LMFDB - Embedded newform 3132.1.o.b.2899.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3132,1,Mod(1855,3132)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3132, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 4, 3]))

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

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

chi := DirichletCharacter("3132.1855");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3132 = 2^{2} \cdot 3^{3} \cdot 29 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3132.o (of order $6$, degree $2$, not 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:	$1.56307161957$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{18})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{3} + 1 $ x^6 - x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 1044)
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.96224799211776.1

Embedding invariants

Embedding label			2899.2
Root			$-0.173648 + 0.984808i$ of defining polynomial
Character	$\chi$	$=$	3132.2899
Dual form			3132.1.o.b.1855.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.173648 - 0.300767i) q^{5} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.173648 - 0.300767i) q^{5} -1.00000 q^{8} +0.347296 q^{10} +(0.173648 + 0.300767i) q^{11} +(-0.766044 + 1.32683i) q^{13} +(-0.500000 - 0.866025i) q^{16} -1.00000 q^{19} +(0.173648 + 0.300767i) q^{20} +(-0.173648 + 0.300767i) q^{22} +(0.439693 + 0.761570i) q^{25} -1.53209 q^{26} +(0.500000 + 0.866025i) q^{29} +(-0.766044 + 1.32683i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-0.500000 - 0.866025i) q^{38} +(-0.173648 + 0.300767i) q^{40} +(0.939693 + 1.62760i) q^{43} -0.347296 q^{44} +(-0.939693 - 1.62760i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-0.439693 + 0.761570i) q^{50} +(-0.766044 - 1.32683i) q^{52} +1.87939 q^{53} +0.120615 q^{55} +(-0.500000 + 0.866025i) q^{58} -1.53209 q^{62} +1.00000 q^{64} +(0.266044 + 0.460802i) q^{65} +(0.500000 - 0.866025i) q^{76} +(-0.766044 - 1.32683i) q^{79} -0.347296 q^{80} +(-0.939693 + 1.62760i) q^{86} +(-0.173648 - 0.300767i) q^{88} +(0.939693 - 1.62760i) q^{94} +(-0.173648 + 0.300767i) q^{95} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{2} - 3 q^{4} - 6 q^{8}+O(q^{10}) $ 6 * q + 3 * q^2 - 3 * q^4 - 6 * q^8	$ 6 q + 3 q^{2} - 3 q^{4} - 6 q^{8} - 3 q^{16} - 6 q^{19} - 3 q^{25} + 3 q^{29} + 3 q^{32} - 3 q^{38} - 3 q^{49} + 3 q^{50} + 12 q^{55} - 3 q^{58} + 6 q^{64} - 3 q^{65} + 3 q^{76} - 6 q^{98}+O(q^{100}) $ 6 * q + 3 * q^2 - 3 * q^4 - 6 * q^8 - 3 * q^16 - 6 * q^19 - 3 * q^25 + 3 * q^29 + 3 * q^32 - 3 * q^38 - 3 * q^49 + 3 * q^50 + 12 * q^55 - 3 * q^58 + 6 * q^64 - 3 * q^65 + 3 * q^76 - 6 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3132\mathbb{Z}\right)^\times$.

$n$	$1567$	$1945$	$1973$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	0.500000	+	0.866025i	0.500000	+	0.866025i
$2$	0.500000	+	0.866025i	0.500000	+	0.866025i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$5$	0.173648	−	0.300767i	0.173648	−	0.300767i	−0.766044	−	0.642788i	$-0.777778\pi$
$5$	0.173648	−	0.300767i	0.173648	−	0.300767i	0.939693	+	0.342020i	$0.111111\pi$
$6$		0			0
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	−1.00000			−1.00000
$9$		0			0
$10$	0.347296			0.347296
$11$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$11$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$12$		0			0
$13$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$13$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$19$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$20$	0.173648	+	0.300767i	0.173648	+	0.300767i
$21$		0			0
$22$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	0.439693	+	0.761570i	0.439693	+	0.761570i
$26$	−1.53209			−1.53209
$27$		0			0
$28$		0			0
$29$	0.500000	+	0.866025i	0.500000	+	0.866025i
$29$	0.500000	+	0.866025i	0.500000	+	0.866025i
$30$		0			0
$31$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$31$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$32$	0.500000	−	0.866025i	0.500000	−	0.866025i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$39$		0			0
$40$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$43$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$44$	−0.347296			−0.347296
$45$		0			0
$46$		0			0
$47$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.766044	−	0.642788i	$-0.777778\pi$
$47$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$48$		0			0
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	−0.439693	+	0.761570i	−0.439693	+	0.761570i
$51$		0			0
$52$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$53$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$53$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$54$		0			0
$55$	0.120615			0.120615
$56$		0			0
$57$		0			0
$58$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$	−1.53209			−1.53209
$63$		0			0
$64$	1.00000			1.00000
$65$	0.266044	+	0.460802i	0.266044	+	0.460802i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$	0.500000	−	0.866025i	0.500000	−	0.866025i
$77$		0			0
$78$		0			0
$79$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$79$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$80$	−0.347296			−0.347296
$81$		0			0
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$87$		0			0
$88$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	0.939693	−	1.62760i	0.939693	−	1.62760i
$95$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$96$		0			0
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$	−1.00000			−1.00000
$99$		0			0
$100$	−0.879385			−0.879385
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$	0.766044	−	1.32683i	0.766044	−	1.32683i
$105$		0			0
$106$	0.939693	+	1.62760i	0.939693	+	1.62760i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$109$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$110$	0.0603074	+	0.104455i	0.0603074	+	0.104455i
$111$		0			0
$112$		0			0
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$		0			0
$116$	−1.00000			−1.00000
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.439693	−	0.761570i	0.439693	−	0.761570i
$122$		0			0
$123$		0			0
$124$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$125$	0.652704			0.652704
$126$		0			0
$127$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$127$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$128$	0.500000	+	0.866025i	0.500000	+	0.866025i
$129$		0			0
$130$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$131$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$131$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		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			0
$143$	−0.532089			−0.532089
$144$		0			0
$145$	0.347296			0.347296
$146$		0			0
$147$		0			0
$148$		0			0
$149$	0.766044	−	1.32683i	0.766044	−	1.32683i	−0.173648	−	0.984808i	$-0.555556\pi$
$149$	0.766044	−	1.32683i	0.766044	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$150$		0			0
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	1.00000			1.00000
$153$		0			0
$154$		0			0
$155$	0.266044	+	0.460802i	0.266044	+	0.460802i
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	0.766044	−	1.32683i	0.766044	−	1.32683i
$159$		0			0
$160$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$161$		0			0
$162$		0			0
$163$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$163$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−0.673648	−	1.16679i	−0.673648	−	1.16679i
$170$		0			0
$171$		0			0
$172$	−1.87939			−1.87939
$173$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$173$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$	0.173648	−	0.300767i	0.173648	−	0.300767i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$181$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$	1.87939			1.87939
$189$		0			0
$190$	−0.347296			−0.347296
$191$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$191$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$192$		0			0
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$197$	−2.00000			−2.00000			−1.00000			$\pi$
$197$	−2.00000			−2.00000			−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.439693	−	0.761570i	−0.439693	−	0.761570i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.53209			1.53209
$209$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$210$		0			0
$211$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$211$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$212$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$213$		0			0
$214$		0			0
$215$	0.652704			0.652704
$216$		0			0
$217$		0			0
$218$	0.173648	+	0.300767i	0.173648	+	0.300767i
$219$		0			0
$220$	−0.0603074	+	0.104455i	−0.0603074	+	0.104455i
$221$		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			0
$226$		0			0
$227$		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
$232$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$233$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$233$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$234$		0			0
$235$	−0.652704			−0.652704
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$241$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$242$	0.879385			0.879385
$243$		0			0
$244$		0			0
$245$	0.173648	+	0.300767i	0.173648	+	0.300767i
$246$		0			0
$247$	0.766044	−	1.32683i	0.766044	−	1.32683i
$248$	0.766044	−	1.32683i	0.766044	−	1.32683i
$249$		0			0
$250$	0.326352	+	0.565258i	0.326352	+	0.565258i
$251$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$251$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$252$		0			0
$253$		0			0
$254$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.173648	+	0.984808i	$0.555556\pi$
$257$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.766044	+	0.642788i	$0.777778\pi$
$258$		0			0
$259$		0			0
$260$	−0.532089			−0.532089
$261$		0			0
$262$	−1.00000			−1.00000
$263$	0.766044	+	1.32683i	0.766044	+	1.32683i	0.939693	+	0.342020i	$0.111111\pi$
$263$	0.766044	+	1.32683i	0.766044	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$264$		0			0
$265$	0.326352	−	0.565258i	0.326352	−	0.565258i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$271$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.152704	+	0.264490i	−0.152704	+	0.264490i
$276$		0			0
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.766044	−	0.642788i	$-0.777778\pi$
$281$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$287$		0			0
$288$		0			0
$289$	1.00000			1.00000
$290$	0.173648	+	0.300767i	0.173648	+	0.300767i
$291$		0			0
$292$		0			0
$293$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$293$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	1.53209			1.53209
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.500000	+	0.866025i	0.500000	+	0.866025i
$305$		0			0
$306$		0			0
$307$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$307$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$308$		0			0
$309$		0			0
$310$	−0.266044	+	0.460802i	−0.266044	+	0.460802i
$311$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$311$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$312$		0			0
$313$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$313$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$314$		0			0
$315$		0			0
$316$	1.53209			1.53209
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$320$	0.173648	−	0.300767i	0.173648	−	0.300767i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.34730			−1.34730
$326$	0.766044	+	1.32683i	0.766044	+	1.32683i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$331$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$337$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$338$	0.673648	−	1.16679i	0.673648	−	1.16679i
$339$		0			0
$340$		0			0
$341$	−0.532089			−0.532089
$342$		0			0
$343$		0			0
$344$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$345$		0			0
$346$	0.500000	−	0.866025i	0.500000	−	0.866025i
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	0.766044	−	0.642788i	$-0.222222\pi$
$349$	−0.173648	−	0.300767i	−0.173648	−	0.300767i	−0.939693	+	0.342020i	$0.888889\pi$
$350$		0			0
$351$		0			0
$352$	0.347296			0.347296
$353$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$353$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.347296			−0.347296			−0.173648	−	0.984808i	$-0.555556\pi$
$359$	−0.347296			−0.347296			−0.173648	+	0.984808i	$0.555556\pi$
$360$		0			0
$361$		0			0
$362$	0.766044	+	1.32683i	0.766044	+	1.32683i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$367$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$373$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$374$		0			0
$375$		0			0
$376$	0.939693	+	1.62760i	0.939693	+	1.62760i
$377$	−1.53209			−1.53209
$378$		0			0
$379$	2.00000			2.00000			1.00000			$0$
$379$	2.00000			2.00000			1.00000			$0$
$380$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$381$		0			0
$382$	0.500000	−	0.866025i	0.500000	−	0.866025i
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	0.500000	−	0.866025i	0.500000	−	0.866025i
$393$		0			0
$394$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$395$	−0.532089			−0.532089
$396$		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
$400$	0.439693	−	0.761570i	0.439693	−	0.761570i
$401$	0.766044	−	1.32683i	0.766044	−	1.32683i	−0.173648	−	0.984808i	$-0.555556\pi$
$401$	0.766044	−	1.32683i	0.766044	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$402$		0			0
$403$	−1.17365	−	2.03282i	−1.17365	−	2.03282i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.766044	+	1.32683i	0.766044	+	1.32683i
$417$		0			0
$418$	0.173648	−	0.300767i	0.173648	−	0.300767i
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$	1.87939			1.87939
$423$		0			0
$424$	−1.87939			−1.87939
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	0.326352	+	0.565258i	0.326352	+	0.565258i
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$	−0.120615			−0.120615
$441$		0			0
$442$		0			0
$443$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$443$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$444$		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			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$457$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$		0			0
$460$		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	$0.666667\pi$
$464$	0.500000	−	0.866025i	0.500000	−	0.866025i
$465$		0			0
$466$	0.939693	+	1.62760i	0.939693	+	1.62760i
$467$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$467$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$468$		0			0
$469$		0			0
$470$	−0.326352	−	0.565258i	−0.326352	−	0.565258i
$471$		0			0
$472$		0			0
$473$	−0.326352	+	0.565258i	−0.326352	+	0.565258i
$474$		0			0
$475$	−0.439693	−	0.761570i	−0.439693	−	0.761570i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.766044	−	0.642788i	$-0.777778\pi$
$479$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$480$		0			0
$481$		0			0
$482$	0.766044	−	1.32683i	0.766044	−	1.32683i
$483$		0			0
$484$	0.439693	+	0.761570i	0.439693	+	0.761570i
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$491$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.173648	+	0.984808i	$0.555556\pi$
$491$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.766044	+	0.642788i	$0.777778\pi$
$492$		0			0
$493$		0			0
$494$	1.53209			1.53209
$495$		0			0
$496$	1.53209			1.53209
$497$		0			0
$498$		0			0
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	−0.326352	+	0.565258i	−0.326352	+	0.565258i
$501$		0			0
$502$	0.939693	+	1.62760i	0.939693	+	1.62760i
$503$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$503$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	0.500000	−	0.866025i	0.500000	−	0.866025i
$509$	0.173648	−	0.300767i	0.173648	−	0.300767i	−0.766044	−	0.642788i	$-0.777778\pi$
$509$	0.173648	−	0.300767i	0.173648	−	0.300767i	0.939693	+	0.342020i	$0.111111\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$		0			0
$514$	−1.87939			−1.87939
$515$		0			0
$516$		0			0
$517$	0.326352	−	0.565258i	0.326352	−	0.565258i
$518$		0			0
$519$		0			0
$520$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$521$	−0.347296			−0.347296			−0.173648	−	0.984808i	$-0.555556\pi$
$521$	−0.347296			−0.347296			−0.173648	+	0.984808i	$0.555556\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$525$		0			0
$526$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$527$		0			0
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$	0.652704			0.652704
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.347296			−0.347296
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$	0.173648	+	0.300767i	0.173648	+	0.300767i
$543$		0			0
$544$		0			0
$545$	0.0603074	−	0.104455i	0.0603074	−	0.104455i
$546$		0			0
$547$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$547$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$548$		0			0
$549$		0			0
$550$	−0.305407			−0.305407
$551$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$552$		0			0
$553$		0			0
$554$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$555$		0			0
$556$		0			0
$557$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$557$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$	−2.87939			−2.87939
$560$		0			0
$561$		0			0
$562$	0.939693	−	1.62760i	0.939693	−	1.62760i
$563$	0.173648	−	0.300767i	0.173648	−	0.300767i	−0.766044	−	0.642788i	$-0.777778\pi$
$563$	0.173648	−	0.300767i	0.173648	−	0.300767i	0.939693	+	0.342020i	$0.111111\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$	0.266044	−	0.460802i	0.266044	−	0.460802i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.500000	+	0.866025i	0.500000	+	0.866025i
$579$		0			0
$580$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$581$		0			0
$582$		0			0
$583$	0.326352	+	0.565258i	0.326352	+	0.565258i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$	0.766044	−	1.32683i	0.766044	−	1.32683i
$590$		0			0
$591$		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
$596$	0.766044	+	1.32683i	0.766044	+	1.32683i
$597$		0			0
$598$		0			0
$599$	0.766044	−	1.32683i	0.766044	−	1.32683i	−0.173648	−	0.984808i	$-0.555556\pi$
$599$	0.766044	−	1.32683i	0.766044	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$600$		0			0
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.152704	−	0.264490i	−0.152704	−	0.264490i
$606$		0			0
$607$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$607$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$608$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$609$		0			0
$610$		0			0
$611$	2.87939			2.87939
$612$		0			0
$613$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$613$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$614$	0.766044	+	1.32683i	0.766044	+	1.32683i
$615$		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
$619$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$619$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$620$	−0.532089			−0.532089
$621$		0			0
$622$	−1.00000			−1.00000
$623$		0			0
$624$		0			0
$625$	−0.326352	+	0.565258i	−0.326352	+	0.565258i
$626$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	0.766044	+	1.32683i	0.766044	+	1.32683i
$633$		0			0
$634$		0			0
$635$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$636$		0			0
$637$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$638$	−0.347296			−0.347296
$639$		0			0
$640$	0.347296			0.347296
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$643$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$	−0.673648	−	1.16679i	−0.673648	−	1.16679i
$651$		0			0
$652$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$	0.173648	+	0.300767i	0.173648	+	0.300767i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$659$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$660$		0			0
$661$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$662$	0.173648	−	0.300767i	0.173648	−	0.300767i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$673$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$674$		0			0
$675$		0			0
$676$	1.34730			1.34730
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.939693	−	1.62760i	0.939693	−	1.62760i
$689$	−1.43969	+	2.49362i	−1.43969	+	2.49362i
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	1.00000			1.00000
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	0.173648	−	0.300767i	0.173648	−	0.300767i
$699$		0			0
$700$		0			0
$701$	−0.347296			−0.347296			−0.173648	−	0.984808i	$-0.555556\pi$
$701$	−0.347296			−0.347296			−0.173648	+	0.984808i	$0.555556\pi$
$702$		0			0
$703$		0			0
$704$	0.173648	+	0.300767i	0.173648	+	0.300767i
$705$		0			0
$706$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$707$		0			0
$708$		0			0
$709$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$709$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−0.0923963	+	0.160035i	−0.0923963	+	0.160035i
$716$		0			0
$717$		0			0
$718$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$725$	−0.439693	+	0.761570i	−0.439693	+	0.761570i
$726$		0			0
$727$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$727$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$739$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$743$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$744$		0			0
$745$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$746$	−1.53209			−1.53209
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$751$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$752$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$753$		0			0
$754$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	1.00000	+	1.73205i	1.00000	+	1.73205i
$759$		0			0
$760$	0.173648	−	0.300767i	0.173648	−	0.300767i
$761$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$761$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$	1.00000			1.00000
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$	−1.34730			−1.34730
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.00000			1.00000
$785$		0			0
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	1.00000	−	1.73205i	1.00000	−	1.73205i
$789$		0			0
$790$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$795$		0			0
$796$		0			0
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$		0			0
$800$	0.879385			0.879385
$801$		0			0
$802$	1.53209			1.53209
$803$		0			0
$804$		0			0
$805$		0			0
$806$	1.17365	−	2.03282i	1.17365	−	2.03282i
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	0.266044	−	0.460802i	0.266044	−	0.460802i
$816$		0			0
$817$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.766044	−	0.642788i	$-0.777778\pi$
$821$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$822$		0			0
$823$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$823$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.53209			−1.53209			−0.766044	−	0.642788i	$-0.777778\pi$
$827$	−1.53209			−1.53209			−0.766044	+	0.642788i	$0.777778\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	0.347296			0.347296
$837$		0			0
$838$		0			0
$839$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.766044	−	0.642788i	$-0.777778\pi$
$839$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$840$		0			0
$841$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$	0.939693	+	1.62760i	0.939693	+	1.62760i
$845$	−0.467911			−0.467911
$846$		0			0
$847$		0			0
$848$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$857$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$858$		0			0
$859$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	−0.939693	−	0.342020i	$-0.888889\pi$
$859$	−0.173648	+	0.300767i	−0.173648	+	0.300767i	0.766044	+	0.642788i	$0.222222\pi$
$860$	−0.326352	+	0.565258i	−0.326352	+	0.565258i
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	−0.347296			−0.347296
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.266044	−	0.460802i	0.266044	−	0.460802i
$870$		0			0
$871$		0			0
$872$	−0.347296			−0.347296
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$877$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$878$		0			0
$879$		0			0
$880$	−0.0603074	−	0.104455i	−0.0603074	−	0.104455i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	0.500000	−	0.866025i	0.500000	−	0.866025i
$887$	0.173648	−	0.300767i	0.173648	−	0.300767i	−0.766044	−	0.642788i	$-0.777778\pi$
$887$	0.173648	−	0.300767i	0.173648	−	0.300767i	0.939693	+	0.342020i	$0.111111\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	0.939693	+	1.62760i	0.939693	+	1.62760i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.53209			−1.53209
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.266044	−	0.460802i	0.266044	−	0.460802i
$906$		0			0
$907$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$907$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$911$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$912$		0			0
$913$		0			0
$914$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	1.00000			1.00000
$929$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$929$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$930$		0			0
$931$	0.500000	−	0.866025i	0.500000	−	0.866025i
$932$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$933$		0			0
$934$	0.939693	+	1.62760i	0.939693	+	1.62760i
$935$		0			0
$936$		0			0
$937$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$937$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$938$		0			0
$939$		0			0
$940$	0.326352	−	0.565258i	0.326352	−	0.565258i
$941$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.173648	+	0.984808i	$0.555556\pi$
$941$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.766044	+	0.642788i	$0.777778\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−0.652704			−0.652704
$947$	0.766044	+	1.32683i	0.766044	+	1.32683i	0.939693	+	0.342020i	$0.111111\pi$
$947$	0.766044	+	1.32683i	0.766044	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$948$		0			0
$949$		0			0
$950$	0.439693	−	0.761570i	0.439693	−	0.761570i
$951$		0			0
$952$		0			0
$953$	−1.53209			−1.53209			−0.766044	−	0.642788i	$-0.777778\pi$
$953$	−1.53209			−1.53209			−0.766044	+	0.642788i	$0.777778\pi$
$954$		0			0
$955$	−0.347296			−0.347296
$956$		0			0
$957$		0			0
$958$	0.939693	−	1.62760i	0.939693	−	1.62760i
$959$		0			0
$960$		0			0
$961$	−0.673648	−	1.16679i	−0.673648	−	1.16679i
$962$		0			0
$963$		0			0
$964$	1.53209			1.53209
$965$		0			0
$966$		0			0
$967$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$967$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$968$	−0.439693	+	0.761570i	−0.439693	+	0.761570i
$969$		0			0
$970$		0			0
$971$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$971$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.766044	−	1.32683i	0.766044	−	1.32683i	−0.173648	−	0.984808i	$-0.555556\pi$
$977$	0.766044	−	1.32683i	0.766044	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$978$		0			0
$979$		0			0
$980$	−0.347296			−0.347296
$981$		0			0
$982$	−1.87939			−1.87939
$983$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$983$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$984$		0			0
$985$	−0.347296	+	0.601535i	−0.347296	+	0.601535i
$986$		0			0
$987$		0			0
$988$	0.766044	+	1.32683i	0.766044	+	1.32683i
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	0.766044	+	1.32683i	0.766044	+	1.32683i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		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	3132.1.o.b.2899.2		6
3.2	odd	2		1044.1.o.a.463.1	yes	6
4.3	odd	2		3132.1.o.a.2899.2		6
9.2	odd	6		1044.1.o.a.115.1	✓	6
9.7	even	3	inner	3132.1.o.b.1855.2		6
12.11	even	2		1044.1.o.b.463.3	yes	6
29.28	even	2		3132.1.o.a.2899.2		6
36.7	odd	6		3132.1.o.a.1855.2		6
36.11	even	6		1044.1.o.b.115.3	yes	6
87.86	odd	2		1044.1.o.b.463.3	yes	6
116.115	odd	2	CM	3132.1.o.b.2899.2		6
261.115	even	6		3132.1.o.a.1855.2		6
261.173	odd	6		1044.1.o.b.115.3	yes	6
348.347	even	2		1044.1.o.a.463.1	yes	6
1044.115	odd	6	inner	3132.1.o.b.1855.2		6
1044.695	even	6		1044.1.o.a.115.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1044.1.o.a.115.1	✓	6	9.2	odd	6
1044.1.o.a.115.1	✓	6	1044.695	even	6
1044.1.o.a.463.1	yes	6	3.2	odd	2
1044.1.o.a.463.1	yes	6	348.347	even	2
1044.1.o.b.115.3	yes	6	36.11	even	6
1044.1.o.b.115.3	yes	6	261.173	odd	6
1044.1.o.b.463.3	yes	6	12.11	even	2
1044.1.o.b.463.3	yes	6	87.86	odd	2
3132.1.o.a.1855.2		6	36.7	odd	6
3132.1.o.a.1855.2		6	261.115	even	6
3132.1.o.a.2899.2		6	4.3	odd	2
3132.1.o.a.2899.2		6	29.28	even	2
3132.1.o.b.1855.2		6	9.7	even	3	inner
3132.1.o.b.1855.2		6	1044.115	odd	6	inner
3132.1.o.b.2899.2		6	1.1	even	1	trivial
3132.1.o.b.2899.2		6	116.115	odd	2	CM

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:	\( N \)	\(=\)	\( 3132 = 2^{2} \cdot 3^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3132.o (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.173648 - 0.300767i) q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.173648 - 0.300767i) q^{5} -1.00000 q^{8} +0.347296 q^{10} +(0.173648 + 0.300767i) q^{11} +(-0.766044 + 1.32683i) q^{13} +(-0.500000 - 0.866025i) q^{16} -1.00000 q^{19} +(0.173648 + 0.300767i) q^{20} +(-0.173648 + 0.300767i) q^{22} +(0.439693 + 0.761570i) q^{25} -1.53209 q^{26} +(0.500000 + 0.866025i) q^{29} +(-0.766044 + 1.32683i) q^{31} +(0.500000 - 0.866025i) q^{32} +(-0.500000 - 0.866025i) q^{38} +(-0.173648 + 0.300767i) q^{40} +(0.939693 + 1.62760i) q^{43} -0.347296 q^{44} +(-0.939693 - 1.62760i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(-0.439693 + 0.761570i) q^{50} +(-0.766044 - 1.32683i) q^{52} +1.87939 q^{53} +0.120615 q^{55} +(-0.500000 + 0.866025i) q^{58} -1.53209 q^{62} +1.00000 q^{64} +(0.266044 + 0.460802i) q^{65} +(0.500000 - 0.866025i) q^{76} +(-0.766044 - 1.32683i) q^{79} -0.347296 q^{80} +(-0.939693 + 1.62760i) q^{86} +(-0.173648 - 0.300767i) q^{88} +(0.939693 - 1.62760i) q^{94} +(-0.173648 + 0.300767i) q^{95} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{2} - 3 q^{4} - 6 q^{8}+O(q^{10}) \) 6 * q + 3 * q^2 - 3 * q^4 - 6 * q^8	\( 6 q + 3 q^{2} - 3 q^{4} - 6 q^{8} - 3 q^{16} - 6 q^{19} - 3 q^{25} + 3 q^{29} + 3 q^{32} - 3 q^{38} - 3 q^{49} + 3 q^{50} + 12 q^{55} - 3 q^{58} + 6 q^{64} - 3 q^{65} + 3 q^{76} - 6 q^{98}+O(q^{100}) \) 6 * q + 3 * q^2 - 3 * q^4 - 6 * q^8 - 3 * q^16 - 6 * q^19 - 3 * q^25 + 3 * q^29 + 3 * q^32 - 3 * q^38 - 3 * q^49 + 3 * q^50 + 12 * q^55 - 3 * q^58 + 6 * q^64 - 3 * q^65 + 3 * q^76 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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