LMFDB - Embedded newform 1444.1.l.a.967.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1444,1,Mod(99,1444)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1444, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 2]))

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

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

chi := DirichletCharacter("1444.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1444 = 2^{2} \cdot 19^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1444.l (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:	$0.720649878242$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	12.0.6053445140625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - 4x^{9} + 17x^{6} + 4x^{3} + 1$ x^12 - 4x^9 + 17x^6 + 4*x^3 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{5}$
Projective field:	Galois closure of 5.1.2085136.1

Embedding invariants

Embedding label			967.1
Root			$-1.23949 - 1.04005i$ of defining polynomial
Character	$\chi$	$=$	1444.967
Dual form			1444.1.l.a.1111.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.173648 - 0.984808i) q^{2} +(-0.939693 - 0.342020i) q^{4} +(-0.580762 + 0.211380i) q^{5} +(-0.500000 + 0.866025i) q^{8} +(0.173648 + 0.984808i) q^{9} +(0.107320 + 0.608645i) q^{10} +(-1.23949 + 1.04005i) q^{13} +(0.766044 + 0.642788i) q^{16} +(-0.280969 + 1.59345i) q^{17} +1.00000 q^{18} +0.618034 q^{20} +(-0.473442 + 0.397265i) q^{25} +(0.809017 + 1.40126i) q^{26} +(-0.280969 - 1.59345i) q^{29} +(0.766044 - 0.642788i) q^{32} +(1.52045 + 0.553400i) q^{34} +(0.173648 - 0.984808i) q^{36} +0.618034 q^{37} +(0.107320 - 0.608645i) q^{40} +(0.473442 + 0.397265i) q^{41} +(-0.309017 - 0.535233i) q^{45} +(-0.500000 + 0.866025i) q^{49} +(0.309017 + 0.535233i) q^{50} +(1.52045 - 0.553400i) q^{52} +(-0.580762 - 0.211380i) q^{53} -1.61803 q^{58} +(1.52045 + 0.553400i) q^{61} +(-0.500000 - 0.866025i) q^{64} +(0.500000 - 0.866025i) q^{65} +(0.809017 - 1.40126i) q^{68} +(-0.939693 - 0.342020i) q^{72} +(0.473442 + 0.397265i) q^{73} +(0.107320 - 0.608645i) q^{74} +(-0.580762 - 0.211380i) q^{80} +(-0.939693 + 0.342020i) q^{81} +(0.473442 - 0.397265i) q^{82} +(-0.173648 - 0.984808i) q^{85} +(0.473442 - 0.397265i) q^{89} +(-0.580762 + 0.211380i) q^{90} +(-0.280969 + 1.59345i) q^{97} +(0.766044 + 0.642788i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q - 6 q^{8} + 12 q^{18} - 6 q^{20} + 3 q^{26} - 6 q^{37} + 3 q^{45} - 6 q^{49} - 3 q^{50} - 6 q^{58} - 6 q^{64} + 6 q^{65} + 3 q^{68}+O(q^{100})$ 12 * q - 6 * q^8 + 12 * q^18 - 6 * q^20 + 3 * q^26 - 6 * q^37 + 3 * q^45 - 6 * q^49 - 3 * q^50 - 6 * q^58 - 6 * q^64 + 6 * q^65 + 3 * q^68

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times$ .

$n$	$723$	$1085$
$\chi(n)$	$-1$	$e\left(\frac{5}{9}\right)$

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

)

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

$2$	0.173648	−	0.984808i	0.173648	−	0.984808i
$2$	0.173648	−	0.984808i	0.173648	−	0.984808i
$3$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$3$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$4$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$5$	−0.580762	+	0.211380i	−0.580762	+	0.211380i	−0.615661	−	0.788011i	$-0.711111\pi$
$5$	−0.580762	+	0.211380i	−0.580762	+	0.211380i	0.0348995	+	0.999391i	$0.488889\pi$
$6$		0			0
$7$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$7$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$8$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$9$	0.173648	+	0.984808i	0.173648	+	0.984808i
$10$	0.107320	+	0.608645i	0.107320	+	0.608645i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	−1.23949	+	1.04005i	−1.23949	+	1.04005i	−0.241922	+	0.970296i	$0.577778\pi$
$13$	−1.23949	+	1.04005i	−1.23949	+	1.04005i	−0.997564	+	0.0697565i	$0.977778\pi$
$14$		0			0
$15$		0			0
$16$	0.766044	+	0.642788i	0.766044	+	0.642788i
$17$	−0.280969	+	1.59345i	−0.280969	+	1.59345i	0.438371	+	0.898794i	$0.355556\pi$
$17$	−0.280969	+	1.59345i	−0.280969	+	1.59345i	−0.719340	+	0.694658i	$0.755556\pi$
$18$	1.00000			1.00000
$19$		0			0
$19$		0			0
$20$	0.618034			0.618034
$21$		0			0
$22$		0			0
$23$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$23$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$24$		0			0
$25$	−0.473442	+	0.397265i	−0.473442	+	0.397265i
$26$	0.809017	+	1.40126i	0.809017	+	1.40126i
$27$		0			0
$28$		0			0
$29$	−0.280969	−	1.59345i	−0.280969	−	1.59345i	−0.719340	−	0.694658i	$-0.755556\pi$
$29$	−0.280969	−	1.59345i	−0.280969	−	1.59345i	0.438371	−	0.898794i	$-0.355556\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	0.766044	−	0.642788i	0.766044	−	0.642788i
$33$		0			0
$34$	1.52045	+	0.553400i	1.52045	+	0.553400i
$35$		0			0
$36$	0.173648	−	0.984808i	0.173648	−	0.984808i
$37$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$37$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$38$		0			0
$39$		0			0
$40$	0.107320	−	0.608645i	0.107320	−	0.608645i
$41$	0.473442	+	0.397265i	0.473442	+	0.397265i	0.848048	−	0.529919i	$-0.177778\pi$
$41$	0.473442	+	0.397265i	0.473442	+	0.397265i	−0.374607	+	0.927184i	$0.622222\pi$
$42$		0			0
$43$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$43$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$44$		0			0
$45$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$46$		0			0
$47$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$47$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$48$		0			0
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	0.309017	+	0.535233i	0.309017	+	0.535233i
$51$		0			0
$52$	1.52045	−	0.553400i	1.52045	−	0.553400i
$53$	−0.580762	−	0.211380i	−0.580762	−	0.211380i	0.0348995	−	0.999391i	$-0.488889\pi$
$53$	−0.580762	−	0.211380i	−0.580762	−	0.211380i	−0.615661	+	0.788011i	$0.711111\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−1.61803			−1.61803
$59$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$59$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$60$		0			0
$61$	1.52045	+	0.553400i	1.52045	+	0.553400i	0.961262	−	0.275637i	$-0.0888889\pi$
$61$	1.52045	+	0.553400i	1.52045	+	0.553400i	0.559193	+	0.829038i	$0.311111\pi$
$62$		0			0
$63$		0			0
$64$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$65$	0.500000	−	0.866025i	0.500000	−	0.866025i
$66$		0			0
$67$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$67$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$68$	0.809017	−	1.40126i	0.809017	−	1.40126i
$69$		0			0
$70$		0			0
$71$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$71$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$72$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$73$	0.473442	+	0.397265i	0.473442	+	0.397265i	0.848048	−	0.529919i	$-0.177778\pi$
$73$	0.473442	+	0.397265i	0.473442	+	0.397265i	−0.374607	+	0.927184i	$0.622222\pi$
$74$	0.107320	−	0.608645i	0.107320	−	0.608645i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$79$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$80$	−0.580762	−	0.211380i	−0.580762	−	0.211380i
$81$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$82$	0.473442	−	0.397265i	0.473442	−	0.397265i
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.473442	−	0.397265i	0.473442	−	0.397265i	−0.374607	−	0.927184i	$-0.622222\pi$
$89$	0.473442	−	0.397265i	0.473442	−	0.397265i	0.848048	+	0.529919i	$0.177778\pi$
$90$	−0.580762	+	0.211380i	−0.580762	+	0.211380i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.280969	+	1.59345i	−0.280969	+	1.59345i	0.438371	+	0.898794i	$0.355556\pi$
$97$	−0.280969	+	1.59345i	−0.280969	+	1.59345i	−0.719340	+	0.694658i	$0.755556\pi$
$98$	0.766044	+	0.642788i	0.766044	+	0.642788i
$99$		0			0
$100$	0.580762	−	0.211380i	0.580762	−	0.211380i
$101$	0.473442	−	0.397265i	0.473442	−	0.397265i	−0.374607	−	0.927184i	$-0.622222\pi$
$101$	0.473442	−	0.397265i	0.473442	−	0.397265i	0.848048	+	0.529919i	$0.177778\pi$
$102$		0			0
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$	−0.280969	−	1.59345i	−0.280969	−	1.59345i
$105$		0			0
$106$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$	−0.580762	+	0.211380i	−0.580762	+	0.211380i	−0.615661	−	0.788011i	$-0.711111\pi$
$109$	−0.580762	+	0.211380i	−0.580762	+	0.211380i	0.0348995	+	0.999391i	$0.488889\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$113$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$114$		0			0
$115$		0			0
$116$	−0.280969	+	1.59345i	−0.280969	+	1.59345i
$117$	−1.23949	−	1.04005i	−1.23949	−	1.04005i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$	0.809017	−	1.40126i	0.809017	−	1.40126i
$123$		0			0
$124$		0			0
$125$	0.500000	−	0.866025i	0.500000	−	0.866025i
$126$		0			0
$127$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$127$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$128$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$129$		0			0
$130$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$131$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$131$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−1.23949	−	1.04005i	−1.23949	−	1.04005i
$137$	−0.580762	−	0.211380i	−0.580762	−	0.211380i	0.0348995	−	0.999391i	$-0.488889\pi$
$137$	−0.580762	−	0.211380i	−0.580762	−	0.211380i	−0.615661	+	0.788011i	$0.711111\pi$
$138$		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
$144$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$145$	0.500000	+	0.866025i	0.500000	+	0.866025i
$146$	0.473442	−	0.397265i	0.473442	−	0.397265i
$147$		0			0
$148$	−0.580762	−	0.211380i	−0.580762	−	0.211380i
$149$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.997564	−	0.0697565i	$-0.977778\pi$
$149$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.241922	−	0.970296i	$-0.577778\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−1.61803			−1.61803
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.52045	−	0.553400i	1.52045	−	0.553400i	0.559193	−	0.829038i	$-0.311111\pi$
$157$	1.52045	−	0.553400i	1.52045	−	0.553400i	0.961262	+	0.275637i	$0.0888889\pi$
$158$		0			0
$159$		0			0
$160$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$161$		0			0
$162$	0.173648	+	0.984808i	0.173648	+	0.984808i
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$165$		0			0
$166$		0			0
$167$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$167$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$168$		0			0
$169$	0.280969	−	1.59345i	0.280969	−	1.59345i
$170$	−1.00000			−1.00000
$171$		0			0
$172$		0			0
$173$	0.107320	−	0.608645i	0.107320	−	0.608645i	−0.882948	−	0.469472i	$-0.844444\pi$
$173$	0.107320	−	0.608645i	0.107320	−	0.608645i	0.990268	−	0.139173i	$-0.0444444\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$	0.107320	+	0.608645i	0.107320	+	0.608645i
$181$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$181$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.358931	+	0.130640i	−0.358931	+	0.130640i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	1.53209	+	1.28558i	1.53209	+	1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$193$	1.53209	+	1.28558i	1.53209	+	1.28558i	0.766044	+	0.642788i	$0.222222\pi$
$194$	1.52045	+	0.553400i	1.52045	+	0.553400i
$195$		0			0
$196$	0.766044	−	0.642788i	0.766044	−	0.642788i
$197$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$197$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$198$		0			0
$199$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$199$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$200$	−0.107320	−	0.608645i	−0.107320	−	0.608645i
$201$		0			0
$202$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$203$		0			0
$204$		0			0
$205$	−0.358931	−	0.130640i	−0.358931	−	0.130640i
$206$		0			0
$207$		0			0
$208$	−1.61803			−1.61803
$209$		0			0
$210$		0			0
$211$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$211$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$212$	0.473442	+	0.397265i	0.473442	+	0.397265i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.107320	+	0.608645i	0.107320	+	0.608645i
$219$		0			0
$220$		0			0
$221$	−1.30902	−	2.26728i	−1.30902	−	2.26728i
$222$		0			0
$223$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$223$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$224$		0			0
$225$	−0.473442	−	0.397265i	−0.473442	−	0.397265i
$226$	−0.280969	+	1.59345i	−0.280969	+	1.59345i
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$229$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$230$		0			0
$231$		0			0
$232$	1.52045	+	0.553400i	1.52045	+	0.553400i
$233$	1.52045	−	0.553400i	1.52045	−	0.553400i	0.559193	−	0.829038i	$-0.311111\pi$
$233$	1.52045	−	0.553400i	1.52045	−	0.553400i	0.961262	+	0.275637i	$0.0888889\pi$
$234$	−1.23949	+	1.04005i	−1.23949	+	1.04005i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$241$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$242$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$243$		0			0
$244$	−1.23949	−	1.04005i	−1.23949	−	1.04005i
$245$	0.107320	−	0.608645i	0.107320	−	0.608645i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$251$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$251$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.173648	+	0.984808i	0.173648	+	0.984808i
$257$	0.107320	+	0.608645i	0.107320	+	0.608645i	0.990268	+	0.139173i	$0.0444444\pi$
$257$	0.107320	+	0.608645i	0.107320	+	0.608645i	−0.882948	+	0.469472i	$0.844444\pi$
$258$		0			0
$259$		0			0
$260$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$261$	1.52045	−	0.553400i	1.52045	−	0.553400i
$262$		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
$265$	0.381966			0.381966
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.997564	−	0.0697565i	$-0.977778\pi$
$269$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.241922	−	0.970296i	$-0.577778\pi$
$270$		0			0
$271$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$271$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$272$	−1.23949	+	1.04005i	−1.23949	+	1.04005i
$273$		0			0
$274$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$275$		0			0
$276$		0			0
$277$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$277$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.52045	+	0.553400i	1.52045	+	0.553400i	0.961262	−	0.275637i	$-0.0888889\pi$
$281$	1.52045	+	0.553400i	1.52045	+	0.553400i	0.559193	+	0.829038i	$0.311111\pi$
$282$		0			0
$283$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$283$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.766044	+	0.642788i	0.766044	+	0.642788i
$289$	−1.52045	−	0.553400i	−1.52045	−	0.553400i
$290$	0.939693	−	0.342020i	0.939693	−	0.342020i
$291$		0			0
$292$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$293$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$293$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$		0			0
$296$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$297$		0			0
$298$	−1.23949	+	1.04005i	−1.23949	+	1.04005i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.00000			−1.00000
$306$	−0.280969	+	1.59345i	−0.280969	+	1.59345i
$307$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$307$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	0.107320	+	0.608645i	0.107320	+	0.608645i	0.990268	+	0.139173i	$0.0444444\pi$
$313$	0.107320	+	0.608645i	0.107320	+	0.608645i	−0.882948	+	0.469472i	$0.844444\pi$
$314$	−0.280969	−	1.59345i	−0.280969	−	1.59345i
$315$		0			0
$316$		0			0
$317$	0.473442	−	0.397265i	0.473442	−	0.397265i	−0.374607	−	0.927184i	$-0.622222\pi$
$317$	0.473442	−	0.397265i	0.473442	−	0.397265i	0.848048	+	0.529919i	$0.177778\pi$
$318$		0			0
$319$		0			0
$320$	0.473442	+	0.397265i	0.473442	+	0.397265i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.00000			1.00000
$325$	0.173648	−	0.984808i	0.173648	−	0.984808i
$326$		0			0
$327$		0			0
$328$	−0.580762	+	0.211380i	−0.580762	+	0.211380i
$329$		0			0
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	0.107320	+	0.608645i	0.107320	+	0.608645i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.580762	+	0.211380i	−0.580762	+	0.211380i	−0.615661	−	0.788011i	$-0.711111\pi$
$337$	−0.580762	+	0.211380i	−0.580762	+	0.211380i	0.0348995	+	0.999391i	$0.488889\pi$
$338$	−1.52045	−	0.553400i	−1.52045	−	0.553400i
$339$		0			0
$340$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.580762	−	0.211380i	−0.580762	−	0.211380i
$347$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$347$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$348$		0			0
$349$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$349$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$353$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$354$		0			0
$355$		0			0
$356$	−0.580762	+	0.211380i	−0.580762	+	0.211380i
$357$		0			0
$358$		0			0
$359$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$359$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$360$	0.618034			0.618034
$361$		0			0
$362$	2.00000			2.00000
$363$		0			0
$364$		0			0
$365$	−0.358931	−	0.130640i	−0.358931	−	0.130640i
$366$		0			0
$367$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$367$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$368$		0			0
$369$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$370$	0.0663277	+	0.376163i	0.0663277	+	0.376163i
$371$		0			0
$372$		0			0
$373$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$373$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.00553	+	1.68284i	2.00553	+	1.68284i
$378$		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		−0.766044	−	0.642788i	$-0.777778\pi$
$383$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$384$		0			0
$385$		0			0
$386$	1.53209	−	1.28558i	1.53209	−	1.28558i
$387$		0			0
$388$	0.809017	−	1.40126i	0.809017	−	1.40126i
$389$	−0.280969	−	1.59345i	−0.280969	−	1.59345i	−0.719340	−	0.694658i	$-0.755556\pi$
$389$	−0.280969	−	1.59345i	−0.280969	−	1.59345i	0.438371	−	0.898794i	$-0.355556\pi$
$390$		0			0
$391$		0			0
$392$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$393$		0			0
$394$	1.52045	−	0.553400i	1.52045	−	0.553400i
$395$		0			0
$396$		0			0
$397$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$397$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$398$		0			0
$399$		0			0
$400$	−0.618034			−0.618034
$401$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$401$	0.347296	−	1.96962i	0.347296	−	1.96962i	0.173648	−	0.984808i	$-0.444444\pi$
$402$		0			0
$403$		0			0
$404$	−0.580762	+	0.211380i	−0.580762	+	0.211380i
$405$	0.473442	−	0.397265i	0.473442	−	0.397265i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.107320	+	0.608645i	0.107320	+	0.608645i	0.990268	+	0.139173i	$0.0444444\pi$
$409$	0.107320	+	0.608645i	0.107320	+	0.608645i	−0.882948	+	0.469472i	$0.844444\pi$
$410$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.280969	+	1.59345i	−0.280969	+	1.59345i
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	0.473442	+	0.397265i	0.473442	+	0.397265i	0.848048	−	0.529919i	$-0.177778\pi$
$421$	0.473442	+	0.397265i	0.473442	+	0.397265i	−0.374607	+	0.927184i	$0.622222\pi$
$422$		0			0
$423$		0			0
$424$	0.473442	−	0.397265i	0.473442	−	0.397265i
$425$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$431$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$432$		0			0
$433$	−0.580762	−	0.211380i	−0.580762	−	0.211380i	0.0348995	−	0.999391i	$-0.488889\pi$
$433$	−0.580762	−	0.211380i	−0.580762	−	0.211380i	−0.615661	+	0.788011i	$0.711111\pi$
$434$		0			0
$435$		0			0
$436$	0.618034			0.618034
$437$		0			0
$438$		0			0
$439$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$439$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$440$		0			0
$441$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$442$	−2.46015	+	0.895420i	−2.46015	+	0.895420i
$443$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$443$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$444$		0			0
$445$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$449$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$450$	−0.473442	+	0.397265i	−0.473442	+	0.397265i
$451$		0			0
$452$	1.52045	+	0.553400i	1.52045	+	0.553400i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$457$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$458$	0.107320	−	0.608645i	0.107320	−	0.608645i
$459$		0			0
$460$		0			0
$461$	−1.87939	+	0.684040i	−1.87939	+	0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$461$	−1.87939	+	0.684040i	−1.87939	+	0.684040i	−0.939693	+	0.342020i	$0.888889\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	0.809017	−	1.40126i	0.809017	−	1.40126i
$465$		0			0
$466$	−0.280969	−	1.59345i	−0.280969	−	1.59345i
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$	0.809017	+	1.40126i	0.809017	+	1.40126i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.107320	−	0.608645i	0.107320	−	0.608645i
$478$		0			0
$479$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$479$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$480$		0			0
$481$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$482$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$483$		0			0
$484$	0.173648	+	0.984808i	0.173648	+	0.984808i
$485$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$486$		0			0
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$	−1.23949	+	1.04005i	−1.23949	+	1.04005i
$489$		0			0
$490$	−0.580762	−	0.211380i	−0.580762	−	0.211380i
$491$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$491$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$492$		0			0
$493$	2.61803			2.61803
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$499$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$500$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$501$		0			0
$502$		0			0
$503$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$503$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$504$		0			0
$505$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.52045	+	0.553400i	1.52045	+	0.553400i	0.961262	−	0.275637i	$-0.0888889\pi$
$509$	1.52045	+	0.553400i	1.52045	+	0.553400i	0.559193	+	0.829038i	$0.311111\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			1.00000
$513$		0			0
$514$	0.618034			0.618034
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.500000	+	0.866025i	0.500000	+	0.866025i
$521$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$521$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$522$	−0.280969	−	1.59345i	−0.280969	−	1.59345i
$523$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$523$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.766044	+	0.642788i	0.766044	+	0.642788i
$530$	0.0663277	−	0.376163i	0.0663277	−	0.376163i
$531$		0			0
$532$		0			0
$533$	−1.00000			−1.00000
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.23949	+	1.04005i	−1.23949	+	1.04005i
$539$		0			0
$540$		0			0
$541$	−0.280969	−	1.59345i	−0.280969	−	1.59345i	−0.719340	−	0.694658i	$-0.755556\pi$
$541$	−0.280969	−	1.59345i	−0.280969	−	1.59345i	0.438371	−	0.898794i	$-0.355556\pi$
$542$		0			0
$543$		0			0
$544$	0.809017	+	1.40126i	0.809017	+	1.40126i
$545$	0.292603	−	0.245523i	0.292603	−	0.245523i
$546$		0			0
$547$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$547$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$548$	0.473442	+	0.397265i	0.473442	+	0.397265i
$549$	−0.280969	+	1.59345i	−0.280969	+	1.59345i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.473442	+	0.397265i	0.473442	+	0.397265i
$555$		0			0
$556$		0			0
$557$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$557$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.809017	−	1.40126i	0.809017	−	1.40126i
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$	0.939693	−	0.342020i	0.939693	−	0.342020i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.618034			0.618034			0.309017	−	0.951057i	$-0.400000\pi$
$569$	0.618034			0.618034			0.309017	+	0.951057i	$0.400000\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.766044	−	0.642788i	0.766044	−	0.642788i
$577$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$577$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$578$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$579$		0			0
$580$	−0.173648	−	0.984808i	−0.173648	−	0.984808i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.580762	+	0.211380i	−0.580762	+	0.211380i
$585$	0.939693	+	0.342020i	0.939693	+	0.342020i
$586$	1.53209	+	1.28558i	1.53209	+	1.28558i
$587$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$587$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.473442	+	0.397265i	0.473442	+	0.397265i
$593$	−0.580762	−	0.211380i	−0.580762	−	0.211380i	0.0348995	−	0.999391i	$-0.488889\pi$
$593$	−0.580762	−	0.211380i	−0.580762	−	0.211380i	−0.615661	+	0.788011i	$0.711111\pi$
$594$		0			0
$595$		0			0
$596$	0.809017	+	1.40126i	0.809017	+	1.40126i
$597$		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
$601$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$601$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.473442	+	0.397265i	0.473442	+	0.397265i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$611$		0			0
$612$	1.52045	+	0.553400i	1.52045	+	0.553400i
$613$	−0.580762	+	0.211380i	−0.580762	+	0.211380i	−0.615661	−	0.788011i	$-0.711111\pi$
$613$	−0.580762	+	0.211380i	−0.580762	+	0.211380i	0.0348995	+	0.999391i	$0.488889\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$617$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$	0.618034			0.618034
$627$		0			0
$628$	−1.61803			−1.61803
$629$	−0.173648	+	0.984808i	−0.173648	+	0.984808i
$630$		0			0
$631$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$631$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$632$		0			0
$633$		0			0
$634$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$635$		0			0
$636$		0			0
$637$	−0.280969	−	1.59345i	−0.280969	−	1.59345i
$638$		0			0
$639$		0			0
$640$	0.473442	−	0.397265i	0.473442	−	0.397265i
$641$	1.52045	−	0.553400i	1.52045	−	0.553400i	0.559193	−	0.829038i	$-0.311111\pi$
$641$	1.52045	−	0.553400i	1.52045	−	0.553400i	0.961262	+	0.275637i	$0.0888889\pi$
$642$		0			0
$643$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$643$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	0.173648	−	0.984808i	0.173648	−	0.984808i
$649$		0			0
$650$	−0.939693	−	0.342020i	−0.939693	−	0.342020i
$651$		0			0
$652$		0			0
$653$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$653$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$654$		0			0
$655$		0			0
$656$	0.107320	+	0.608645i	0.107320	+	0.608645i
$657$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$658$		0			0
$659$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$659$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$660$		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	$-0.888889\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.618034			0.618034
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$673$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$674$	0.107320	+	0.608645i	0.107320	+	0.608645i
$675$		0			0
$676$	−0.809017	+	1.40126i	−0.809017	+	1.40126i
$677$	0.809017	+	1.40126i	0.809017	+	1.40126i	0.913545	+	0.406737i	$0.133333\pi$
$677$	0.809017	+	1.40126i	0.809017	+	1.40126i	−0.104528	+	0.994522i	$0.533333\pi$
$678$		0			0
$679$		0			0
$680$	0.939693	+	0.342020i	0.939693	+	0.342020i
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		0			0
$685$	0.381966			0.381966
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.939693	−	0.342020i	0.939693	−	0.342020i
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$698$	1.52045	−	0.553400i	1.52045	−	0.553400i
$699$		0			0
$700$		0			0
$701$	0.107320	−	0.608645i	0.107320	−	0.608645i	−0.882948	−	0.469472i	$-0.844444\pi$
$701$	0.107320	−	0.608645i	0.107320	−	0.608645i	0.990268	−	0.139173i	$-0.0444444\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−1.23949	−	1.04005i	−1.23949	−	1.04005i
$707$		0			0
$708$		0			0
$709$	−1.23949	+	1.04005i	−1.23949	+	1.04005i	−0.241922	+	0.970296i	$0.577778\pi$
$709$	−1.23949	+	1.04005i	−1.23949	+	1.04005i	−0.997564	+	0.0697565i	$0.977778\pi$
$710$		0			0
$711$		0			0
$712$	0.107320	+	0.608645i	0.107320	+	0.608645i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$719$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$720$	0.107320	−	0.608645i	0.107320	−	0.608645i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	0.347296	−	1.96962i	0.347296	−	1.96962i
$725$	0.766044	+	0.642788i	0.766044	+	0.642788i
$726$		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
$729$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$730$	−0.190983	+	0.330792i	−0.190983	+	0.330792i
$731$		0			0
$732$		0			0
$733$	0.809017	−	1.40126i	0.809017	−	1.40126i	−0.104528	−	0.994522i	$-0.533333\pi$
$733$	0.809017	−	1.40126i	0.809017	−	1.40126i	0.913545	−	0.406737i	$-0.133333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	0.473442	+	0.397265i	0.473442	+	0.397265i
$739$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$739$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$740$	0.381966			0.381966
$741$		0			0
$742$		0			0
$743$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$743$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$744$		0			0
$745$	0.939693	+	0.342020i	0.939693	+	0.342020i
$746$	1.52045	−	0.553400i	1.52045	−	0.553400i
$747$		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
$754$	2.00553	−	1.68284i	2.00553	−	1.68284i
$755$		0			0
$756$		0			0
$757$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.997564	−	0.0697565i	$-0.977778\pi$
$757$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.241922	−	0.970296i	$-0.577778\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.00000			2.00000			1.00000			$0$
$761$	2.00000			2.00000			1.00000			$0$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.939693	−	0.342020i	0.939693	−	0.342020i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.107320	+	0.608645i	0.107320	+	0.608645i	0.990268	+	0.139173i	$0.0444444\pi$
$769$	0.107320	+	0.608645i	0.107320	+	0.608645i	−0.882948	+	0.469472i	$0.844444\pi$
$770$		0			0
$771$		0			0
$772$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$773$	−1.23949	+	1.04005i	−1.23949	+	1.04005i	−0.241922	+	0.970296i	$0.577778\pi$
$773$	−1.23949	+	1.04005i	−1.23949	+	1.04005i	−0.997564	+	0.0697565i	$0.977778\pi$
$774$		0			0
$775$		0			0
$776$	−1.23949	−	1.04005i	−1.23949	−	1.04005i
$777$		0			0
$778$	−1.61803			−1.61803
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$785$	−0.766044	+	0.642788i	−0.766044	+	0.642788i
$786$		0			0
$787$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$787$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$788$	−0.280969	−	1.59345i	−0.280969	−	1.59345i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.46015	+	0.895420i	−2.46015	+	0.895420i
$794$	−1.87939	−	0.684040i	−1.87939	−	0.684040i
$795$		0			0
$796$		0			0
$797$	−1.61803			−1.61803			−0.809017	−	0.587785i	$-0.800000\pi$
$797$	−1.61803			−1.61803			−0.809017	+	0.587785i	$0.800000\pi$
$798$		0			0
$799$		0			0
$800$	−0.107320	+	0.608645i	−0.107320	+	0.608645i
$801$	0.473442	+	0.397265i	0.473442	+	0.397265i
$802$	−1.87939	−	0.684040i	−1.87939	−	0.684040i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.107320	+	0.608645i	0.107320	+	0.608645i
$809$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$809$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$810$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$811$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$811$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.618034			0.618034
$819$		0			0
$820$	0.292603	+	0.245523i	0.292603	+	0.245523i
$821$	1.52045	+	0.553400i	1.52045	+	0.553400i	0.961262	−	0.275637i	$-0.0888889\pi$
$821$	1.52045	+	0.553400i	1.52045	+	0.553400i	0.559193	+	0.829038i	$0.311111\pi$
$822$		0			0
$823$		0			0		0.766044	−	0.642788i	$-0.222222\pi$
$823$		0			0		−0.766044	+	0.642788i	$0.777778\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$827$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$828$		0			0
$829$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$829$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$830$		0			0
$831$		0			0
$832$	1.52045	+	0.553400i	1.52045	+	0.553400i
$833$	−1.23949	−	1.04005i	−1.23949	−	1.04005i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$839$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$840$		0			0
$841$	−1.52045	+	0.553400i	−1.52045	+	0.553400i
$842$	0.473442	−	0.397265i	0.473442	−	0.397265i
$843$		0			0
$844$		0			0
$845$	0.173648	+	0.984808i	0.173648	+	0.984808i
$846$		0			0
$847$		0			0
$848$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$849$		0			0
$850$	−0.939693	+	0.342020i	−0.939693	+	0.342020i
$851$		0			0
$852$		0			0
$853$	−0.280969	+	1.59345i	−0.280969	+	1.59345i	0.438371	+	0.898794i	$0.355556\pi$
$853$	−0.280969	+	1.59345i	−0.280969	+	1.59345i	−0.719340	+	0.694658i	$0.755556\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.280969	+	1.59345i	−0.280969	+	1.59345i	0.438371	+	0.898794i	$0.355556\pi$
$857$	−0.280969	+	1.59345i	−0.280969	+	1.59345i	−0.719340	+	0.694658i	$0.755556\pi$
$858$		0			0
$859$		0			0		−0.939693	−	0.342020i	$-0.888889\pi$
$859$		0			0		0.939693	+	0.342020i	$0.111111\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$	0.0663277	+	0.376163i	0.0663277	+	0.376163i
$866$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.107320	−	0.608645i	0.107320	−	0.608645i
$873$	−1.61803			−1.61803
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.997564	−	0.0697565i	$-0.977778\pi$
$877$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.241922	−	0.970296i	$-0.577778\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	0.669131	−	0.743145i	$-0.266667\pi$
$881$	−0.309017	−	0.535233i	−0.309017	−	0.535233i	−0.978148	+	0.207912i	$0.933333\pi$
$882$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$883$		0			0		−0.173648	−	0.984808i	$-0.555556\pi$
$883$		0			0		0.173648	+	0.984808i	$0.444444\pi$
$884$	0.454617	+	2.57826i	0.454617	+	2.57826i
$885$		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
$890$	0.292603	+	0.245523i	0.292603	+	0.245523i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.52045	−	0.553400i	1.52045	−	0.553400i
$899$		0			0
$900$	0.309017	+	0.535233i	0.309017	+	0.535233i
$901$	0.500000	−	0.866025i	0.500000	−	0.866025i
$902$		0			0
$903$		0			0
$904$	0.809017	−	1.40126i	0.809017	−	1.40126i
$905$	−0.618034	−	1.07047i	−0.618034	−	1.07047i
$906$		0			0
$907$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$907$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$908$		0			0
$909$	0.473442	+	0.397265i	0.473442	+	0.397265i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	0.107320	−	0.608645i	0.107320	−	0.608645i
$915$		0			0
$916$	−0.580762	−	0.211380i	−0.580762	−	0.211380i
$917$		0			0
$918$		0			0
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$		0			0
$922$	0.347296	+	1.96962i	0.347296	+	1.96962i
$923$		0			0
$924$		0			0
$925$	−0.292603	+	0.245523i	−0.292603	+	0.245523i
$926$		0			0
$927$		0			0
$928$	−1.23949	−	1.04005i	−1.23949	−	1.04005i
$929$	−0.280969	+	1.59345i	−0.280969	+	1.59345i	0.438371	+	0.898794i	$0.355556\pi$
$929$	−0.280969	+	1.59345i	−0.280969	+	1.59345i	−0.719340	+	0.694658i	$0.755556\pi$
$930$		0			0
$931$		0			0
$932$	−1.61803			−1.61803
$933$		0			0
$934$		0			0
$935$		0			0
$936$	1.52045	−	0.553400i	1.52045	−	0.553400i
$937$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$937$	1.53209	−	1.28558i	1.53209	−	1.28558i	0.766044	−	0.642788i	$-0.222222\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$941$	0.347296	+	1.96962i	0.347296	+	1.96962i	0.173648	+	0.984808i	$0.444444\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$947$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$948$		0			0
$949$	−1.00000			−1.00000
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.997564	−	0.0697565i	$-0.977778\pi$
$953$	−1.23949	−	1.04005i	−1.23949	−	1.04005i	−0.241922	−	0.970296i	$-0.577778\pi$
$954$	−0.580762	−	0.211380i	−0.580762	−	0.211380i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$	0.500000	+	0.866025i	0.500000	+	0.866025i
$963$		0			0
$964$	−1.87939	+	0.684040i	−1.87939	+	0.684040i
$965$	−1.16152	−	0.422760i	−1.16152	−	0.422760i
$966$		0			0
$967$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$967$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$968$	1.00000			1.00000
$969$		0			0
$970$	−1.00000			−1.00000
$971$		0			0		0.173648	−	0.984808i	$-0.444444\pi$
$971$		0			0		−0.173648	+	0.984808i	$0.555556\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.809017	+	1.40126i	0.809017	+	1.40126i
$977$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	−0.978148	−	0.207912i	$-0.933333\pi$
$977$	−0.309017	+	0.535233i	−0.309017	+	0.535233i	0.669131	+	0.743145i	$0.266667\pi$
$978$		0			0
$979$		0			0
$980$	−0.309017	+	0.535233i	−0.309017	+	0.535233i
$981$	−0.309017	−	0.535233i	−0.309017	−	0.535233i
$982$		0			0
$983$		0			0		0.939693	−	0.342020i	$-0.111111\pi$
$983$		0			0		−0.939693	+	0.342020i	$0.888889\pi$
$984$		0			0
$985$	−0.766044	−	0.642788i	−0.766044	−	0.642788i
$986$	0.454617	−	2.57826i	0.454617	−	2.57826i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.766044	−	0.642788i	$-0.777778\pi$
$991$		0			0		0.766044	+	0.642788i	$0.222222\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.107320	+	0.608645i	0.107320	+	0.608645i	0.990268	+	0.139173i	$0.0444444\pi$
$997$	0.107320	+	0.608645i	0.107320	+	0.608645i	−0.882948	+	0.469472i	$0.844444\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1444.1.l.a.967.1		12
4.3	odd	2	CM	1444.1.l.a.967.1		12
19.2	odd	18		1444.1.g.b.1151.1		4
19.3	odd	18		1444.1.b.a.723.2	✓	2
19.4	even	9	inner	1444.1.l.a.595.2		12
19.5	even	9		1444.1.g.a.1375.1		4
19.6	even	9	inner	1444.1.l.a.99.2		12
19.7	even	3	inner	1444.1.l.a.415.2		12
19.8	odd	6		1444.1.l.b.423.2		12
19.9	even	9	inner	1444.1.l.a.1111.1		12
19.10	odd	18		1444.1.l.b.1111.1		12
19.11	even	3	inner	1444.1.l.a.423.2		12
19.12	odd	6		1444.1.l.b.415.2		12
19.13	odd	18		1444.1.l.b.99.2		12
19.14	odd	18		1444.1.g.b.1375.1		4
19.15	odd	18		1444.1.l.b.595.2		12
19.16	even	9		1444.1.b.b.723.2	yes	2
19.17	even	9		1444.1.g.a.1151.1		4
19.18	odd	2		1444.1.l.b.967.1		12
76.3	even	18		1444.1.b.a.723.2	✓	2
76.7	odd	6	inner	1444.1.l.a.415.2		12
76.11	odd	6	inner	1444.1.l.a.423.2		12
76.15	even	18		1444.1.l.b.595.2		12
76.23	odd	18	inner	1444.1.l.a.595.2		12
76.27	even	6		1444.1.l.b.423.2		12
76.31	even	6		1444.1.l.b.415.2		12
76.35	odd	18		1444.1.b.b.723.2	yes	2
76.43	odd	18		1444.1.g.a.1375.1		4
76.47	odd	18	inner	1444.1.l.a.1111.1		12
76.51	even	18		1444.1.l.b.99.2		12
76.55	odd	18		1444.1.g.a.1151.1		4
76.59	even	18		1444.1.g.b.1151.1		4
76.63	odd	18	inner	1444.1.l.a.99.2		12
76.67	even	18		1444.1.l.b.1111.1		12
76.71	even	18		1444.1.g.b.1375.1		4
76.75	even	2		1444.1.l.b.967.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1444.1.b.a.723.2	✓	2	19.3	odd	18
1444.1.b.a.723.2	✓	2	76.3	even	18
1444.1.b.b.723.2	yes	2	19.16	even	9
1444.1.b.b.723.2	yes	2	76.35	odd	18
1444.1.g.a.1151.1		4	19.17	even	9
1444.1.g.a.1151.1		4	76.55	odd	18
1444.1.g.a.1375.1		4	19.5	even	9
1444.1.g.a.1375.1		4	76.43	odd	18
1444.1.g.b.1151.1		4	19.2	odd	18
1444.1.g.b.1151.1		4	76.59	even	18
1444.1.g.b.1375.1		4	19.14	odd	18
1444.1.g.b.1375.1		4	76.71	even	18
1444.1.l.a.99.2		12	19.6	even	9	inner
1444.1.l.a.99.2		12	76.63	odd	18	inner
1444.1.l.a.415.2		12	19.7	even	3	inner
1444.1.l.a.415.2		12	76.7	odd	6	inner
1444.1.l.a.423.2		12	19.11	even	3	inner
1444.1.l.a.423.2		12	76.11	odd	6	inner
1444.1.l.a.595.2		12	19.4	even	9	inner
1444.1.l.a.595.2		12	76.23	odd	18	inner
1444.1.l.a.967.1		12	1.1	even	1	trivial
1444.1.l.a.967.1		12	4.3	odd	2	CM
1444.1.l.a.1111.1		12	19.9	even	9	inner
1444.1.l.a.1111.1		12	76.47	odd	18	inner
1444.1.l.b.99.2		12	19.13	odd	18
1444.1.l.b.99.2		12	76.51	even	18
1444.1.l.b.415.2		12	19.12	odd	6
1444.1.l.b.415.2		12	76.31	even	6
1444.1.l.b.423.2		12	19.8	odd	6
1444.1.l.b.423.2		12	76.27	even	6
1444.1.l.b.595.2		12	19.15	odd	18
1444.1.l.b.595.2		12	76.15	even	18
1444.1.l.b.967.1		12	19.18	odd	2
1444.1.l.b.967.1		12	76.75	even	2
1444.1.l.b.1111.1		12	19.10	odd	18
1444.1.l.b.1111.1		12	76.67	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 $\F_{q}$

Introduction

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Modular forms

Varieties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1444.1.l.a.967.1

Level	$1444$
Weight	$1$
Character	1444.967
Analytic conductor	$0.721$
Analytic rank	$0$
Dimension	$12$
Projective image	$D_{5}$
CM discriminant	-4
Inner twists	$12$