LMFDB - Embedded newform 2916.1.j.f.163.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2916,1,Mod(163,2916)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2916, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 16]))

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

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

chi := DirichletCharacter("2916.163");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2916 = 2^{2} \cdot 3^{6} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2916.j (of order $18$, degree $6$, 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:	$1.45527357684$
Analytic rank:	$0$
Dimension:	$6$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{9}$
Projective field:	Galois closure of 9.1.8033551259904.2

Embedding invariants

Embedding label			163.1
Root			$-0.766044 - 0.642788i$ of defining polynomial
Character	$\chi$	$=$	2916.163
Dual form			2916.1.j.f.2755.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.939693 - 0.342020i) q^{2} +(0.766044 - 0.642788i) q^{4} +(0.326352 + 1.85083i) q^{5} +(0.500000 - 0.866025i) q^{8} +O(q^{10})$	$q+(0.939693 - 0.342020i) q^{2} +(0.766044 - 0.642788i) q^{4} +(0.326352 + 1.85083i) q^{5} +(0.500000 - 0.866025i) q^{8} +(0.939693 + 1.62760i) q^{10} +(-0.326352 - 0.118782i) q^{13} +(0.173648 - 0.984808i) q^{16} +(0.766044 + 1.32683i) q^{17} +(1.43969 + 1.20805i) q^{20} +(-2.37939 + 0.866025i) q^{25} -0.347296 q^{26} +(0.326352 - 0.118782i) q^{29} +(-0.173648 - 0.984808i) q^{32} +(1.17365 + 0.984808i) q^{34} +(-0.766044 - 1.32683i) q^{37} +(1.76604 + 0.642788i) q^{40} +(-0.939693 - 0.342020i) q^{41} +(0.173648 + 0.984808i) q^{49} +(-1.93969 + 1.62760i) q^{50} +(-0.326352 + 0.118782i) q^{52} +1.00000 q^{53} +(0.266044 - 0.223238i) q^{58} +(0.266044 + 0.223238i) q^{61} +(-0.500000 - 0.866025i) q^{64} +(0.113341 - 0.642788i) q^{65} +(1.43969 + 0.524005i) q^{68} +(0.939693 - 1.62760i) q^{73} +(-1.17365 - 0.984808i) q^{74} +1.87939 q^{80} -1.00000 q^{82} +(-2.20574 + 1.85083i) q^{85} +(0.766044 - 1.32683i) q^{89} +(-0.173648 + 0.984808i) q^{97} +(0.500000 + 0.866025i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{5} + 3 q^{8}+O(q^{10}) $ 6 * q + 3 * q^5 + 3 * q^8	$ 6 q + 3 q^{5} + 3 q^{8} - 3 q^{13} + 3 q^{20} - 3 q^{25} + 3 q^{29} + 6 q^{34} + 6 q^{40} - 6 q^{50} - 3 q^{52} + 6 q^{53} - 3 q^{58} - 3 q^{61} - 3 q^{64} - 6 q^{65} + 3 q^{68} - 6 q^{74} - 6 q^{82} - 3 q^{85} + 3 q^{98}+O(q^{100}) $ 6 * q + 3 * q^5 + 3 * q^8 - 3 * q^13 + 3 * q^20 - 3 * q^25 + 3 * q^29 + 6 * q^34 + 6 * q^40 - 6 * q^50 - 3 * q^52 + 6 * q^53 - 3 * q^58 - 3 * q^61 - 3 * q^64 - 6 * q^65 + 3 * q^68 - 6 * q^74 - 6 * q^82 - 3 * q^85 + 3 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2916\mathbb{Z}\right)^\times$.

$n$	$1459$	$2189$
$\chi(n)$	$-1$	$e\left(\frac{8}{9}\right)$

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 $

$2$	0.939693	−	0.342020i	0.939693	−	0.342020i
$2$	0.939693	−	0.342020i	0.939693	−	0.342020i
$3$		0			0
$3$		0			0
$4$	0.766044	−	0.642788i	0.766044	−	0.642788i
$5$	0.326352	+	1.85083i	0.326352	+	1.85083i	0.500000	+	0.866025i	$0.333333\pi$
$5$	0.326352	+	1.85083i	0.326352	+	1.85083i	−0.173648	+	0.984808i	$0.555556\pi$
$6$		0			0
$7$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$7$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$8$	0.500000	−	0.866025i	0.500000	−	0.866025i
$9$		0			0
$10$	0.939693	+	1.62760i	0.939693	+	1.62760i
$11$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$11$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$12$		0			0
$13$	−0.326352	−	0.118782i	−0.326352	−	0.118782i	0.173648	−	0.984808i	$-0.444444\pi$
$13$	−0.326352	−	0.118782i	−0.326352	−	0.118782i	−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$		0			0
$16$	0.173648	−	0.984808i	0.173648	−	0.984808i
$17$	0.766044	+	1.32683i	0.766044	+	1.32683i	0.939693	+	0.342020i	$0.111111\pi$
$17$	0.766044	+	1.32683i	0.766044	+	1.32683i	−0.173648	+	0.984808i	$0.555556\pi$
$18$		0			0
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$	1.43969	+	1.20805i	1.43969	+	1.20805i
$21$		0			0
$22$		0			0
$23$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$23$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$24$		0			0
$25$	−2.37939	+	0.866025i	−2.37939	+	0.866025i
$26$	−0.347296			−0.347296
$27$		0			0
$28$		0			0
$29$	0.326352	−	0.118782i	0.326352	−	0.118782i	−0.173648	−	0.984808i	$-0.555556\pi$
$29$	0.326352	−	0.118782i	0.326352	−	0.118782i	0.500000	+	0.866025i	$0.333333\pi$
$30$		0			0
$31$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$31$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$32$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$33$		0			0
$34$	1.17365	+	0.984808i	1.17365	+	0.984808i
$35$		0			0
$36$		0			0
$37$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	−0.939693	−	0.342020i	$-0.888889\pi$
$37$	−0.766044	−	1.32683i	−0.766044	−	1.32683i	0.173648	−	0.984808i	$-0.444444\pi$
$38$		0			0
$39$		0			0
$40$	1.76604	+	0.642788i	1.76604	+	0.642788i
$41$	−0.939693	−	0.342020i	−0.939693	−	0.342020i	−0.173648	−	0.984808i	$-0.555556\pi$
$41$	−0.939693	−	0.342020i	−0.939693	−	0.342020i	−0.766044	+	0.642788i	$0.777778\pi$
$42$		0			0
$43$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$43$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$47$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$48$		0			0
$49$	0.173648	+	0.984808i	0.173648	+	0.984808i
$50$	−1.93969	+	1.62760i	−1.93969	+	1.62760i
$51$		0			0
$52$	−0.326352	+	0.118782i	−0.326352	+	0.118782i
$53$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$53$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.266044	−	0.223238i	0.266044	−	0.223238i
$59$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$59$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$60$		0			0
$61$	0.266044	+	0.223238i	0.266044	+	0.223238i	0.766044	−	0.642788i	$-0.222222\pi$
$61$	0.266044	+	0.223238i	0.266044	+	0.223238i	−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$65$	0.113341	−	0.642788i	0.113341	−	0.642788i
$66$		0			0
$67$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$67$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$68$	1.43969	+	0.524005i	1.43969	+	0.524005i
$69$		0			0
$70$		0			0
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$		0			0
$73$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$73$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$74$	−1.17365	−	0.984808i	−1.17365	−	0.984808i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$79$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$80$	1.87939			1.87939
$81$		0			0
$82$	−1.00000			−1.00000
$83$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$83$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$84$		0			0
$85$	−2.20574	+	1.85083i	−2.20574	+	1.85083i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.766044	−	1.32683i	0.766044	−	1.32683i	−0.173648	−	0.984808i	$-0.555556\pi$
$89$	0.766044	−	1.32683i	0.766044	−	1.32683i	0.939693	−	0.342020i	$-0.111111\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$97$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$98$	0.500000	+	0.866025i	0.500000	+	0.866025i
$99$		0			0
$100$	−1.26604	+	2.19285i	−1.26604	+	2.19285i
$101$	0.766044	+	0.642788i	0.766044	+	0.642788i	0.939693	−	0.342020i	$-0.111111\pi$
$101$	0.766044	+	0.642788i	0.766044	+	0.642788i	−0.173648	+	0.984808i	$0.555556\pi$
$102$		0			0
$103$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$103$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$104$	−0.266044	+	0.223238i	−0.266044	+	0.223238i
$105$		0			0
$106$	0.939693	−	0.342020i	0.939693	−	0.342020i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−1.87939			−1.87939			−0.939693	−	0.342020i	$-0.888889\pi$
$109$	−1.87939			−1.87939			−0.939693	+	0.342020i	$0.888889\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i	0.939693	−	0.342020i	$-0.111111\pi$
$113$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i	−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$	0.173648	−	0.300767i	0.173648	−	0.300767i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$122$	0.326352	+	0.118782i	0.326352	+	0.118782i
$123$		0			0
$124$		0			0
$125$	−1.43969	−	2.49362i	−1.43969	−	2.49362i
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$128$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$129$		0			0
$130$	−0.113341	−	0.642788i	−0.113341	−	0.642788i
$131$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$131$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.53209			1.53209
$137$	1.43969	−	0.524005i	1.43969	−	0.524005i	0.500000	−	0.866025i	$-0.333333\pi$
$137$	1.43969	−	0.524005i	1.43969	−	0.524005i	0.939693	+	0.342020i	$0.111111\pi$
$138$		0			0
$139$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$139$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	0.326352	+	0.565258i	0.326352	+	0.565258i
$146$	0.326352	−	1.85083i	0.326352	−	1.85083i
$147$		0			0
$148$	−1.43969	−	0.524005i	−1.43969	−	0.524005i
$149$	−1.76604	−	0.642788i	−1.76604	−	0.642788i	−0.766044	−	0.642788i	$-0.777778\pi$
$149$	−1.76604	−	0.642788i	−1.76604	−	0.642788i	−1.00000			$\pi$
$150$		0			0
$151$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$151$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.326352	−	1.85083i	−0.326352	−	1.85083i	−0.500000	−	0.866025i	$-0.666667\pi$
$157$	−0.326352	−	1.85083i	−0.326352	−	1.85083i	0.173648	−	0.984808i	$-0.444444\pi$
$158$		0			0
$159$		0			0
$160$	1.76604	−	0.642788i	1.76604	−	0.642788i
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$165$		0			0
$166$		0			0
$167$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$167$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$168$		0			0
$169$	−0.673648	−	0.565258i	−0.673648	−	0.565258i
$170$	−1.43969	+	2.49362i	−1.43969	+	2.49362i
$171$		0			0
$172$		0			0
$173$	−0.266044	+	1.50881i	−0.266044	+	1.50881i	0.500000	+	0.866025i	$0.333333\pi$
$173$	−0.266044	+	1.50881i	−0.266044	+	1.50881i	−0.766044	+	0.642788i	$0.777778\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.266044	−	1.50881i	0.266044	−	1.50881i
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$181$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.20574	−	1.85083i	2.20574	−	1.85083i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$191$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$192$		0			0
$193$	1.17365	−	0.984808i	1.17365	−	0.984808i	0.173648	−	0.984808i	$-0.444444\pi$
$193$	1.17365	−	0.984808i	1.17365	−	0.984808i	1.00000			$0$
$194$	0.173648	+	0.984808i	0.173648	+	0.984808i
$195$		0			0
$196$	0.766044	+	0.642788i	0.766044	+	0.642788i
$197$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.173648	+	0.984808i	$0.555556\pi$
$197$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.766044	+	0.642788i	$0.777778\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	−0.439693	+	2.49362i	−0.439693	+	2.49362i
$201$		0			0
$202$	0.939693	+	0.342020i	0.939693	+	0.342020i
$203$		0			0
$204$		0			0
$205$	0.326352	−	1.85083i	0.326352	−	1.85083i
$206$		0			0
$207$		0			0
$208$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$209$		0			0
$210$		0			0
$211$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$211$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$212$	0.766044	−	0.642788i	0.766044	−	0.642788i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.76604	+	0.642788i	−1.76604	+	0.642788i
$219$		0			0
$220$		0			0
$221$	−0.0923963	−	0.524005i	−0.0923963	−	0.524005i
$222$		0			0
$223$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$223$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$224$		0			0
$225$		0			0
$226$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$227$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$227$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$228$		0			0
$229$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.500000	−	0.866025i	$-0.666667\pi$
$229$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.939693	+	0.342020i	$0.888889\pi$
$230$		0			0
$231$		0			0
$232$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$233$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$233$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$239$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$240$		0			0
$241$	1.76604	−	0.642788i	1.76604	−	0.642788i	0.766044	−	0.642788i	$-0.222222\pi$
$241$	1.76604	−	0.642788i	1.76604	−	0.642788i	1.00000			$0$
$242$	−1.00000			−1.00000
$243$		0			0
$244$	0.347296			0.347296
$245$	−1.76604	+	0.642788i	−1.76604	+	0.642788i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−2.20574	−	1.85083i	−2.20574	−	1.85083i
$251$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$251$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$257$	0.326352	+	0.118782i	0.326352	+	0.118782i	0.500000	−	0.866025i	$-0.333333\pi$
$257$	0.326352	+	0.118782i	0.326352	+	0.118782i	−0.173648	+	0.984808i	$0.555556\pi$
$258$		0			0
$259$		0			0
$260$	−0.326352	−	0.565258i	−0.326352	−	0.565258i
$261$		0			0
$262$		0			0
$263$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$263$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$264$		0			0
$265$	0.326352	+	1.85083i	0.326352	+	1.85083i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.53209			−1.53209			−0.766044	−	0.642788i	$-0.777778\pi$
$269$	−1.53209			−1.53209			−0.766044	+	0.642788i	$0.777778\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	1.43969	−	0.524005i	1.43969	−	0.524005i
$273$		0			0
$274$	1.17365	−	0.984808i	1.17365	−	0.984808i
$275$		0			0
$276$		0			0
$277$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$277$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.326352	−	1.85083i	0.326352	−	1.85083i	−0.173648	−	0.984808i	$-0.555556\pi$
$281$	0.326352	−	1.85083i	0.326352	−	1.85083i	0.500000	−	0.866025i	$-0.333333\pi$
$282$		0			0
$283$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$283$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.673648	+	1.16679i	−0.673648	+	1.16679i
$290$	0.500000	+	0.419550i	0.500000	+	0.419550i
$291$		0			0
$292$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$293$	−0.266044	+	0.223238i	−0.266044	+	0.223238i	−0.766044	−	0.642788i	$-0.777778\pi$
$293$	−0.266044	+	0.223238i	−0.266044	+	0.223238i	0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$	−1.53209			−1.53209
$297$		0			0
$298$	−1.87939			−1.87939
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.326352	+	0.565258i	−0.326352	+	0.565258i
$306$		0			0
$307$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$307$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$311$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$312$		0			0
$313$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	0.173648	+	0.984808i	$0.444444\pi$
$313$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	−0.500000	+	0.866025i	$0.666667\pi$
$314$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$315$		0			0
$316$		0			0
$317$	−0.266044	−	0.223238i	−0.266044	−	0.223238i	0.500000	−	0.866025i	$-0.333333\pi$
$317$	−0.266044	−	0.223238i	−0.266044	−	0.223238i	−0.766044	+	0.642788i	$0.777778\pi$
$318$		0			0
$319$		0			0
$320$	1.43969	−	1.20805i	1.43969	−	1.20805i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	0.879385			0.879385
$326$		0			0
$327$		0			0
$328$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$329$		0			0
$330$		0			0
$331$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$331$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$337$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$338$	−0.826352	−	0.300767i	−0.826352	−	0.300767i
$339$		0			0
$340$	−0.500000	+	2.83564i	−0.500000	+	2.83564i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.266044	+	1.50881i	0.266044	+	1.50881i
$347$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$347$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$348$		0			0
$349$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$349$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.939693	+	0.342020i	−0.939693	+	0.342020i	−0.766044	−	0.642788i	$-0.777778\pi$
$353$	−0.939693	+	0.342020i	−0.939693	+	0.342020i	−0.173648	+	0.984808i	$0.555556\pi$
$354$		0			0
$355$		0			0
$356$	−0.266044	−	1.50881i	−0.266044	−	1.50881i
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$	0.173648	−	0.984808i	0.173648	−	0.984808i
$363$		0			0
$364$		0			0
$365$	3.31908	+	1.20805i	3.31908	+	1.20805i
$366$		0			0
$367$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$367$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$368$		0			0
$369$		0			0
$370$	1.43969	−	2.49362i	1.43969	−	2.49362i
$371$		0			0
$372$		0			0
$373$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$373$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.120615			−0.120615
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$383$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$384$		0			0
$385$		0			0
$386$	0.766044	−	1.32683i	0.766044	−	1.32683i
$387$		0			0
$388$	0.500000	+	0.866025i	0.500000	+	0.866025i
$389$	−0.347296	+	1.96962i	−0.347296	+	1.96962i	−0.173648	+	0.984808i	$0.555556\pi$
$389$	−0.347296	+	1.96962i	−0.347296	+	1.96962i	−0.173648	+	0.984808i	$0.555556\pi$
$390$		0			0
$391$		0			0
$392$	0.939693	+	0.342020i	0.939693	+	0.342020i
$393$		0			0
$394$	−0.326352	+	1.85083i	−0.326352	+	1.85083i
$395$		0			0
$396$		0			0
$397$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	0.173648	+	0.984808i	$0.444444\pi$
$397$	−0.766044	+	1.32683i	−0.766044	+	1.32683i	−0.939693	+	0.342020i	$0.888889\pi$
$398$		0			0
$399$		0			0
$400$	0.439693	+	2.49362i	0.439693	+	2.49362i
$401$	−1.17365	+	0.984808i	−1.17365	+	0.984808i	−0.173648	+	0.984808i	$0.555556\pi$
$401$	−1.17365	+	0.984808i	−1.17365	+	0.984808i	−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	1.00000			1.00000
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.266044	−	0.223238i	0.266044	−	0.223238i	−0.500000	−	0.866025i	$-0.666667\pi$
$409$	0.266044	−	0.223238i	0.266044	−	0.223238i	0.766044	+	0.642788i	$0.222222\pi$
$410$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.0603074	+	0.342020i	−0.0603074	+	0.342020i
$417$		0			0
$418$		0			0
$419$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$419$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$420$		0			0
$421$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	0.173648	+	0.984808i	$0.444444\pi$
$421$	−0.326352	+	1.85083i	−0.326352	+	1.85083i	−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$	0.500000	−	0.866025i	0.500000	−	0.866025i
$425$	−2.97178	−	2.49362i	−2.97178	−	2.49362i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	0.347296			0.347296			0.173648	−	0.984808i	$-0.444444\pi$
$433$	0.347296			0.347296			0.173648	+	0.984808i	$0.444444\pi$
$434$		0			0
$435$		0			0
$436$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$437$		0			0
$438$		0			0
$439$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$439$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$440$		0			0
$441$		0			0
$442$	−0.266044	−	0.460802i	−0.266044	−	0.460802i
$443$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$443$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$444$		0			0
$445$	2.70574	+	0.984808i	2.70574	+	0.984808i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$449$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$	−0.266044	−	0.223238i	−0.266044	−	0.223238i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	−0.500000	−	0.866025i	$-0.666667\pi$
$457$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	0.173648	+	0.984808i	$0.444444\pi$
$458$	−1.53209			−1.53209
$459$		0			0
$460$		0			0
$461$	−0.939693	+	0.342020i	−0.939693	+	0.342020i	−0.766044	−	0.642788i	$-0.777778\pi$
$461$	−0.939693	+	0.342020i	−0.939693	+	0.342020i	−0.173648	+	0.984808i	$0.555556\pi$
$462$		0			0
$463$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$463$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$464$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i
$465$		0			0
$466$	0.266044	+	0.223238i	0.266044	+	0.223238i
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$479$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$480$		0			0
$481$	0.0923963	+	0.524005i	0.0923963	+	0.524005i
$482$	1.43969	−	1.20805i	1.43969	−	1.20805i
$483$		0			0
$484$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$485$	−1.87939			−1.87939
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	0.326352	−	0.118782i	0.326352	−	0.118782i
$489$		0			0
$490$	−1.43969	+	1.20805i	−1.43969	+	1.20805i
$491$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$491$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$492$		0			0
$493$	0.407604	+	0.342020i	0.407604	+	0.342020i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$499$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$500$	−2.70574	−	0.984808i	−2.70574	−	0.984808i
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.53209	+	1.28558i	−1.53209	+	1.28558i	−0.766044	+	0.642788i	$0.777778\pi$
$509$	−1.53209	+	1.28558i	−1.53209	+	1.28558i	−0.766044	+	0.642788i	$0.777778\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$		0			0
$514$	0.347296			0.347296
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.500000	−	0.419550i	−0.500000	−	0.419550i
$521$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$521$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$522$		0			0
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.173648	−	0.984808i	0.173648	−	0.984808i
$530$	0.939693	+	1.62760i	0.939693	+	1.62760i
$531$		0			0
$532$		0			0
$533$	0.266044	+	0.223238i	0.266044	+	0.223238i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$539$		0			0
$540$		0			0
$541$	1.53209			1.53209			0.766044	−	0.642788i	$-0.222222\pi$
$541$	1.53209			1.53209			0.766044	+	0.642788i	$0.222222\pi$
$542$		0			0
$543$		0			0
$544$	1.17365	−	0.984808i	1.17365	−	0.984808i
$545$	−0.613341	−	3.47843i	−0.613341	−	3.47843i
$546$		0			0
$547$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$547$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$548$	0.766044	−	1.32683i	0.766044	−	1.32683i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$555$		0			0
$556$		0			0
$557$	0.173648	+	0.300767i	0.173648	+	0.300767i	0.939693	−	0.342020i	$-0.111111\pi$
$557$	0.173648	+	0.300767i	0.173648	+	0.300767i	−0.766044	+	0.642788i	$0.777778\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.326352	−	1.85083i	−0.326352	−	1.85083i
$563$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$563$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$564$		0			0
$565$	0.613341	−	0.223238i	0.613341	−	0.223238i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.76604	+	0.642788i	−1.76604	+	0.642788i	−0.766044	+	0.642788i	$0.777778\pi$
$569$	−1.76604	+	0.642788i	−1.76604	+	0.642788i	−1.00000			$1.00000\pi$
$570$		0			0
$571$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$571$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.766044	+	0.642788i	$0.222222\pi$
$577$	0.939693	+	1.62760i	0.939693	+	1.62760i	0.173648	+	0.984808i	$0.444444\pi$
$578$	−0.233956	+	1.32683i	−0.233956	+	1.32683i
$579$		0			0
$580$	0.613341	+	0.223238i	0.613341	+	0.223238i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$585$		0			0
$586$	−0.173648	+	0.300767i	−0.173648	+	0.300767i
$587$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$587$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$593$	1.87939			1.87939			0.939693	−	0.342020i	$-0.111111\pi$
$593$	1.87939			1.87939			0.939693	+	0.342020i	$0.111111\pi$
$594$		0			0
$595$		0			0
$596$	−1.76604	+	0.642788i	−1.76604	+	0.642788i
$597$		0			0
$598$		0			0
$599$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$599$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$600$		0			0
$601$	1.17365	+	0.984808i	1.17365	+	0.984808i	1.00000			$0$
$601$	1.17365	+	0.984808i	1.17365	+	0.984808i	0.173648	+	0.984808i	$0.444444\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.326352	−	1.85083i	0.326352	−	1.85083i
$606$		0			0
$607$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$607$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$608$		0			0
$609$		0			0
$610$	−0.113341	+	0.642788i	−0.113341	+	0.642788i
$611$		0			0
$612$		0			0
$613$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$613$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.43969	−	1.20805i	1.43969	−	1.20805i	0.500000	−	0.866025i	$-0.333333\pi$
$617$	1.43969	−	1.20805i	1.43969	−	1.20805i	0.939693	−	0.342020i	$-0.111111\pi$
$618$		0			0
$619$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$619$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	2.20574	−	1.85083i	2.20574	−	1.85083i
$626$	0.326352	+	1.85083i	0.326352	+	1.85083i
$627$		0			0
$628$	−1.43969	−	1.20805i	−1.43969	−	1.20805i
$629$	1.17365	−	2.03282i	1.17365	−	2.03282i
$630$		0			0
$631$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$631$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$		0			0
$634$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$635$		0			0
$636$		0			0
$637$	0.0603074	−	0.342020i	0.0603074	−	0.342020i
$638$		0			0
$639$		0			0
$640$	0.939693	−	1.62760i	0.939693	−	1.62760i
$641$	1.43969	+	1.20805i	1.43969	+	1.20805i	0.939693	+	0.342020i	$0.111111\pi$
$641$	1.43969	+	1.20805i	1.43969	+	1.20805i	0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$643$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$644$		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
$650$	0.826352	−	0.300767i	0.826352	−	0.300767i
$651$		0			0
$652$		0			0
$653$	0.173648	+	0.984808i	0.173648	+	0.984808i	0.939693	+	0.342020i	$0.111111\pi$
$653$	0.173648	+	0.984808i	0.173648	+	0.984808i	−0.766044	+	0.642788i	$0.777778\pi$
$654$		0			0
$655$		0			0
$656$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$657$		0			0
$658$		0			0
$659$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$659$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$660$		0			0
$661$	1.76604	+	0.642788i	1.76604	+	0.642788i	1.00000			$0$
$661$	1.76604	+	0.642788i	1.76604	+	0.642788i	0.766044	+	0.642788i	$0.222222\pi$
$662$		0			0
$663$		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
$673$	1.76604	−	0.642788i	1.76604	−	0.642788i	0.766044	−	0.642788i	$-0.222222\pi$
$673$	1.76604	−	0.642788i	1.76604	−	0.642788i	1.00000			$0$
$674$	1.00000			1.00000
$675$		0			0
$676$	−0.879385			−0.879385
$677$	1.87939	−	0.684040i	1.87939	−	0.684040i	0.939693	−	0.342020i	$-0.111111\pi$
$677$	1.87939	−	0.684040i	1.87939	−	0.684040i	0.939693	−	0.342020i	$-0.111111\pi$
$678$		0			0
$679$		0			0
$680$	0.500000	+	2.83564i	0.500000	+	2.83564i
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$	1.43969	+	2.49362i	1.43969	+	2.49362i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$690$		0			0
$691$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$691$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$692$	0.766044	+	1.32683i	0.766044	+	1.32683i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.266044	−	1.50881i	−0.266044	−	1.50881i
$698$	0.766044	−	0.642788i	0.766044	−	0.642788i
$699$		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			0
$705$		0			0
$706$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$707$		0			0
$708$		0			0
$709$	−1.43969	−	1.20805i	−1.43969	−	1.20805i	−0.939693	−	0.342020i	$-0.888889\pi$
$709$	−1.43969	−	1.20805i	−1.43969	−	1.20805i	−0.500000	−	0.866025i	$-0.666667\pi$
$710$		0			0
$711$		0			0
$712$	−0.766044	−	1.32683i	−0.766044	−	1.32683i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$723$		0			0
$724$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$725$	−0.673648	+	0.565258i	−0.673648	+	0.565258i
$726$		0			0
$727$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$727$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$728$		0			0
$729$		0			0
$730$	3.53209			3.53209
$731$		0			0
$732$		0			0
$733$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$733$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$	0.500000	−	2.83564i	0.500000	−	2.83564i
$741$		0			0
$742$		0			0
$743$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$743$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$744$		0			0
$745$	0.613341	−	3.47843i	0.613341	−	3.47843i
$746$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$751$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$752$		0			0
$753$		0			0
$754$	−0.113341	+	0.0412527i	−0.113341	+	0.0412527i
$755$		0			0
$756$		0			0
$757$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$757$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i	0.939693	−	0.342020i	$-0.111111\pi$
$761$	−0.0603074	−	0.342020i	−0.0603074	−	0.342020i	−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.500000	−	0.866025i	$-0.666667\pi$
$769$	−1.43969	−	0.524005i	−1.43969	−	0.524005i	−0.939693	+	0.342020i	$0.888889\pi$
$770$		0			0
$771$		0			0
$772$	0.266044	−	1.50881i	0.266044	−	1.50881i
$773$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.766044	−	0.642788i	$-0.777778\pi$
$773$	−0.939693	−	1.62760i	−0.939693	−	1.62760i	−0.173648	−	0.984808i	$-0.555556\pi$
$774$		0			0
$775$		0			0
$776$	0.766044	+	0.642788i	0.766044	+	0.642788i
$777$		0			0
$778$	0.347296	+	1.96962i	0.347296	+	1.96962i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.00000			1.00000
$785$	3.31908	−	1.20805i	3.31908	−	1.20805i
$786$		0			0
$787$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$787$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$788$	0.326352	+	1.85083i	0.326352	+	1.85083i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.0603074	−	0.104455i	−0.0603074	−	0.104455i
$794$	−0.266044	+	1.50881i	−0.266044	+	1.50881i
$795$		0			0
$796$		0			0
$797$	0.326352	+	0.118782i	0.326352	+	0.118782i	0.500000	−	0.866025i	$-0.333333\pi$
$797$	0.326352	+	0.118782i	0.326352	+	0.118782i	−0.173648	+	0.984808i	$0.555556\pi$
$798$		0			0
$799$		0			0
$800$	1.26604	+	2.19285i	1.26604	+	2.19285i
$801$		0			0
$802$	−0.766044	+	1.32683i	−0.766044	+	1.32683i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.939693	−	0.342020i	0.939693	−	0.342020i
$809$	−0.347296			−0.347296			−0.173648	−	0.984808i	$-0.555556\pi$
$809$	−0.347296			−0.347296			−0.173648	+	0.984808i	$0.555556\pi$
$810$		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			0
$816$		0			0
$817$		0			0
$818$	0.173648	−	0.300767i	0.173648	−	0.300767i
$819$		0			0
$820$	−0.939693	−	1.62760i	−0.939693	−	1.62760i
$821$	−0.266044	+	1.50881i	−0.266044	+	1.50881i	0.500000	+	0.866025i	$0.333333\pi$
$821$	−0.266044	+	1.50881i	−0.266044	+	1.50881i	−0.766044	+	0.642788i	$0.777778\pi$
$822$		0			0
$823$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$823$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$827$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$828$		0			0
$829$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$829$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$	0.0603074	+	0.342020i	0.0603074	+	0.342020i
$833$	−1.17365	+	0.984808i	−1.17365	+	0.984808i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$839$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$840$		0			0
$841$	−0.673648	+	0.565258i	−0.673648	+	0.565258i
$842$	0.326352	+	1.85083i	0.326352	+	1.85083i
$843$		0			0
$844$		0			0
$845$	0.826352	−	1.43128i	0.826352	−	1.43128i
$846$		0			0
$847$		0			0
$848$	0.173648	−	0.984808i	0.173648	−	0.984808i
$849$		0			0
$850$	−3.64543	−	1.32683i	−3.64543	−	1.32683i
$851$		0			0
$852$		0			0
$853$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$853$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.43969	+	1.20805i	1.43969	+	1.20805i	0.939693	+	0.342020i	$0.111111\pi$
$857$	1.43969	+	1.20805i	1.43969	+	1.20805i	0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$859$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	−2.87939			−2.87939
$866$	0.326352	−	0.118782i	0.326352	−	0.118782i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.76604	+	0.642788i	1.76604	+	0.642788i	1.00000			$0$
$877$	1.76604	+	0.642788i	1.76604	+	0.642788i	0.766044	+	0.642788i	$0.222222\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$881$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$882$		0			0
$883$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$883$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$884$	−0.407604	−	0.342020i	−0.407604	−	0.342020i
$885$		0			0
$886$		0			0
$887$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$887$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$888$		0			0
$889$		0			0
$890$	2.87939			2.87939
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.53209	+	1.28558i	1.53209	+	1.28558i
$899$		0			0
$900$		0			0
$901$	0.766044	+	1.32683i	0.766044	+	1.32683i
$902$		0			0
$903$		0			0
$904$	−0.326352	−	0.118782i	−0.326352	−	0.118782i
$905$	1.76604	+	0.642788i	1.76604	+	0.642788i
$906$		0			0
$907$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$907$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$911$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$912$		0			0
$913$		0			0
$914$	−0.266044	+	0.223238i	−0.266044	+	0.223238i
$915$		0			0
$916$	−1.43969	+	0.524005i	−1.43969	+	0.524005i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$923$		0			0
$924$		0			0
$925$	2.97178	+	2.49362i	2.97178	+	2.49362i
$926$		0			0
$927$		0			0
$928$	−0.173648	−	0.300767i	−0.173648	−	0.300767i
$929$	−0.0603074	+	0.342020i	−0.0603074	+	0.342020i	0.939693	+	0.342020i	$0.111111\pi$
$929$	−0.0603074	+	0.342020i	−0.0603074	+	0.342020i	−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	0.326352	+	0.118782i	0.326352	+	0.118782i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.173648	−	0.984808i	$-0.444444\pi$
$937$	0.939693	−	1.62760i	0.939693	−	1.62760i	0.766044	−	0.642788i	$-0.222222\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.266044	+	0.223238i	−0.266044	+	0.223238i	−0.766044	−	0.642788i	$-0.777778\pi$
$941$	−0.266044	+	0.223238i	−0.266044	+	0.223238i	0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$947$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$948$		0			0
$949$	−0.500000	+	0.419550i	−0.500000	+	0.419550i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.173648	+	0.984808i	$0.555556\pi$
$953$	−0.939693	+	1.62760i	−0.939693	+	1.62760i	−0.766044	+	0.642788i	$0.777778\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.173648	−	0.984808i	0.173648	−	0.984808i
$962$	0.266044	+	0.460802i	0.266044	+	0.460802i
$963$		0			0
$964$	0.939693	−	1.62760i	0.939693	−	1.62760i
$965$	2.20574	+	1.85083i	2.20574	+	1.85083i
$966$		0			0
$967$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$967$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$968$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$969$		0			0
$970$	−1.76604	+	0.642788i	−1.76604	+	0.642788i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.266044	−	0.223238i	0.266044	−	0.223238i
$977$	−0.347296	−	1.96962i	−0.347296	−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$977$	−0.347296	−	1.96962i	−0.347296	−	1.96962i	−0.173648	−	0.984808i	$-0.555556\pi$
$978$		0			0
$979$		0			0
$980$	−0.939693	+	1.62760i	−0.939693	+	1.62760i
$981$		0			0
$982$		0			0
$983$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$983$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$984$		0			0
$985$	−3.31908	−	1.20805i	−3.31908	−	1.20805i
$986$	0.500000	+	0.181985i	0.500000	+	0.181985i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	−0.500000	−	0.866025i	$-0.666667\pi$
$997$	−0.326352	+	0.118782i	−0.326352	+	0.118782i	0.173648	+	0.984808i	$0.444444\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2916.1.j.f.163.1		6
3.2	odd	2		2916.1.j.b.163.1		6
4.3	odd	2	CM	2916.1.j.f.163.1		6
9.2	odd	6		2916.1.j.h.1135.1		6
9.4	even	3		2916.1.j.g.2107.1		6
9.5	odd	6		2916.1.j.c.2107.1		6
9.7	even	3		2916.1.j.a.1135.1		6
12.11	even	2		2916.1.j.b.163.1		6
27.2	odd	18		2916.1.d.b.1459.1	yes	3
27.4	even	9	inner	2916.1.j.f.2755.1		6
27.5	odd	18		2916.1.j.c.811.1		6
27.7	even	9		2916.1.f.b.487.1		6
27.11	odd	18		2916.1.f.a.2431.3		6
27.13	even	9		2916.1.j.a.1783.1		6
27.14	odd	18		2916.1.j.h.1783.1		6
27.16	even	9		2916.1.f.b.2431.1		6
27.20	odd	18		2916.1.f.a.487.3		6
27.22	even	9		2916.1.j.g.811.1		6
27.23	odd	18		2916.1.j.b.2755.1		6
27.25	even	9		2916.1.d.a.1459.3	✓	3
36.7	odd	6		2916.1.j.a.1135.1		6
36.11	even	6		2916.1.j.h.1135.1		6
36.23	even	6		2916.1.j.c.2107.1		6
36.31	odd	6		2916.1.j.g.2107.1		6
108.7	odd	18		2916.1.f.b.487.1		6
108.11	even	18		2916.1.f.a.2431.3		6
108.23	even	18		2916.1.j.b.2755.1		6
108.31	odd	18	inner	2916.1.j.f.2755.1		6
108.43	odd	18		2916.1.f.b.2431.1		6
108.47	even	18		2916.1.f.a.487.3		6
108.59	even	18		2916.1.j.c.811.1		6
108.67	odd	18		2916.1.j.a.1783.1		6
108.79	odd	18		2916.1.d.a.1459.3	✓	3
108.83	even	18		2916.1.d.b.1459.1	yes	3
108.95	even	18		2916.1.j.h.1783.1		6
108.103	odd	18		2916.1.j.g.811.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2916.1.d.a.1459.3	✓	3	27.25	even	9
2916.1.d.a.1459.3	✓	3	108.79	odd	18
2916.1.d.b.1459.1	yes	3	27.2	odd	18
2916.1.d.b.1459.1	yes	3	108.83	even	18
2916.1.f.a.487.3		6	27.20	odd	18
2916.1.f.a.487.3		6	108.47	even	18
2916.1.f.a.2431.3		6	27.11	odd	18
2916.1.f.a.2431.3		6	108.11	even	18
2916.1.f.b.487.1		6	27.7	even	9
2916.1.f.b.487.1		6	108.7	odd	18
2916.1.f.b.2431.1		6	27.16	even	9
2916.1.f.b.2431.1		6	108.43	odd	18
2916.1.j.a.1135.1		6	9.7	even	3
2916.1.j.a.1135.1		6	36.7	odd	6
2916.1.j.a.1783.1		6	27.13	even	9
2916.1.j.a.1783.1		6	108.67	odd	18
2916.1.j.b.163.1		6	3.2	odd	2
2916.1.j.b.163.1		6	12.11	even	2
2916.1.j.b.2755.1		6	27.23	odd	18
2916.1.j.b.2755.1		6	108.23	even	18
2916.1.j.c.811.1		6	27.5	odd	18
2916.1.j.c.811.1		6	108.59	even	18
2916.1.j.c.2107.1		6	9.5	odd	6
2916.1.j.c.2107.1		6	36.23	even	6
2916.1.j.f.163.1		6	1.1	even	1	trivial
2916.1.j.f.163.1		6	4.3	odd	2	CM
2916.1.j.f.2755.1		6	27.4	even	9	inner
2916.1.j.f.2755.1		6	108.31	odd	18	inner
2916.1.j.g.811.1		6	27.22	even	9
2916.1.j.g.811.1		6	108.103	odd	18
2916.1.j.g.2107.1		6	9.4	even	3
2916.1.j.g.2107.1		6	36.31	odd	6
2916.1.j.h.1135.1		6	9.2	odd	6
2916.1.j.h.1135.1		6	36.11	even	6
2916.1.j.h.1783.1		6	27.14	odd	18
2916.1.j.h.1783.1		6	108.95	even	18

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:	\( N \)	\(=\)	\( 2916 = 2^{2} \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2916.j (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.939693 - 0.342020i) q^{2} +(0.766044 - 0.642788i) q^{4} +(0.326352 + 1.85083i) q^{5} +(0.500000 - 0.866025i) q^{8} +O(q^{10})\)	\(q+(0.939693 - 0.342020i) q^{2} +(0.766044 - 0.642788i) q^{4} +(0.326352 + 1.85083i) q^{5} +(0.500000 - 0.866025i) q^{8} +(0.939693 + 1.62760i) q^{10} +(-0.326352 - 0.118782i) q^{13} +(0.173648 - 0.984808i) q^{16} +(0.766044 + 1.32683i) q^{17} +(1.43969 + 1.20805i) q^{20} +(-2.37939 + 0.866025i) q^{25} -0.347296 q^{26} +(0.326352 - 0.118782i) q^{29} +(-0.173648 - 0.984808i) q^{32} +(1.17365 + 0.984808i) q^{34} +(-0.766044 - 1.32683i) q^{37} +(1.76604 + 0.642788i) q^{40} +(-0.939693 - 0.342020i) q^{41} +(0.173648 + 0.984808i) q^{49} +(-1.93969 + 1.62760i) q^{50} +(-0.326352 + 0.118782i) q^{52} +1.00000 q^{53} +(0.266044 - 0.223238i) q^{58} +(0.266044 + 0.223238i) q^{61} +(-0.500000 - 0.866025i) q^{64} +(0.113341 - 0.642788i) q^{65} +(1.43969 + 0.524005i) q^{68} +(0.939693 - 1.62760i) q^{73} +(-1.17365 - 0.984808i) q^{74} +1.87939 q^{80} -1.00000 q^{82} +(-2.20574 + 1.85083i) q^{85} +(0.766044 - 1.32683i) q^{89} +(-0.173648 + 0.984808i) q^{97} +(0.500000 + 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{5} + 3 q^{8}+O(q^{10}) \) 6 * q + 3 * q^5 + 3 * q^8	\( 6 q + 3 q^{5} + 3 q^{8} - 3 q^{13} + 3 q^{20} - 3 q^{25} + 3 q^{29} + 6 q^{34} + 6 q^{40} - 6 q^{50} - 3 q^{52} + 6 q^{53} - 3 q^{58} - 3 q^{61} - 3 q^{64} - 6 q^{65} + 3 q^{68} - 6 q^{74} - 6 q^{82} - 3 q^{85} + 3 q^{98}+O(q^{100}) \) 6 * q + 3 * q^5 + 3 * q^8 - 3 * q^13 + 3 * q^20 - 3 * q^25 + 3 * q^29 + 6 * q^34 + 6 * q^40 - 6 * q^50 - 3 * q^52 + 6 * q^53 - 3 * q^58 - 3 * q^61 - 3 * q^64 - 6 * q^65 + 3 * q^68 - 6 * q^74 - 6 * q^82 - 3 * q^85 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more