LMFDB - Embedded newform 2280.1.cs.a.179.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2280,1,Mod(179,2280)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2280, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 3, 3, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2280.179"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2280.cs (of order $6$ , degree $2$ , minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$1.13786822880$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{2} + 1$ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.1426233024000.1

Embedding invariants

Embedding label			179.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2280.179
Dual form			2280.1.cs.a.1019.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +(0.866025 + 0.500000i) q^{3} -1.00000 q^{4} +(0.866025 + 0.500000i) q^{5} +(-0.500000 + 0.866025i) q^{6} -1.00000i q^{8} +(0.500000 + 0.866025i) q^{9} +(-0.500000 + 0.866025i) q^{10} +(-0.866025 - 0.500000i) q^{12} +(0.500000 + 0.866025i) q^{15} +1.00000 q^{16} +(0.866025 - 1.50000i) q^{17} +(-0.866025 + 0.500000i) q^{18} +(-0.500000 + 0.866025i) q^{19} +(-0.866025 - 0.500000i) q^{20} +(-1.73205 + 1.00000i) q^{23} +(0.500000 - 0.866025i) q^{24} +(0.500000 + 0.866025i) q^{25} +1.00000i q^{27} +(-0.866025 + 0.500000i) q^{30} +1.00000 q^{31} +1.00000i q^{32} +(1.50000 + 0.866025i) q^{34} +(-0.500000 - 0.866025i) q^{36} +(-0.866025 - 0.500000i) q^{38} +(0.500000 - 0.866025i) q^{40} +1.00000i q^{45} +(-1.00000 - 1.73205i) q^{46} +(0.866025 - 0.500000i) q^{47} +(0.866025 + 0.500000i) q^{48} -1.00000 q^{49} +(-0.866025 + 0.500000i) q^{50} +(1.50000 - 0.866025i) q^{51} +(-0.866025 - 1.50000i) q^{53} -1.00000 q^{54} +(-0.866025 + 0.500000i) q^{57} +(-0.500000 - 0.866025i) q^{60} +1.00000i q^{62} -1.00000 q^{64} +(-0.866025 + 1.50000i) q^{68} -2.00000 q^{69} +(0.866025 - 0.500000i) q^{72} +1.00000i q^{75} +(0.500000 - 0.866025i) q^{76} +(1.00000 - 1.73205i) q^{79} +(0.866025 + 0.500000i) q^{80} +(-0.500000 + 0.866025i) q^{81} -1.73205 q^{83} +(1.50000 - 0.866025i) q^{85} -1.00000 q^{90} +(1.73205 - 1.00000i) q^{92} +(0.866025 + 0.500000i) q^{93} +(0.500000 + 0.866025i) q^{94} +(-0.866025 + 0.500000i) q^{95} +(-0.500000 + 0.866025i) q^{96} -1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{4} - 2 q^{6} + 2 q^{9} - 2 q^{10} + 2 q^{15} + 4 q^{16} - 2 q^{19} + 2 q^{24} + 2 q^{25} + 4 q^{31} + 6 q^{34} - 2 q^{36} + 2 q^{40} - 4 q^{46} - 4 q^{49} + 6 q^{51} - 4 q^{54} - 2 q^{60} - 4 q^{64}+ \cdots - 2 q^{96}+O(q^{100})$ 4 * q - 4 * q^4 - 2 * q^6 + 2 * q^9 - 2 * q^10 + 2 * q^15 + 4 * q^16 - 2 * q^19 + 2 * q^24 + 2 * q^25 + 4 * q^31 + 6 * q^34 - 2 * q^36 + 2 * q^40 - 4 * q^46 - 4 * q^49 + 6 * q^51 - 4 * q^54 - 2 * q^60 - 4 * q^64 - 8 * q^69 + 2 * q^76 + 4 * q^79 - 2 * q^81 + 6 * q^85 - 4 * q^90 + 2 * q^94 - 2 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2280\mathbb{Z}\right)^\times$ .

$n$	$457$	$761$	$1141$	$1711$	$1921$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

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

)

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

$2$			1.00000i			1.00000i
$2$			1.00000i			1.00000i
$3$	0.866025	+	0.500000i	0.866025	+	0.500000i
$3$	0.866025	+	0.500000i	0.866025	+	0.500000i
$4$	−1.00000			−1.00000
$5$	0.866025	+	0.500000i	0.866025	+	0.500000i
$5$	0.866025	+	0.500000i	0.866025	+	0.500000i
$6$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		−	1.00000i		−	1.00000i
$9$	0.500000	+	0.866025i	0.500000	+	0.866025i
$10$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$	0.500000	+	0.866025i	0.500000	+	0.866025i
$16$	1.00000			1.00000
$17$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$17$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$18$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$19$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$19$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$20$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$21$		0			0
$22$		0			0
$23$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$23$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$24$	0.500000	−	0.866025i	0.500000	−	0.866025i
$25$	0.500000	+	0.866025i	0.500000	+	0.866025i
$26$		0			0
$27$			1.00000i			1.00000i
$28$		0			0
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$31$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$31$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$32$			1.00000i			1.00000i
$33$		0			0
$34$	1.50000	+	0.866025i	1.50000	+	0.866025i
$35$		0			0
$36$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$39$		0			0
$40$	0.500000	−	0.866025i	0.500000	−	0.866025i
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		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
$45$			1.00000i			1.00000i
$46$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$47$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$47$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$48$	0.866025	+	0.500000i	0.866025	+	0.500000i
$49$	−1.00000			−1.00000
$50$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$51$	1.50000	−	0.866025i	1.50000	−	0.866025i
$52$		0			0
$53$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$53$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$54$	−1.00000			−1.00000
$55$		0			0
$56$		0			0
$57$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$			1.00000i			1.00000i
$63$		0			0
$64$	−1.00000			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$67$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$68$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$69$	−2.00000			−2.00000
$70$		0			0
$71$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$71$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$72$	0.866025	−	0.500000i	0.866025	−	0.500000i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$			1.00000i			1.00000i
$76$	0.500000	−	0.866025i	0.500000	−	0.866025i
$77$		0			0
$78$		0			0
$79$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$79$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$80$	0.866025	+	0.500000i	0.866025	+	0.500000i
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$83$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$84$		0			0
$85$	1.50000	−	0.866025i	1.50000	−	0.866025i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$	−1.00000			−1.00000
$91$		0			0
$92$	1.73205	−	1.00000i	1.73205	−	1.00000i
$93$	0.866025	+	0.500000i	0.866025	+	0.500000i
$94$	0.500000	+	0.866025i	0.500000	+	0.866025i
$95$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$96$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$97$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$97$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$98$		−	1.00000i		−	1.00000i
$99$		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	$0.833333\pi$
$102$	0.866025	+	1.50000i	0.866025	+	1.50000i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	1.50000	−	0.866025i	1.50000	−	0.866025i
$107$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$107$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$108$		−	1.00000i		−	1.00000i
$109$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$109$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$113$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$114$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$115$	−2.00000			−2.00000
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	0.866025	−	0.500000i	0.866025	−	0.500000i
$121$	1.00000			1.00000
$122$		0			0
$123$		0			0
$124$	−1.00000			−1.00000
$125$			1.00000i			1.00000i
$126$		0			0
$127$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$127$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$128$		−	1.00000i		−	1.00000i
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$136$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$137$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$137$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$138$		−	2.00000i		−	2.00000i
$139$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$139$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$	1.00000			1.00000
$142$		0			0
$143$		0			0
$144$	0.500000	+	0.866025i	0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$	−1.00000			−1.00000
$151$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$151$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$152$	0.866025	+	0.500000i	0.866025	+	0.500000i
$153$	1.73205			1.73205
$154$		0			0
$155$	0.866025	+	0.500000i	0.866025	+	0.500000i
$156$		0			0
$157$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$157$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$158$	1.73205	+	1.00000i	1.73205	+	1.00000i
$159$		−	1.73205i		−	1.73205i
$160$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$161$		0			0
$162$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		−	1.73205i		−	1.73205i
$167$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$167$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$	0.866025	+	1.50000i	0.866025	+	1.50000i
$171$	−1.00000			−1.00000
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		−	1.00000i		−	1.00000i
$181$	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
$184$	1.00000	+	1.73205i	1.00000	+	1.73205i
$185$		0			0
$186$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$187$		0			0
$188$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$189$		0			0
$190$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$193$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$193$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$194$		0			0
$195$		0			0
$196$	1.00000			1.00000
$197$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$197$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	0.866025	−	0.500000i	0.866025	−	0.500000i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$205$		0			0
$206$		0			0
$207$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$211$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$212$	0.866025	+	1.50000i	0.866025	+	1.50000i
$213$		0			0
$214$	−1.00000			−1.00000
$215$		0			0
$216$	1.00000			1.00000
$217$		0			0
$218$	0.866025	+	0.500000i	0.866025	+	0.500000i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$226$	1.00000			1.00000
$227$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$227$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$228$	0.866025	−	0.500000i	0.866025	−	0.500000i
$229$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$229$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$230$		−	2.00000i		−	2.00000i
$231$		0			0
$232$		0			0
$233$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$233$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$234$		0			0
$235$	1.00000			1.00000
$236$		0			0
$237$	1.73205	−	1.00000i	1.73205	−	1.00000i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	0.500000	+	0.866025i	0.500000	+	0.866025i
$241$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$241$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$242$			1.00000i			1.00000i
$243$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$244$		0			0
$245$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$246$		0			0
$247$		0			0
$248$		−	1.00000i		−	1.00000i
$249$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$250$	−1.00000			−1.00000
$251$		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
$255$	1.73205			1.73205
$256$	1.00000			1.00000
$257$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$257$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$263$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$264$		0			0
$265$		−	1.73205i		−	1.73205i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$271$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$271$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$272$	0.866025	−	1.50000i	0.866025	−	1.50000i
$273$		0			0
$274$	1.50000	−	0.866025i	1.50000	−	0.866025i
$275$		0			0
$276$	2.00000			2.00000
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$279$	0.500000	+	0.866025i	0.500000	+	0.866025i
$280$		0			0
$281$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$281$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$282$			1.00000i			1.00000i
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$	−1.00000			−1.00000
$286$		0			0
$287$		0			0
$288$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$289$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$293$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$294$	0.500000	−	0.866025i	0.500000	−	0.866025i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		−	1.00000i		−	1.00000i
$301$		0			0
$302$		−	1.00000i		−	1.00000i
$303$		0			0
$304$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$305$		0			0
$306$			1.73205i			1.73205i
$307$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$307$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$308$		0			0
$309$		0			0
$310$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$311$		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
$316$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$317$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$317$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$318$	1.73205			1.73205
$319$		0			0
$320$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$321$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$322$		0			0
$323$	0.866025	+	1.50000i	0.866025	+	1.50000i
$324$	0.500000	−	0.866025i	0.500000	−	0.866025i
$325$		0			0
$326$		0			0
$327$	0.866025	−	0.500000i	0.866025	−	0.500000i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$	1.73205			1.73205
$333$		0			0
$334$	1.50000	−	0.866025i	1.50000	−	0.866025i
$335$		0			0
$336$		0			0
$337$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$337$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$338$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$339$	0.500000	−	0.866025i	0.500000	−	0.866025i
$340$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$341$		0			0
$342$		−	1.00000i		−	1.00000i
$343$		0			0
$344$		0			0
$345$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$346$		0			0
$347$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$347$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$349$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		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	$0.166667\pi$
$360$	1.00000			1.00000
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$363$	0.866025	+	0.500000i	0.866025	+	0.500000i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$376$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	0.866025	−	0.500000i	0.866025	−	0.500000i
$381$		0			0
$382$		0			0
$383$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$383$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$384$	0.500000	−	0.866025i	0.500000	−	0.866025i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$389$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$390$		0			0
$391$			3.46410i			3.46410i
$392$			1.00000i			1.00000i
$393$		0			0
$394$	1.00000			1.00000
$395$	1.73205	−	1.00000i	1.73205	−	1.00000i
$396$		0			0
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$	0.500000	+	0.866025i	0.500000	+	0.866025i
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$406$		0			0
$407$		0			0
$408$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		−	1.73205i		−	1.73205i
$412$		0			0
$413$		0			0
$414$	1.00000	−	1.73205i	1.00000	−	1.73205i
$415$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$416$		0			0
$417$			1.00000i			1.00000i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$421$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$422$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$423$	0.866025	+	0.500000i	0.866025	+	0.500000i
$424$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$425$	1.73205			1.73205
$426$		0			0
$427$		0			0
$428$		−	1.00000i		−	1.00000i
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$			1.00000i			1.00000i
$433$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$433$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$434$		0			0
$435$		0			0
$436$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$437$		−	2.00000i		−	2.00000i
$438$		0			0
$439$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$439$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$440$		0			0
$441$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$442$		0			0
$443$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$443$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$451$		0			0
$452$			1.00000i			1.00000i
$453$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$454$	2.00000			2.00000
$455$		0			0
$456$	0.500000	+	0.866025i	0.500000	+	0.866025i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−1.73205			−1.73205
$459$	1.50000	+	0.866025i	1.50000	+	0.866025i
$460$	2.00000			2.00000
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$	0.500000	+	0.866025i	0.500000	+	0.866025i
$466$	1.50000	+	0.866025i	1.50000	+	0.866025i
$467$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$467$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$468$		0			0
$469$		0			0
$470$			1.00000i			1.00000i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	1.00000	+	1.73205i	1.00000	+	1.73205i
$475$	−1.00000			−1.00000
$476$		0			0
$477$	0.866025	−	1.50000i	0.866025	−	1.50000i
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$481$		0			0
$482$	0.866025	+	1.50000i	0.866025	+	1.50000i
$483$		0			0
$484$	−1.00000			−1.00000
$485$		0			0
$486$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$	0.500000	−	0.866025i	0.500000	−	0.866025i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.00000			1.00000
$497$		0			0
$498$	0.866025	−	1.50000i	0.866025	−	1.50000i
$499$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$499$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$500$		−	1.00000i		−	1.00000i
$501$		−	1.73205i		−	1.73205i
$502$		0			0
$503$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$503$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$			1.73205i			1.73205i
$511$		0			0
$512$			1.00000i			1.00000i
$513$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$514$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$523$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$		0			0
$526$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$527$	0.866025	−	1.50000i	0.866025	−	1.50000i
$528$		0			0
$529$	1.50000	−	2.59808i	1.50000	−	2.59808i
$530$	1.73205			1.73205
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	0.500000	−	0.866025i	0.500000	−	0.866025i
$541$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$541$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$542$	0.866025	−	1.50000i	0.866025	−	1.50000i
$543$			1.00000i			1.00000i
$544$	1.50000	+	0.866025i	1.50000	+	0.866025i
$545$	0.866025	−	0.500000i	0.866025	−	0.500000i
$546$		0			0
$547$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$547$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$548$	0.866025	+	1.50000i	0.866025	+	1.50000i
$549$		0			0
$550$		0			0
$551$		0			0
$552$			2.00000i			2.00000i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$557$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$557$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$558$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$563$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$564$	−1.00000			−1.00000
$565$	0.500000	−	0.866025i	0.500000	−	0.866025i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		−	1.00000i		−	1.00000i
$571$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$576$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	1.73205	−	1.00000i	1.73205	−	1.00000i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$			1.73205i			1.73205i
$587$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$587$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$588$	0.866025	+	0.500000i	0.866025	+	0.500000i
$589$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$590$		0			0
$591$	0.500000	−	0.866025i	0.500000	−	0.866025i
$592$		0			0
$593$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$593$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	1.00000			1.00000
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	1.00000			1.00000
$605$	0.866025	+	0.500000i	0.866025	+	0.500000i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−1.73205			−1.73205
$613$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$613$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$617$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$618$		0			0
$619$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$619$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$620$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$621$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$631$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$632$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$633$	0.866025	+	1.50000i	0.866025	+	1.50000i
$634$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$635$		0			0
$636$			1.73205i			1.73205i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.500000	−	0.866025i	0.500000	−	0.866025i
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$643$		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
$646$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$647$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$647$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$648$	0.866025	+	0.500000i	0.866025	+	0.500000i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$653$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$654$	0.500000	+	0.866025i	0.500000	+	0.866025i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$659$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$660$		0			0
$661$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$661$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$			1.73205i			1.73205i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	0.866025	+	1.50000i	0.866025	+	1.50000i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$676$	0.500000	−	0.866025i	0.500000	−	0.866025i
$677$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$677$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$678$	0.866025	+	0.500000i	0.866025	+	0.500000i
$679$		0			0
$680$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$681$	1.00000	−	1.73205i	1.00000	−	1.73205i
$682$		0			0
$683$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$683$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$684$	1.00000			1.00000
$685$		−	1.73205i		−	1.73205i
$686$		0			0
$687$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$688$		0			0
$689$		0			0
$690$	1.00000	−	1.73205i	1.00000	−	1.73205i
$691$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$691$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$695$			1.00000i			1.00000i
$696$		0			0
$697$		0			0
$698$	−1.73205			−1.73205
$699$	1.50000	−	0.866025i	1.50000	−	0.866025i
$700$		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
$705$	0.866025	+	0.500000i	0.866025	+	0.500000i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$709$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$710$		0			0
$711$	2.00000			2.00000
$712$		0			0
$713$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$			1.00000i			1.00000i
$721$		0			0
$722$	0.866025	−	0.500000i	0.866025	−	0.500000i
$723$	1.73205			1.73205
$724$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$725$		0			0
$726$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$727$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$727$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$728$		0			0
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$736$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$737$		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
$743$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$743$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$744$	0.500000	−	0.866025i	0.500000	−	0.866025i
$745$		0			0
$746$		0			0
$747$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$748$		0			0
$749$		0			0
$750$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$751$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$751$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$752$	0.866025	−	0.500000i	0.866025	−	0.500000i
$753$		0			0
$754$		0			0
$755$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$756$		0			0
$757$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$757$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$758$		0			0
$759$		0			0
$760$	0.500000	+	0.866025i	0.500000	+	0.866025i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.50000	+	0.866025i	1.50000	+	0.866025i
$766$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$767$		0			0
$768$	0.866025	+	0.500000i	0.866025	+	0.500000i
$769$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$769$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$770$		0			0
$771$	−2.00000			−2.00000
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		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
$782$	−3.46410			−3.46410
$783$		0			0
$784$	−1.00000			−1.00000
$785$		0			0
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$			1.00000i			1.00000i
$789$	0.500000	+	0.866025i	0.500000	+	0.866025i
$790$	1.00000	+	1.73205i	1.00000	+	1.73205i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	0.866025	−	1.50000i	0.866025	−	1.50000i
$796$		0			0
$797$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$797$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		−	1.73205i		−	1.73205i
$800$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$801$		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			$\pi$
$810$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$811$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$811$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$812$		0			0
$813$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$814$		0			0
$815$		0			0
$816$	1.50000	−	0.866025i	1.50000	−	0.866025i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$821$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$822$	1.73205			1.73205
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$827$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$828$	1.73205	+	1.00000i	1.73205	+	1.00000i
$829$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$829$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$830$	0.866025	−	1.50000i	0.866025	−	1.50000i
$831$		0			0
$832$		0			0
$833$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$834$	−1.00000			−1.00000
$835$		−	1.73205i		−	1.73205i
$836$		0			0
$837$			1.00000i			1.00000i
$838$		0			0
$839$		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
$842$	1.73205	+	1.00000i	1.73205	+	1.00000i
$843$		0			0
$844$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$845$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$846$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$847$		0			0
$848$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$849$		0			0
$850$			1.73205i			1.73205i
$851$		0			0
$852$		0			0
$853$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$853$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$854$		0			0
$855$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$856$	1.00000			1.00000
$857$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$857$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$859$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	−1.00000			−1.00000
$865$		0			0
$866$		0			0
$867$		−	2.00000i		−	2.00000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$873$		0			0
$874$	2.00000			2.00000
$875$		0			0
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$	0.866025	+	0.500000i	0.866025	+	0.500000i
$879$	1.50000	+	0.866025i	1.50000	+	0.866025i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	0.866025	−	0.500000i	0.866025	−	0.500000i
$883$		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
$886$	1.50000	−	0.866025i	1.50000	−	0.866025i
$887$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$887$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$			1.00000i			1.00000i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.500000	−	0.866025i	0.500000	−	0.866025i
$901$	−3.00000			−3.00000
$902$		0			0
$903$		0			0
$904$	−1.00000			−1.00000
$905$			1.00000i			1.00000i
$906$	0.500000	−	0.866025i	0.500000	−	0.866025i
$907$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$907$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$908$			2.00000i			2.00000i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$913$		0			0
$914$		0			0
$915$		0			0
$916$		−	1.73205i		−	1.73205i
$917$		0			0
$918$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$919$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$919$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$920$			2.00000i			2.00000i
$921$		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		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$931$	0.500000	−	0.866025i	0.500000	−	0.866025i
$932$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$933$		0			0
$934$			1.73205i			1.73205i
$935$		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
$940$	−1.00000			−1.00000
$941$		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		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$949$		0			0
$950$		−	1.00000i		−	1.00000i
$951$			1.73205i			1.73205i
$952$		0			0
$953$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$953$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$954$	1.50000	+	0.866025i	1.50000	+	0.866025i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$961$		0			0
$962$		0			0
$963$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$964$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$965$		0			0
$966$		0			0
$967$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$967$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$968$		−	1.00000i		−	1.00000i
$969$			1.73205i			1.73205i
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$	0.866025	−	0.500000i	0.866025	−	0.500000i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$977$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$978$		0			0
$979$		0			0
$980$	0.866025	+	0.500000i	0.866025	+	0.500000i
$981$	1.00000			1.00000
$982$		0			0
$983$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$983$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$	0.500000	−	0.866025i	0.500000	−	0.866025i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$991$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$992$			1.00000i			1.00000i
$993$		0			0
$994$		0			0
$995$		0			0
$996$	1.50000	+	0.866025i	1.50000	+	0.866025i
$997$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$997$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$998$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$999$		0			0

Display

a_p

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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	2280.1.cs.a.179.2	yes	4
3.2	odd	2	inner	2280.1.cs.a.179.1	✓	4
5.4	even	2	inner	2280.1.cs.a.179.1	✓	4
8.3	odd	2		2280.1.cs.b.179.2	yes	4
15.14	odd	2	CM	2280.1.cs.a.179.2	yes	4
19.12	odd	6		2280.1.cs.b.1019.2	yes	4
24.11	even	2		2280.1.cs.b.179.1	yes	4
40.19	odd	2		2280.1.cs.b.179.1	yes	4
57.50	even	6		2280.1.cs.b.1019.1	yes	4
95.69	odd	6		2280.1.cs.b.1019.1	yes	4
120.59	even	2		2280.1.cs.b.179.2	yes	4
152.107	even	6	inner	2280.1.cs.a.1019.1	yes	4
285.164	even	6		2280.1.cs.b.1019.2	yes	4
456.107	odd	6	inner	2280.1.cs.a.1019.2	yes	4
760.259	even	6	inner	2280.1.cs.a.1019.2	yes	4
2280.1019	odd	6	inner	2280.1.cs.a.1019.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.1.cs.a.179.1	✓	4	3.2	odd	2	inner
2280.1.cs.a.179.1	✓	4	5.4	even	2	inner
2280.1.cs.a.179.2	yes	4	1.1	even	1	trivial
2280.1.cs.a.179.2	yes	4	15.14	odd	2	CM
2280.1.cs.a.1019.1	yes	4	152.107	even	6	inner
2280.1.cs.a.1019.1	yes	4	2280.1019	odd	6	inner
2280.1.cs.a.1019.2	yes	4	456.107	odd	6	inner
2280.1.cs.a.1019.2	yes	4	760.259	even	6	inner
2280.1.cs.b.179.1	yes	4	24.11	even	2
2280.1.cs.b.179.1	yes	4	40.19	odd	2
2280.1.cs.b.179.2	yes	4	8.3	odd	2
2280.1.cs.b.179.2	yes	4	120.59	even	2
2280.1.cs.b.1019.1	yes	4	57.50	even	6
2280.1.cs.b.1019.1	yes	4	95.69	odd	6
2280.1.cs.b.1019.2	yes	4	19.12	odd	6
2280.1.cs.b.1019.2	yes	4	285.164	even	6

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 $\F_{q}$
Belyi maps

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	2280.1.cs.a.179.2

Level	$2280$
Weight	$1$
Character	2280.179
Analytic conductor	$1.138$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{6}$
CM discriminant	-15
Inner twists	$8$