LMFDB - Embedded newform 2151.1.f.b.238.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.238");

S:= CuspForms(chi, 1);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.07348884217$
Analytic rank:	$0$
Dimension:	$24$
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			238.2
Root			$0.0348995 + 0.999391i$ of defining polynomial
Character	$\chi$	$=$	2151.238
Dual form			2151.1.f.b.1672.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.961262 - 1.66495i) q^{2} +(-0.104528 - 0.994522i) q^{3} +(-1.34805 + 2.33489i) q^{4} +(-0.438371 + 0.759281i) q^{5} +(-1.55535 + 1.13003i) q^{6} +3.26078 q^{8} +(-0.978148 + 0.207912i) q^{9} +O(q^{10})$	\(q+(-0.961262 - 1.66495i) q^{2} +(-0.104528 - 0.994522i) q^{3} +(-1.34805 + 2.33489i) q^{4} +(-0.438371 + 0.759281i) q^{5} +(-1.55535 + 1.13003i) q^{6} +3.26078 q^{8} +(-0.978148 + 0.207912i) q^{9} +1.68556 q^{10} +(-0.990268 - 1.71519i) q^{11} +(2.46301 + 1.09660i) q^{12} +(0.800944 + 0.356603i) q^{15} +(-1.78642 - 3.09417i) q^{16} -1.87939 q^{17} +(1.28642 + 1.42871i) q^{18} +(-1.18189 - 2.04709i) q^{20} +(-1.90381 + 3.29750i) q^{22} +(-0.340845 - 3.24292i) q^{24} +(0.115661 + 0.200332i) q^{25} +(0.309017 + 0.951057i) q^{27} +(0.882948 + 1.52931i) q^{29} +(-0.176189 - 1.67632i) q^{30} +(0.241922 - 0.419021i) q^{31} +(-1.80404 + 3.12469i) q^{32} +(-1.60229 + 1.16413i) q^{33} +(1.80658 + 3.12909i) q^{34} +(0.833140 - 2.56414i) q^{36} +(-1.42943 + 2.47585i) q^{40} +5.33972 q^{44} +(0.270928 - 0.833831i) q^{45} +(-2.89049 + 2.10006i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(0.222362 - 0.385142i) q^{50} +(0.196449 + 1.86909i) q^{51} +(1.28642 - 1.42871i) q^{54} +1.73642 q^{55} +(1.69749 - 2.94013i) q^{58} +(-1.91234 + 1.38940i) q^{60} +(0.939693 + 1.62760i) q^{61} -0.930201 q^{62} +3.36378 q^{64} +(3.47844 + 1.54870i) q^{66} +(-0.766044 + 1.32683i) q^{67} +(2.53350 - 4.38815i) q^{68} +0.618034 q^{71} +(-3.18953 + 0.677955i) q^{72} +(0.187144 - 0.135968i) q^{75} +3.13246 q^{80} +(0.913545 - 0.406737i) q^{81} +(-0.0348995 - 0.0604477i) q^{83} +(0.823868 - 1.42698i) q^{85} +(1.42864 - 1.03797i) q^{87} +(-3.22905 - 5.59288i) q^{88} +(-1.64872 + 0.350447i) q^{90} +(-0.442013 - 0.196797i) q^{93} +(3.29615 + 1.46754i) q^{96} +1.92252 q^{98} +(1.32524 + 1.47183i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.719340	−	0.694658i	$-0.755556\pi$
$2$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.241922	−	0.970296i	$-0.577778\pi$
$3$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$3$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$4$	−1.34805	+	2.33489i	−1.34805	+	2.33489i
$5$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	−0.997564	−	0.0697565i	$-0.977778\pi$
$5$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	0.559193	+	0.829038i	$0.311111\pi$
$6$	−1.55535	+	1.13003i	−1.55535	+	1.13003i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	3.26078			3.26078
$9$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$10$	1.68556			1.68556
$11$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.615661	−	0.788011i	$-0.711111\pi$
$11$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.374607	−	0.927184i	$-0.622222\pi$
$12$	2.46301	+	1.09660i	2.46301	+	1.09660i
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$	0.800944	+	0.356603i	0.800944	+	0.356603i
$16$	−1.78642	−	3.09417i	−1.78642	−	3.09417i
$17$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$17$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$18$	1.28642	+	1.42871i	1.28642	+	1.42871i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	−1.18189	−	2.04709i	−1.18189	−	2.04709i
$21$		0			0
$22$	−1.90381	+	3.29750i	−1.90381	+	3.29750i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	−0.340845	−	3.24292i	−0.340845	−	3.24292i
$25$	0.115661	+	0.200332i	0.115661	+	0.200332i
$26$		0			0
$27$	0.309017	+	0.951057i	0.309017	+	0.951057i
$28$		0			0
$29$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.848048	+	0.529919i	$0.177778\pi$
$29$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.0348995	+	0.999391i	$0.488889\pi$
$30$	−0.176189	−	1.67632i	−0.176189	−	1.67632i
$31$	0.241922	−	0.419021i	0.241922	−	0.419021i	−0.719340	−	0.694658i	$-0.755556\pi$
$31$	0.241922	−	0.419021i	0.241922	−	0.419021i	0.961262	+	0.275637i	$0.0888889\pi$
$32$	−1.80404	+	3.12469i	−1.80404	+	3.12469i
$33$	−1.60229	+	1.16413i	−1.60229	+	1.16413i
$34$	1.80658	+	3.12909i	1.80658	+	3.12909i
$35$		0			0
$36$	0.833140	−	2.56414i	0.833140	−	2.56414i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	−1.42943	+	2.47585i	−1.42943	+	2.47585i
$41$		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	$0.333333\pi$
$44$	5.33972			5.33972
$45$	0.270928	−	0.833831i	0.270928	−	0.833831i
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	−2.89049	+	2.10006i	−2.89049	+	2.10006i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	0.222362	−	0.385142i	0.222362	−	0.385142i
$51$	0.196449	+	1.86909i	0.196449	+	1.86909i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	1.28642	−	1.42871i	1.28642	−	1.42871i
$55$	1.73642			1.73642
$56$		0			0
$57$		0			0
$58$	1.69749	−	2.94013i	1.69749	−	2.94013i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	−1.91234	+	1.38940i	−1.91234	+	1.38940i
$61$	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	$0.444444\pi$
$62$	−0.930201			−0.930201
$63$		0			0
$64$	3.36378			3.36378
$65$		0			0
$66$	3.47844	+	1.54870i	3.47844	+	1.54870i
$67$	−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	$0.888889\pi$
$68$	2.53350	−	4.38815i	2.53350	−	4.38815i
$69$		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	$0.400000\pi$
$72$	−3.18953	+	0.677955i	−3.18953	+	0.677955i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	0.187144	−	0.135968i	0.187144	−	0.135968i
$76$		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	$0.333333\pi$
$80$	3.13246			3.13246
$81$	0.913545	−	0.406737i	0.913545	−	0.406737i
$82$		0			0
$83$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	0.848048	−	0.529919i	$-0.177778\pi$
$83$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	−0.882948	+	0.469472i	$0.844444\pi$
$84$		0			0
$85$	0.823868	−	1.42698i	0.823868	−	1.42698i
$86$		0			0
$87$	1.42864	−	1.03797i	1.42864	−	1.03797i
$88$	−3.22905	−	5.59288i	−3.22905	−	5.59288i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−1.64872	+	0.350447i	−1.64872	+	0.350447i
$91$		0			0
$92$		0			0
$93$	−0.442013	−	0.196797i	−0.442013	−	0.196797i
$94$		0			0
$95$		0			0
$96$	3.29615	+	1.46754i	3.29615	+	1.46754i
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$	1.92252			1.92252
$99$	1.32524	+	1.47183i	1.32524	+	1.47183i
$100$	−0.623669			−0.623669
$101$	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	$0.533333\pi$
$102$	2.92311	−	2.12376i	2.92311	−	2.12376i
$103$		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			$\pi$
$108$	−2.63718	−	0.560550i	−2.63718	−	0.560550i
$109$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$109$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$110$	−1.66915	−	2.89106i	−1.66915	−	2.89106i
$111$		0			0
$112$		0			0
$113$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.104528	+	0.994522i	$0.533333\pi$
$113$	−0.913545	+	1.58231i	−0.913545	+	1.58231i	−0.809017	+	0.587785i	$0.800000\pi$
$114$		0			0
$115$		0			0
$116$	−4.76102			−4.76102
$117$		0			0
$118$		0			0
$119$		0			0
$120$	2.61171	+	1.16281i	2.61171	+	1.16281i
$121$	−1.46126	+	2.53098i	−1.46126	+	2.53098i
$122$	1.80658	−	3.12909i	1.80658	−	3.12909i
$123$		0			0
$124$	0.652245	+	1.12972i	0.652245	+	1.12972i
$125$	−1.07955			−1.07955
$126$		0			0
$127$	−0.749213			−0.749213			−0.374607	−	0.927184i	$-0.622222\pi$
$127$	−0.749213			−0.749213			−0.374607	+	0.927184i	$0.622222\pi$
$128$	−1.42943	−	2.47585i	−1.42943	−	2.47585i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	−0.558152	−	5.31046i	−0.558152	−	5.31046i
$133$		0			0
$134$	2.94548			2.94548
$135$	−0.857583	−	0.182285i	−0.857583	−	0.182285i
$136$	−6.12827			−6.12827
$137$		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
$142$	−0.594092	−	1.02900i	−0.594092	−	1.02900i
$143$		0			0
$144$	2.39070	+	2.65514i	2.39070	+	2.65514i
$145$	−1.54823			−1.54823
$146$		0			0
$147$	0.913545	+	0.406737i	0.913545	+	0.406737i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	−0.406275	−	0.180885i	−0.406275	−	0.180885i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$	1.83832	−	0.390746i	1.83832	−	0.390746i
$154$		0			0
$155$	0.212103	+	0.367373i	0.212103	+	0.367373i
$156$		0			0
$157$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.0348995	−	0.999391i	$-0.488889\pi$
$157$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.848048	−	0.529919i	$-0.177778\pi$
$158$		0			0
$159$		0			0
$160$	−1.58168	−	2.73955i	−1.58168	−	2.73955i
$161$		0			0
$162$	−1.55535	−	1.13003i	−1.55535	−	1.13003i
$163$	−1.99513			−1.99513			−0.997564	−	0.0697565i	$-0.977778\pi$
$163$	−1.99513			−1.99513			−0.997564	+	0.0697565i	$0.977778\pi$
$164$		0			0
$165$	−0.181505	−	1.72691i	−0.181505	−	1.72691i
$166$	−0.0670951	+	0.116212i	−0.0670951	+	0.116212i
$167$		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
$170$	−3.16781			−3.16781
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$	−3.10146	−	1.38086i	−3.10146	−	1.38086i
$175$		0			0
$176$	−3.53807	+	6.12811i	−3.53807	+	6.12811i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	1.58168	+	1.75663i	1.58168	+	1.75663i
$181$		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
$186$	0.0972325	+	0.925105i	0.0972325	+	0.925105i
$187$	1.86110	+	3.22351i	1.86110	+	3.22351i
$188$		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	$0.333333\pi$
$192$	−0.351611	−	3.34535i	−0.351611	−	3.34535i
$193$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.438371	−	0.898794i	$-0.355556\pi$
$193$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.559193	−	0.829038i	$-0.311111\pi$
$194$		0			0
$195$		0			0
$196$	−1.34805	−	2.33489i	−1.34805	−	2.33489i
$197$	−1.23132			−1.23132			−0.615661	−	0.788011i	$-0.711111\pi$
$197$	−1.23132			−1.23132			−0.615661	+	0.788011i	$0.711111\pi$
$198$	1.17662	−	3.62127i	1.17662	−	3.62127i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	0.377147	+	0.653238i	0.377147	+	0.653238i
$201$	1.39963	+	0.623157i	1.39963	+	0.623157i
$202$	1.55535	−	2.69395i	1.55535	−	2.69395i
$203$		0			0
$204$	−4.62894	−	2.06094i	−4.62894	−	2.06094i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	−0.882948	−	0.469472i	$-0.844444\pi$
$211$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	0.848048	+	0.529919i	$0.177778\pi$
$212$		0			0
$213$	−0.0646021	−	0.614648i	−0.0646021	−	0.614648i
$214$		0			0
$215$		0			0
$216$	1.00764	+	3.10119i	1.00764	+	3.10119i
$217$		0			0
$218$	1.88051	+	3.25714i	1.88051	+	3.25714i
$219$		0			0
$220$	−2.34078	+	4.05435i	−2.34078	+	4.05435i
$221$		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
$225$	−0.154785	−	0.171906i	−0.154785	−	0.171906i
$226$	3.51263			3.51263
$227$		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
$232$	2.87910	+	4.98675i	2.87910	+	4.98675i
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$239$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$240$	−0.327431	−	3.11530i	−0.327431	−	3.11530i
$241$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$241$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$242$	5.61862			5.61862
$243$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$244$	−5.06700			−5.06700
$245$	−0.438371	−	0.759281i	−0.438371	−	0.759281i
$246$		0			0
$247$		0			0
$248$	0.788855	−	1.36634i	0.788855	−	1.36634i
$249$	−0.0564686	+	0.0410268i	−0.0564686	+	0.0410268i
$250$	1.03773	+	1.79741i	1.03773	+	1.79741i
$251$	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
$254$	0.720190	+	1.24741i	0.720190	+	1.24741i
$255$	−1.50528	−	0.670195i	−1.50528	−	0.670195i
$256$	−1.06623	+	1.84676i	−1.06623	+	1.84676i
$257$	0.374607	−	0.648838i	0.374607	−	0.648838i	−0.615661	−	0.788011i	$-0.711111\pi$
$257$	0.374607	−	0.648838i	0.374607	−	0.648838i	0.990268	+	0.139173i	$0.0444444\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−1.18161	−	1.31232i	−1.18161	−	1.31232i
$262$		0			0
$263$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	−0.882948	−	0.469472i	$-0.844444\pi$
$263$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	0.0348995	−	0.999391i	$-0.488889\pi$
$264$	−5.22471	+	3.79598i	−5.22471	+	3.79598i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−2.06533	−	3.57726i	−2.06533	−	3.57726i
$269$	−0.483844			−0.483844			−0.241922	−	0.970296i	$-0.577778\pi$
$269$	−0.483844			−0.483844			−0.241922	+	0.970296i	$0.577778\pi$
$270$	0.520866	+	1.60306i	0.520866	+	1.60306i
$271$	1.98054			1.98054			0.990268	−	0.139173i	$-0.0444444\pi$
$271$	1.98054			1.98054			0.990268	+	0.139173i	$0.0444444\pi$
$272$	3.35737	+	5.81514i	3.35737	+	5.81514i
$273$		0			0
$274$		0			0
$275$	0.229072	−	0.396764i	0.229072	−	0.396764i
$276$		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
$279$	−0.149516	+	0.460163i	−0.149516	+	0.460163i
$280$		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
$283$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.241922	+	0.970296i	$0.577778\pi$
$283$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.719340	+	0.694658i	$0.755556\pi$
$284$	−0.833140	+	1.44304i	−0.833140	+	1.44304i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.11496	−	3.43149i	1.11496	−	3.43149i
$289$	2.53209			2.53209
$290$	1.48826	+	2.57774i	1.48826	+	2.57774i
$291$		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	$0.266667\pi$
$294$	−0.200958	−	1.91199i	−0.200958	−	1.91199i
$295$		0			0
$296$		0			0
$297$	1.32524	−	1.47183i	1.32524	−	1.47183i
$298$		0			0
$299$		0			0
$300$	0.0651912	+	0.620252i	0.0651912	+	0.620252i
$301$		0			0
$302$		0			0
$303$	1.30902	−	0.951057i	1.30902	−	0.951057i
$304$		0			0
$305$	−1.64774			−1.64774
$306$	−2.41768	−	2.68510i	−2.41768	−	2.68510i
$307$	0.876742			0.876742			0.438371	−	0.898794i	$-0.355556\pi$
$307$	0.876742			0.876742			0.438371	+	0.898794i	$0.355556\pi$
$308$		0			0
$309$		0			0
$310$	0.407773	−	0.706284i	0.407773	−	0.706284i
$311$	−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	$0.333333\pi$
$314$	−3.39497			−3.39497
$315$		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
$320$	−1.47458	+	2.55406i	−1.47458	+	2.55406i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.281819	+	2.68133i	−0.281819	+	2.68133i
$325$		0			0
$326$	1.91784	+	3.32180i	1.91784	+	3.32180i
$327$	0.204489	+	1.94558i	0.204489	+	1.94558i
$328$		0			0
$329$		0			0
$330$	−2.70075	+	1.96221i	−2.70075	+	1.96221i
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	0.188185			0.188185
$333$		0			0
$334$		0			0
$335$	−0.671624	−	1.16329i	−0.671624	−	1.16329i
$336$		0			0
$337$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	−0.997564	−	0.0697565i	$-0.977778\pi$
$337$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	0.559193	+	0.829038i	$0.311111\pi$
$338$	−0.961262	+	1.66495i	−0.961262	+	1.66495i
$339$	1.66913	+	0.743145i	1.66913	+	0.743145i
$340$	2.22123	+	3.84728i	2.22123	+	3.84728i
$341$	−0.958270			−0.958270
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.241922	−	0.419021i	0.241922	−	0.419021i	−0.719340	−	0.694658i	$-0.755556\pi$
$347$	0.241922	−	0.419021i	0.241922	−	0.419021i	0.961262	+	0.275637i	$0.0888889\pi$
$348$	0.497662	+	4.73494i	0.497662	+	4.73494i
$349$	0.374607	+	0.648838i	0.374607	+	0.648838i	0.990268	−	0.139173i	$-0.0444444\pi$
$349$	0.374607	+	0.648838i	0.374607	+	0.648838i	−0.615661	+	0.788011i	$0.711111\pi$
$350$		0			0
$351$		0			0
$352$	7.14593			7.14593
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$	−0.270928	+	0.469262i	−0.270928	+	0.469262i
$356$		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	$0.800000\pi$
$360$	0.883439	−	2.71894i	0.883439	−	2.71894i
$361$	1.00000			1.00000
$362$		0			0
$363$	2.66986	+	1.18870i	2.66986	+	1.18870i
$364$		0			0
$365$		0			0
$366$	−3.30079	−	1.46961i	−3.30079	−	1.46961i
$367$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	0.848048	−	0.529919i	$-0.177778\pi$
$367$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	−0.882948	+	0.469472i	$0.844444\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	1.05535	−	0.766760i	1.05535	−	0.766760i
$373$	−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	$0.266667\pi$
$374$	3.57800	−	6.19728i	3.57800	−	6.19728i
$375$	0.112844	+	1.07364i	0.112844	+	1.07364i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	0.0783141	+	0.745109i	0.0783141	+	0.745109i
$382$		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			$0$
$384$	−2.31287	+	1.68040i	−2.31287	+	1.68040i
$385$		0			0
$386$	−3.83568			−3.83568
$387$		0			0
$388$		0			0
$389$	0.719340	+	1.24593i	0.719340	+	1.24593i	0.961262	+	0.275637i	$0.0888889\pi$
$389$	0.719340	+	1.24593i	0.719340	+	1.24593i	−0.241922	+	0.970296i	$0.577778\pi$
$390$		0			0
$391$		0			0
$392$	−1.63039	+	2.82392i	−1.63039	+	2.82392i
$393$		0			0
$394$	1.18362	+	2.05010i	1.18362	+	2.05010i
$395$		0			0
$396$	−5.22303	+	1.11019i	−5.22303	+	1.11019i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	0.413240	−	0.715752i	0.413240	−	0.715752i
$401$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	−0.882948	−	0.469472i	$-0.844444\pi$
$401$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	0.848048	+	0.529919i	$0.177778\pi$
$402$	−0.307886	−	2.92934i	−0.307886	−	2.92934i
$403$		0			0
$404$	−4.36238			−4.36238
$405$	−0.0916445	+	0.871939i	−0.0916445	+	0.871939i
$406$		0			0
$407$		0			0
$408$	0.640579	+	6.09470i	0.640579	+	6.09470i
$409$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	0.438371	+	0.898794i	$0.355556\pi$
$409$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	−0.997564	+	0.0697565i	$0.977778\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	0.0611957			0.0611957
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	0.0348995	+	0.999391i	$0.488889\pi$
$419$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	−0.882948	+	0.469472i	$0.844444\pi$
$420$		0			0
$421$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.809017	−	0.587785i	$-0.800000\pi$
$421$	−0.913545	−	1.58231i	−0.913545	−	1.58231i	−0.104528	−	0.994522i	$-0.533333\pi$
$422$	0.134190			0.134190
$423$		0			0
$424$		0			0
$425$	−0.217372	−	0.376500i	−0.217372	−	0.376500i
$426$	−0.961262	+	0.698398i	−0.961262	+	0.698398i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.99513			−1.99513			−0.997564	−	0.0697565i	$-0.977778\pi$
$431$	−1.99513			−1.99513			−0.997564	+	0.0697565i	$0.977778\pi$
$432$	2.39070	−	2.65514i	2.39070	−	2.65514i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	0.161835	+	1.53975i	0.161835	+	1.53975i
$436$	2.63718	−	4.56773i	2.63718	−	4.56773i
$437$		0			0
$438$		0			0
$439$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$439$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$440$	5.66209			5.66209
$441$	0.309017	−	0.951057i	0.309017	−	0.951057i
$442$		0			0
$443$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	−0.997564	−	0.0697565i	$-0.977778\pi$
$443$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	0.438371	−	0.898794i	$-0.355556\pi$
$444$		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			$\pi$
$450$	−0.137427	+	0.422957i	−0.137427	+	0.422957i
$451$		0			0
$452$	−2.46301	−	4.26605i	−2.46301	−	4.26605i
$453$		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	$0.666667\pi$
$464$	3.15463	−	5.46398i	3.15463	−	5.46398i
$465$	0.343190	−	0.249342i	0.343190	−	0.249342i
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1.61323	−	0.718254i	−1.61323	−	0.718254i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1.92252			1.92252
$479$	0.719340	+	1.24593i	0.719340	+	1.24593i	0.961262	+	0.275637i	$0.0888889\pi$
$479$	0.719340	+	1.24593i	0.719340	+	1.24593i	−0.241922	+	0.970296i	$0.577778\pi$
$480$	−2.55921	+	1.85937i	−2.55921	+	1.85937i
$481$		0			0
$482$	−1.75631	+	3.04202i	−1.75631	+	3.04202i
$483$		0			0
$484$	−3.93970	−	6.82376i	−3.93970	−	6.82376i
$485$		0			0
$486$	−0.961262	+	1.66495i	−0.961262	+	1.66495i
$487$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$487$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$488$	3.06414	+	5.30724i	3.06414	+	5.30724i
$489$	0.208548	+	1.98420i	0.208548	+	1.98420i
$490$	−0.842779	+	1.45974i	−0.842779	+	1.45974i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$	−1.65940	−	2.87416i	−1.65940	−	2.87416i
$494$		0			0
$495$	−1.69847	+	0.361022i	−1.69847	+	0.361022i
$496$	−1.72870			−1.72870
$497$		0			0
$498$	0.122589	+	0.0545801i	0.122589	+	0.0545801i
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	1.45529	−	2.52063i	1.45529	−	2.52063i
$501$		0			0
$502$	−0.333843	−	0.578232i	−0.333843	−	0.578232i
$503$	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
$505$	−1.41860			−1.41860
$506$		0			0
$507$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$508$	1.00998	−	1.74933i	1.00998	−	1.74933i
$509$	−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	$0.888889\pi$
$510$	0.331127	+	3.15046i	0.331127	+	3.15046i
$511$		0			0
$512$	1.24083			1.24083
$513$		0			0
$514$	−1.44038			−1.44038
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	−1.04910	+	3.22881i	−1.04910	+	3.22881i
$523$	−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
$526$	−1.63039	+	2.82392i	−1.63039	+	2.82392i
$527$	−0.454664	+	0.787502i	−0.454664	+	0.787502i
$528$	6.46437	+	2.87812i	6.46437	+	2.87812i
$529$	−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
$536$	−2.49791	+	4.32650i	−2.49791	+	4.32650i
$537$		0			0
$538$	0.465101	+	0.805578i	0.465101	+	0.805578i
$539$	1.98054			1.98054
$540$	1.58168	−	1.75663i	1.58168	−	1.75663i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	−1.90381	−	3.29750i	−1.90381	−	3.29750i
$543$		0			0
$544$	3.39049	−	5.87250i	3.39049	−	5.87250i
$545$	0.857583	−	1.48538i	0.857583	−	1.48538i
$546$		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
$549$	−1.25755	−	1.39666i	−1.25755	−	1.39666i
$550$	−0.880792			−0.880792
$551$		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			$\pi$
$558$	0.909874	−	0.193400i	0.909874	−	0.193400i
$559$		0			0
$560$		0			0
$561$	3.01132	−	2.18785i	3.01132	−	2.18785i
$562$		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
$565$	−0.800944	−	1.38728i	−0.800944	−	1.38728i
$566$	3.69610			3.69610
$567$		0			0
$568$	2.01528			2.01528
$569$	0.241922	+	0.419021i	0.241922	+	0.419021i	0.961262	−	0.275637i	$-0.0888889\pi$
$569$	0.241922	+	0.419021i	0.241922	+	0.419021i	−0.719340	+	0.694658i	$0.755556\pi$
$570$		0			0
$571$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$571$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−3.29027	+	0.699370i	−3.29027	+	0.699370i
$577$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$577$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$578$	−2.43400	−	4.21581i	−2.43400	−	4.21581i
$579$	−1.82264	−	0.811492i	−1.82264	−	0.811492i
$580$	2.08710	−	3.61495i	2.08710	−	3.61495i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	1.18818			1.18818
$587$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.559193	+	0.829038i	$0.311111\pi$
$587$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.438371	+	0.898794i	$0.355556\pi$
$588$	−2.18119	+	1.58473i	−2.18119	+	1.58473i
$589$		0			0
$590$		0			0
$591$	0.128708	+	1.22458i	0.128708	+	1.22458i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−3.72442	−	0.791650i	−3.72442	−	0.791650i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	0.104528	−	0.181049i	0.104528	−	0.181049i	−0.809017	−	0.587785i	$-0.800000\pi$
$599$	0.104528	−	0.181049i	0.104528	−	0.181049i	0.913545	+	0.406737i	$0.133333\pi$
$600$	0.610237	−	0.443363i	0.610237	−	0.443363i
$601$		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
$605$	−1.28115	−	2.21902i	−1.28115	−	2.21902i
$606$	−2.84177	−	1.26524i	−2.84177	−	1.26524i
$607$		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
$610$	1.58391	+	2.74341i	1.58391	+	2.74341i
$611$		0			0
$612$	−1.56579	+	4.81901i	−1.56579	+	4.81901i
$613$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$613$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$614$	−0.842779	−	1.45974i	−0.842779	−	1.45974i
$615$		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	$0.333333\pi$
$620$	−1.14370			−1.14370
$621$		0			0
$622$	0.667685			0.667685
$623$		0			0
$624$		0			0
$625$	0.357583	−	0.619353i	0.357583	−	0.619353i
$626$		0			0
$627$		0			0
$628$	2.38051	+	4.12317i	2.38051	+	4.12317i
$629$		0			0
$630$		0			0
$631$	0.876742			0.876742			0.438371	−	0.898794i	$-0.355556\pi$
$631$	0.876742			0.876742			0.438371	+	0.898794i	$0.355556\pi$
$632$		0			0
$633$	0.0637646	+	0.0283898i	0.0637646	+	0.0283898i
$634$		0			0
$635$	0.328433	−	0.568863i	0.328433	−	0.568863i
$636$		0			0
$637$		0			0
$638$	−6.72387			−6.72387
$639$	−0.604528	+	0.128496i	−0.604528	+	0.128496i
$640$	2.50649			2.50649
$641$	0.615661	+	1.06636i	0.615661	+	1.06636i	0.990268	+	0.139173i	$0.0444444\pi$
$641$	0.615661	+	1.06636i	0.615661	+	1.06636i	−0.374607	+	0.927184i	$0.622222\pi$
$642$		0			0
$643$	0.374607	−	0.648838i	0.374607	−	0.648838i	−0.615661	−	0.788011i	$-0.711111\pi$
$643$	0.374607	−	0.648838i	0.374607	−	0.648838i	0.990268	+	0.139173i	$0.0444444\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.43868			−1.43868			−0.719340	−	0.694658i	$-0.755556\pi$
$647$	−1.43868			−1.43868			−0.719340	+	0.694658i	$0.755556\pi$
$648$	2.97887	−	1.32628i	2.97887	−	1.32628i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	2.68953	−	4.65840i	2.68953	−	4.65840i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	3.04273	−	2.21067i	3.04273	−	2.21067i
$655$		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	$0.333333\pi$
$660$	4.27681	+	1.90416i	4.27681	+	1.90416i
$661$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.374607	+	0.927184i	$0.622222\pi$
$661$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.615661	+	0.788011i	$0.711111\pi$
$662$		0			0
$663$		0			0
$664$	−0.113800	−	0.197107i	−0.113800	−	0.197107i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	−1.29121	+	2.23644i	−1.29121	+	2.23644i
$671$	1.86110	−	3.22351i	1.86110	−	3.22351i
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$	1.68556			1.68556
$675$	−0.154785	+	0.171906i	−0.154785	+	0.171906i
$676$	2.69610			2.69610
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$	−0.367169	−	3.49338i	−0.367169	−	3.49338i
$679$		0			0
$680$	2.68646	−	4.65308i	2.68646	−	4.65308i
$681$		0			0
$682$	0.921148	+	1.59548i	0.921148	+	1.59548i
$683$		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
$694$	−0.930201			−0.930201
$695$		0			0
$696$	4.65848	−	3.38459i	4.65848	−	3.38459i
$697$		0			0
$698$	0.720190	−	1.24741i	0.720190	−	1.24741i
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	−3.33105	−	5.76954i	−3.33105	−	5.76954i
$705$		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	$0.333333\pi$
$710$	1.04173			1.04173
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	0.913545	+	0.406737i	0.913545	+	0.406737i
$718$	1.55535	+	2.69395i	1.55535	+	2.69395i
$719$	1.11839			1.11839			0.559193	−	0.829038i	$-0.311111\pi$
$719$	1.11839			1.11839			0.559193	+	0.829038i	$0.311111\pi$
$720$	−3.06401	+	0.651275i	−3.06401	+	0.651275i
$721$		0			0
$722$	−0.961262	−	1.66495i	−0.961262	−	1.66495i
$723$	−1.47815	+	1.07394i	−1.47815	+	1.07394i
$724$		0			0
$725$	−0.204246	+	0.353765i	−0.204246	+	0.353765i
$726$	−0.587306	−	5.58784i	−0.587306	−	5.58784i
$727$	0.104528	+	0.181049i	0.104528	+	0.181049i	0.913545	−	0.406737i	$-0.133333\pi$
$727$	0.104528	+	0.181049i	0.104528	+	0.181049i	−0.809017	+	0.587785i	$0.800000\pi$
$728$		0			0
$729$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$730$		0			0
$731$		0			0
$732$	0.529646	+	5.03925i	0.529646	+	5.03925i
$733$	0.241922	−	0.419021i	0.241922	−	0.419021i	−0.719340	−	0.694658i	$-0.755556\pi$
$733$	0.241922	−	0.419021i	0.241922	−	0.419021i	0.961262	+	0.275637i	$0.0888889\pi$
$734$	−0.0670951	+	0.116212i	−0.0670951	+	0.116212i
$735$	−0.709299	+	0.515336i	−0.709299	+	0.515336i
$736$		0			0
$737$	3.03436			3.03436
$738$		0			0
$739$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$739$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$	−1.44131	−	0.641713i	−1.44131	−	0.641713i
$745$		0			0
$746$	1.18818			1.18818
$747$	0.0467046	+	0.0518708i	0.0467046	+	0.0518708i
$748$	−10.0354			−10.0354
$749$		0			0
$750$	1.67909	−	1.21993i	1.67909	−	1.21993i
$751$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$751$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$752$		0			0
$753$	−0.0363024	−	0.345394i	−0.0363024	−	0.345394i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.92252			1.92252			0.961262	−	0.275637i	$-0.0888889\pi$
$757$	1.92252			1.92252			0.961262	+	0.275637i	$0.0888889\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.309017	−	0.951057i	$-0.400000\pi$
$761$	0.978148	−	1.69420i	0.978148	−	1.69420i	0.669131	−	0.743145i	$-0.266667\pi$
$762$	1.16529	−	0.846634i	1.16529	−	0.846634i
$763$		0			0
$764$		0			0
$765$	−0.509179	+	1.56709i	−0.509179	+	1.56709i
$766$	−1.92252			−1.92252
$767$		0			0
$768$	1.94810	+	0.867349i	1.94810	+	0.867349i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$	−0.684440	−	0.304732i	−0.684440	−	0.304732i
$772$	2.68953	+	4.65840i	2.68953	+	4.65840i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	0.111924			0.111924
$776$		0			0
$777$		0			0
$778$	1.38295	−	2.39534i	1.38295	−	2.39534i
$779$		0			0
$780$		0			0
$781$	−0.612019	−	1.06005i	−0.612019	−	1.06005i
$782$		0			0
$783$	−1.18161	+	1.31232i	−1.18161	+	1.31232i
$784$	3.57284			3.57284
$785$	0.774117	+	1.34081i	0.774117	+	1.34081i
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	1.65988	−	2.87500i	1.65988	−	2.87500i
$789$	−1.37217	+	0.996940i	−1.37217	+	0.996940i
$790$		0			0
$791$		0			0
$792$	4.32131	+	4.79930i	4.32131	+	4.79930i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.719340	−	1.24593i	0.719340	−	1.24593i	−0.241922	−	0.970296i	$-0.577778\pi$
$797$	0.719340	−	1.24593i	0.719340	−	1.24593i	0.961262	−	0.275637i	$-0.0888889\pi$
$798$		0			0
$799$		0			0
$800$	−0.834632			−0.834632
$801$		0			0
$802$	0.134190			0.134190
$803$		0			0
$804$	−3.34177	+	2.42794i	−3.34177	+	2.42794i
$805$		0			0
$806$		0			0
$807$	0.0505754	+	0.481193i	0.0505754	+	0.481193i
$808$	2.63803	+	4.56920i	2.63803	+	4.56920i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	1.53983	−	0.685578i	1.53983	−	0.685578i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−0.207022	−	1.96969i	−0.207022	−	1.96969i
$814$		0			0
$815$	0.874607	−	1.51486i	0.874607	−	1.51486i
$816$	5.43234	−	3.94683i	5.43234	−	3.94683i
$817$		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
$825$	−0.418535	−	0.186344i	−0.418535	−	0.186344i
$826$		0			0
$827$	−1.76590			−1.76590			−0.882948	−	0.469472i	$-0.844444\pi$
$827$	−1.76590			−1.76590			−0.882948	+	0.469472i	$0.844444\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$	−0.0588251	−	0.101888i	−0.0588251	−	0.101888i
$831$		0			0
$832$		0			0
$833$	0.939693	−	1.62760i	0.939693	−	1.62760i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.473271	+	0.100597i	0.473271	+	0.100597i
$838$	3.26078			3.26078
$839$	0.615661	+	1.06636i	0.615661	+	1.06636i	0.990268	+	0.139173i	$0.0444444\pi$
$839$	0.615661	+	1.06636i	0.615661	+	1.06636i	−0.374607	+	0.927184i	$0.622222\pi$
$840$		0			0
$841$	−1.05919	+	1.83458i	−1.05919	+	1.83458i
$842$	−1.75631	+	3.04202i	−1.75631	+	3.04202i
$843$		0			0
$844$	−0.0940924	−	0.162973i	−0.0940924	−	0.162973i
$845$	0.876742			0.876742
$846$		0			0
$847$		0			0
$848$		0			0
$849$	1.75631	+	0.781961i	1.75631	+	0.781961i
$850$	−0.417904	+	0.723830i	−0.417904	+	0.723830i
$851$		0			0
$852$	1.52222	+	0.677737i	1.52222	+	0.677737i
$853$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	−0.997564	−	0.0697565i	$-0.977778\pi$
$853$	−0.559193	−	0.968551i	−0.559193	−	0.968551i	0.438371	−	0.898794i	$-0.355556\pi$
$854$		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
$859$	0.615661	−	1.06636i	0.615661	−	1.06636i	−0.374607	−	0.927184i	$-0.622222\pi$
$859$	0.615661	−	1.06636i	0.615661	−	1.06636i	0.990268	−	0.139173i	$-0.0444444\pi$
$860$		0			0
$861$		0			0
$862$	1.91784	+	3.32180i	1.91784	+	3.32180i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−3.52924	−	0.750162i	−3.52924	−	0.750162i
$865$		0			0
$866$		0			0
$867$	−0.264675	−	2.51822i	−0.264675	−	2.51822i
$868$		0			0
$869$		0			0
$870$	2.40805	−	1.74955i	2.40805	−	1.74955i
$871$		0			0
$872$	−6.37906			−6.37906
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	−0.882948	−	0.469472i	$-0.844444\pi$
$877$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	0.848048	+	0.529919i	$0.177778\pi$
$878$	0.200958	−	0.348070i	0.200958	−	0.348070i
$879$	0.564602	+	0.251377i	0.564602	+	0.251377i
$880$	−3.10197	−	5.37278i	−3.10197	−	5.37278i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.88051	+	0.399715i	−1.88051	+	0.399715i
$883$	−1.43868			−1.43868			−0.719340	−	0.694658i	$-0.755556\pi$
$883$	−1.43868			−1.43868			−0.719340	+	0.694658i	$0.755556\pi$
$884$		0			0
$885$		0			0
$886$	−1.07506	+	1.86206i	−1.07506	+	1.86206i
$887$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$887$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1.60229	−	1.16413i	−1.60229	−	1.16413i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	0.854417			0.854417
$900$	0.610040	−	0.129668i	0.610040	−	0.129668i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−2.97887	+	5.15956i	−2.97887	+	5.15956i
$905$		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
$909$	−1.08268	−	1.20243i	−1.08268	−	1.20243i
$910$		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
$913$	−0.0691197	+	0.119719i	−0.0691197	+	0.119719i
$914$	−0.594092	+	1.02900i	−0.594092	+	1.02900i
$915$	0.172235	+	1.63871i	0.172235	+	1.63871i
$916$		0			0
$917$		0			0
$918$	−2.41768	+	2.68510i	−2.41768	+	2.68510i
$919$	−1.99513			−1.99513			−0.997564	−	0.0697565i	$-0.977778\pi$
$919$	−1.99513			−1.99513			−0.997564	+	0.0697565i	$0.977778\pi$
$920$		0			0
$921$	−0.0916445	−	0.871939i	−0.0916445	−	0.871939i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−6.37149			−6.37149
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$	−0.745039	−	0.331713i	−0.745039	−	0.331713i
$931$		0			0
$932$		0			0
$933$	0.317271	+	0.141258i	0.317271	+	0.141258i
$934$		0			0
$935$	−3.26340			−3.26340
$936$		0			0
$937$	−1.43868			−1.43868			−0.719340	−	0.694658i	$-0.755556\pi$
$937$	−1.43868			−1.43868			−0.719340	+	0.694658i	$0.755556\pi$
$938$		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	$0.666667\pi$
$942$	0.354871	+	3.37638i	0.354871	+	3.37638i
$943$		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
$956$	−1.34805	−	2.33489i	−1.34805	−	2.33489i
$957$	−3.19505	−	1.42253i	−3.19505	−	1.42253i
$958$	1.38295	−	2.39534i	1.38295	−	2.39534i
$959$		0			0
$960$	2.69420	+	1.19954i	2.69420	+	1.19954i
$961$	0.382948	+	0.663285i	0.382948	+	0.663285i
$962$		0			0
$963$		0			0
$964$	4.92601			4.92601
$965$	0.874607	+	1.51486i	0.874607	+	1.51486i
$966$		0			0
$967$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	0.309017	+	0.951057i	$0.400000\pi$
$967$	−0.669131	+	1.15897i	−0.669131	+	1.15897i	−0.978148	+	0.207912i	$0.933333\pi$
$968$	−4.76486	+	8.25298i	−4.76486	+	8.25298i
$969$		0			0
$970$		0			0
$971$	1.11839			1.11839			0.559193	−	0.829038i	$-0.311111\pi$
$971$	1.11839			1.11839			0.559193	+	0.829038i	$0.311111\pi$
$972$	2.69610			2.69610
$973$		0			0
$974$	−0.333843	−	0.578232i	−0.333843	−	0.578232i
$975$		0			0
$976$	3.35737	−	5.81514i	3.35737	−	5.81514i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$	3.10313	−	2.25456i	3.10313	−	2.25456i
$979$		0			0
$980$	2.36378			2.36378
$981$	1.91355	−	0.406737i	1.91355	−	0.406737i
$982$		0			0
$983$	0.374607	+	0.648838i	0.374607	+	0.648838i	0.990268	−	0.139173i	$-0.0444444\pi$
$983$	0.374607	+	0.648838i	0.374607	+	0.648838i	−0.615661	+	0.788011i	$0.711111\pi$
$984$		0			0
$985$	0.539776	−	0.934920i	0.539776	−	0.934920i
$986$	−3.19023	+	5.52565i	−3.19023	+	5.52565i
$987$		0			0
$988$		0			0
$989$		0			0
$990$	2.23376	+	2.48085i	2.23376	+	2.48085i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	0.872874	+	1.51186i	0.872874	+	1.51186i
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−0.0196707	−	0.187154i	−0.0196707	−	0.187154i
$997$		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.238.2	✓	24
9.7	even	3	inner	2151.1.f.b.1672.2	yes	24
239.238	odd	2	CM	2151.1.f.b.238.2	✓	24
2151.1672	odd	6	inner	2151.1.f.b.1672.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.1.f.b.238.2	✓	24	1.1	even	1	trivial
2151.1.f.b.238.2	✓	24	239.238	odd	2	CM
2151.1.f.b.1672.2	yes	24	9.7	even	3	inner
2151.1.f.b.1672.2	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)

\(f(q)\)	\(=\)	\(q+(-0.961262 - 1.66495i) q^{2} +(-0.104528 - 0.994522i) q^{3} +(-1.34805 + 2.33489i) q^{4} +(-0.438371 + 0.759281i) q^{5} +(-1.55535 + 1.13003i) q^{6} +3.26078 q^{8} +(-0.978148 + 0.207912i) q^{9} +O(q^{10})\)	\(q+(-0.961262 - 1.66495i) q^{2} +(-0.104528 - 0.994522i) q^{3} +(-1.34805 + 2.33489i) q^{4} +(-0.438371 + 0.759281i) q^{5} +(-1.55535 + 1.13003i) q^{6} +3.26078 q^{8} +(-0.978148 + 0.207912i) q^{9} +1.68556 q^{10} +(-0.990268 - 1.71519i) q^{11} +(2.46301 + 1.09660i) q^{12} +(0.800944 + 0.356603i) q^{15} +(-1.78642 - 3.09417i) q^{16} -1.87939 q^{17} +(1.28642 + 1.42871i) q^{18} +(-1.18189 - 2.04709i) q^{20} +(-1.90381 + 3.29750i) q^{22} +(-0.340845 - 3.24292i) q^{24} +(0.115661 + 0.200332i) q^{25} +(0.309017 + 0.951057i) q^{27} +(0.882948 + 1.52931i) q^{29} +(-0.176189 - 1.67632i) q^{30} +(0.241922 - 0.419021i) q^{31} +(-1.80404 + 3.12469i) q^{32} +(-1.60229 + 1.16413i) q^{33} +(1.80658 + 3.12909i) q^{34} +(0.833140 - 2.56414i) q^{36} +(-1.42943 + 2.47585i) q^{40} +5.33972 q^{44} +(0.270928 - 0.833831i) q^{45} +(-2.89049 + 2.10006i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(0.222362 - 0.385142i) q^{50} +(0.196449 + 1.86909i) q^{51} +(1.28642 - 1.42871i) q^{54} +1.73642 q^{55} +(1.69749 - 2.94013i) q^{58} +(-1.91234 + 1.38940i) q^{60} +(0.939693 + 1.62760i) q^{61} -0.930201 q^{62} +3.36378 q^{64} +(3.47844 + 1.54870i) q^{66} +(-0.766044 + 1.32683i) q^{67} +(2.53350 - 4.38815i) q^{68} +0.618034 q^{71} +(-3.18953 + 0.677955i) q^{72} +(0.187144 - 0.135968i) q^{75} +3.13246 q^{80} +(0.913545 - 0.406737i) q^{81} +(-0.0348995 - 0.0604477i) q^{83} +(0.823868 - 1.42698i) q^{85} +(1.42864 - 1.03797i) q^{87} +(-3.22905 - 5.59288i) q^{88} +(-1.64872 + 0.350447i) q^{90} +(-0.442013 - 0.196797i) q^{93} +(3.29615 + 1.46754i) q^{96} +1.92252 q^{98} +(1.32524 + 1.47183i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 3 q^{3} - 12 q^{4} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) 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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