LMFDB - Embedded newform 2151.1.f.b.238.7

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			238.7
Root			$0.848048 - 0.529919i$ of defining polynomial
Character	$\chi$	$=$	2151.238
Dual form			2151.1.f.b.1672.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.241922 + 0.419021i) q^{2} +(-0.104528 - 0.994522i) q^{3} +(0.382948 - 0.663285i) q^{4} +(-0.559193 + 0.968551i) q^{5} +(0.391438 - 0.284396i) q^{6} +0.854417 q^{8} +(-0.978148 + 0.207912i) q^{9} +O(q^{10})$	\(q+(0.241922 + 0.419021i) q^{2} +(-0.104528 - 0.994522i) q^{3} +(0.382948 - 0.663285i) q^{4} +(-0.559193 + 0.968551i) q^{5} +(0.391438 - 0.284396i) q^{6} +0.854417 q^{8} +(-0.978148 + 0.207912i) q^{9} -0.541124 q^{10} +(0.615661 + 1.06636i) q^{11} +(-0.699680 - 0.311518i) q^{12} +(1.02170 + 0.454888i) q^{15} +(-0.176245 - 0.305266i) q^{16} +1.53209 q^{17} +(-0.323755 - 0.359566i) q^{18} +(0.428283 + 0.741808i) q^{20} +(-0.297884 + 0.515950i) q^{22} +(-0.0893109 - 0.849737i) q^{24} +(-0.125393 - 0.217188i) q^{25} +(0.309017 + 0.951057i) q^{27} +(-0.0348995 - 0.0604477i) q^{29} +(0.0565629 + 0.538160i) q^{30} +(0.719340 - 1.24593i) q^{31} +(0.512484 - 0.887648i) q^{32} +(0.996161 - 0.723753i) q^{33} +(0.370646 + 0.641977i) q^{34} +(-0.236675 + 0.728410i) q^{36} +(-0.477784 + 0.827546i) q^{40} +0.943064 q^{44} +(0.345600 - 1.06365i) q^{45} +(-0.285171 + 0.207189i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(0.0606708 - 0.105085i) q^{50} +(-0.160147 - 1.52370i) q^{51} +(-0.323755 + 0.359566i) q^{54} -1.37709 q^{55} +(0.0168859 - 0.0292472i) q^{58} +(0.692977 - 0.503477i) q^{60} +(-0.766044 - 1.32683i) q^{61} +0.696096 q^{62} +0.143434 q^{64} +(0.544261 + 0.242321i) q^{66} +(-0.173648 + 0.300767i) q^{67} +(0.586710 - 1.01621i) q^{68} +0.618034 q^{71} +(-0.835746 + 0.177643i) q^{72} +(-0.202891 + 0.147409i) q^{75} +0.394221 q^{80} +(0.913545 - 0.406737i) q^{81} +(-0.848048 - 1.46886i) q^{83} +(-0.856733 + 1.48391i) q^{85} +(-0.0564686 + 0.0410268i) q^{87} +(0.526032 + 0.911114i) q^{88} +(0.529299 - 0.112506i) q^{90} +(-1.31430 - 0.585164i) q^{93} +(-0.936355 - 0.416892i) q^{96} -0.483844 q^{98} +(-0.823916 - 0.915051i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.241922	+	0.419021i	0.241922	+	0.419021i	0.961262	−	0.275637i	$-0.0888889\pi$
$2$	0.241922	+	0.419021i	0.241922	+	0.419021i	−0.719340	+	0.694658i	$0.755556\pi$
$3$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$3$	−0.104528	−	0.994522i	−0.104528	−	0.994522i
$4$	0.382948	−	0.663285i	0.382948	−	0.663285i
$5$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	0.438371	+	0.898794i	$0.355556\pi$
$5$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	−0.997564	+	0.0697565i	$0.977778\pi$
$6$	0.391438	−	0.284396i	0.391438	−	0.284396i
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	0.854417			0.854417
$9$	−0.978148	+	0.207912i	−0.978148	+	0.207912i
$10$	−0.541124			−0.541124
$11$	0.615661	+	1.06636i	0.615661	+	1.06636i	0.990268	+	0.139173i	$0.0444444\pi$
$11$	0.615661	+	1.06636i	0.615661	+	1.06636i	−0.374607	+	0.927184i	$0.622222\pi$
$12$	−0.699680	−	0.311518i	−0.699680	−	0.311518i
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$	1.02170	+	0.454888i	1.02170	+	0.454888i
$16$	−0.176245	−	0.305266i	−0.176245	−	0.305266i
$17$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$17$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$18$	−0.323755	−	0.359566i	−0.323755	−	0.359566i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	0.428283	+	0.741808i	0.428283	+	0.741808i
$21$		0			0
$22$	−0.297884	+	0.515950i	−0.297884	+	0.515950i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	−0.0893109	−	0.849737i	−0.0893109	−	0.849737i
$25$	−0.125393	−	0.217188i	−0.125393	−	0.217188i
$26$		0			0
$27$	0.309017	+	0.951057i	0.309017	+	0.951057i
$28$		0			0
$29$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	0.848048	−	0.529919i	$-0.177778\pi$
$29$	−0.0348995	−	0.0604477i	−0.0348995	−	0.0604477i	−0.882948	+	0.469472i	$0.844444\pi$
$30$	0.0565629	+	0.538160i	0.0565629	+	0.538160i
$31$	0.719340	−	1.24593i	0.719340	−	1.24593i	−0.241922	−	0.970296i	$-0.577778\pi$
$31$	0.719340	−	1.24593i	0.719340	−	1.24593i	0.961262	−	0.275637i	$-0.0888889\pi$
$32$	0.512484	−	0.887648i	0.512484	−	0.887648i
$33$	0.996161	−	0.723753i	0.996161	−	0.723753i
$34$	0.370646	+	0.641977i	0.370646	+	0.641977i
$35$		0			0
$36$	−0.236675	+	0.728410i	−0.236675	+	0.728410i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	−0.477784	+	0.827546i	−0.477784	+	0.827546i
$41$		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	$0.333333\pi$
$44$	0.943064			0.943064
$45$	0.345600	−	1.06365i	0.345600	−	1.06365i
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	−0.285171	+	0.207189i	−0.285171	+	0.207189i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	0.0606708	−	0.105085i	0.0606708	−	0.105085i
$51$	−0.160147	−	1.52370i	−0.160147	−	1.52370i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−0.323755	+	0.359566i	−0.323755	+	0.359566i
$55$	−1.37709			−1.37709
$56$		0			0
$57$		0			0
$58$	0.0168859	−	0.0292472i	0.0168859	−	0.0292472i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	0.692977	−	0.503477i	0.692977	−	0.503477i
$61$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$61$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$62$	0.696096			0.696096
$63$		0			0
$64$	0.143434			0.143434
$65$		0			0
$66$	0.544261	+	0.242321i	0.544261	+	0.242321i
$67$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$67$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$68$	0.586710	−	1.01621i	0.586710	−	1.01621i
$69$		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	$0.400000\pi$
$72$	−0.835746	+	0.177643i	−0.835746	+	0.177643i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$	−0.202891	+	0.147409i	−0.202891	+	0.147409i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$	0.394221			0.394221
$81$	0.913545	−	0.406737i	0.913545	−	0.406737i
$82$		0			0
$83$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	−0.882948	−	0.469472i	$-0.844444\pi$
$83$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	0.0348995	−	0.999391i	$-0.488889\pi$
$84$		0			0
$85$	−0.856733	+	1.48391i	−0.856733	+	1.48391i
$86$		0			0
$87$	−0.0564686	+	0.0410268i	−0.0564686	+	0.0410268i
$88$	0.526032	+	0.911114i	0.526032	+	0.911114i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	0.529299	−	0.112506i	0.529299	−	0.112506i
$91$		0			0
$92$		0			0
$93$	−1.31430	−	0.585164i	−1.31430	−	0.585164i
$94$		0			0
$95$		0			0
$96$	−0.936355	−	0.416892i	−0.936355	−	0.416892i
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$	−0.483844			−0.483844
$99$	−0.823916	−	0.915051i	−0.823916	−	0.915051i
$100$	−0.192076			−0.192076
$101$	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	$0.533333\pi$
$102$	0.599718	−	0.435720i	0.599718	−	0.435720i
$103$		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			$\pi$
$108$	0.749159	+	0.159239i	0.749159	+	0.159239i
$109$	−1.95630			−1.95630			−0.978148	−	0.207912i	$-0.933333\pi$
$109$	−1.95630			−1.95630			−0.978148	+	0.207912i	$0.933333\pi$
$110$	−0.333149	−	0.577031i	−0.333149	−	0.577031i
$111$		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
$116$	−0.0534587			−0.0534587
$117$		0			0
$118$		0			0
$119$		0			0
$120$	0.872955	+	0.388665i	0.872955	+	0.388665i
$121$	−0.258078	+	0.447004i	−0.258078	+	0.447004i
$122$	0.370646	−	0.641977i	0.370646	−	0.641977i
$123$		0			0
$124$	−0.550939	−	0.954254i	−0.550939	−	0.954254i
$125$	−0.837909			−0.837909
$126$		0			0
$127$	1.98054			1.98054			0.990268	−	0.139173i	$-0.0444444\pi$
$127$	1.98054			1.98054			0.990268	+	0.139173i	$0.0444444\pi$
$128$	−0.477784	−	0.827546i	−0.477784	−	0.827546i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	−0.0985771	−	0.937898i	−0.0985771	−	0.937898i
$133$		0			0
$134$	−0.168037			−0.168037
$135$	−1.09395	−	0.232525i	−1.09395	−	0.232525i
$136$	1.30904			1.30904
$137$		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
$142$	0.149516	+	0.258969i	0.149516	+	0.258969i
$143$		0			0
$144$	0.235862	+	0.261952i	0.235862	+	0.261952i
$145$	0.0780622			0.0780622
$146$		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	$0.666667\pi$
$150$	−0.110851	−	0.0493541i	−0.110851	−	0.0493541i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$	−1.49861	+	0.318539i	−1.49861	+	0.318539i
$154$		0			0
$155$	0.804499	+	1.39343i	0.804499	+	1.39343i
$156$		0			0
$157$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	−0.882948	−	0.469472i	$-0.844444\pi$
$157$	−0.0348995	+	0.0604477i	−0.0348995	+	0.0604477i	0.848048	+	0.529919i	$0.177778\pi$
$158$		0			0
$159$		0			0
$160$	0.573155	+	0.992733i	0.573155	+	0.992733i
$161$		0			0
$162$	0.391438	+	0.284396i	0.391438	+	0.284396i
$163$	0.876742			0.876742			0.438371	−	0.898794i	$-0.355556\pi$
$163$	0.876742			0.876742			0.438371	+	0.898794i	$0.355556\pi$
$164$		0			0
$165$	0.143946	+	1.36955i	0.143946	+	1.36955i
$166$	0.410323	−	0.710700i	0.410323	−	0.710700i
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$	−0.829050			−0.829050
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$	−0.0308521	−	0.0137362i	−0.0308521	−	0.0137362i
$175$		0			0
$176$	0.217015	−	0.375881i	0.217015	−	0.375881i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−0.573155	−	0.636553i	−0.573155	−	0.636553i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	−1.23949	+	0.900539i	−1.23949	+	0.900539i
$184$		0			0
$185$		0			0
$186$	−0.0727619	−	0.692283i	−0.0727619	−	0.692283i
$187$	0.943248	+	1.63375i	0.943248	+	1.63375i
$188$		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	$0.333333\pi$
$192$	−0.0149929	−	0.142648i	−0.0149929	−	0.142648i
$193$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	−0.997564	−	0.0697565i	$-0.977778\pi$
$193$	−0.438371	+	0.759281i	−0.438371	+	0.759281i	0.559193	+	0.829038i	$0.311111\pi$
$194$		0			0
$195$		0			0
$196$	0.382948	+	0.663285i	0.382948	+	0.663285i
$197$	−0.749213			−0.749213			−0.374607	−	0.927184i	$-0.622222\pi$
$197$	−0.749213			−0.749213			−0.374607	+	0.927184i	$0.622222\pi$
$198$	0.184102	−	0.566609i	0.184102	−	0.566609i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.107138	−	0.185569i	−0.107138	−	0.185569i
$201$	0.317271	+	0.141258i	0.317271	+	0.141258i
$202$	−0.391438	+	0.677990i	−0.391438	+	0.677990i
$203$		0			0
$204$	−1.07197	−	0.477273i	−1.07197	−	0.477273i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	0.0348995	+	0.999391i	$0.488889\pi$
$211$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	−0.882948	+	0.469472i	$0.844444\pi$
$212$		0			0
$213$	−0.0646021	−	0.614648i	−0.0646021	−	0.614648i
$214$		0			0
$215$		0			0
$216$	0.264030	+	0.812599i	0.264030	+	0.812599i
$217$		0			0
$218$	−0.473271	−	0.819729i	−0.473271	−	0.819729i
$219$		0			0
$220$	−0.527355	+	0.913405i	−0.527355	+	0.913405i
$221$		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
$225$	0.167809	+	0.186371i	0.167809	+	0.186371i
$226$	−0.884027			−0.884027
$227$		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
$232$	−0.0298187	−	0.0516476i	−0.0298187	−	0.0516476i
$233$		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
$240$	−0.0412073	−	0.392061i	−0.0412073	−	0.392061i
$241$	−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	$-0.533333\pi$
$242$	−0.249739			−0.249739
$243$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$244$	−1.17342			−1.17342
$245$	−0.559193	−	0.968551i	−0.559193	−	0.968551i
$246$		0			0
$247$		0			0
$248$	0.614616	−	1.06455i	0.614616	−	1.06455i
$249$	−1.37217	+	0.996940i	−1.37217	+	0.996940i
$250$	−0.202709	−	0.351102i	−0.202709	−	0.351102i
$251$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$251$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$252$		0			0
$253$		0			0
$254$	0.479135	+	0.829886i	0.479135	+	0.829886i
$255$	1.56533	+	0.696930i	1.56533	+	0.696930i
$256$	0.302890	−	0.524620i	0.302890	−	0.524620i
$257$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.374607	+	0.927184i	$0.622222\pi$
$257$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.615661	+	0.788011i	$0.711111\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.0467046	+	0.0518708i	0.0467046	+	0.0518708i
$262$		0			0
$263$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.848048	+	0.529919i	$0.177778\pi$
$263$	0.882948	+	1.52931i	0.882948	+	1.52931i	0.0348995	+	0.999391i	$0.488889\pi$
$264$	0.851137	−	0.618388i	0.851137	−	0.618388i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.132996	+	0.230356i	0.132996	+	0.230356i
$269$	−1.43868			−1.43868			−0.719340	−	0.694658i	$-0.755556\pi$
$269$	−1.43868			−1.43868			−0.719340	+	0.694658i	$0.755556\pi$
$270$	−0.167217	−	0.514640i	−0.167217	−	0.514640i
$271$	−1.23132			−1.23132			−0.615661	−	0.788011i	$-0.711111\pi$
$271$	−1.23132			−1.23132			−0.615661	+	0.788011i	$0.711111\pi$
$272$	−0.270023	−	0.467694i	−0.270023	−	0.467694i
$273$		0			0
$274$		0			0
$275$	0.154400	−	0.267428i	0.154400	−	0.267428i
$276$		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
$279$	−0.444576	+	1.36827i	−0.444576	+	1.36827i
$280$		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
$283$	0.241922	−	0.419021i	0.241922	−	0.419021i	−0.719340	−	0.694658i	$-0.755556\pi$
$283$	0.241922	−	0.419021i	0.241922	−	0.419021i	0.961262	+	0.275637i	$0.0888889\pi$
$284$	0.236675	−	0.409932i	0.236675	−	0.409932i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.316732	+	0.974802i	−0.316732	+	0.974802i
$289$	1.34730			1.34730
$290$	0.0188850	+	0.0327097i	0.0188850	+	0.0327097i
$291$		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	$0.266667\pi$
$294$	0.0505754	+	0.481193i	0.0505754	+	0.481193i
$295$		0			0
$296$		0			0
$297$	−0.823916	+	0.915051i	−0.823916	+	0.915051i
$298$		0			0
$299$		0			0
$300$	0.0200775	+	0.191024i	0.0200775	+	0.191024i
$301$		0			0
$302$		0			0
$303$	1.30902	−	0.951057i	1.30902	−	0.951057i
$304$		0			0
$305$	1.71347			1.71347
$306$	−0.496021	−	0.550887i	−0.496021	−	0.550887i
$307$	1.11839			1.11839			0.559193	−	0.829038i	$-0.311111\pi$
$307$	1.11839			1.11839			0.559193	+	0.829038i	$0.311111\pi$
$308$		0			0
$309$		0			0
$310$	−0.389252	+	0.674204i	−0.389252	+	0.674204i
$311$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$311$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$	−0.0337718			−0.0337718
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$	0.0429726	−	0.0744306i	0.0429726	−	0.0744306i
$320$	−0.0802071	+	0.138923i	−0.0802071	+	0.138923i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.0800578	−	0.761700i	0.0800578	−	0.761700i
$325$		0			0
$326$	0.212103	+	0.367373i	0.212103	+	0.367373i
$327$	0.204489	+	1.94558i	0.204489	+	1.94558i
$328$		0			0
$329$		0			0
$330$	−0.539047	+	0.391640i	−0.539047	+	0.391640i
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	−1.29903			−1.29903
$333$		0			0
$334$		0			0
$335$	−0.194206	−	0.336374i	−0.194206	−	0.336374i
$336$		0			0
$337$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	0.438371	+	0.898794i	$0.355556\pi$
$337$	−0.559193	+	0.968551i	−0.559193	+	0.968551i	−0.997564	+	0.0697565i	$0.977778\pi$
$338$	0.241922	−	0.419021i	0.241922	−	0.419021i
$339$	1.66913	+	0.743145i	1.66913	+	0.743145i
$340$	0.656168	+	1.13652i	0.656168	+	1.13652i
$341$	1.77148			1.77148
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.719340	−	1.24593i	0.719340	−	1.24593i	−0.241922	−	0.970296i	$-0.577778\pi$
$347$	0.719340	−	1.24593i	0.719340	−	1.24593i	0.961262	−	0.275637i	$-0.0888889\pi$
$348$	0.00558796	+	0.0531659i	0.00558796	+	0.0531659i
$349$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.615661	−	0.788011i	$-0.711111\pi$
$349$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.374607	−	0.927184i	$-0.622222\pi$
$350$		0			0
$351$		0			0
$352$	1.26207			1.26207
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$	−0.345600	+	0.598597i	−0.345600	+	0.598597i
$356$		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	$0.800000\pi$
$360$	0.295287	−	0.908799i	0.295287	−	0.908799i
$361$	1.00000			1.00000
$362$		0			0
$363$	0.471532	+	0.209940i	0.471532	+	0.209940i
$364$		0			0
$365$		0			0
$366$	−0.677204	−	0.301510i	−0.677204	−	0.301510i
$367$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	−0.882948	−	0.469472i	$-0.844444\pi$
$367$	−0.848048	−	1.46886i	−0.848048	−	1.46886i	0.0348995	−	0.999391i	$-0.488889\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.891438	+	0.647668i	−0.891438	+	0.647668i
$373$	−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	$0.266667\pi$
$374$	−0.456385	+	0.790482i	−0.456385	+	0.790482i
$375$	0.0875854	+	0.833319i	0.0875854	+	0.833319i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−0.207022	−	1.96969i	−0.207022	−	1.96969i
$382$		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			$0$
$384$	−0.773071	+	0.561669i	−0.773071	+	0.561669i
$385$		0			0
$386$	−0.424206			−0.424206
$387$		0			0
$388$		0			0
$389$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.719340	−	0.694658i	$-0.755556\pi$
$389$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.241922	−	0.970296i	$-0.577778\pi$
$390$		0			0
$391$		0			0
$392$	−0.427209	+	0.739947i	−0.427209	+	0.739947i
$393$		0			0
$394$	−0.181251	−	0.313936i	−0.181251	−	0.313936i
$395$		0			0
$396$	−0.922456	+	0.196074i	−0.922456	+	0.196074i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−0.0442000	+	0.0765566i	−0.0442000	+	0.0765566i
$401$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	0.0348995	+	0.999391i	$0.488889\pi$
$401$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	−0.882948	+	0.469472i	$0.844444\pi$
$402$	0.0175647	+	0.167117i	0.0175647	+	0.167117i
$403$		0			0
$404$	1.23924			1.23924
$405$	−0.116903	+	1.11226i	−0.116903	+	1.11226i
$406$		0			0
$407$		0			0
$408$	−0.136832	−	1.30187i	−0.136832	−	1.30187i
$409$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.438371	−	0.898794i	$-0.355556\pi$
$409$	0.997564	−	1.72783i	0.997564	−	1.72783i	0.559193	−	0.829038i	$-0.311111\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1.89689			1.89689
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.0348995	−	0.999391i	$-0.488889\pi$
$419$	0.882948	−	1.52931i	0.882948	−	1.52931i	0.848048	−	0.529919i	$-0.177778\pi$
$420$		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	$-0.533333\pi$
$422$	−0.820646			−0.820646
$423$		0			0
$424$		0			0
$425$	−0.192114	−	0.332751i	−0.192114	−	0.332751i
$426$	0.241922	−	0.175767i	0.241922	−	0.175767i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	0.876742			0.876742			0.438371	−	0.898794i	$-0.355556\pi$
$431$	0.876742			0.876742			0.438371	+	0.898794i	$0.355556\pi$
$432$	0.235862	−	0.261952i	0.235862	−	0.261952i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−0.00815972	−	0.0776346i	−0.00815972	−	0.0776346i
$436$	−0.749159	+	1.29758i	−0.749159	+	1.29758i
$437$		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	$0.800000\pi$
$440$	−1.17661			−1.17661
$441$	0.309017	−	0.951057i	0.309017	−	0.951057i
$442$		0			0
$443$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.559193	+	0.829038i	$0.311111\pi$
$443$	0.997564	+	1.72783i	0.997564	+	1.72783i	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			$\pi$
$450$	−0.0374966	+	0.115403i	−0.0374966	+	0.115403i
$451$		0			0
$452$	0.699680	+	1.21188i	0.699680	+	1.21188i
$453$		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
$459$	0.473442	+	1.45710i	0.473442	+	1.45710i
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$	−0.0123017	+	0.0213072i	−0.0123017	+	0.0213072i
$465$	1.30171	−	0.945746i	1.30171	−	0.945746i
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.0637646	+	0.0283898i	0.0637646	+	0.0283898i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−0.483844			−0.483844
$479$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.719340	−	0.694658i	$-0.755556\pi$
$479$	−0.961262	−	1.66495i	−0.961262	−	1.66495i	−0.241922	−	0.970296i	$-0.577778\pi$
$480$	0.927384	−	0.673784i	0.927384	−	0.673784i
$481$		0			0
$482$	0.442013	−	0.765589i	0.442013	−	0.765589i
$483$		0			0
$484$	0.197661	+	0.342359i	0.197661	+	0.342359i
$485$		0			0
$486$	0.241922	−	0.419021i	0.241922	−	0.419021i
$487$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$487$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$488$	−0.654522	−	1.13366i	−0.654522	−	1.13366i
$489$	−0.0916445	−	0.871939i	−0.0916445	−	0.871939i
$490$	0.270562	−	0.468627i	0.270562	−	0.468627i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$	−0.0534691	−	0.0926113i	−0.0534691	−	0.0926113i
$494$		0			0
$495$	1.34700	−	0.286314i	1.34700	−	0.286314i
$496$	−0.507121			−0.507121
$497$		0			0
$498$	−0.749697	−	0.333787i	−0.749697	−	0.333787i
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	−0.320875	+	0.555772i	−0.320875	+	0.555772i
$501$		0			0
$502$	−0.454664	−	0.787502i	−0.454664	−	0.787502i
$503$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$503$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$504$		0			0
$505$	−1.80959			−1.80959
$506$		0			0
$507$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$508$	0.758442	−	1.31366i	0.758442	−	1.31366i
$509$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$509$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$510$	0.0866593	+	0.824508i	0.0866593	+	0.824508i
$511$		0			0
$512$	−0.662466			−0.662466
$513$		0			0
$514$	−0.958270			−0.958270
$515$		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			$\pi$
$522$	−0.0104361	+	0.0321189i	−0.0104361	+	0.0321189i
$523$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$523$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$524$		0			0
$525$		0			0
$526$	−0.427209	+	0.739947i	−0.427209	+	0.739947i
$527$	1.10209	−	1.90888i	1.10209	−	1.90888i
$528$	−0.396506	−	0.176536i	−0.396506	−	0.176536i
$529$	−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
$536$	−0.148368	+	0.256981i	−0.148368	+	0.256981i
$537$		0			0
$538$	−0.348048	−	0.602837i	−0.348048	−	0.602837i
$539$	−1.23132			−1.23132
$540$	−0.573155	+	0.636553i	−0.573155	+	0.636553i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	−0.297884	−	0.515950i	−0.297884	−	0.515950i
$543$		0			0
$544$	0.785171	−	1.35996i	0.785171	−	1.35996i
$545$	1.09395	−	1.89477i	1.09395	−	1.89477i
$546$		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
$549$	1.02517	+	1.13856i	1.02517	+	1.13856i
$550$	0.149411			0.149411
$551$		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			$\pi$
$558$	−0.680885	+	0.144727i	−0.680885	+	0.144727i
$559$		0			0
$560$		0			0
$561$	1.52621	−	1.10885i	1.52621	−	1.10885i
$562$		0			0
$563$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$563$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$	−1.02170	−	1.76963i	−1.02170	−	1.76963i
$566$	0.234105			0.234105
$567$		0			0
$568$	0.528059			0.528059
$569$	0.719340	+	1.24593i	0.719340	+	1.24593i	0.961262	+	0.275637i	$0.0888889\pi$
$569$	0.719340	+	1.24593i	0.719340	+	1.24593i	−0.241922	+	0.970296i	$0.577778\pi$
$570$		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
$576$	−0.140299	+	0.0298215i	−0.140299	+	0.0298215i
$577$	1.33826			1.33826			0.669131	−	0.743145i	$-0.266667\pi$
$577$	1.33826			1.33826			0.669131	+	0.743145i	$0.266667\pi$
$578$	0.325940	+	0.564545i	0.325940	+	0.564545i
$579$	0.800944	+	0.356603i	0.800944	+	0.356603i
$580$	0.0298937	−	0.0517775i	0.0298937	−	0.0517775i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.299032			−0.299032
$587$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	0.559193	−	0.829038i	$-0.311111\pi$
$587$	−0.438371	−	0.759281i	−0.438371	−	0.759281i	−0.997564	+	0.0697565i	$0.977778\pi$
$588$	0.619622	−	0.450182i	0.619622	−	0.450182i
$589$		0			0
$590$		0			0
$591$	0.0783141	+	0.745109i	0.0783141	+	0.745109i
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−0.582749	−	0.123867i	−0.582749	−	0.123867i
$595$		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	$0.133333\pi$
$600$	−0.173353	+	0.125949i	−0.173353	+	0.125949i
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$	0.107320	−	0.330298i	0.107320	−	0.330298i
$604$		0			0
$605$	−0.288631	−	0.499923i	−0.288631	−	0.499923i
$606$	0.715193	+	0.318424i	0.715193	+	0.318424i
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$	0.414525	+	0.717978i	0.414525	+	0.717978i
$611$		0			0
$612$	−0.362607	+	1.11599i	−0.362607	+	1.11599i
$613$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$613$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$614$	0.270562	+	0.468627i	0.270562	+	0.468627i
$615$		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	$0.333333\pi$
$620$	1.23232			1.23232
$621$		0			0
$622$	0.909329			0.909329
$623$		0			0
$624$		0			0
$625$	0.593946	−	1.02875i	0.593946	−	1.02875i
$626$		0			0
$627$		0			0
$628$	0.0267294	+	0.0462966i	0.0267294	+	0.0462966i
$629$		0			0
$630$		0			0
$631$	1.11839			1.11839			0.559193	−	0.829038i	$-0.311111\pi$
$631$	1.11839			1.11839			0.559193	+	0.829038i	$0.311111\pi$
$632$		0			0
$633$	1.54946	+	0.689864i	1.54946	+	0.689864i
$634$		0			0
$635$	−1.10750	+	1.91825i	−1.10750	+	1.91825i
$636$		0			0
$637$		0			0
$638$	0.0415840			0.0415840
$639$	−0.604528	+	0.128496i	−0.604528	+	0.128496i
$640$	1.06869			1.06869
$641$	0.374607	+	0.648838i	0.374607	+	0.648838i	0.990268	−	0.139173i	$-0.0444444\pi$
$641$	0.374607	+	0.648838i	0.374607	+	0.648838i	−0.615661	+	0.788011i	$0.711111\pi$
$642$		0			0
$643$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.374607	+	0.927184i	$0.622222\pi$
$643$	−0.990268	+	1.71519i	−0.990268	+	1.71519i	−0.615661	+	0.788011i	$0.711111\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	1.92252			1.92252			0.961262	−	0.275637i	$-0.0888889\pi$
$647$	1.92252			1.92252			0.961262	+	0.275637i	$0.0888889\pi$
$648$	0.780549	−	0.347523i	0.780549	−	0.347523i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.335746	−	0.581530i	0.335746	−	0.581530i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	−0.765768	+	0.556363i	−0.765768	+	0.556363i
$655$		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	$0.333333\pi$
$660$	0.963525	+	0.428989i	0.963525	+	0.428989i
$661$	0.615661	−	1.06636i	0.615661	−	1.06636i	−0.374607	−	0.927184i	$-0.622222\pi$
$661$	0.615661	−	1.06636i	0.615661	−	1.06636i	0.990268	−	0.139173i	$-0.0444444\pi$
$662$		0			0
$663$		0			0
$664$	−0.724587	−	1.25502i	−0.724587	−	1.25502i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	0.0939652	−	0.162753i	0.0939652	−	0.162753i
$671$	0.943248	−	1.63375i	0.943248	−	1.63375i
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$	−0.541124			−0.541124
$675$	0.167809	−	0.186371i	0.167809	−	0.186371i
$676$	−0.765895			−0.765895
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$	0.0924059	+	0.879184i	0.0924059	+	0.879184i
$679$		0			0
$680$	−0.732008	+	1.26787i	−0.732008	+	1.26787i
$681$		0			0
$682$	0.428560	+	0.742287i	0.428560	+	0.742287i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$691$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$692$		0			0
$693$		0			0
$694$	0.696096			0.696096
$695$		0			0
$696$	−0.0482477	+	0.0350540i	−0.0482477	+	0.0350540i
$697$		0			0
$698$	0.479135	−	0.829886i	0.479135	−	0.829886i
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$	0.0883066	+	0.152952i	0.0883066	+	0.152952i
$705$		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	$0.333333\pi$
$710$	−0.334433			−0.334433
$711$		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
$718$	−0.391438	−	0.677990i	−0.391438	−	0.677990i
$719$	−1.99513			−1.99513			−0.997564	−	0.0697565i	$-0.977778\pi$
$719$	−1.99513			−1.99513			−0.997564	+	0.0697565i	$0.977778\pi$
$720$	−0.385606	+	0.0819631i	−0.385606	+	0.0819631i
$721$		0			0
$722$	0.241922	+	0.419021i	0.241922	+	0.419021i
$723$	−1.47815	+	1.07394i	−1.47815	+	1.07394i
$724$		0			0
$725$	−0.00875233	+	0.0151595i	−0.00875233	+	0.0151595i
$726$	0.0261048	+	0.248371i	0.0261048	+	0.248371i
$727$	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
$732$	0.122656	+	1.16699i	0.122656	+	1.16699i
$733$	0.719340	−	1.24593i	0.719340	−	1.24593i	−0.241922	−	0.970296i	$-0.577778\pi$
$733$	0.719340	−	1.24593i	0.719340	−	1.24593i	0.961262	−	0.275637i	$-0.0888889\pi$
$734$	0.410323	−	0.710700i	0.410323	−	0.710700i
$735$	−0.904793	+	0.657371i	−0.904793	+	0.657371i
$736$		0			0
$737$	−0.427634			−0.427634
$738$		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	$0.666667\pi$
$744$	−1.12296	−	0.499974i	−1.12296	−	0.499974i
$745$		0			0
$746$	−0.299032			−0.299032
$747$	1.13491	+	1.26045i	1.13491	+	1.26045i
$748$	1.44486			1.44486
$749$		0			0
$750$	−0.327989	+	0.238298i	−0.327989	+	0.238298i
$751$	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
$753$	0.196449	+	1.86909i	0.196449	+	1.86909i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.483844			−0.483844			−0.241922	−	0.970296i	$-0.577778\pi$
$757$	−0.483844			−0.483844			−0.241922	+	0.970296i	$0.577778\pi$
$758$		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	$-0.266667\pi$
$762$	0.775257	−	0.563257i	0.775257	−	0.563257i
$763$		0			0
$764$		0			0
$765$	0.529490	−	1.62960i	0.529490	−	1.62960i
$766$	0.483844			0.483844
$767$		0			0
$768$	−0.553407	−	0.246393i	−0.553407	−	0.246393i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$	1.80931	+	0.805557i	1.80931	+	0.805557i
$772$	0.335746	+	0.581530i	0.335746	+	0.581530i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−0.360802			−0.360802
$776$		0			0
$777$		0			0
$778$	0.465101	−	0.805578i	0.465101	−	0.805578i
$779$		0			0
$780$		0			0
$781$	0.380500	+	0.659045i	0.380500	+	0.659045i
$782$		0			0
$783$	0.0467046	−	0.0518708i	0.0467046	−	0.0518708i
$784$	0.352491			0.352491
$785$	−0.0390311	−	0.0676039i	−0.0390311	−	0.0676039i
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	−0.286909	+	0.496942i	−0.286909	+	0.496942i
$789$	1.42864	−	1.03797i	1.42864	−	1.03797i
$790$		0			0
$791$		0			0
$792$	−0.703968	−	0.781836i	−0.703968	−	0.781836i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.241922	+	0.970296i	$0.577778\pi$
$797$	−0.961262	+	1.66495i	−0.961262	+	1.66495i	−0.719340	+	0.694658i	$0.755556\pi$
$798$		0			0
$799$		0			0
$800$	−0.257048			−0.257048
$801$		0			0
$802$	−0.820646			−0.820646
$803$		0			0
$804$	0.215193	−	0.156347i	0.215193	−	0.156347i
$805$		0			0
$806$		0			0
$807$	0.150383	+	1.43080i	0.150383	+	1.43080i
$808$	0.691238	+	1.19726i	0.691238	+	1.19726i
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−0.494341	+	0.220095i	−0.494341	+	0.220095i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	0.128708	+	1.22458i	0.128708	+	1.22458i
$814$		0			0
$815$	−0.490268	+	0.849169i	−0.490268	+	0.849169i
$816$	−0.436907	+	0.317432i	−0.436907	+	0.317432i
$817$		0			0
$818$	0.965330			0.965330
$819$		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
$825$	−0.282102	−	0.125600i	−0.282102	−	0.125600i
$826$		0			0
$827$	0.0697990			0.0697990			0.0348995	−	0.999391i	$-0.488889\pi$
$827$	0.0697990			0.0697990			0.0348995	+	0.999391i	$0.488889\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$	0.458899	+	0.794837i	0.458899	+	0.794837i
$831$		0			0
$832$		0			0
$833$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.40724	+	0.299118i	1.40724	+	0.299118i
$838$	0.854417			0.854417
$839$	0.374607	+	0.648838i	0.374607	+	0.648838i	0.990268	−	0.139173i	$-0.0444444\pi$
$839$	0.374607	+	0.648838i	0.374607	+	0.648838i	−0.615661	+	0.788011i	$0.711111\pi$
$840$		0			0
$841$	0.497564	−	0.861806i	0.497564	−	0.861806i
$842$	0.442013	−	0.765589i	0.442013	−	0.765589i
$843$		0			0
$844$	0.649516	+	1.12499i	0.649516	+	1.12499i
$845$	1.11839			1.11839
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−0.442013	−	0.196797i	−0.442013	−	0.196797i
$850$	0.0929531	−	0.160999i	0.0929531	−	0.160999i
$851$		0			0
$852$	−0.432426	−	0.192528i	−0.432426	−	0.192528i
$853$	0.997564	+	1.72783i	0.997564	+	1.72783i	0.559193	+	0.829038i	$0.311111\pi$
$853$	0.997564	+	1.72783i	0.997564	+	1.72783i	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
$859$	0.374607	−	0.648838i	0.374607	−	0.648838i	−0.615661	−	0.788011i	$-0.711111\pi$
$859$	0.374607	−	0.648838i	0.374607	−	0.648838i	0.990268	+	0.139173i	$0.0444444\pi$
$860$		0			0
$861$		0			0
$862$	0.212103	+	0.367373i	0.212103	+	0.367373i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	1.00257	+	0.213103i	1.00257	+	0.213103i
$865$		0			0
$866$		0			0
$867$	−0.140831	−	1.33992i	−0.140831	−	1.33992i
$868$		0			0
$869$		0			0
$870$	0.0305565	−	0.0222006i	0.0305565	−	0.0222006i
$871$		0			0
$872$	−1.67149			−1.67149
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	0.0348995	+	0.999391i	$0.488889\pi$
$877$	−0.848048	+	1.46886i	−0.848048	+	1.46886i	−0.882948	+	0.469472i	$0.844444\pi$
$878$	−0.0505754	+	0.0875992i	−0.0505754	+	0.0875992i
$879$	0.564602	+	0.251377i	0.564602	+	0.251377i
$880$	0.242706	+	0.420380i	0.242706	+	0.420380i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.473271	−	0.100597i	0.473271	−	0.100597i
$883$	1.92252			1.92252			0.961262	−	0.275637i	$-0.0888889\pi$
$883$	1.92252			1.92252			0.961262	+	0.275637i	$0.0888889\pi$
$884$		0			0
$885$		0			0
$886$	−0.482665	+	0.836001i	−0.482665	+	0.836001i
$887$	−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
$891$	0.996161	+	0.723753i	0.996161	+	0.723753i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.100418			−0.100418
$900$	0.187879	−	0.0399349i	0.187879	−	0.0399349i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.780549	+	1.35195i	−0.780549	+	1.35195i
$905$		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
$913$	1.04422	−	1.80864i	1.04422	−	1.80864i
$914$	0.149516	−	0.258969i	0.149516	−	0.258969i
$915$	−0.179106	−	1.70408i	−0.179106	−	1.70408i
$916$		0			0
$917$		0			0
$918$	−0.496021	+	0.550887i	−0.496021	+	0.550887i
$919$	0.876742			0.876742			0.438371	−	0.898794i	$-0.355556\pi$
$919$	0.876742			0.876742			0.438371	+	0.898794i	$0.355556\pi$
$920$		0			0
$921$	−0.116903	−	1.11226i	−0.116903	−	1.11226i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−0.0715417			−0.0715417
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$	0.711199	+	0.316646i	0.711199	+	0.316646i
$931$		0			0
$932$		0			0
$933$	−1.71690	−	0.764415i	−1.71690	−	0.764415i
$934$		0			0
$935$	−2.10983			−2.10983
$936$		0			0
$937$	1.92252			1.92252			0.961262	−	0.275637i	$-0.0888889\pi$
$937$	1.92252			1.92252			0.961262	+	0.275637i	$0.0888889\pi$
$938$		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	$0.666667\pi$
$942$	0.00353012	+	0.0335868i	0.00353012	+	0.0335868i
$943$		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
$956$	0.382948	+	0.663285i	0.382948	+	0.663285i
$957$	−0.0785148	−	0.0349570i	−0.0785148	−	0.0349570i
$958$	0.465101	−	0.805578i	0.465101	−	0.805578i
$959$		0			0
$960$	0.146546	+	0.0652463i	0.146546	+	0.0652463i
$961$	−0.534899	−	0.926473i	−0.534899	−	0.926473i
$962$		0			0
$963$		0			0
$964$	−1.39936			−1.39936
$965$	−0.490268	−	0.849169i	−0.490268	−	0.849169i
$966$		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	$0.933333\pi$
$968$	−0.220506	+	0.381928i	−0.220506	+	0.381928i
$969$		0			0
$970$		0			0
$971$	−1.99513			−1.99513			−0.997564	−	0.0697565i	$-0.977778\pi$
$971$	−1.99513			−1.99513			−0.997564	+	0.0697565i	$0.977778\pi$
$972$	−0.765895			−0.765895
$973$		0			0
$974$	−0.454664	−	0.787502i	−0.454664	−	0.787502i
$975$		0			0
$976$	−0.270023	+	0.467694i	−0.270023	+	0.467694i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$	0.343190	−	0.249342i	0.343190	−	0.249342i
$979$		0			0
$980$	−0.856566			−0.856566
$981$	1.91355	−	0.406737i	1.91355	−	0.406737i
$982$		0			0
$983$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.615661	−	0.788011i	$-0.711111\pi$
$983$	−0.990268	−	1.71519i	−0.990268	−	1.71519i	−0.374607	−	0.927184i	$-0.622222\pi$
$984$		0			0
$985$	0.418955	−	0.725651i	0.418955	−	0.725651i
$986$	0.0258707	−	0.0448094i	0.0258707	−	0.0448094i
$987$		0			0
$988$		0			0
$989$		0			0
$990$	0.445841	+	0.495156i	0.445841	+	0.495156i
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−0.737300	−	1.27704i	−0.737300	−	1.27704i
$993$		0			0
$994$		0			0
$995$		0			0
$996$	0.135786	+	1.29192i	0.135786	+	1.29192i
$997$		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.238.7	✓	24
9.7	even	3	inner	2151.1.f.b.1672.7	yes	24
239.238	odd	2	CM	2151.1.f.b.238.7	✓	24
2151.1672	odd	6	inner	2151.1.f.b.1672.7	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.1.f.b.238.7	✓	24	1.1	even	1	trivial
2151.1.f.b.238.7	✓	24	239.238	odd	2	CM
2151.1.f.b.1672.7	yes	24	9.7	even	3	inner
2151.1.f.b.1672.7	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)

\(f(q)\)	\(=\)	\(q+(0.241922 + 0.419021i) q^{2} +(-0.104528 - 0.994522i) q^{3} +(0.382948 - 0.663285i) q^{4} +(-0.559193 + 0.968551i) q^{5} +(0.391438 - 0.284396i) q^{6} +0.854417 q^{8} +(-0.978148 + 0.207912i) q^{9} +O(q^{10})\)	\(q+(0.241922 + 0.419021i) q^{2} +(-0.104528 - 0.994522i) q^{3} +(0.382948 - 0.663285i) q^{4} +(-0.559193 + 0.968551i) q^{5} +(0.391438 - 0.284396i) q^{6} +0.854417 q^{8} +(-0.978148 + 0.207912i) q^{9} -0.541124 q^{10} +(0.615661 + 1.06636i) q^{11} +(-0.699680 - 0.311518i) q^{12} +(1.02170 + 0.454888i) q^{15} +(-0.176245 - 0.305266i) q^{16} +1.53209 q^{17} +(-0.323755 - 0.359566i) q^{18} +(0.428283 + 0.741808i) q^{20} +(-0.297884 + 0.515950i) q^{22} +(-0.0893109 - 0.849737i) q^{24} +(-0.125393 - 0.217188i) q^{25} +(0.309017 + 0.951057i) q^{27} +(-0.0348995 - 0.0604477i) q^{29} +(0.0565629 + 0.538160i) q^{30} +(0.719340 - 1.24593i) q^{31} +(0.512484 - 0.887648i) q^{32} +(0.996161 - 0.723753i) q^{33} +(0.370646 + 0.641977i) q^{34} +(-0.236675 + 0.728410i) q^{36} +(-0.477784 + 0.827546i) q^{40} +0.943064 q^{44} +(0.345600 - 1.06365i) q^{45} +(-0.285171 + 0.207189i) q^{48} +(-0.500000 + 0.866025i) q^{49} +(0.0606708 - 0.105085i) q^{50} +(-0.160147 - 1.52370i) q^{51} +(-0.323755 + 0.359566i) q^{54} -1.37709 q^{55} +(0.0168859 - 0.0292472i) q^{58} +(0.692977 - 0.503477i) q^{60} +(-0.766044 - 1.32683i) q^{61} +0.696096 q^{62} +0.143434 q^{64} +(0.544261 + 0.242321i) q^{66} +(-0.173648 + 0.300767i) q^{67} +(0.586710 - 1.01621i) q^{68} +0.618034 q^{71} +(-0.835746 + 0.177643i) q^{72} +(-0.202891 + 0.147409i) q^{75} +0.394221 q^{80} +(0.913545 - 0.406737i) q^{81} +(-0.848048 - 1.46886i) q^{83} +(-0.856733 + 1.48391i) q^{85} +(-0.0564686 + 0.0410268i) q^{87} +(0.526032 + 0.911114i) q^{88} +(0.529299 - 0.112506i) q^{90} +(-1.31430 - 0.585164i) q^{93} +(-0.936355 - 0.416892i) q^{96} -0.483844 q^{98} +(-0.823916 - 0.915051i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 3 q^{3} - 12 q^{4} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) 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

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