	$0.666667\pi$
$464$	1.17633	−	2.03747i	1.17633	−	2.03747i
$465$	−2.32216	−	1.68715i	−2.32216	−	1.68715i
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.207022	+	1.96969i	−0.207022	+	1.96969i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.11839			1.11839
$479$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	0.559193	−	0.829038i	$-0.311111\pi$
$479$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	−0.997564	+	0.0697565i	$0.977778\pi$
$480$	−0.571032	−	0.414879i	−0.571032	−	0.414879i
$481$		0			0
$482$	0.116903	−	0.202482i	0.116903	−	0.202482i
$483$		0			0
$484$	−0.265632	−	0.460087i	−0.265632	−	0.460087i
$485$		0			0
$486$	−0.559193	+	0.968551i	−0.559193	+	0.968551i
$487$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$487$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$488$	−0.787377	−	1.36378i	−0.787377	−	1.36378i
$489$	−0.442013	+	0.196797i	−0.442013	+	0.196797i
$490$	0.804499	−	1.39343i	0.804499	−	1.39343i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$	1.86110	+	3.22351i	1.86110	+	3.22351i
$494$		0			0
$495$	1.69996	−	1.88800i	1.69996	−	1.88800i
$496$	2.37000			2.37000
$497$		0			0
$498$	0.143946	−	1.36955i	0.143946	−	1.36955i
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	0.0125918	−	0.0218096i	0.0125918	−	0.0218096i
$501$		0			0
$502$	−0.194206	−	0.336374i	−0.194206	−	0.336374i
$503$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$503$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$504$		0			0
$505$	2.32783			2.32783
$506$		0			0
$507$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$508$	−0.212679	+	0.368371i	−0.212679	+	0.368371i
$509$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$509$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$510$	2.76250	−	1.22994i	2.76250	−	1.22994i
$511$		0			0
$512$	−0.412551			−0.412551
$513$		0			0
$514$	1.89689			1.89689
$515$		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			$\pi$
$522$	0.684474	+	2.10659i	0.684474	+	2.10659i
$523$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$523$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$524$		0			0
$525$		0			0
$526$	0.418955	−	0.725651i	0.418955	−	0.725651i
$527$	−1.87481	+	3.24726i	−1.87481	+	3.24726i
$528$	−0.219269	+	2.08620i	−0.219269	+	2.08620i
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.641876	−	1.11176i	0.641876	−	1.11176i
$537$		0			0
$538$	1.11566	+	1.93238i	1.11566	+	1.93238i
$539$	−1.76590			−1.76590
$540$	0.352917	−	0.0750149i	0.352917	−	0.0750149i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	0.987476	+	1.71036i	0.987476	+	1.71036i
$543$		0			0
$544$	−0.461025	+	0.798520i	−0.461025	+	0.798520i
$545$	0.962665	−	1.66738i	0.962665	−	1.66738i
$546$		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
$549$	1.83832	+	0.390746i	1.83832	+	0.390746i
$550$	−2.11280			−2.11280
$551$		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			$\pi$
$558$	−1.49305	+	1.65820i	−1.49305	+	1.65820i
$559$		0			0
$560$		0			0
$561$	−2.68496	−	1.95074i	−2.68496	−	1.95074i
$562$		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
$565$	−0.150383	−	0.260471i	−0.150383	−	0.260471i
$566$	1.25079			1.25079
$567$		0			0
$568$	−0.517856			−0.517856
$569$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.559193	+	0.829038i	$0.311111\pi$
$569$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.438371	+	0.898794i	$0.355556\pi$
$570$		0			0
$571$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$571$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.427707	−	0.475017i	0.427707	−	0.475017i
$577$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$577$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$578$	−1.41593	−	2.45246i	−1.41593	−	2.45246i
$579$	0.0505754	−	0.481193i	0.0505754	−	0.481193i
$580$	0.357291	−	0.618845i	0.357291	−	0.618845i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.691200			0.691200
$587$	0.241922	+	0.419021i	0.241922	+	0.419021i	0.961262	−	0.275637i	$-0.0888889\pi$
$587$	0.241922	+	0.419021i	0.241922	+	0.419021i	−0.719340	+	0.694658i	$0.755556\pi$
$588$	−0.202891	−	0.147409i	−0.202891	−	0.147409i
$589$		0			0
$590$		0			0
$591$	0.0637646	−	0.0283898i	0.0637646	−	0.0283898i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−1.32150	−	1.46768i	−1.32150	−	1.46768i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$599$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$600$	0.725198	+	0.526888i	0.725198	+	0.526888i
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$	0.473442	+	1.45710i	0.473442	+	1.45710i
$604$		0			0
$605$	1.52384	+	2.63937i	1.52384	+	2.63937i
$606$	0.189153	−	1.79967i	0.189153	−	1.79967i
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$	−1.51196	−	2.61880i	−1.51196	−	2.61880i
$611$		0			0
$612$	−0.145647	−	0.448257i	−0.145647	−	0.448257i
$613$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$613$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$614$	0.804499	+	1.39343i	0.804499	+	1.39343i
$615$		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	$0.333333\pi$
$620$	0.719846			0.719846
$621$		0			0
$622$	0.388411			0.388411
$623$		0			0
$624$		0			0
$625$	0.462665	−	0.801359i	0.462665	−	0.801359i
$626$		0			0
$627$		0			0
$628$	−0.248346	−	0.430148i	−0.248346	−	0.430148i
$629$		0			0
$630$		0			0
$631$	−1.43868			−1.43868			−0.719340	−	0.694658i	$-0.755556\pi$
$631$	−1.43868			−1.43868			−0.719340	+	0.694658i	$0.755556\pi$
$632$		0			0
$633$	0.128708	−	1.22458i	0.128708	−	1.22458i
$634$		0			0
$635$	1.22007	−	2.11322i	1.22007	−	2.11322i
$636$		0			0
$637$		0			0
$638$	−3.91146			−3.91146
$639$	0.413545	−	0.459289i	0.413545	−	0.459289i
$640$	−1.73430			−1.73430
$641$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	0.848048	−	0.529919i	$-0.177778\pi$
$641$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	−0.882948	+	0.469472i	$0.844444\pi$
$642$		0			0
$643$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	0.0348995	+	0.999391i	$0.488889\pi$
$643$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	−0.882948	+	0.469472i	$0.844444\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.876742			0.876742			0.438371	−	0.898794i	$-0.355556\pi$
$647$	0.876742			0.876742			0.438371	+	0.898794i	$0.355556\pi$
$648$	0.0875854	+	0.833319i	0.0875854	+	0.833319i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.0606708	−	0.105085i	0.0606708	−	0.105085i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	−1.21085	−	0.879734i	−1.21085	−	0.879734i
$655$		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	$0.333333\pi$
$660$	−0.0665991	+	0.633648i	−0.0665991	+	0.633648i
$661$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.0348995	−	0.999391i	$-0.488889\pi$
$661$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.848048	−	0.529919i	$-0.177778\pi$
$662$		0			0
$663$		0			0
$664$	−0.515869	−	0.893511i	−0.515869	−	0.893511i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	1.23256	−	2.13486i	1.23256	−	2.13486i
$671$	−1.65940	+	2.87416i	−1.65940	+	2.87416i
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$	−1.60900			−1.60900
$675$	−1.04642	+	0.222424i	−1.04642	+	0.222424i
$676$	0.250787			0.250787
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$	−0.213593	+	0.0950976i	−0.213593	+	0.0950976i
$679$		0			0
$680$	1.13278	−	1.96204i	1.13278	−	1.96204i
$681$		0			0
$682$	−1.97014	−	3.41238i	−1.97014	−	3.41238i
$683$		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
$691$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$691$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$692$		0			0
$693$		0			0
$694$	−2.23132			−2.23132
$695$		0			0
$696$	1.34257	+	0.975435i	1.34257	+	0.975435i
$697$		0			0
$698$	−0.948445	+	1.64275i	−0.948445	+	1.64275i
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	0.564378	+	0.977532i	0.564378	+	0.977532i
$705$		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	$0.333333\pi$
$710$	−0.994416			−0.994416
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−0.104528	+	0.994522i	−0.104528	+	0.994522i
$718$	0.904793	+	1.56715i	0.904793	+	1.56715i
$719$	1.92252			1.92252			0.961262	−	0.275637i	$-0.0888889\pi$
$719$	1.92252			1.92252			0.961262	+	0.275637i	$0.0888889\pi$
$720$	1.14354	−	1.27003i	1.14354	−	1.27003i
$721$		0			0
$722$	−0.559193	−	0.968551i	−0.559193	−	0.968551i
$723$	0.169131	+	0.122881i	0.169131	+	0.122881i
$724$		0			0
$725$	−1.05939	+	1.83491i	−1.05939	+	1.83491i
$726$	2.16435	−	0.963629i	2.16435	−	0.963629i
$727$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$727$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$728$		0			0
$729$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$730$		0			0
$731$		0			0
$732$	−0.430577	+	0.191705i	−0.430577	+	0.191705i
$733$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.438371	−	0.898794i	$-0.355556\pi$
$733$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.559193	−	0.829038i	$-0.311111\pi$
$734$	0.688547	−	1.19260i	0.688547	−	1.19260i
$735$	1.16392	+	0.845635i	1.16392	+	0.845635i
$736$		0			0
$737$	−2.70551			−2.70551
$738$		0			0
$739$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$739$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$740$		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	$0.666667\pi$
$744$	−0.174744	+	1.66258i	−0.174744	+	1.66258i
$745$		0			0
$746$	0.691200			0.691200
$747$	1.20442	+	0.256006i	1.20442	+	0.256006i
$748$	0.832311			0.832311
$749$		0			0
$750$	0.0908579	+	0.0660121i	0.0908579	+	0.0660121i
$751$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$751$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$752$		0			0
$753$	0.317271	−	0.141258i	0.317271	−	0.141258i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.11839			1.11839			0.559193	−	0.829038i	$-0.311111\pi$
$757$	1.11839			1.11839			0.559193	+	0.829038i	$0.311111\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$761$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$762$	−1.53462	−	1.11496i	−1.53462	−	1.11496i
$763$		0			0
$764$		0			0
$765$	0.835530	+	2.57150i	0.835530	+	2.57150i
$766$	−1.11839			−1.11839
$767$		0			0
$768$	−0.0741104	+	0.705113i	−0.0741104	+	0.705113i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$	−0.177290	+	1.68680i	−0.177290	+	1.68680i
$772$	0.0606708	+	0.105085i	0.0606708	+	0.105085i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−2.13439			−2.13439
$776$		0			0
$777$		0			0
$778$	−0.490268	+	0.849169i	−0.490268	+	0.849169i
$779$		0			0
$780$		0			0
$781$	0.545692	+	0.945166i	0.545692	+	0.945166i
$782$		0			0
$783$	−1.93726	+	0.411777i	−1.93726	+	0.411777i
$784$	−1.18789			−1.18789
$785$	1.42468	+	2.46762i	1.42468	+	2.46762i
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	−0.00875233	+	0.0151595i	−0.00875233	+	0.0151595i
$789$	0.606126	+	0.440376i	0.606126	+	0.440376i
$790$		0			0
$791$		0			0
$792$	−1.44733	−	0.307639i	−1.44733	−	0.307639i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	−0.997564	−	0.0697565i	$-0.977778\pi$
$797$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	0.559193	+	0.829038i	$0.311111\pi$
$798$		0			0
$799$		0			0
$800$	−0.524857			−0.524857
$801$		0			0
$802$	−1.37709			−1.37709
$803$		0			0
$804$	−0.310847	−	0.225843i	−0.310847	−	0.225843i
$805$		0			0
$806$		0			0
$807$	−1.82264	+	0.811492i	−1.82264	+	0.811492i
$808$	−0.677883	−	1.17413i	−0.677883	−	1.17413i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	0.168186	+	1.60018i	0.168186	+	1.60018i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−1.61323	+	0.718254i	−1.61323	+	0.718254i
$814$		0			0
$815$	−0.348048	+	0.602837i	−0.348048	+	0.602837i
$816$	−1.80614	−	1.31224i	−1.80614	−	1.31224i
$817$		0			0
$818$	2.15012			2.15012
$819$		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
$825$	0.197470	−	1.87880i	0.197470	−	1.87880i
$826$		0			0
$827$	1.98054			1.98054			0.990268	−	0.139173i	$-0.0444444\pi$
$827$	1.98054			1.98054			0.990268	+	0.139173i	$0.0444444\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$	−0.990599	−	1.71577i	−0.990599	−	1.71577i
$831$		0			0
$832$		0			0
$833$	0.939693	−	1.62760i	0.939693	−	1.62760i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.33500	−	1.48267i	−1.33500	−	1.48267i
$838$	−0.837909			−0.837909
$839$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	0.848048	−	0.529919i	$-0.177778\pi$
$839$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	−0.882948	+	0.469472i	$0.844444\pi$
$840$		0			0
$841$	−1.46126	+	2.53098i	−1.46126	+	2.53098i
$842$	0.116903	−	0.202482i	0.116903	−	0.202482i
$843$		0			0
$844$	0.154400	+	0.267428i	0.154400	+	0.267428i
$845$	−1.43868			−1.43868
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−0.116903	+	1.11226i	−0.116903	+	1.11226i
$850$	1.12429	−	1.94733i	1.12429	−	1.94733i
$851$		0			0
$852$	−0.0162014	+	0.154146i	−0.0162014	+	0.154146i
$853$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.719340	−	0.694658i	$-0.755556\pi$
$853$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.241922	−	0.970296i	$-0.577778\pi$
$854$		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
$859$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	−0.882948	−	0.469472i	$-0.844444\pi$
$859$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	0.848048	+	0.529919i	$0.177778\pi$
$860$		0			0
$861$		0			0
$862$	0.270562	+	0.468627i	0.270562	+	0.468627i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.328284	−	0.364597i	−0.328284	−	0.364597i
$865$		0			0
$866$		0			0
$867$	2.31318	−	1.02989i	2.31318	−	1.02989i
$868$		0			0
$869$		0			0
$870$	2.57808	+	1.87308i	2.57808	+	1.87308i
$871$		0			0
$872$	−1.12134			−1.12134
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.615661	−	1.06636i	0.615661	−	1.06636i	−0.374607	−	0.927184i	$-0.622222\pi$
$877$	0.615661	−	1.06636i	0.615661	−	1.06636i	0.990268	−	0.139173i	$-0.0444444\pi$
$878$	−1.02170	+	1.76963i	−1.02170	+	1.76963i
$879$	−0.0646021	+	0.614648i	−0.0646021	+	0.614648i
$880$	1.50895	+	2.61359i	1.50895	+	2.61359i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.748346	−	0.831123i	0.748346	−	0.831123i
$883$	0.876742			0.876742			0.438371	−	0.898794i	$-0.355556\pi$
$883$	0.876742			0.876742			0.438371	+	0.898794i	$0.355556\pi$
$884$		0			0
$885$		0			0
$886$	−1.07506	+	1.86206i	−1.07506	+	1.86206i
$887$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$887$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.42864	−	1.03797i	1.42864	−	1.03797i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.95142			−3.95142
$900$	0.179522	−	0.199379i	0.179522	−	0.199379i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.0875854	+	0.151702i	−0.0875854	+	0.151702i
$905$		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
$909$	1.58268	+	0.336408i	1.58268	+	0.336408i
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$		0			0
$913$	−1.08719	+	1.88307i	−1.08719	+	1.88307i
$914$	−0.345600	+	0.598597i	−0.345600	+	0.598597i
$915$	2.47007	−	1.09975i	2.47007	−	1.09975i
$916$		0			0
$917$		0			0
$918$	2.05595	−	0.437005i	2.05595	−	0.437005i
$919$	−0.483844			−0.483844			−0.241922	−	0.970296i	$-0.577778\pi$
$919$	−0.483844			−0.483844			−0.241922	+	0.970296i	$0.577778\pi$
$920$		0			0
$921$	−1.31430	+	0.585164i	−1.31430	+	0.585164i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−0.971677			−0.971677
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$	−0.335553	+	3.19257i	−0.335553	+	3.19257i
$931$		0			0
$932$		0			0
$933$	−0.0363024	+	0.345394i	−0.0363024	+	0.345394i
$934$		0			0
$935$	−4.77469			−4.77469
$936$		0			0
$937$	0.876742			0.876742			0.438371	−	0.898794i	$-0.355556\pi$
$937$	0.876742			0.876742			0.438371	+	0.898794i	$0.355556\pi$
$938$		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	$0.666667\pi$
$942$	2.02351	−	0.900923i	2.02351	−	0.900923i
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−0.125393	−	0.217188i	−0.125393	−	0.217188i
$957$	0.365580	−	3.47826i	0.365580	−	3.47826i
$958$	−0.490268	+	0.849169i	−0.490268	+	0.849169i
$959$		0			0
$960$	0.0961245	−	0.914564i	0.0961245	−	0.914564i
$961$	−1.49027	−	2.58122i	−1.49027	−	2.58122i
$962$		0			0
$963$		0			0
$964$	−0.0524287			−0.0524287
$965$	−0.348048	−	0.602837i	−0.348048	−	0.602837i
$966$		0			0
$967$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$967$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$968$	0.887508	−	1.53721i	0.887508	−	1.53721i
$969$		0			0
$970$		0			0
$971$	1.92252			1.92252			0.961262	−	0.275637i	$-0.0888889\pi$
$971$	1.92252			1.92252			0.961262	+	0.275637i	$0.0888889\pi$
$972$	0.250787			0.250787
$973$		0			0
$974$	−0.194206	−	0.336374i	−0.194206	−	0.336374i
$975$		0			0
$976$	−1.11625	+	1.93341i	−1.11625	+	1.93341i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$	0.437779	+	0.318065i	0.437779	+	0.318065i
$979$		0			0
$980$	−0.360802			−0.360802
$981$	0.895472	−	0.994522i	0.895472	−	0.994522i
$982$		0			0
$983$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	−0.882948	−	0.469472i	$-0.844444\pi$
$983$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	0.0348995	−	0.999391i	$-0.488889\pi$
$984$		0			0
$985$	0.0502092	−	0.0869649i	0.0502092	−	0.0869649i
$986$	2.08142	−	3.60513i	2.08142	−	3.60513i
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−2.77923	−	0.590744i	−2.77923	−	0.590744i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−0.489418	−	0.847697i	−0.489418	−	0.847697i
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−0.282102	+	0.125600i	−0.282102	+	0.125600i
$997$		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.238.4	✓	24
9.7	even	3	inner	2151.1.f.b.1672.4	yes	24
239.238	odd	2	CM	2151.1.f.b.238.4	✓	24
2151.1672	odd	6	inner	2151.1.f.b.1672.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.1.f.b.238.4	✓	24	1.1	even	1	trivial
2151.1.f.b.238.4	✓	24	239.238	odd	2	CM
2151.1.f.b.1672.4	yes	24	9.7	even	3	inner
2151.1.f.b.1672.4	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)

\(f(q)\)	\(=\)	\(q+(-0.559193 - 0.968551i) q^{2} +(0.913545 - 0.406737i) q^{3} +(-0.125393 + 0.217188i) q^{4} +(0.719340 - 1.24593i) q^{5} +(-0.904793 - 0.657371i) q^{6} -0.837909 q^{8} +(0.669131 - 0.743145i) q^{9} +O(q^{10})\)	\(q+(-0.559193 - 0.968551i) q^{2} +(0.913545 - 0.406737i) q^{3} +(-0.125393 + 0.217188i) q^{4} +(0.719340 - 1.24593i) q^{5} +(-0.904793 - 0.657371i) q^{6} -0.837909 q^{8} +(0.669131 - 0.743145i) q^{9} -1.60900 q^{10} +(0.882948 + 1.52931i) q^{11} +(-0.0262144 + 0.249413i) q^{12} +(0.150383 - 1.43080i) q^{15} +(0.593946 + 1.02875i) q^{16} -1.87939 q^{17} +(-1.09395 - 0.232525i) q^{18} +(0.180401 + 0.312464i) q^{20} +(0.987476 - 1.71036i) q^{22} +(-0.765468 + 0.340808i) q^{24} +(-0.534899 - 0.926473i) q^{25} +(0.309017 - 0.951057i) q^{27} +(-0.990268 - 1.71519i) q^{29} +(-1.46989 + 0.654439i) q^{30} +(0.997564 - 1.72783i) q^{31} +(0.245307 - 0.424883i) q^{32} +(1.42864 + 1.03797i) q^{33} +(1.05094 + 1.82028i) q^{34} +(0.0774974 + 0.238512i) q^{36} +(-0.602742 + 1.04398i) q^{40} -0.442863 q^{44} +(-0.444576 - 1.36827i) q^{45} +(0.961025 + 0.698226i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.598224 + 1.03615i) q^{50} +(-1.71690 + 0.764415i) q^{51} +(-1.09395 + 0.232525i) q^{54} +2.54056 q^{55} +(-1.10750 + 1.91825i) q^{58} +(0.291895 + 0.212074i) q^{60} +(0.939693 + 1.62760i) q^{61} -2.23132 q^{62} +0.639198 q^{64} +(0.206439 - 1.96413i) q^{66} +(-0.766044 + 1.32683i) q^{67} +(0.235663 - 0.408179i) q^{68} +0.618034 q^{71} +(-0.560671 + 0.622688i) q^{72} +(-0.865486 - 0.628812i) q^{75} +1.70900 q^{80} +(-0.104528 - 0.994522i) q^{81} +(0.615661 + 1.06636i) q^{83} +(-1.35192 + 2.34159i) q^{85} +(-1.60229 - 1.16413i) q^{87} +(-0.739830 - 1.28142i) q^{88} +(-1.07663 + 1.19572i) q^{90} +(0.208548 - 1.98420i) q^{93} +(0.0512830 - 0.487925i) q^{96} +1.11839 q^{98} +(1.72731 + 0.367150i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 3 q^{3} - 12 q^{4} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more