LMFDB - Embedded newform 3751.1.t.e.3657.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3751,1,Mod(2138,3751)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3751, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([2, 5])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3751.2138"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 3751 = 11^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3751.t (of order $10$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,-3,0,0,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(8)] == traces)

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

Self dual:	no
Analytic conductor:	$1.87199286239$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$3$ over $\Q(\zeta_{10})$
Coefficient field:	12.0.84075626953125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 3x^{10} - x^{9} + 9x^{8} + 9x^{7} + 28x^{6} + 18x^{5} + 75x^{4} + 26x^{3} + 9x^{2} + 3x + 1 $ x^12 + 3x^10 - x^9 + 9x^8 + 9x^7 + 28x^6 + 18x^5 + 75x^4 + 26x^3 + 9x^2 + 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.1636073786281.1

Embedding invariants

Embedding label			3657.2
Root			$0.473442 + 1.45710i$ of defining polynomial
Character	$\chi$	$=$	3751.3657
Dual form			3751.1.t.e.2913.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.107320 - 0.330298i) q^{2} +(0.711438 - 0.516890i) q^{4} +(0.107320 - 0.330298i) q^{5} +(1.23949 - 0.900539i) q^{7} +(-0.528048 - 0.383650i) q^{8} +(0.309017 + 0.951057i) q^{9} -0.120615 q^{10} +(-0.430469 - 0.312754i) q^{14} +(0.201697 - 0.620758i) q^{16} +(0.280969 - 0.204136i) q^{18} +(0.280969 + 0.204136i) q^{19} +(-0.0943760 - 0.290460i) q^{20} +(0.711438 + 0.516890i) q^{25} +(0.416337 - 1.28135i) q^{28} +(0.309017 + 0.951057i) q^{31} -0.879385 q^{32} +(-0.164425 - 0.506047i) q^{35} +(0.711438 + 0.516890i) q^{36} +(0.0372720 - 0.114711i) q^{38} +(-0.183389 + 0.133240i) q^{40} +(-1.52045 - 1.10467i) q^{41} +0.347296 q^{45} +(0.809017 + 0.587785i) q^{47} +(0.416337 - 1.28135i) q^{49} +(0.0943760 - 0.290460i) q^{50} -1.00000 q^{56} +(-1.23949 + 0.900539i) q^{59} +(0.280969 - 0.204136i) q^{62} +(1.23949 + 0.900539i) q^{63} +(-0.107320 - 0.330298i) q^{64} -1.00000 q^{67} +(-0.149500 + 0.108618i) q^{70} +(0.473442 - 1.45710i) q^{71} +(0.201697 - 0.620758i) q^{72} +0.305407 q^{76} +(-0.183389 - 0.133240i) q^{80} +(-0.809017 + 0.587785i) q^{81} +(-0.201697 + 0.620758i) q^{82} +(-0.0372720 - 0.114711i) q^{90} +(0.107320 - 0.330298i) q^{94} +(0.0975794 - 0.0708956i) q^{95} +(0.473442 + 1.45710i) q^{97} -0.467911 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{4} - 3 q^{8} - 3 q^{9} - 24 q^{10} + 3 q^{14} - 3 q^{16} + 3 q^{20} - 3 q^{25} - 3 q^{28} - 3 q^{31} + 12 q^{32} - 3 q^{35} - 3 q^{36} - 6 q^{38} + 6 q^{40} + 3 q^{47} - 3 q^{49} - 3 q^{50}+ \cdots - 24 q^{98}+O(q^{100}) $ 12 * q - 3 * q^4 - 3 * q^8 - 3 * q^9 - 24 * q^10 + 3 * q^14 - 3 * q^16 + 3 * q^20 - 3 * q^25 - 3 * q^28 - 3 * q^31 + 12 * q^32 - 3 * q^35 - 3 * q^36 - 6 * q^38 + 6 * q^40 + 3 * q^47 - 3 * q^49 - 3 * q^50 - 12 * q^56 - 12 * q^67 + 3 * q^70 - 3 * q^72 + 12 * q^76 + 6 * q^80 - 3 * q^81 + 3 * q^82 + 6 * q^90 + 6 * q^95 - 24 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3751\mathbb{Z}\right)^\times$.

$n$	$2421$	$2543$
$\chi(n)$	$-1$	$e\left(\frac{2}{5}\right)$

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 $

$2$	−0.107320	−	0.330298i	−0.107320	−	0.330298i	0.882948	−	0.469472i	$-0.155556\pi$
$2$	−0.107320	−	0.330298i	−0.107320	−	0.330298i	−0.990268	+	0.139173i	$0.955556\pi$
$3$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$3$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$4$	0.711438	−	0.516890i	0.711438	−	0.516890i
$5$	0.107320	−	0.330298i	0.107320	−	0.330298i	−0.882948	−	0.469472i	$-0.844444\pi$
$5$	0.107320	−	0.330298i	0.107320	−	0.330298i	0.990268	+	0.139173i	$0.0444444\pi$
$6$		0			0
$7$	1.23949	−	0.900539i	1.23949	−	0.900539i	0.241922	−	0.970296i	$-0.422222\pi$
$7$	1.23949	−	0.900539i	1.23949	−	0.900539i	0.997564	+	0.0697565i	$0.0222222\pi$
$8$	−0.528048	−	0.383650i	−0.528048	−	0.383650i
$9$	0.309017	+	0.951057i	0.309017	+	0.951057i
$10$	−0.120615			−0.120615
$11$		0			0
$11$		0			0
$12$		0			0
$13$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$13$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$14$	−0.430469	−	0.312754i	−0.430469	−	0.312754i
$15$		0			0
$16$	0.201697	−	0.620758i	0.201697	−	0.620758i
$17$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$17$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$18$	0.280969	−	0.204136i	0.280969	−	0.204136i
$19$	0.280969	+	0.204136i	0.280969	+	0.204136i	0.719340	−	0.694658i	$-0.244444\pi$
$19$	0.280969	+	0.204136i	0.280969	+	0.204136i	−0.438371	+	0.898794i	$0.644444\pi$
$20$	−0.0943760	−	0.290460i	−0.0943760	−	0.290460i
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	0.711438	+	0.516890i	0.711438	+	0.516890i
$26$		0			0
$27$		0			0
$28$	0.416337	−	1.28135i	0.416337	−	1.28135i
$29$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$29$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$30$		0			0
$31$	0.309017	+	0.951057i	0.309017	+	0.951057i
$31$	0.309017	+	0.951057i	0.309017	+	0.951057i
$32$	−0.879385			−0.879385
$33$		0			0
$34$		0			0
$35$	−0.164425	−	0.506047i	−0.164425	−	0.506047i
$36$	0.711438	+	0.516890i	0.711438	+	0.516890i
$37$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$37$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$38$	0.0372720	−	0.114711i	0.0372720	−	0.114711i
$39$		0			0
$40$	−0.183389	+	0.133240i	−0.183389	+	0.133240i
$41$	−1.52045	−	1.10467i	−1.52045	−	1.10467i	−0.961262	−	0.275637i	$-0.911111\pi$
$41$	−1.52045	−	1.10467i	−1.52045	−	1.10467i	−0.559193	−	0.829038i	$-0.688889\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	0.347296			0.347296
$46$		0			0
$47$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$47$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$48$		0			0
$49$	0.416337	−	1.28135i	0.416337	−	1.28135i
$50$	0.0943760	−	0.290460i	0.0943760	−	0.290460i
$51$		0			0
$52$		0			0
$53$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$53$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$54$		0			0
$55$		0			0
$56$	−1.00000			−1.00000
$57$		0			0
$58$		0			0
$59$	−1.23949	+	0.900539i	−1.23949	+	0.900539i	−0.997564	−	0.0697565i	$-0.977778\pi$
$59$	−1.23949	+	0.900539i	−1.23949	+	0.900539i	−0.241922	+	0.970296i	$0.577778\pi$
$60$		0			0
$61$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$61$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$62$	0.280969	−	0.204136i	0.280969	−	0.204136i
$63$	1.23949	+	0.900539i	1.23949	+	0.900539i
$64$	−0.107320	−	0.330298i	−0.107320	−	0.330298i
$65$		0			0
$66$		0			0
$67$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$67$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$	−0.149500	+	0.108618i	−0.149500	+	0.108618i
$71$	0.473442	−	1.45710i	0.473442	−	1.45710i	−0.374607	−	0.927184i	$-0.622222\pi$
$71$	0.473442	−	1.45710i	0.473442	−	1.45710i	0.848048	−	0.529919i	$-0.177778\pi$
$72$	0.201697	−	0.620758i	0.201697	−	0.620758i
$73$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$73$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$74$		0			0
$75$		0			0
$76$	0.305407			0.305407
$77$		0			0
$78$		0			0
$79$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$79$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$80$	−0.183389	−	0.133240i	−0.183389	−	0.133240i
$81$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$82$	−0.201697	+	0.620758i	−0.201697	+	0.620758i
$83$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$83$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−0.0372720	−	0.114711i	−0.0372720	−	0.114711i
$91$		0			0
$92$		0			0
$93$		0			0
$94$	0.107320	−	0.330298i	0.107320	−	0.330298i
$95$	0.0975794	−	0.0708956i	0.0975794	−	0.0708956i
$96$		0			0
$97$	0.473442	+	1.45710i	0.473442	+	1.45710i	0.848048	+	0.529919i	$0.177778\pi$
$97$	0.473442	+	1.45710i	0.473442	+	1.45710i	−0.374607	+	0.927184i	$0.622222\pi$
$98$	−0.467911			−0.467911
$99$		0			0
$100$	0.773318			0.773318
$101$	0.580762	+	1.78740i	0.580762	+	1.78740i	0.615661	+	0.788011i	$0.288889\pi$
$101$	0.580762	+	1.78740i	0.580762	+	1.78740i	−0.0348995	+	0.999391i	$0.511111\pi$
$102$		0			0
$103$	−0.280969	+	0.204136i	−0.280969	+	0.204136i	−0.719340	−	0.694658i	$-0.755556\pi$
$103$	−0.280969	+	0.204136i	−0.280969	+	0.204136i	0.438371	+	0.898794i	$0.355556\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.52045	−	1.10467i	−1.52045	−	1.10467i	−0.961262	−	0.275637i	$-0.911111\pi$
$107$	−1.52045	−	1.10467i	−1.52045	−	1.10467i	−0.559193	−	0.829038i	$-0.688889\pi$
$108$		0			0
$109$	−1.53209			−1.53209			−0.766044	−	0.642788i	$-0.777778\pi$
$109$	−1.53209			−1.53209			−0.766044	+	0.642788i	$0.777778\pi$
$110$		0			0
$111$		0			0
$112$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$113$	−1.23949	−	0.900539i	−1.23949	−	0.900539i	−0.241922	−	0.970296i	$-0.577778\pi$
$113$	−1.23949	−	0.900539i	−1.23949	−	0.900539i	−0.997564	+	0.0697565i	$0.977778\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$	0.430469	+	0.312754i	0.430469	+	0.312754i
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$	0.711438	+	0.516890i	0.711438	+	0.516890i
$125$	0.528048	−	0.383650i	0.528048	−	0.383650i
$126$	0.164425	−	0.506047i	0.164425	−	0.506047i
$127$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$127$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$128$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$129$		0			0
$130$		0			0
$131$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$131$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	0.532089			0.532089
$134$	0.107320	+	0.330298i	0.107320	+	0.330298i
$135$		0			0
$136$		0			0
$137$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$137$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$138$		0			0
$139$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$139$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$140$	−0.378548	−	0.275031i	−0.378548	−	0.275031i
$141$		0			0
$142$	−0.532089			−0.532089
$143$		0			0
$144$	0.652704			0.652704
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$149$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$150$		0			0
$151$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$151$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$152$	−0.0700485	−	0.215587i	−0.0700485	−	0.215587i
$153$		0			0
$154$		0			0
$155$	0.347296			0.347296
$156$		0			0
$157$	−1.23949	−	0.900539i	−1.23949	−	0.900539i	−0.241922	−	0.970296i	$-0.577778\pi$
$157$	−1.23949	−	0.900539i	−1.23949	−	0.900539i	−0.997564	+	0.0697565i	$0.977778\pi$
$158$		0			0
$159$		0			0
$160$	−0.0943760	+	0.290460i	−0.0943760	+	0.290460i
$161$		0			0
$162$	0.280969	+	0.204136i	0.280969	+	0.204136i
$163$	−0.580762	−	1.78740i	−0.580762	−	1.78740i	−0.615661	−	0.788011i	$-0.711111\pi$
$163$	−0.580762	−	1.78740i	−0.580762	−	1.78740i	0.0348995	−	0.999391i	$-0.488889\pi$
$164$	−1.65270			−1.65270
$165$		0			0
$166$		0			0
$167$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$167$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$168$		0			0
$169$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$170$		0			0
$171$	−0.107320	+	0.330298i	−0.107320	+	0.330298i
$172$		0			0
$173$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$173$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$174$		0			0
$175$	1.34730			1.34730
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$179$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$180$	0.247080	−	0.179514i	0.247080	−	0.179514i
$181$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$181$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.879385			0.879385
$189$		0			0
$190$	−0.0338890	−	0.0246218i	−0.0338890	−	0.0246218i
$191$	1.52045	−	1.10467i	1.52045	−	1.10467i	0.559193	−	0.829038i	$-0.311111\pi$
$191$	1.52045	−	1.10467i	1.52045	−	1.10467i	0.961262	−	0.275637i	$-0.0888889\pi$
$192$		0			0
$193$	0.580762	−	1.78740i	0.580762	−	1.78740i	−0.0348995	−	0.999391i	$-0.511111\pi$
$193$	0.580762	−	1.78740i	0.580762	−	1.78740i	0.615661	−	0.788011i	$-0.288889\pi$
$194$	0.430469	−	0.312754i	0.430469	−	0.312754i
$195$		0			0
$196$	−0.366121	−	1.12680i	−0.366121	−	1.12680i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.177369	−	0.545885i	−0.177369	−	0.545885i
$201$		0			0
$202$	0.528048	−	0.383650i	0.528048	−	0.383650i
$203$		0			0
$204$		0			0
$205$	−0.528048	+	0.383650i	−0.528048	+	0.383650i
$206$	0.0975794	+	0.0708956i	0.0975794	+	0.0708956i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.580762	+	1.78740i	0.580762	+	1.78740i	0.615661	+	0.788011i	$0.288889\pi$
$211$	0.580762	+	1.78740i	0.580762	+	1.78740i	−0.0348995	+	0.999391i	$0.511111\pi$
$212$		0			0
$213$		0			0
$214$	−0.201697	+	0.620758i	−0.201697	+	0.620758i
$215$		0			0
$216$		0			0
$217$	1.23949	+	0.900539i	1.23949	+	0.900539i
$218$	0.164425	+	0.506047i	0.164425	+	0.506047i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$223$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$224$	−1.08999	+	0.791921i	−1.08999	+	0.791921i
$225$	−0.271745	+	0.836345i	−0.271745	+	0.836345i
$226$	−0.164425	+	0.506047i	−0.164425	+	0.506047i
$227$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$227$	1.61803	−	1.17557i	1.61803	−	1.17557i	0.809017	−	0.587785i	$-0.200000\pi$
$228$		0			0
$229$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$229$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.107320	−	0.330298i	−0.107320	−	0.330298i	0.882948	−	0.469472i	$-0.155556\pi$
$233$	−0.107320	−	0.330298i	−0.107320	−	0.330298i	−0.990268	+	0.139173i	$0.955556\pi$
$234$		0			0
$235$	0.280969	−	0.204136i	0.280969	−	0.204136i
$236$	−0.416337	+	1.28135i	−0.416337	+	1.28135i
$237$		0			0
$238$		0			0
$239$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$239$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.378548	−	0.275031i	−0.378548	−	0.275031i
$246$		0			0
$247$		0			0
$248$	0.201697	−	0.620758i	0.201697	−	0.620758i
$249$		0			0
$250$	−0.183389	−	0.133240i	−0.183389	−	0.133240i
$251$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$251$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$252$	1.34730			1.34730
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.52045	−	1.10467i	1.52045	−	1.10467i	0.559193	−	0.829038i	$-0.311111\pi$
$257$	1.52045	−	1.10467i	1.52045	−	1.10467i	0.961262	−	0.275637i	$-0.0888889\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−0.107320	−	0.330298i	−0.107320	−	0.330298i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$	−0.0571040	−	0.175748i	−0.0571040	−	0.175748i
$267$		0			0
$268$	−0.711438	+	0.516890i	−0.711438	+	0.516890i
$269$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$269$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$270$		0			0
$271$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$271$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$277$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$278$		0			0
$279$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$280$	−0.107320	+	0.330298i	−0.107320	+	0.330298i
$281$	−0.473442	+	1.45710i	−0.473442	+	1.45710i	0.374607	+	0.927184i	$0.377778\pi$
$281$	−0.473442	+	1.45710i	−0.473442	+	1.45710i	−0.848048	+	0.529919i	$0.822222\pi$
$282$		0			0
$283$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$283$	1.61803	+	1.17557i	1.61803	+	1.17557i	0.809017	+	0.587785i	$0.200000\pi$
$284$	−0.416337	−	1.28135i	−0.416337	−	1.28135i
$285$		0			0
$286$		0			0
$287$	−2.87939			−2.87939
$288$	−0.271745	−	0.836345i	−0.271745	−	0.836345i
$289$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	−0.913545	−	0.406737i	$-0.866667\pi$
$293$	−0.809017	+	0.587785i	−0.809017	+	0.587785i	0.104528	+	0.994522i	$0.466667\pi$
$294$		0			0
$295$	0.164425	+	0.506047i	0.164425	+	0.506047i
$296$		0			0
$297$		0			0
$298$	0.694593			0.694593
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.183389	−	0.133240i	0.183389	−	0.133240i
$305$		0			0
$306$		0			0
$307$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$307$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$308$		0			0
$309$		0			0
$310$	−0.0372720	−	0.114711i	−0.0372720	−	0.114711i
$311$	1.52045	+	1.10467i	1.52045	+	1.10467i	0.961262	+	0.275637i	$0.0888889\pi$
$311$	1.52045	+	1.10467i	1.52045	+	1.10467i	0.559193	+	0.829038i	$0.311111\pi$
$312$		0			0
$313$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$313$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$314$	−0.164425	+	0.506047i	−0.164425	+	0.506047i
$315$	0.430469	−	0.312754i	0.430469	−	0.312754i
$316$		0			0
$317$	0.107320	+	0.330298i	0.107320	+	0.330298i	0.990268	−	0.139173i	$-0.0444444\pi$
$317$	0.107320	+	0.330298i	0.107320	+	0.330298i	−0.882948	+	0.469472i	$0.844444\pi$
$318$		0			0
$319$		0			0
$320$	−0.120615			−0.120615
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.271745	+	0.836345i	−0.271745	+	0.836345i
$325$		0			0
$326$	−0.528048	+	0.383650i	−0.528048	+	0.383650i
$327$		0			0
$328$	0.379065	+	1.16664i	0.379065	+	1.16664i
$329$	1.53209			1.53209
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.107320	+	0.330298i	−0.107320	+	0.330298i
$336$		0			0
$337$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$337$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$338$	0.280969	+	0.204136i	0.280969	+	0.204136i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	0.120615			0.120615
$343$	−0.164425	−	0.506047i	−0.164425	−	0.506047i
$344$		0			0
$345$		0			0
$346$	−0.107320	+	0.330298i	−0.107320	+	0.330298i
$347$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$347$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$348$		0			0
$349$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$349$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$350$	−0.144592	−	0.445010i	−0.144592	−	0.445010i
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$	−0.430469	−	0.312754i	−0.430469	−	0.312754i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.52045	+	1.10467i	−1.52045	+	1.10467i	−0.559193	+	0.829038i	$0.688889\pi$
$359$	−1.52045	+	1.10467i	−1.52045	+	1.10467i	−0.961262	+	0.275637i	$0.911111\pi$
$360$	−0.183389	−	0.133240i	−0.183389	−	0.133240i
$361$	−0.271745	−	0.836345i	−0.271745	−	0.836345i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$367$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$368$		0			0
$369$	0.580762	−	1.78740i	0.580762	−	1.78740i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$373$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$374$		0			0
$375$		0			0
$376$	−0.201697	−	0.620758i	−0.201697	−	0.620758i
$377$		0			0
$378$		0			0
$379$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$379$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$380$	0.0327765	−	0.100876i	0.0327765	−	0.100876i
$381$		0			0
$382$	−0.528048	−	0.383650i	−0.528048	−	0.383650i
$383$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$383$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$384$		0			0
$385$		0			0
$386$	−0.652704			−0.652704
$387$		0			0
$388$	1.08999	+	0.791921i	1.08999	+	0.791921i
$389$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$389$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$390$		0			0
$391$		0			0
$392$	−0.711438	+	0.516890i	−0.711438	+	0.516890i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$397$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$398$		0			0
$399$		0			0
$400$	0.464358	−	0.337376i	0.464358	−	0.337376i
$401$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$401$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$402$		0			0
$403$		0			0
$404$	1.33707	+	0.971435i	1.33707	+	0.971435i
$405$	0.107320	+	0.330298i	0.107320	+	0.330298i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$409$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$410$	0.183389	+	0.133240i	0.183389	+	0.133240i
$411$		0			0
$412$	−0.0943760	+	0.290460i	−0.0943760	+	0.290460i
$413$	−0.725354	+	2.23241i	−0.725354	+	2.23241i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$419$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$420$		0			0
$421$	−0.280969	−	0.204136i	−0.280969	−	0.204136i	0.438371	−	0.898794i	$-0.355556\pi$
$421$	−0.280969	−	0.204136i	−0.280969	−	0.204136i	−0.719340	+	0.694658i	$0.755556\pi$
$422$	0.528048	−	0.383650i	0.528048	−	0.383650i
$423$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.65270			−1.65270
$429$		0			0
$430$		0			0
$431$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$431$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$432$		0			0
$433$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$433$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$434$	0.164425	−	0.506047i	0.164425	−	0.506047i
$435$		0			0
$436$	−1.08999	+	0.791921i	−1.08999	+	0.791921i
$437$		0			0
$438$		0			0
$439$	−1.53209			−1.53209			−0.766044	−	0.642788i	$-0.777778\pi$
$439$	−1.53209			−1.53209			−0.766044	+	0.642788i	$0.777778\pi$
$440$		0			0
$441$	1.34730			1.34730
$442$		0			0
$443$	−0.280969	−	0.204136i	−0.280969	−	0.204136i	0.438371	−	0.898794i	$-0.355556\pi$
$443$	−0.280969	−	0.204136i	−0.280969	−	0.204136i	−0.719340	+	0.694658i	$0.755556\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.430469	−	0.312754i	−0.430469	−	0.312754i
$449$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$449$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$450$	0.305407			0.305407
$451$		0			0
$452$	−1.34730			−1.34730
$453$		0			0
$454$	−0.561937	−	0.408271i	−0.561937	−	0.408271i
$455$		0			0
$456$		0			0
$457$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$457$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	−0.0975794	+	0.0708956i	−0.0975794	+	0.0708956i
$467$	−0.580762	+	1.78740i	−0.580762	+	1.78740i	0.0348995	+	0.999391i	$0.488889\pi$
$467$	−0.580762	+	1.78740i	−0.580762	+	1.78740i	−0.615661	+	0.788011i	$0.711111\pi$
$468$		0			0
$469$	−1.23949	+	0.900539i	−1.23949	+	0.900539i
$470$	−0.0975794	−	0.0708956i	−0.0975794	−	0.0708956i
$471$		0			0
$472$	1.00000			1.00000
$473$		0			0
$474$		0			0
$475$	0.0943760	+	0.290460i	0.0943760	+	0.290460i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.107320	+	0.330298i	−0.107320	+	0.330298i	−0.990268	−	0.139173i	$-0.955556\pi$
$479$	−0.107320	+	0.330298i	−0.107320	+	0.330298i	0.882948	+	0.469472i	$0.155556\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.532089			0.532089
$486$		0			0
$487$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$487$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$488$		0			0
$489$		0			0
$490$	−0.0502164	+	0.154550i	−0.0502164	+	0.154550i
$491$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$491$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.652704			0.652704
$497$	−0.725354	−	2.23241i	−0.725354	−	2.23241i
$498$		0			0
$499$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$499$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$500$	0.177369	−	0.545885i	0.177369	−	0.545885i
$501$		0			0
$502$		0			0
$503$	1.23949	+	0.900539i	1.23949	+	0.900539i	0.997564	−	0.0697565i	$-0.0222222\pi$
$503$	1.23949	+	0.900539i	1.23949	+	0.900539i	0.241922	+	0.970296i	$0.422222\pi$
$504$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$505$	0.652704			0.652704
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$509$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$510$		0			0
$511$		0			0
$512$	−0.309017	+	0.951057i	−0.309017	+	0.951057i
$513$		0			0
$514$	−0.528048	−	0.383650i	−0.528048	−	0.383650i
$515$	0.0372720	+	0.114711i	0.0372720	+	0.114711i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$521$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$522$		0			0
$523$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$523$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$524$	0.711438	−	0.516890i	0.711438	−	0.516890i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	1.00000			1.00000
$530$		0			0
$531$	−1.23949	−	0.900539i	−1.23949	−	0.900539i
$532$	0.378548	−	0.275031i	0.378548	−	0.275031i
$533$		0			0
$534$		0			0
$535$	−0.528048	+	0.383650i	−0.528048	+	0.383650i
$536$	0.528048	+	0.383650i	0.528048	+	0.383650i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−0.473442	−	1.45710i	−0.473442	−	1.45710i	−0.848048	−	0.529919i	$-0.822222\pi$
$541$	−0.473442	−	1.45710i	−0.473442	−	1.45710i	0.374607	−	0.927184i	$-0.377778\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−0.164425	+	0.506047i	−0.164425	+	0.506047i
$546$		0			0
$547$	0.280969	+	0.204136i	0.280969	+	0.204136i	0.719340	−	0.694658i	$-0.244444\pi$
$547$	0.280969	+	0.204136i	0.280969	+	0.204136i	−0.438371	+	0.898794i	$0.644444\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$557$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$558$	0.280969	+	0.204136i	0.280969	+	0.204136i
$559$		0			0
$560$	−0.347296			−0.347296
$561$		0			0
$562$	0.532089			0.532089
$563$	0.580762	+	1.78740i	0.580762	+	1.78740i	0.615661	+	0.788011i	$0.288889\pi$
$563$	0.580762	+	1.78740i	0.580762	+	1.78740i	−0.0348995	+	0.999391i	$0.511111\pi$
$564$		0			0
$565$	−0.430469	+	0.312754i	−0.430469	+	0.312754i
$566$	0.214641	−	0.660597i	0.214641	−	0.660597i
$567$	−0.473442	+	1.45710i	−0.473442	+	1.45710i
$568$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$569$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$569$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$	0.309017	+	0.951057i	0.309017	+	0.951057i
$575$		0			0
$576$	0.280969	−	0.204136i	0.280969	−	0.204136i
$577$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	0.669131	+	0.743145i	$0.266667\pi$
$577$	−0.309017	+	0.951057i	−0.309017	+	0.951057i	−0.978148	+	0.207912i	$0.933333\pi$
$578$	−0.107320	+	0.330298i	−0.107320	+	0.330298i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	0.280969	+	0.204136i	0.280969	+	0.204136i
$587$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$587$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$588$		0			0
$589$	−0.107320	+	0.330298i	−0.107320	+	0.330298i
$590$	0.149500	−	0.108618i	0.149500	−	0.108618i
$591$		0			0
$592$		0			0
$593$	−1.53209			−1.53209			−0.766044	−	0.642788i	$-0.777778\pi$
$593$	−1.53209			−1.53209			−0.766044	+	0.642788i	$0.777778\pi$
$594$		0			0
$595$		0			0
$596$	0.543490	+	1.67269i	0.543490	+	1.67269i
$597$		0			0
$598$		0			0
$599$	0.107320	−	0.330298i	0.107320	−	0.330298i	−0.882948	−	0.469472i	$-0.844444\pi$
$599$	0.107320	−	0.330298i	0.107320	−	0.330298i	0.990268	+	0.139173i	$0.0444444\pi$
$600$		0			0
$601$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$601$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$602$		0			0
$603$	−0.309017	−	0.951057i	−0.309017	−	0.951057i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	0.309017	+	0.951057i	0.309017	+	0.951057i	0.978148	+	0.207912i	$0.0666667\pi$
$607$	0.309017	+	0.951057i	0.309017	+	0.951057i	−0.669131	+	0.743145i	$0.733333\pi$
$608$	−0.247080	−	0.179514i	−0.247080	−	0.179514i
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$613$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$614$	−0.201697	−	0.620758i	−0.201697	−	0.620758i
$615$		0			0
$616$		0			0
$617$	2.00000			2.00000			1.00000			$0$
$617$	2.00000			2.00000			1.00000			$0$
$618$		0			0
$619$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$619$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$620$	0.247080	−	0.179514i	0.247080	−	0.179514i
$621$		0			0
$622$	0.201697	−	0.620758i	0.201697	−	0.620758i
$623$		0			0
$624$		0			0
$625$	0.201697	+	0.620758i	0.201697	+	0.620758i
$626$		0			0
$627$		0			0
$628$	−1.34730			−1.34730
$629$		0			0
$630$	−0.149500	−	0.108618i	−0.149500	−	0.108618i
$631$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$631$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$632$		0			0
$633$		0			0
$634$	0.0975794	−	0.0708956i	0.0975794	−	0.0708956i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	1.53209			1.53209
$640$	0.107320	+	0.330298i	0.107320	+	0.330298i
$641$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$641$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$642$		0			0
$643$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$643$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$647$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$648$	0.652704			0.652704
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.33707	−	0.971435i	−1.33707	−	0.971435i
$653$	0.809017	−	0.587785i	0.809017	−	0.587785i	−0.104528	−	0.994522i	$-0.533333\pi$
$653$	0.809017	−	0.587785i	0.809017	−	0.587785i	0.913545	+	0.406737i	$0.133333\pi$
$654$		0			0
$655$	0.107320	−	0.330298i	0.107320	−	0.330298i
$656$	−0.992406	+	0.721025i	−0.992406	+	0.721025i
$657$		0			0
$658$	−0.164425	−	0.506047i	−0.164425	−	0.506047i
$659$	−0.347296			−0.347296			−0.173648	−	0.984808i	$-0.555556\pi$
$659$	−0.347296			−0.347296			−0.173648	+	0.984808i	$0.555556\pi$
$660$		0			0
$661$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$661$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	0.0571040	−	0.175748i	0.0571040	−	0.175748i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	0.120615			0.120615
$671$		0			0
$672$		0			0
$673$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$673$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$674$		0			0
$675$		0			0
$676$	−0.271745	+	0.836345i	−0.271745	+	0.836345i
$677$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$677$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$678$		0			0
$679$	1.89900	+	1.37971i	1.89900	+	1.37971i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$683$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$684$	0.0943760	+	0.290460i	0.0943760	+	0.290460i
$685$		0			0
$686$	−0.149500	+	0.108618i	−0.149500	+	0.108618i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.107320	+	0.330298i	0.107320	+	0.330298i	0.990268	−	0.139173i	$-0.0444444\pi$
$691$	0.107320	+	0.330298i	0.107320	+	0.330298i	−0.882948	+	0.469472i	$0.844444\pi$
$692$	−0.879385			−0.879385
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−0.107320	+	0.330298i	−0.107320	+	0.330298i
$699$		0			0
$700$	0.958517	−	0.696404i	0.958517	−	0.696404i
$701$	1.23949	+	0.900539i	1.23949	+	0.900539i	0.997564	−	0.0697565i	$-0.0222222\pi$
$701$	1.23949	+	0.900539i	1.23949	+	0.900539i	0.241922	+	0.970296i	$0.422222\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2.32947	+	1.69246i	2.32947	+	1.69246i
$708$		0			0
$709$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$709$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$710$	−0.0571040	+	0.175748i	−0.0571040	+	0.175748i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	0.528048	+	0.383650i	0.528048	+	0.383650i
$719$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$719$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$720$	0.0700485	−	0.215587i	0.0700485	−	0.215587i
$721$	−0.164425	+	0.506047i	−0.164425	+	0.506047i
$722$	−0.247080	+	0.179514i	−0.247080	+	0.179514i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$727$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$728$		0			0
$729$	−0.809017	−	0.587785i	−0.809017	−	0.587785i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1.23949	−	0.900539i	1.23949	−	0.900539i	0.241922	−	0.970296i	$-0.422222\pi$
$733$	1.23949	−	0.900539i	1.23949	−	0.900539i	0.997564	+	0.0697565i	$0.0222222\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−0.652704			−0.652704
$739$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$739$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$743$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$744$		0			0
$745$	0.561937	+	0.408271i	0.561937	+	0.408271i
$746$	−0.201697	−	0.620758i	−0.201697	−	0.620758i
$747$		0			0
$748$		0			0
$749$	−2.87939			−2.87939
$750$		0			0
$751$	−0.280969	−	0.204136i	−0.280969	−	0.204136i	0.438371	−	0.898794i	$-0.355556\pi$
$751$	−0.280969	−	0.204136i	−0.280969	−	0.204136i	−0.719340	+	0.694658i	$0.755556\pi$
$752$	0.528048	−	0.383650i	0.528048	−	0.383650i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$757$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$758$	0.347296			0.347296
$759$		0			0
$760$	−0.0787257			−0.0787257
$761$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$761$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$762$		0			0
$763$	−1.89900	+	1.37971i	−1.89900	+	1.37971i
$764$	0.510714	−	1.57181i	0.510714	−	1.57181i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.347296			−0.347296			−0.173648	−	0.984808i	$-0.555556\pi$
$769$	−0.347296			−0.347296			−0.173648	+	0.984808i	$0.555556\pi$
$770$		0			0
$771$		0			0
$772$	−0.510714	−	1.57181i	−0.510714	−	1.57181i
$773$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$773$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$774$		0			0
$775$	−0.271745	+	0.836345i	−0.271745	+	0.836345i
$776$	0.309017	−	0.951057i	0.309017	−	0.951057i
$777$		0			0
$778$		0			0
$779$	−0.201697	−	0.620758i	−0.201697	−	0.620758i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.711438	−	0.516890i	−0.711438	−	0.516890i
$785$	−0.430469	+	0.312754i	−0.430469	+	0.312754i
$786$		0			0
$787$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$787$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.34730			−2.34730
$792$		0			0
$793$		0			0
$794$	0.201697	+	0.620758i	0.201697	+	0.620758i
$795$		0			0
$796$		0			0
$797$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$797$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$798$		0			0
$799$		0			0
$800$	−0.625628	−	0.454545i	−0.625628	−	0.454545i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.379065	−	1.16664i	0.379065	−	1.16664i
$809$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$809$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$810$	0.0975794	−	0.0708956i	0.0975794	−	0.0708956i
$811$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	0.104528	−	0.994522i	$-0.466667\pi$
$811$	−0.809017	−	0.587785i	−0.809017	−	0.587785i	−0.913545	+	0.406737i	$0.866667\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.652704			−0.652704
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$	−0.177369	+	0.545885i	−0.177369	+	0.545885i
$821$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$821$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$822$		0			0
$823$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$823$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$824$	0.226682			0.226682
$825$		0			0
$826$	0.815207			0.815207
$827$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$827$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$828$		0			0
$829$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$829$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	0.201697	+	0.620758i	0.201697	+	0.620758i
$839$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$839$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$840$		0			0
$841$	0.309017	−	0.951057i	0.309017	−	0.951057i
$842$	−0.0372720	+	0.114711i	−0.0372720	+	0.114711i
$843$		0			0
$844$	1.33707	+	0.971435i	1.33707	+	0.971435i
$845$	0.107320	+	0.330298i	0.107320	+	0.330298i
$846$	0.347296			0.347296
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$853$	−0.618034	+	1.90211i	−0.618034	+	1.90211i	−0.309017	+	0.951057i	$0.600000\pi$
$854$		0			0
$855$	0.0975794	+	0.0708956i	0.0975794	+	0.0708956i
$856$	0.379065	+	1.16664i	0.379065	+	1.16664i
$857$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$857$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	0.280969	−	0.204136i	0.280969	−	0.204136i
$863$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$863$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$864$		0			0
$865$	−0.280969	+	0.204136i	−0.280969	+	0.204136i
$866$		0			0
$867$		0			0
$868$	1.34730			1.34730
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.809017	+	0.587785i	0.809017	+	0.587785i
$873$	−1.23949	+	0.900539i	−1.23949	+	0.900539i
$874$		0			0
$875$	0.309017	−	0.951057i	0.309017	−	0.951057i
$876$		0			0
$877$	−1.52045	−	1.10467i	−1.52045	−	1.10467i	−0.961262	−	0.275637i	$-0.911111\pi$
$877$	−1.52045	−	1.10467i	−1.52045	−	1.10467i	−0.559193	−	0.829038i	$-0.688889\pi$
$878$	0.164425	+	0.506047i	0.164425	+	0.506047i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.144592	−	0.445010i	−0.144592	−	0.445010i
$883$		0			0		−0.809017	−	0.587785i	$-0.800000\pi$
$883$		0			0		0.809017	+	0.587785i	$0.200000\pi$
$884$		0			0
$885$		0			0
$886$	−0.0372720	+	0.114711i	−0.0372720	+	0.114711i
$887$	0.280969	−	0.204136i	0.280969	−	0.204136i	−0.438371	−	0.898794i	$-0.644444\pi$
$887$	0.280969	−	0.204136i	0.280969	−	0.204136i	0.719340	+	0.694658i	$0.244444\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.107320	+	0.330298i	0.107320	+	0.330298i
$894$		0			0
$895$		0			0
$896$	−0.473442	+	1.45710i	−0.473442	+	1.45710i
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.238969	+	0.735470i	0.238969	+	0.735470i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.309017	+	0.951057i	0.309017	+	0.951057i
$905$		0			0
$906$		0			0
$907$	0.107320	−	0.330298i	0.107320	−	0.330298i	−0.882948	−	0.469472i	$-0.844444\pi$
$907$	0.107320	−	0.330298i	0.107320	−	0.330298i	0.990268	+	0.139173i	$0.0444444\pi$
$908$	0.543490	−	1.67269i	0.543490	−	1.67269i
$909$	−1.52045	+	1.10467i	−1.52045	+	1.10467i
$910$		0			0
$911$		0			0		−0.309017	−	0.951057i	$-0.600000\pi$
$911$		0			0		0.309017	+	0.951057i	$0.400000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.23949	−	0.900539i	1.23949	−	0.900539i
$918$		0			0
$919$	0.309017	−	0.951057i	0.309017	−	0.951057i	−0.669131	−	0.743145i	$-0.733333\pi$
$919$	0.309017	−	0.951057i	0.309017	−	0.951057i	0.978148	−	0.207912i	$-0.0666667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−0.280969	−	0.204136i	−0.280969	−	0.204136i
$928$		0			0
$929$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$929$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$930$		0			0
$931$	0.378548	−	0.275031i	0.378548	−	0.275031i
$932$	−0.247080	−	0.179514i	−0.247080	−	0.179514i
$933$		0			0
$934$	0.652704			0.652704
$935$		0			0
$936$		0			0
$937$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$937$	−0.618034	−	1.90211i	−0.618034	−	1.90211i	−0.309017	−	0.951057i	$-0.600000\pi$
$938$	0.430469	+	0.312754i	0.430469	+	0.312754i
$939$		0			0
$940$	0.0943760	−	0.290460i	0.0943760	−	0.290460i
$941$		0			0		0.309017	−	0.951057i	$-0.400000\pi$
$941$		0			0		−0.309017	+	0.951057i	$0.600000\pi$
$942$		0			0
$943$		0			0
$944$	0.309017	+	0.951057i	0.309017	+	0.951057i
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$	0.0858099	−	0.0623445i	0.0858099	−	0.0623445i
$951$		0			0
$952$		0			0
$953$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$953$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$954$		0			0
$955$	−0.201697	−	0.620758i	−0.201697	−	0.620758i
$956$		0			0
$957$		0			0
$958$	0.120615			0.120615
$959$		0			0
$960$		0			0
$961$	−0.809017	+	0.587785i	−0.809017	+	0.587785i
$962$		0			0
$963$	0.580762	−	1.78740i	0.580762	−	1.78740i
$964$		0			0
$965$	−0.528048	−	0.383650i	−0.528048	−	0.383650i
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$	−0.0571040	−	0.175748i	−0.0571040	−	0.175748i
$971$	0.809017	+	0.587785i	0.809017	+	0.587785i	0.913545	−	0.406737i	$-0.133333\pi$
$971$	0.809017	+	0.587785i	0.809017	+	0.587785i	−0.104528	+	0.994522i	$0.533333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.580762	−	1.78740i	−0.580762	−	1.78740i	−0.615661	−	0.788011i	$-0.711111\pi$
$977$	−0.580762	−	1.78740i	−0.580762	−	1.78740i	0.0348995	−	0.999391i	$-0.488889\pi$
$978$		0			0
$979$		0			0
$980$	−0.411474			−0.411474
$981$	−0.473442	−	1.45710i	−0.473442	−	1.45710i
$982$		0			0
$983$		0			0		0.809017	−	0.587785i	$-0.200000\pi$
$983$		0			0		−0.809017	+	0.587785i	$0.800000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−0.271745	−	0.836345i	−0.271745	−	0.836345i
$993$		0			0
$994$	−0.659517	+	0.479167i	−0.659517	+	0.479167i
$995$		0			0
$996$		0			0
$997$	0.280969	−	0.204136i	0.280969	−	0.204136i	−0.438371	−	0.898794i	$-0.644444\pi$
$997$	0.280969	−	0.204136i	0.280969	−	0.204136i	0.719340	+	0.694658i	$0.244444\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3751.1.t.e.3657.2		12
11.2	odd	10		3751.1.t.f.2913.2		12
11.3	even	5		3751.1.d.e.1332.2	yes	3
11.4	even	5	inner	3751.1.t.e.2665.2		12
11.5	even	5	inner	3751.1.t.e.2138.2		12
11.6	odd	10		3751.1.t.f.2138.2		12
11.7	odd	10		3751.1.t.f.2665.2		12
11.8	odd	10		3751.1.d.d.1332.2	✓	3
11.9	even	5	inner	3751.1.t.e.2913.2		12
11.10	odd	2		3751.1.t.f.3657.2		12
31.30	odd	2	CM	3751.1.t.e.3657.2		12
341.30	even	10		3751.1.d.d.1332.2	✓	3
341.61	even	10		3751.1.t.f.2138.2		12
341.92	odd	10	inner	3751.1.t.e.2665.2		12
341.123	even	10		3751.1.t.f.2913.2		12
341.185	odd	10	inner	3751.1.t.e.2913.2		12
341.216	even	10		3751.1.t.f.2665.2		12
341.247	odd	10	inner	3751.1.t.e.2138.2		12
341.278	odd	10		3751.1.d.e.1332.2	yes	3
341.340	even	2		3751.1.t.f.3657.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3751.1.d.d.1332.2	✓	3	11.8	odd	10
3751.1.d.d.1332.2	✓	3	341.30	even	10
3751.1.d.e.1332.2	yes	3	11.3	even	5
3751.1.d.e.1332.2	yes	3	341.278	odd	10
3751.1.t.e.2138.2		12	11.5	even	5	inner
3751.1.t.e.2138.2		12	341.247	odd	10	inner
3751.1.t.e.2665.2		12	11.4	even	5	inner
3751.1.t.e.2665.2		12	341.92	odd	10	inner
3751.1.t.e.2913.2		12	11.9	even	5	inner
3751.1.t.e.2913.2		12	341.185	odd	10	inner
3751.1.t.e.3657.2		12	1.1	even	1	trivial
3751.1.t.e.3657.2		12	31.30	odd	2	CM
3751.1.t.f.2138.2		12	11.6	odd	10
3751.1.t.f.2138.2		12	341.61	even	10
3751.1.t.f.2665.2		12	11.7	odd	10
3751.1.t.f.2665.2		12	341.216	even	10
3751.1.t.f.2913.2		12	11.2	odd	10
3751.1.t.f.2913.2		12	341.123	even	10
3751.1.t.f.3657.2		12	11.10	odd	2
3751.1.t.f.3657.2		12	341.340	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3751 = 11^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3751.t (of order \(10\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.107320 - 0.330298i) q^{2} +(0.711438 - 0.516890i) q^{4} +(0.107320 - 0.330298i) q^{5} +(1.23949 - 0.900539i) q^{7} +(-0.528048 - 0.383650i) q^{8} +(0.309017 + 0.951057i) q^{9} -0.120615 q^{10} +(-0.430469 - 0.312754i) q^{14} +(0.201697 - 0.620758i) q^{16} +(0.280969 - 0.204136i) q^{18} +(0.280969 + 0.204136i) q^{19} +(-0.0943760 - 0.290460i) q^{20} +(0.711438 + 0.516890i) q^{25} +(0.416337 - 1.28135i) q^{28} +(0.309017 + 0.951057i) q^{31} -0.879385 q^{32} +(-0.164425 - 0.506047i) q^{35} +(0.711438 + 0.516890i) q^{36} +(0.0372720 - 0.114711i) q^{38} +(-0.183389 + 0.133240i) q^{40} +(-1.52045 - 1.10467i) q^{41} +0.347296 q^{45} +(0.809017 + 0.587785i) q^{47} +(0.416337 - 1.28135i) q^{49} +(0.0943760 - 0.290460i) q^{50} -1.00000 q^{56} +(-1.23949 + 0.900539i) q^{59} +(0.280969 - 0.204136i) q^{62} +(1.23949 + 0.900539i) q^{63} +(-0.107320 - 0.330298i) q^{64} -1.00000 q^{67} +(-0.149500 + 0.108618i) q^{70} +(0.473442 - 1.45710i) q^{71} +(0.201697 - 0.620758i) q^{72} +0.305407 q^{76} +(-0.183389 - 0.133240i) q^{80} +(-0.809017 + 0.587785i) q^{81} +(-0.201697 + 0.620758i) q^{82} +(-0.0372720 - 0.114711i) q^{90} +(0.107320 - 0.330298i) q^{94} +(0.0975794 - 0.0708956i) q^{95} +(0.473442 + 1.45710i) q^{97} -0.467911 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{4} - 3 q^{8} - 3 q^{9} - 24 q^{10} + 3 q^{14} - 3 q^{16} + 3 q^{20} - 3 q^{25} - 3 q^{28} - 3 q^{31} + 12 q^{32} - 3 q^{35} - 3 q^{36} - 6 q^{38} + 6 q^{40} + 3 q^{47} - 3 q^{49} - 3 q^{50}+ \cdots - 24 q^{98}+O(q^{100}) \) 12 * q - 3 * q^4 - 3 * q^8 - 3 * q^9 - 24 * q^10 + 3 * q^14 - 3 * q^16 + 3 * q^20 - 3 * q^25 - 3 * q^28 - 3 * q^31 + 12 * q^32 - 3 * q^35 - 3 * q^36 - 6 * q^38 + 6 * q^40 + 3 * q^47 - 3 * q^49 - 3 * q^50 - 12 * q^56 - 12 * q^67 + 3 * q^70 - 3 * q^72 + 12 * q^76 + 6 * q^80 - 3 * q^81 + 3 * q^82 + 6 * q^90 + 6 * q^95 - 24 * q^98

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