LMFDB - Embedded newform 1805.1.h.b.654.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1805,1,Mod(69,1805)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1805, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("1805.69");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1805 = 5 \cdot 19^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1805.h (of order $6$ , degree $2$ , not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$0.900812347803$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 2x^{2} + 4$ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 95)
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.475.1
Artin image:	$C_3\times D_8$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{24} - \cdots)$

Embedding invariants

Embedding label			654.1
Root			$0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1805.654
Dual form			1805.1.h.b.69.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 1.22474i) q^{2} +(0.707107 - 1.22474i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(1.00000 + 1.73205i) q^{6} +(-0.500000 - 0.866025i) q^{9} +(0.707107 + 1.22474i) q^{10} -1.41421 q^{12} +(-0.707107 - 1.22474i) q^{13} +(-0.707107 - 1.22474i) q^{15} +(0.500000 - 0.866025i) q^{16} +1.41421 q^{18} -1.00000 q^{20} +(-0.500000 - 0.866025i) q^{25} +2.00000 q^{26} +2.00000 q^{30} +(0.707107 + 1.22474i) q^{32} +(-0.500000 + 0.866025i) q^{36} -1.41421 q^{37} -2.00000 q^{39} -1.00000 q^{45} +(-0.707107 - 1.22474i) q^{48} +1.00000 q^{49} +1.41421 q^{50} +(-0.707107 + 1.22474i) q^{52} +(0.707107 + 1.22474i) q^{53} +(-0.707107 + 1.22474i) q^{60} -1.00000 q^{64} -1.41421 q^{65} +(-0.707107 - 1.22474i) q^{67} +(1.00000 - 1.73205i) q^{74} -1.41421 q^{75} +(1.41421 - 2.44949i) q^{78} +(-0.500000 - 0.866025i) q^{80} +(0.500000 - 0.866025i) q^{81} +(0.707107 - 1.22474i) q^{90} +2.00000 q^{96} +(0.707107 - 1.22474i) q^{97} +(-0.707107 + 1.22474i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{4} + 2 q^{5} + 4 q^{6} - 2 q^{9} + 2 q^{16} - 4 q^{20} - 2 q^{25} + 8 q^{26} + 8 q^{30} - 2 q^{36} - 8 q^{39} - 4 q^{45} + 4 q^{49} - 4 q^{64} + 4 q^{74} - 2 q^{80} + 2 q^{81} + 8 q^{96}+O(q^{100})$ 4 * q - 2 * q^4 + 2 * q^5 + 4 * q^6 - 2 * q^9 + 2 * q^16 - 4 * q^20 - 2 * q^25 + 8 * q^26 + 8 * q^30 - 2 * q^36 - 8 * q^39 - 4 * q^45 + 4 * q^49 - 4 * q^64 + 4 * q^74 - 2 * q^80 + 2 * q^81 + 8 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1805\mathbb{Z}\right)^\times$ .

$n$	$362$	$1446$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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

)

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

$2$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$2$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$3$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$3$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$4$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$	0.500000	−	0.866025i	0.500000	−	0.866025i
$5$	0.500000	−	0.866025i	0.500000	−	0.866025i
$6$	1.00000	+	1.73205i	1.00000	+	1.73205i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$	0.707107	+	1.22474i	0.707107	+	1.22474i
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−1.41421			−1.41421
$13$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$13$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$14$		0			0
$15$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$16$	0.500000	−	0.866025i	0.500000	−	0.866025i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$	1.41421			1.41421
$19$		0			0
$19$		0			0
$20$	−1.00000			−1.00000
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$	2.00000			2.00000
$27$		0			0
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$	2.00000			2.00000
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.707107	+	1.22474i	0.707107	+	1.22474i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$37$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$37$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$	−2.00000			−2.00000
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$	−1.00000			−1.00000
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$49$	1.00000			1.00000
$50$	1.41421			1.41421
$51$		0			0
$52$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$53$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$53$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$61$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$61$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$62$		0			0
$63$		0			0
$64$	−1.00000			−1.00000
$65$	−1.41421			−1.41421
$66$		0			0
$67$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$67$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$68$		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	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$	1.00000	−	1.73205i	1.00000	−	1.73205i
$75$	−1.41421			−1.41421
$76$		0			0
$77$		0			0
$78$	1.41421	−	2.44949i	1.41421	−	2.44949i
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$81$	0.500000	−	0.866025i	0.500000	−	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		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	$0.333333\pi$
$90$	0.707107	−	1.22474i	0.707107	−	1.22474i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	2.00000			2.00000
$97$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$97$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$98$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$99$		0			0
$100$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$103$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$104$		0			0
$105$		0			0
$106$	−2.00000			−2.00000
$107$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$107$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$108$		0			0
$109$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$109$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$112$		0			0
$113$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$113$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−1.00000
$126$		0			0
$127$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$127$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$128$		0			0
$129$		0			0
$130$	1.00000	−	1.73205i	1.00000	−	1.73205i
$131$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$131$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$	2.00000			2.00000
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$139$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1.00000			−1.00000
$145$		0			0
$146$		0			0
$147$	0.707107	−	1.22474i	0.707107	−	1.22474i
$148$	0.707107	+	1.22474i	0.707107	+	1.22474i
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$	1.00000	−	1.73205i	1.00000	−	1.73205i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.00000	+	1.73205i	1.00000	+	1.73205i
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$	2.00000			2.00000
$160$	1.41421			1.41421
$161$		0			0
$162$	0.707107	+	1.22474i	0.707107	+	1.22474i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$167$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$173$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	0.500000	+	0.866025i	0.500000	+	0.866025i
$181$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$181$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.00000			−2.00000			−1.00000			$\pi$
$191$	−2.00000			−2.00000			−1.00000			$\pi$
$192$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$193$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$193$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$194$	1.00000	+	1.73205i	1.00000	+	1.73205i
$195$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$196$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$199$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$200$		0			0
$201$	−2.00000			−2.00000
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$207$		0			0
$208$	−1.41421			−1.41421
$209$		0			0
$210$		0			0
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$	0.707107	−	1.22474i	0.707107	−	1.22474i
$213$		0			0
$214$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	−1.41421	−	2.44949i	−1.41421	−	2.44949i
$223$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$223$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$224$		0			0
$225$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$226$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$227$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$227$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2.00000			2.00000			1.00000			$0$
$239$	2.00000			2.00000			1.00000			$0$
$240$	−1.41421			−1.41421
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	0.707107	−	1.22474i	0.707107	−	1.22474i
$243$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$244$		0			0
$245$	0.500000	−	0.866025i	0.500000	−	0.866025i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.707107	−	1.22474i	0.707107	−	1.22474i
$251$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$251$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$252$		0			0
$253$		0			0
$254$	−2.00000			−2.00000
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$257$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$258$		0			0
$259$		0			0
$260$	0.707107	+	1.22474i	0.707107	+	1.22474i
$261$		0			0
$262$	−1.41421	−	2.44949i	−1.41421	−	2.44949i
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$	1.41421			1.41421
$266$		0			0
$267$		0			0
$268$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.707107	−	1.22474i	0.707107	−	1.22474i
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$292$		0			0
$293$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$293$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$294$	1.00000	+	1.73205i	1.00000	+	1.73205i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.707107	+	1.22474i	0.707107	+	1.22474i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$307$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$308$		0			0
$309$	1.00000	−	1.73205i	1.00000	−	1.73205i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$317$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$318$	−1.41421	+	2.44949i	−1.41421	+	2.44949i
$319$		0			0
$320$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$321$	1.00000	−	1.73205i	1.00000	−	1.73205i
$322$		0			0
$323$		0			0
$324$	−1.00000			−1.00000
$325$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	0.707107	+	1.22474i	0.707107	+	1.22474i
$334$	−2.00000			−2.00000
$335$	−1.41421			−1.41421
$336$		0			0
$337$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$337$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$338$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$339$	1.00000	−	1.73205i	1.00000	−	1.73205i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	−2.00000			−2.00000			−1.00000			$\pi$
$349$	−2.00000			−2.00000			−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$		0			0
$369$		0			0
$370$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$371$		0			0
$372$		0			0
$373$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$373$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$374$		0			0
$375$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	2.00000			2.00000
$382$	1.41421	−	2.44949i	1.41421	−	2.44949i
$383$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$383$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$384$		0			0
$385$		0			0
$386$	1.00000	+	1.73205i	1.00000	+	1.73205i
$387$		0			0
$388$	−1.41421			−1.41421
$389$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$389$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$390$	−1.41421	−	2.44949i	−1.41421	−	2.44949i
$391$		0			0
$392$		0			0
$393$	1.41421	+	2.44949i	1.41421	+	2.44949i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$	−2.82843			−2.82843
$399$		0			0
$400$	−1.00000			−1.00000
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$	1.41421	−	2.44949i	1.41421	−	2.44949i
$403$		0			0
$404$		0			0
$405$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		0			0
$412$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	1.00000	−	1.73205i	1.00000	−	1.73205i
$417$		0			0
$418$		0			0
$419$	2.00000			2.00000			1.00000			$0$
$419$	2.00000			2.00000			1.00000			$0$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$433$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	2.00000			2.00000
$445$		0			0
$446$	1.00000	+	1.73205i	1.00000	+	1.73205i
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$451$		0			0
$452$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$453$		0			0
$454$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$461$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	1.41421			1.41421
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.707107	−	1.22474i	0.707107	−	1.22474i
$478$	−1.41421	+	2.44949i	−1.41421	+	2.44949i
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$	1.00000	−	1.73205i	1.00000	−	1.73205i
$481$	1.00000	+	1.73205i	1.00000	+	1.73205i
$482$		0			0
$483$		0			0
$484$	0.500000	+	0.866025i	0.500000	+	0.866025i
$485$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$486$	2.00000			2.00000
$487$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$487$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$488$		0			0
$489$		0			0
$490$	0.707107	+	1.22474i	0.707107	+	1.22474i
$491$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$491$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$499$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$500$	0.500000	+	0.866025i	0.500000	+	0.866025i
$501$	2.00000			2.00000
$502$	−2.82843			−2.82843
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.707107	+	1.22474i	0.707107	+	1.22474i
$508$	0.707107	−	1.22474i	0.707107	−	1.22474i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	1.41421			1.41421
$513$		0			0
$514$	−2.00000			−2.00000
$515$	0.707107	−	1.22474i	0.707107	−	1.22474i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	1.00000	+	1.73205i	1.00000	+	1.73205i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$523$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$524$	2.00000			2.00000
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.707107	−	1.22474i	0.707107	−	1.22474i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$541$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$547$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	1.00000	+	1.73205i	1.00000	+	1.73205i
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$563$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$	0.707107	−	1.22474i	0.707107	−	1.22474i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		0			0
$573$	−1.41421	+	2.44949i	−1.41421	+	2.44949i
$574$		0			0
$575$		0			0
$576$	0.500000	+	0.866025i	0.500000	+	0.866025i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	1.41421			1.41421
$579$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$580$		0			0
$581$		0			0
$582$	2.82843			2.82843
$583$		0			0
$584$		0			0
$585$	0.707107	+	1.22474i	0.707107	+	1.22474i
$586$	1.00000	−	1.73205i	1.00000	−	1.73205i
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$	−1.41421			−1.41421
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.82843			2.82843
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$604$		0			0
$605$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$606$		0			0
$607$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$607$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$	1.41421	+	2.44949i	1.41421	+	2.44949i
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$631$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$632$		0			0
$633$		0			0
$634$	2.00000			2.00000
$635$	1.41421			1.41421
$636$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$637$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	1.41421	+	2.44949i	1.41421	+	2.44949i
$643$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$643$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$	1.00000	+	1.73205i	1.00000	+	1.73205i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−2.00000			−2.00000
$667$		0			0
$668$	0.707107	−	1.22474i	0.707107	−	1.22474i
$669$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$670$	1.00000	−	1.73205i	1.00000	−	1.73205i
$671$		0			0
$672$		0			0
$673$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$673$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$674$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$675$		0			0
$676$	1.00000			1.00000
$677$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$677$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$678$	1.41421	+	2.44949i	1.41421	+	2.44949i
$679$		0			0
$680$		0			0
$681$	1.00000	−	1.73205i	1.00000	−	1.73205i
$682$		0			0
$683$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$683$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.00000	−	1.73205i	1.00000	−	1.73205i
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$	1.41421			1.41421
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$	1.41421	−	2.44949i	1.41421	−	2.44949i
$699$		0			0
$700$		0			0
$701$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$701$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$709$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	1.41421	−	2.44949i	1.41421	−	2.44949i
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$739$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$740$	1.41421			1.41421
$741$		0			0
$742$		0			0
$743$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$743$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$744$		0			0
$745$		0			0
$746$	1.00000	−	1.73205i	1.00000	−	1.73205i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$	2.82843			2.82843
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$757$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$	−1.41421	+	2.44949i	−1.41421	+	2.44949i
$763$		0			0
$764$	1.00000	+	1.73205i	1.00000	+	1.73205i
$765$		0			0
$766$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$767$		0			0
$768$	−1.41421			−1.41421
$769$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$769$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$770$		0			0
$771$	2.00000			2.00000
$772$	−1.41421			−1.41421
$773$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$773$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	2.82843			2.82843
$779$		0			0
$780$	2.00000			2.00000
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.500000	−	0.866025i	0.500000	−	0.866025i
$785$		0			0
$786$	−4.00000			−4.00000
$787$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$787$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	1.00000	−	1.73205i	1.00000	−	1.73205i
$796$	1.00000	−	1.73205i	1.00000	−	1.73205i
$797$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$797$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$798$		0			0
$799$		0			0
$800$	0.707107	−	1.22474i	0.707107	−	1.22474i
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.00000	+	1.73205i	1.00000	+	1.73205i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.00000			−2.00000			−1.00000			$\pi$
$809$	−2.00000			−2.00000			−1.00000			$\pi$
$810$	1.41421			1.41421
$811$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$811$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$821$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$827$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	0.707107	+	1.22474i	0.707107	+	1.22474i
$833$		0			0
$834$		0			0
$835$	1.41421			1.41421
$836$		0			0
$837$		0			0
$838$	−1.41421	+	2.44949i	−1.41421	+	2.44949i
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.500000	+	0.866025i	0.500000	+	0.866025i
$846$		0			0
$847$		0			0
$848$	1.41421			1.41421
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$857$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$858$		0			0
$859$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$859$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$863$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$	0.707107	+	1.22474i	0.707107	+	1.22474i
$866$	2.00000			2.00000
$867$	−1.41421			−1.41421
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$872$		0			0
$873$	−1.41421			−1.41421
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$877$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$878$		0			0
$879$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	1.41421			1.41421
$883$		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			0
$887$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$887$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−1.41421			−1.41421
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	1.00000			1.00000
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$907$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$908$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		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			0
$921$	1.00000	+	1.73205i	1.00000	+	1.73205i
$922$	−1.41421	−	2.44949i	−1.41421	−	2.44949i
$923$		0			0
$924$		0			0
$925$	0.707107	+	1.22474i	0.707107	+	1.22474i
$926$		0			0
$927$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$928$		0			0
$929$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$929$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−2.00000			−2.00000
$952$		0			0
$953$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$953$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$954$	1.00000	+	1.73205i	1.00000	+	1.73205i
$955$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$956$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$957$		0			0
$958$		0			0
$959$		0			0
$960$	0.707107	+	1.22474i	0.707107	+	1.22474i
$961$	1.00000			1.00000
$962$	−2.82843			−2.82843
$963$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$964$		0			0
$965$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$966$		0			0
$967$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$		0			0
$969$		0			0
$970$	2.00000			2.00000
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$973$		0			0
$974$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$975$	1.00000	+	1.73205i	1.00000	+	1.73205i
$976$		0			0
$977$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$977$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$		0			0
$980$	−1.00000			−1.00000
$981$		0			0
$982$	1.41421	+	2.44949i	1.41421	+	2.44949i
$983$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$983$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2.00000			2.00000
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$		0			0

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	1805.1.h.b.654.1		4
5.4	even	2	inner	1805.1.h.b.654.2		4
19.2	odd	18		1805.1.o.b.849.1		12
19.3	odd	18		1805.1.o.b.984.2		12
19.4	even	9		1805.1.o.b.299.1		12
19.5	even	9		1805.1.o.b.694.2		12
19.6	even	9		1805.1.o.b.1029.2		12
19.7	even	3	inner	1805.1.h.b.69.1		4
19.8	odd	6		95.1.d.b.94.1	✓	2
19.9	even	9		1805.1.o.b.1199.2		12
19.10	odd	18		1805.1.o.b.1199.1		12
19.11	even	3		95.1.d.b.94.2	yes	2
19.12	odd	6	inner	1805.1.h.b.69.2		4
19.13	odd	18		1805.1.o.b.1029.1		12
19.14	odd	18		1805.1.o.b.694.1		12
19.15	odd	18		1805.1.o.b.299.2		12
19.16	even	9		1805.1.o.b.984.1		12
19.17	even	9		1805.1.o.b.849.2		12
19.18	odd	2	inner	1805.1.h.b.654.2		4
57.8	even	6		855.1.g.c.379.2		2
57.11	odd	6		855.1.g.c.379.1		2
76.11	odd	6		1520.1.m.b.1329.2		2
76.27	even	6		1520.1.m.b.1329.1		2
95.4	even	18		1805.1.o.b.299.2		12
95.8	even	12		475.1.c.b.151.2		2
95.9	even	18		1805.1.o.b.1199.1		12
95.14	odd	18		1805.1.o.b.694.2		12
95.24	even	18		1805.1.o.b.694.1		12
95.27	even	12		475.1.c.b.151.1		2
95.29	odd	18		1805.1.o.b.1199.2		12
95.34	odd	18		1805.1.o.b.299.1		12
95.44	even	18		1805.1.o.b.1029.1		12
95.49	even	6		95.1.d.b.94.1	✓	2
95.54	even	18		1805.1.o.b.984.2		12
95.59	odd	18		1805.1.o.b.849.2		12
95.64	even	6	inner	1805.1.h.b.69.2		4
95.68	odd	12		475.1.c.b.151.1		2
95.69	odd	6	inner	1805.1.h.b.69.1		4
95.74	even	18		1805.1.o.b.849.1		12
95.79	odd	18		1805.1.o.b.984.1		12
95.84	odd	6		95.1.d.b.94.2	yes	2
95.87	odd	12		475.1.c.b.151.2		2
95.89	odd	18		1805.1.o.b.1029.2		12
95.94	odd	2	CM	1805.1.h.b.654.1		4
285.179	even	6		855.1.g.c.379.1		2
285.239	odd	6		855.1.g.c.379.2		2
380.179	even	6		1520.1.m.b.1329.2		2
380.239	odd	6		1520.1.m.b.1329.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.1.d.b.94.1	✓	2	19.8	odd	6
95.1.d.b.94.1	✓	2	95.49	even	6
95.1.d.b.94.2	yes	2	19.11	even	3
95.1.d.b.94.2	yes	2	95.84	odd	6
475.1.c.b.151.1		2	95.27	even	12
475.1.c.b.151.1		2	95.68	odd	12
475.1.c.b.151.2		2	95.8	even	12
475.1.c.b.151.2		2	95.87	odd	12
855.1.g.c.379.1		2	57.11	odd	6
855.1.g.c.379.1		2	285.179	even	6
855.1.g.c.379.2		2	57.8	even	6
855.1.g.c.379.2		2	285.239	odd	6
1520.1.m.b.1329.1		2	76.27	even	6
1520.1.m.b.1329.1		2	380.239	odd	6
1520.1.m.b.1329.2		2	76.11	odd	6
1520.1.m.b.1329.2		2	380.179	even	6
1805.1.h.b.69.1		4	19.7	even	3	inner
1805.1.h.b.69.1		4	95.69	odd	6	inner
1805.1.h.b.69.2		4	19.12	odd	6	inner
1805.1.h.b.69.2		4	95.64	even	6	inner
1805.1.h.b.654.1		4	1.1	even	1	trivial
1805.1.h.b.654.1		4	95.94	odd	2	CM
1805.1.h.b.654.2		4	5.4	even	2	inner
1805.1.h.b.654.2		4	19.18	odd	2	inner
1805.1.o.b.299.1		12	19.4	even	9
1805.1.o.b.299.1		12	95.34	odd	18
1805.1.o.b.299.2		12	19.15	odd	18
1805.1.o.b.299.2		12	95.4	even	18
1805.1.o.b.694.1		12	19.14	odd	18
1805.1.o.b.694.1		12	95.24	even	18
1805.1.o.b.694.2		12	19.5	even	9
1805.1.o.b.694.2		12	95.14	odd	18
1805.1.o.b.849.1		12	19.2	odd	18
1805.1.o.b.849.1		12	95.74	even	18
1805.1.o.b.849.2		12	19.17	even	9
1805.1.o.b.849.2		12	95.59	odd	18
1805.1.o.b.984.1		12	19.16	even	9
1805.1.o.b.984.1		12	95.79	odd	18
1805.1.o.b.984.2		12	19.3	odd	18
1805.1.o.b.984.2		12	95.54	even	18
1805.1.o.b.1029.1		12	19.13	odd	18
1805.1.o.b.1029.1		12	95.44	even	18
1805.1.o.b.1029.2		12	19.6	even	9
1805.1.o.b.1029.2		12	95.89	odd	18
1805.1.o.b.1199.1		12	19.10	odd	18
1805.1.o.b.1199.1		12	95.9	even	18
1805.1.o.b.1199.2		12	19.9	even	9
1805.1.o.b.1199.2		12	95.29	odd	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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	1805.1.h.b.654.1

Level	$1805$
Weight	$1$
Character	1805.654
Analytic conductor	$0.901$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{4}$
CM discriminant	-95
Inner twists	$8$