LMFDB - Embedded newform 3645.1.n.d.1619.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3645,1,Mod(404,3645)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3645, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([5, 9]))

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

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

chi := DirichletCharacter("3645.404");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3645 = 3^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3645.n (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:	$1.81909197105$
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:	no (minimal twist has level 135)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.135.1
Artin image:	$S_3\times C_9$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{18} + \cdots)$

Embedding invariants

Embedding label			1619.1
Root			$0.939693 - 0.342020i$ of defining polynomial
Character	$\chi$	$=$	3645.1619
Dual form			3645.1.n.d.2024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.939693 + 0.342020i) q^{2} +(0.173648 - 0.984808i) q^{5} +(-0.500000 - 0.866025i) q^{8} +(0.500000 - 0.866025i) q^{10} +(-0.173648 - 0.984808i) q^{16} +(0.500000 - 0.866025i) q^{17} +(0.500000 + 0.866025i) q^{19} +(-0.766044 - 0.642788i) q^{23} +(-0.939693 - 0.342020i) q^{25} +(-0.766044 - 0.642788i) q^{31} +(0.766044 - 0.642788i) q^{34} +(0.173648 + 0.984808i) q^{38} +(-0.939693 + 0.342020i) q^{40} +(-0.500000 - 0.866025i) q^{46} +(1.53209 - 1.28558i) q^{47} +(0.173648 - 0.984808i) q^{49} +(-0.766044 - 0.642788i) q^{50} -1.00000 q^{53} +(-0.766044 + 0.642788i) q^{61} +(-0.500000 - 0.866025i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(0.939693 + 0.342020i) q^{79} -1.00000 q^{80} +(0.939693 + 0.342020i) q^{83} +(-0.766044 - 0.642788i) q^{85} +(1.87939 - 0.684040i) q^{94} +(0.939693 - 0.342020i) q^{95} +(0.500000 - 0.866025i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{8} + 3 q^{10} + 3 q^{17} + 3 q^{19} - 3 q^{46} - 6 q^{53} - 3 q^{62} - 3 q^{64} - 6 q^{80} + 3 q^{98}+O(q^{100}) $ 6 * q - 3 * q^8 + 3 * q^10 + 3 * q^17 + 3 * q^19 - 3 * q^46 - 6 * q^53 - 3 * q^62 - 3 * q^64 - 6 * q^80 + 3 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3645\mathbb{Z}\right)^\times$.

$n$	$731$	$2917$
$\chi(n)$	$e\left(\frac{11}{18}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$2$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.173648	−	0.984808i	0.173648	−	0.984808i
$5$	0.173648	−	0.984808i	0.173648	−	0.984808i
$6$		0			0
$7$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$7$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$8$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$9$		0			0
$10$	0.500000	−	0.866025i	0.500000	−	0.866025i
$11$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$11$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$12$		0			0
$13$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$13$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$14$		0			0
$15$		0			0
$16$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$17$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$17$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$18$		0			0
$19$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$19$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$23$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$24$		0			0
$25$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$29$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$30$		0			0
$31$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$31$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$32$		0			0
$33$		0			0
$34$	0.766044	−	0.642788i	0.766044	−	0.642788i
$35$		0			0
$36$		0			0
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$	0.173648	+	0.984808i	0.173648	+	0.984808i
$39$		0			0
$40$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$41$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$41$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$42$		0			0
$43$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$43$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$44$		0			0
$45$		0			0
$46$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$47$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$47$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$48$		0			0
$49$	0.173648	−	0.984808i	0.173648	−	0.984808i
$50$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$51$		0			0
$52$		0			0
$53$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$53$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$59$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$60$		0			0
$61$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$61$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$62$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$63$		0			0
$64$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$65$		0			0
$66$		0			0
$67$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$67$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$71$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$72$		0			0
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$79$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$80$	−1.00000			−1.00000
$81$		0			0
$82$		0			0
$83$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$83$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$84$		0			0
$85$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	1.87939	−	0.684040i	1.87939	−	0.684040i
$95$	0.939693	−	0.342020i	0.939693	−	0.342020i
$96$		0			0
$97$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$97$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$98$	0.500000	−	0.866025i	0.500000	−	0.866025i
$99$		0			0
$100$		0			0
$101$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$101$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$102$		0			0
$103$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$103$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$104$		0			0
$105$		0			0
$106$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$107$	2.00000			2.00000			1.00000			$0$
$107$	2.00000			2.00000			1.00000			$0$
$108$		0			0
$109$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$109$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$113$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$114$		0			0
$115$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$122$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$123$		0			0
$124$		0			0
$125$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$126$		0			0
$127$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$127$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$128$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$129$		0			0
$130$		0			0
$131$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$131$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−1.00000			−1.00000
$137$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$137$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$138$		0			0
$139$	1.53209	+	1.28558i	1.53209	+	1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$139$	1.53209	+	1.28558i	1.53209	+	1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$149$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$150$		0			0
$151$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$151$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$152$	0.500000	−	0.866025i	0.500000	−	0.866025i
$153$		0			0
$154$		0			0
$155$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$156$		0			0
$157$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$157$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$158$	0.766044	+	0.642788i	0.766044	+	0.642788i
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	0.766044	+	0.642788i	0.766044	+	0.642788i
$167$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$167$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$168$		0			0
$169$	0.766044	−	0.642788i	0.766044	−	0.642788i
$170$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$171$		0			0
$172$		0			0
$173$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$173$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$181$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	1.00000			1.00000
$191$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$191$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$192$		0			0
$193$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$193$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$197$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$199$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$200$	0.173648	+	0.984808i	0.173648	+	0.984808i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$211$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$212$		0			0
$213$		0			0
$214$	1.87939	+	0.684040i	1.87939	+	0.684040i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$223$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$224$		0			0
$225$		0			0
$226$	1.00000	−	1.73205i	1.00000	−	1.73205i
$227$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$227$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$228$		0			0
$229$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$229$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$230$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$231$		0			0
$232$		0			0
$233$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$233$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$239$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$240$		0			0
$241$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$241$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$242$	−1.00000			−1.00000
$243$		0			0
$244$		0			0
$245$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$246$		0			0
$247$		0			0
$248$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$249$		0			0
$250$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$251$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$257$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$263$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$264$		0			0
$265$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$271$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$272$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$273$		0			0
$274$	0.766044	+	0.642788i	0.766044	+	0.642788i
$275$		0			0
$276$		0			0
$277$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$277$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$278$	1.00000	+	1.73205i	1.00000	+	1.73205i
$279$		0			0
$280$		0			0
$281$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$281$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$282$		0			0
$283$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$283$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		0			0
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$293$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−0.347296	+	1.96962i	−0.347296	+	1.96962i
$303$		0			0
$304$	0.766044	−	0.642788i	0.766044	−	0.642788i
$305$	0.500000	+	0.866025i	0.500000	+	0.866025i
$306$		0			0
$307$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$307$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$308$		0			0
$309$		0			0
$310$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$311$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$311$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$312$		0			0
$313$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$313$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$317$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$318$		0			0
$319$		0			0
$320$	0.766044	+	0.642788i	0.766044	+	0.642788i
$321$		0			0
$322$		0			0
$323$	1.00000			1.00000
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$331$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$332$		0			0
$333$		0			0
$334$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$335$		0			0
$336$		0			0
$337$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$337$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$338$	0.939693	−	0.342020i	0.939693	−	0.342020i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.173648	−	0.984808i	0.173648	−	0.984808i
$347$	1.53209	+	1.28558i	1.53209	+	1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$347$	1.53209	+	1.28558i	1.53209	+	1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$348$		0			0
$349$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$349$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.87939	−	0.684040i	−1.87939	−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$353$	−1.87939	−	0.684040i	−1.87939	−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$		0			0
$362$	0.173648	+	0.984808i	0.173648	+	0.984808i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$367$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$368$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$373$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$374$		0			0
$375$		0			0
$376$	−1.87939	−	0.684040i	−1.87939	−	0.684040i
$377$		0			0
$378$		0			0
$379$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$383$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$389$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$390$		0			0
$391$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$392$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$393$		0			0
$394$	0.173648	+	0.984808i	0.173648	+	0.984808i
$395$	0.500000	−	0.866025i	0.500000	−	0.866025i
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$	−1.53209	+	1.28558i	−1.53209	+	1.28558i
$399$		0			0
$400$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$401$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$401$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$409$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	0.500000	−	0.866025i	0.500000	−	0.866025i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$419$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$420$		0			0
$421$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$421$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$422$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$423$		0			0
$424$	0.500000	+	0.866025i	0.500000	+	0.866025i
$425$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$426$		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			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.173648	−	0.984808i	0.173648	−	0.984808i
$438$		0			0
$439$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$439$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$443$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$449$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$	0.173648	−	0.984808i	0.173648	−	0.984808i
$455$		0			0
$456$		0			0
$457$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$457$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$458$	1.00000			1.00000
$459$		0			0
$460$		0			0
$461$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$461$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$462$		0			0
$463$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$463$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$464$		0			0
$465$		0			0
$466$	−1.53209	+	1.28558i	−1.53209	+	1.28558i
$467$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$467$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$		0			0
$470$	−0.347296	−	1.96962i	−0.347296	−	1.96962i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$479$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$480$		0			0
$481$		0			0
$482$	0.766044	+	0.642788i	0.766044	+	0.642788i
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	0.939693	+	0.342020i	0.939693	+	0.342020i
$489$		0			0
$490$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$491$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$491$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$497$		0			0
$498$		0			0
$499$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$499$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$503$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$509$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			1.00000
$513$		0			0
$514$	1.00000			1.00000
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$	1.87939	−	0.684040i	1.87939	−	0.684040i
$527$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$528$		0			0
$529$		0			0
$530$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.347296	−	1.96962i	0.347296	−	1.96962i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	2.00000			2.00000			1.00000			$0$
$541$	2.00000			2.00000			1.00000			$0$
$542$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$543$		0			0
$544$		0			0
$545$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$546$		0			0
$547$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$547$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$548$		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
$557$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$557$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.53209	+	1.28558i	1.53209	+	1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$563$	1.53209	+	1.28558i	1.53209	+	1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$564$		0			0
$565$	−1.87939	−	0.684040i	−1.87939	−	0.684040i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$569$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$570$		0			0
$571$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$571$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.500000	+	0.866025i	0.500000	+	0.866025i
$576$		0			0
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$587$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$587$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$588$		0			0
$589$	0.173648	−	0.984808i	0.173648	−	0.984808i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$593$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$599$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$600$		0			0
$601$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$601$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.173648	+	0.984808i	0.173648	+	0.984808i
$606$		0			0
$607$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$607$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$608$		0			0
$609$		0			0
$610$	0.173648	+	0.984808i	0.173648	+	0.984808i
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$617$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$618$		0			0
$619$	−1.87939	−	0.684040i	−1.87939	−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$619$	−1.87939	−	0.684040i	−1.87939	−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.766044	+	0.642788i	0.766044	+	0.642788i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$631$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$632$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$633$		0			0
$634$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.500000	+	0.866025i	0.500000	+	0.866025i
$641$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$641$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$642$		0			0
$643$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$643$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$644$		0			0
$645$		0			0
$646$	0.939693	+	0.342020i	0.939693	+	0.342020i
$647$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$653$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$659$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$660$		0			0
$661$	−1.87939	+	0.684040i	−1.87939	+	0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$661$	−1.87939	+	0.684040i	−1.87939	+	0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$662$	1.87939	−	0.684040i	1.87939	−	0.684040i
$663$		0			0
$664$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$665$		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			0		−0.939693	−	0.342020i	$-0.888889\pi$
$673$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.87939	−	0.684040i	−1.87939	−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$677$	−1.87939	−	0.684040i	−1.87939	−	0.684040i	−0.939693	−	0.342020i	$-0.888889\pi$
$678$		0			0
$679$		0			0
$680$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$681$		0			0
$682$		0			0
$683$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$683$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$	0.500000	−	0.866025i	0.500000	−	0.866025i
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$691$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$692$		0			0
$693$		0			0
$694$	1.00000	+	1.73205i	1.00000	+	1.73205i
$695$	1.53209	−	1.28558i	1.53209	−	1.28558i
$696$		0			0
$697$		0			0
$698$	0.766044	+	0.642788i	0.766044	+	0.642788i
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−1.53209	−	1.28558i	−1.53209	−	1.28558i
$707$		0			0
$708$		0			0
$709$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$709$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.173648	+	0.984808i	0.173648	+	0.984808i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$727$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$733$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$739$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.87939	+	0.684040i	−1.87939	+	0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$743$	−1.87939	+	0.684040i	−1.87939	+	0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$751$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$752$	−1.53209	−	1.28558i	−1.53209	−	1.28558i
$753$		0			0
$754$		0			0
$755$	2.00000			2.00000
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$759$		0			0
$760$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$761$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$761$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$767$		0			0
$768$		0			0
$769$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$769$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$773$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$774$		0			0
$775$	0.500000	+	0.866025i	0.500000	+	0.866025i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−1.00000			−1.00000
$783$		0			0
$784$	−1.00000			−1.00000
$785$		0			0
$786$		0			0
$787$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$787$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$788$		0			0
$789$		0			0
$790$	0.766044	−	0.642788i	0.766044	−	0.642788i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.173648	−	0.984808i	$-0.444444\pi$
$797$	0.939693	−	0.342020i	0.939693	−	0.342020i	0.766044	+	0.642788i	$0.222222\pi$
$798$		0			0
$799$	−0.347296	−	1.96962i	−0.347296	−	1.96962i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	2.00000			2.00000			1.00000			$0$
$811$	2.00000			2.00000			1.00000			$0$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$819$		0			0
$820$		0			0
$821$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$821$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$822$		0			0
$823$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$823$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$827$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$828$		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	$-0.666667\pi$
$830$	0.766044	−	0.642788i	0.766044	−	0.642788i
$831$		0			0
$832$		0			0
$833$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$834$		0			0
$835$	0.939693	+	0.342020i	0.939693	+	0.342020i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$839$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$840$		0			0
$841$	0.766044	+	0.642788i	0.766044	+	0.642788i
$842$	0.173648	−	0.984808i	0.173648	−	0.984808i
$843$		0			0
$844$		0			0
$845$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$846$		0			0
$847$		0			0
$848$	0.173648	+	0.984808i	0.173648	+	0.984808i
$849$		0			0
$850$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$851$		0			0
$852$		0			0
$853$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$853$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$854$		0			0
$855$		0			0
$856$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$857$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	−0.939693	−	0.342020i	$-0.888889\pi$
$857$	−0.766044	+	0.642788i	−0.766044	+	0.642788i	0.173648	+	0.984808i	$0.444444\pi$
$858$		0			0
$859$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	0.766044	+	0.642788i	$0.222222\pi$
$859$	−0.173648	+	0.984808i	−0.173648	+	0.984808i	−0.939693	+	0.342020i	$0.888889\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$863$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$	−1.00000			−1.00000
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.500000	+	0.866025i	0.500000	+	0.866025i
$873$		0			0
$874$	0.500000	−	0.866025i	0.500000	−	0.866025i
$875$		0			0
$876$		0			0
$877$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$877$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$878$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$879$		0			0
$880$		0			0
$881$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$881$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$883$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$884$		0			0
$885$		0			0
$886$	0.173648	−	0.984808i	0.173648	−	0.984808i
$887$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	0.173648	−	0.984808i	$-0.444444\pi$
$887$	−0.766044	−	0.642788i	−0.766044	−	0.642788i	−0.939693	+	0.342020i	$0.888889\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	1.87939	+	0.684040i	1.87939	+	0.684040i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$902$		0			0
$903$		0			0
$904$	−1.87939	+	0.684040i	−1.87939	+	0.684040i
$905$	0.939693	−	0.342020i	0.939693	−	0.342020i
$906$		0			0
$907$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$907$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$911$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	2.00000			2.00000			1.00000			$0$
$919$	2.00000			2.00000			1.00000			$0$
$920$	0.939693	+	0.342020i	0.939693	+	0.342020i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$929$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$930$		0			0
$931$	0.939693	−	0.342020i	0.939693	−	0.342020i
$932$		0			0
$933$		0			0
$934$	0.173648	+	0.984808i	0.173648	+	0.984808i
$935$		0			0
$936$		0			0
$937$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$937$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$941$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.766044	−	0.642788i	$-0.222222\pi$
$947$	0.939693	+	0.342020i	0.939693	+	0.342020i	0.173648	+	0.984808i	$0.444444\pi$
$948$		0			0
$949$		0			0
$950$	0.173648	−	0.984808i	0.173648	−	0.984808i
$951$		0			0
$952$		0			0
$953$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$953$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		0			0
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$967$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$968$	0.766044	+	0.642788i	0.766044	+	0.642788i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.766044	+	0.642788i	0.766044	+	0.642788i
$977$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$977$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	−0.939693	−	0.342020i	$-0.888889\pi$
$983$	−0.173648	−	0.984808i	−0.173648	−	0.984808i	0.766044	−	0.642788i	$-0.222222\pi$
$984$		0			0
$985$	0.939693	−	0.342020i	0.939693	−	0.342020i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$991$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.53209	+	1.28558i	1.53209	+	1.28558i
$996$		0			0
$997$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$997$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$998$	1.00000			1.00000
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3645.1.n.d.1619.1		6
3.2	odd	2		3645.1.n.e.1619.1		6
5.4	even	2		3645.1.n.e.1619.1		6
9.2	odd	6		3645.1.n.e.2834.1		6
9.4	even	3	inner	3645.1.n.d.404.1		6
9.5	odd	6		3645.1.n.e.404.1		6
9.7	even	3	inner	3645.1.n.d.2834.1		6
15.14	odd	2	CM	3645.1.n.d.1619.1		6
27.2	odd	18		3645.1.n.e.3239.1		6
27.4	even	9		405.1.h.b.134.1		2
27.5	odd	18		405.1.h.a.269.1		2
27.7	even	9	inner	3645.1.n.d.2024.1		6
27.11	odd	18		3645.1.n.e.809.1		6
27.13	even	9		135.1.d.a.134.1	✓	1
27.14	odd	18		135.1.d.b.134.1	yes	1
27.16	even	9	inner	3645.1.n.d.809.1		6
27.20	odd	18		3645.1.n.e.2024.1		6
27.22	even	9		405.1.h.b.269.1		2
27.23	odd	18		405.1.h.a.134.1		2
27.25	even	9	inner	3645.1.n.d.3239.1		6
45.4	even	6		3645.1.n.e.404.1		6
45.14	odd	6	inner	3645.1.n.d.404.1		6
45.29	odd	6	inner	3645.1.n.d.2834.1		6
45.34	even	6		3645.1.n.e.2834.1		6
108.67	odd	18		2160.1.c.b.1889.1		1
108.95	even	18		2160.1.c.a.1889.1		1
135.4	even	18		405.1.h.a.134.1		2
135.13	odd	36		675.1.c.c.26.2		2
135.14	odd	18		135.1.d.a.134.1	✓	1
135.22	odd	36		2025.1.j.c.26.2		4
135.23	even	36		2025.1.j.c.701.1		4
135.29	odd	18	inner	3645.1.n.d.3239.1		6
135.32	even	36		2025.1.j.c.26.1		4
135.34	even	18		3645.1.n.e.2024.1		6
135.49	even	18		405.1.h.a.269.1		2
135.58	odd	36		2025.1.j.c.701.2		4
135.59	odd	18		405.1.h.b.269.1		2
135.67	odd	36		675.1.c.c.26.1		2
135.68	even	36		675.1.c.c.26.1		2
135.74	odd	18	inner	3645.1.n.d.2024.1		6
135.77	even	36		2025.1.j.c.701.2		4
135.79	even	18		3645.1.n.e.3239.1		6
135.94	even	18		135.1.d.b.134.1	yes	1
135.103	odd	36		2025.1.j.c.26.1		4
135.104	odd	18		405.1.h.b.134.1		2
135.112	odd	36		2025.1.j.c.701.1		4
135.113	even	36		2025.1.j.c.26.2		4
135.119	odd	18	inner	3645.1.n.d.809.1		6
135.122	even	36		675.1.c.c.26.2		2
135.124	even	18		3645.1.n.e.809.1		6
540.419	even	18		2160.1.c.b.1889.1		1
540.499	odd	18		2160.1.c.a.1889.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.1.d.a.134.1	✓	1	27.13	even	9
135.1.d.a.134.1	✓	1	135.14	odd	18
135.1.d.b.134.1	yes	1	27.14	odd	18
135.1.d.b.134.1	yes	1	135.94	even	18
405.1.h.a.134.1		2	27.23	odd	18
405.1.h.a.134.1		2	135.4	even	18
405.1.h.a.269.1		2	27.5	odd	18
405.1.h.a.269.1		2	135.49	even	18
405.1.h.b.134.1		2	27.4	even	9
405.1.h.b.134.1		2	135.104	odd	18
405.1.h.b.269.1		2	27.22	even	9
405.1.h.b.269.1		2	135.59	odd	18
675.1.c.c.26.1		2	135.67	odd	36
675.1.c.c.26.1		2	135.68	even	36
675.1.c.c.26.2		2	135.13	odd	36
675.1.c.c.26.2		2	135.122	even	36
2025.1.j.c.26.1		4	135.32	even	36
2025.1.j.c.26.1		4	135.103	odd	36
2025.1.j.c.26.2		4	135.22	odd	36
2025.1.j.c.26.2		4	135.113	even	36
2025.1.j.c.701.1		4	135.23	even	36
2025.1.j.c.701.1		4	135.112	odd	36
2025.1.j.c.701.2		4	135.58	odd	36
2025.1.j.c.701.2		4	135.77	even	36
2160.1.c.a.1889.1		1	108.95	even	18
2160.1.c.a.1889.1		1	540.499	odd	18
2160.1.c.b.1889.1		1	108.67	odd	18
2160.1.c.b.1889.1		1	540.419	even	18
3645.1.n.d.404.1		6	9.4	even	3	inner
3645.1.n.d.404.1		6	45.14	odd	6	inner
3645.1.n.d.809.1		6	27.16	even	9	inner
3645.1.n.d.809.1		6	135.119	odd	18	inner
3645.1.n.d.1619.1		6	1.1	even	1	trivial
3645.1.n.d.1619.1		6	15.14	odd	2	CM
3645.1.n.d.2024.1		6	27.7	even	9	inner
3645.1.n.d.2024.1		6	135.74	odd	18	inner
3645.1.n.d.2834.1		6	9.7	even	3	inner
3645.1.n.d.2834.1		6	45.29	odd	6	inner
3645.1.n.d.3239.1		6	27.25	even	9	inner
3645.1.n.d.3239.1		6	135.29	odd	18	inner
3645.1.n.e.404.1		6	9.5	odd	6
3645.1.n.e.404.1		6	45.4	even	6
3645.1.n.e.809.1		6	27.11	odd	18
3645.1.n.e.809.1		6	135.124	even	18
3645.1.n.e.1619.1		6	3.2	odd	2
3645.1.n.e.1619.1		6	5.4	even	2
3645.1.n.e.2024.1		6	27.20	odd	18
3645.1.n.e.2024.1		6	135.34	even	18
3645.1.n.e.2834.1		6	9.2	odd	6
3645.1.n.e.2834.1		6	45.34	even	6
3645.1.n.e.3239.1		6	27.2	odd	18
3645.1.n.e.3239.1		6	135.79	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:	\( N \)	\(=\)	\( 3645 = 3^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3645.n (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.939693 + 0.342020i) q^{2} +(0.173648 - 0.984808i) q^{5} +(-0.500000 - 0.866025i) q^{8} +(0.500000 - 0.866025i) q^{10} +(-0.173648 - 0.984808i) q^{16} +(0.500000 - 0.866025i) q^{17} +(0.500000 + 0.866025i) q^{19} +(-0.766044 - 0.642788i) q^{23} +(-0.939693 - 0.342020i) q^{25} +(-0.766044 - 0.642788i) q^{31} +(0.766044 - 0.642788i) q^{34} +(0.173648 + 0.984808i) q^{38} +(-0.939693 + 0.342020i) q^{40} +(-0.500000 - 0.866025i) q^{46} +(1.53209 - 1.28558i) q^{47} +(0.173648 - 0.984808i) q^{49} +(-0.766044 - 0.642788i) q^{50} -1.00000 q^{53} +(-0.766044 + 0.642788i) q^{61} +(-0.500000 - 0.866025i) q^{62} +(-0.500000 + 0.866025i) q^{64} +(0.939693 + 0.342020i) q^{79} -1.00000 q^{80} +(0.939693 + 0.342020i) q^{83} +(-0.766044 - 0.642788i) q^{85} +(1.87939 - 0.684040i) q^{94} +(0.939693 - 0.342020i) q^{95} +(0.500000 - 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{8} + 3 q^{10} + 3 q^{17} + 3 q^{19} - 3 q^{46} - 6 q^{53} - 3 q^{62} - 3 q^{64} - 6 q^{80} + 3 q^{98}+O(q^{100}) \) 6 * q - 3 * q^8 + 3 * q^10 + 3 * q^17 + 3 * q^19 - 3 * q^46 - 6 * q^53 - 3 * q^62 - 3 * q^64 - 6 * q^80 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more