LMFDB - Embedded newform 3420.1.bn.b.2279.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3420,1,Mod(1139,3420)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3420.1139"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3420, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5, 3, 3])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3420.bn (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(24)] == traces)

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

Self dual:	no
Analytic conductor:	$1.70680234320$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{48})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{8} + 1 $ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{24}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{24} - \cdots)$

Embedding invariants

Embedding label			2279.5
Root			$0.608761 + 0.793353i$ of defining polynomial
Character	$\chi$	$=$	3420.2279
Dual form			3420.1.bn.b.1139.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.382683 - 0.923880i) q^{2} +(0.130526 - 0.991445i) q^{3} +(-0.707107 - 0.707107i) q^{4} +(-0.866025 + 0.500000i) q^{5} +(-0.866025 - 0.500000i) q^{6} +(-0.923880 + 0.382683i) q^{8} +(-0.965926 - 0.258819i) q^{9} +(0.130526 + 0.991445i) q^{10} +(-0.965926 + 1.67303i) q^{11} +(-0.793353 + 0.608761i) q^{12} +(0.608761 + 1.05441i) q^{13} +(0.382683 + 0.923880i) q^{15} +1.00000i q^{16} +(-0.608761 + 0.793353i) q^{18} -1.00000i q^{19} +(0.965926 + 0.258819i) q^{20} +(1.17604 + 1.53264i) q^{22} +(0.258819 + 0.965926i) q^{24} +(0.500000 - 0.866025i) q^{25} +(1.20711 - 0.158919i) q^{26} +(-0.382683 + 0.923880i) q^{27} +1.00000 q^{30} +(0.923880 + 0.382683i) q^{32} +(1.53264 + 1.17604i) q^{33} +(0.500000 + 0.866025i) q^{36} +1.58671 q^{37} +(-0.923880 - 0.382683i) q^{38} +(1.12484 - 0.465926i) q^{39} +(0.608761 - 0.793353i) q^{40} +(1.86603 - 0.500000i) q^{44} +(0.965926 - 0.258819i) q^{45} +(0.991445 + 0.130526i) q^{48} +(0.500000 + 0.866025i) q^{49} +(-0.608761 - 0.793353i) q^{50} +(0.315118 - 1.17604i) q^{52} +1.98289i q^{53} +(0.707107 + 0.707107i) q^{54} -1.93185i q^{55} +(-0.991445 - 0.130526i) q^{57} +(0.382683 - 0.923880i) q^{60} +(-0.707107 + 1.22474i) q^{61} +(0.707107 - 0.707107i) q^{64} +(-1.05441 - 0.608761i) q^{65} +(1.67303 - 0.965926i) q^{66} +(-1.60021 + 0.923880i) q^{67} +(0.991445 - 0.130526i) q^{72} +(0.607206 - 1.46593i) q^{74} +(-0.793353 - 0.608761i) q^{75} +(-0.707107 + 0.707107i) q^{76} -1.21752i q^{78} +(-0.500000 - 0.866025i) q^{80} +(0.866025 + 0.500000i) q^{81} +(0.252157 - 1.91532i) q^{88} +(0.130526 - 0.991445i) q^{90} +(0.500000 + 0.866025i) q^{95} +(0.500000 - 0.866025i) q^{96} +(-0.923880 + 1.60021i) q^{97} +(0.991445 - 0.130526i) q^{98} +(1.36603 - 1.36603i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{25} + 8 q^{26} + 16 q^{30} + 8 q^{36} + 16 q^{44} + 8 q^{49} - 8 q^{80} + 8 q^{95} + 8 q^{96} + 8 q^{99}+O(q^{100}) $ 16 * q + 8 * q^25 + 8 * q^26 + 16 * q^30 + 8 * q^36 + 16 * q^44 + 8 * q^49 - 8 * q^80 + 8 * q^95 + 8 * q^96 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$.

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0.382683	−	0.923880i	0.382683	−	0.923880i
$2$	0.382683	−	0.923880i	0.382683	−	0.923880i
$3$	0.130526	−	0.991445i	0.130526	−	0.991445i
$3$	0.130526	−	0.991445i	0.130526	−	0.991445i
$4$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$5$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$5$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$6$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$9$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$10$	0.130526	+	0.991445i	0.130526	+	0.991445i
$11$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$11$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$12$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$13$	0.608761	+	1.05441i	0.608761	+	1.05441i	0.991445	+	0.130526i	$0.0416667\pi$
$13$	0.608761	+	1.05441i	0.608761	+	1.05441i	−0.382683	+	0.923880i	$0.625000\pi$
$14$		0			0
$15$	0.382683	+	0.923880i	0.382683	+	0.923880i
$16$			1.00000i			1.00000i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$19$		−	1.00000i		−	1.00000i
$19$		−	1.00000i		−	1.00000i
$20$	0.965926	+	0.258819i	0.965926	+	0.258819i
$21$		0			0
$22$	1.17604	+	1.53264i	1.17604	+	1.53264i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	0.258819	+	0.965926i	0.258819	+	0.965926i
$25$	0.500000	−	0.866025i	0.500000	−	0.866025i
$26$	1.20711	−	0.158919i	1.20711	−	0.158919i
$27$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$	1.00000			1.00000
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	0.923880	+	0.382683i	0.923880	+	0.382683i
$33$	1.53264	+	1.17604i	1.53264	+	1.17604i
$34$		0			0
$35$		0			0
$36$	0.500000	+	0.866025i	0.500000	+	0.866025i
$37$	1.58671			1.58671			0.793353	−	0.608761i	$-0.208333\pi$
$37$	1.58671			1.58671			0.793353	+	0.608761i	$0.208333\pi$
$38$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$39$	1.12484	−	0.465926i	1.12484	−	0.465926i
$40$	0.608761	−	0.793353i	0.608761	−	0.793353i
$41$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$41$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$42$		0			0
$43$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$43$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$44$	1.86603	−	0.500000i	1.86603	−	0.500000i
$45$	0.965926	−	0.258819i	0.965926	−	0.258819i
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	0.991445	+	0.130526i	0.991445	+	0.130526i
$49$	0.500000	+	0.866025i	0.500000	+	0.866025i
$50$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$51$		0			0
$52$	0.315118	−	1.17604i	0.315118	−	1.17604i
$53$			1.98289i			1.98289i	0.130526	+	0.991445i	$0.458333\pi$
$53$			1.98289i			1.98289i	−0.130526	+	0.991445i	$0.541667\pi$
$54$	0.707107	+	0.707107i	0.707107	+	0.707107i
$55$		−	1.93185i		−	1.93185i
$56$		0			0
$57$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$	0.382683	−	0.923880i	0.382683	−	0.923880i
$61$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$61$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$62$		0			0
$63$		0			0
$64$	0.707107	−	0.707107i	0.707107	−	0.707107i
$65$	−1.05441	−	0.608761i	−1.05441	−	0.608761i
$66$	1.67303	−	0.965926i	1.67303	−	0.965926i
$67$	−1.60021	+	0.923880i	−1.60021	+	0.923880i	−0.608761	+	0.793353i	$0.708333\pi$
$67$	−1.60021	+	0.923880i	−1.60021	+	0.923880i	−0.991445	+	0.130526i	$0.958333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.991445	−	0.130526i	0.991445	−	0.130526i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	0.607206	−	1.46593i	0.607206	−	1.46593i
$75$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$76$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$77$		0			0
$78$		−	1.21752i		−	1.21752i
$79$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$79$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$80$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$81$	0.866025	+	0.500000i	0.866025	+	0.500000i
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.252157	−	1.91532i	0.252157	−	1.91532i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	0.130526	−	0.991445i	0.130526	−	0.991445i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	0.500000	+	0.866025i	0.500000	+	0.866025i
$96$	0.500000	−	0.866025i	0.500000	−	0.866025i
$97$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$97$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$98$	0.991445	−	0.130526i	0.991445	−	0.130526i
$99$	1.36603	−	1.36603i	1.36603	−	1.36603i
$100$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$101$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$101$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$102$		0			0
$103$	0.226078	−	0.130526i	0.226078	−	0.130526i	−0.382683	−	0.923880i	$-0.625000\pi$
$103$	0.226078	−	0.130526i	0.226078	−	0.130526i	0.608761	+	0.793353i	$0.291667\pi$
$104$	−0.965926	−	0.741181i	−0.965926	−	0.741181i
$105$		0			0
$106$	1.83195	+	0.758819i	1.83195	+	0.758819i
$107$	1.21752			1.21752			0.608761	−	0.793353i	$-0.291667\pi$
$107$	1.21752			1.21752			0.608761	+	0.793353i	$0.291667\pi$
$108$	0.923880	−	0.382683i	0.923880	−	0.382683i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	−1.78480	−	0.739288i	−1.78480	−	0.739288i
$111$	0.207107	−	1.57313i	0.207107	−	1.57313i
$112$		0			0
$113$	−1.60021	+	0.923880i	−1.60021	+	0.923880i	−0.608761	+	0.793353i	$0.708333\pi$
$113$	−1.60021	+	0.923880i	−1.60021	+	0.923880i	−0.991445	+	0.130526i	$0.958333\pi$
$114$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$115$		0			0
$116$		0			0
$117$	−0.315118	−	1.17604i	−0.315118	−	1.17604i
$118$		0			0
$119$		0			0
$120$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$121$	−1.36603	−	2.36603i	−1.36603	−	2.36603i
$122$	0.860919	+	1.12197i	0.860919	+	1.12197i
$123$		0			0
$124$		0			0
$125$			1.00000i			1.00000i
$126$		0			0
$127$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$127$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$128$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$129$		0			0
$130$	−0.965926	+	0.741181i	−0.965926	+	0.741181i
$131$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$131$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$132$	−0.252157	−	1.91532i	−0.252157	−	1.91532i
$133$		0			0
$134$	0.241181	+	1.83195i	0.241181	+	1.83195i
$135$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$139$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.35207			−2.35207
$144$	0.258819	−	0.965926i	0.258819	−	0.965926i
$145$		0			0
$146$		0			0
$147$	0.923880	−	0.382683i	0.923880	−	0.382683i
$148$	−1.12197	−	1.12197i	−1.12197	−	1.12197i
$149$	1.67303	−	0.965926i	1.67303	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$149$	1.67303	−	0.965926i	1.67303	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$150$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$151$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$151$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$152$	0.382683	+	0.923880i	0.382683	+	0.923880i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.12484	−	0.465926i	−1.12484	−	0.465926i
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$	1.96593	+	0.258819i	1.96593	+	0.258819i
$160$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$161$		0			0
$162$	0.793353	−	0.608761i	0.793353	−	0.608761i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$	−1.91532	−	0.252157i	−1.91532	−	0.252157i
$166$		0			0
$167$	−0.130526	−	0.226078i	−0.130526	−	0.226078i	0.793353	−	0.608761i	$-0.208333\pi$
$167$	−0.130526	−	0.226078i	−0.130526	−	0.226078i	−0.923880	+	0.382683i	$0.875000\pi$
$168$		0			0
$169$	−0.241181	+	0.417738i	−0.241181	+	0.417738i
$170$		0			0
$171$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$172$		0			0
$173$	1.37413	+	0.793353i	1.37413	+	0.793353i	0.991445	−	0.130526i	$-0.0416667\pi$
$173$	1.37413	+	0.793353i	1.37413	+	0.793353i	0.382683	+	0.923880i	$0.375000\pi$
$174$		0			0
$175$		0			0
$176$	−1.67303	−	0.965926i	−1.67303	−	0.965926i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	1.12197	+	0.860919i	1.12197	+	0.860919i
$184$		0			0
$185$	−1.37413	+	0.793353i	−1.37413	+	0.793353i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	0.991445	−	0.130526i	0.991445	−	0.130526i
$191$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$191$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$192$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$193$	0.130526	+	0.226078i	0.130526	+	0.226078i	0.923880	−	0.382683i	$-0.125000\pi$
$193$	0.130526	+	0.226078i	0.130526	+	0.226078i	−0.793353	+	0.608761i	$0.791667\pi$
$194$	1.12484	+	1.46593i	1.12484	+	1.46593i
$195$	−0.741181	+	0.965926i	−0.741181	+	0.965926i
$196$	0.258819	−	0.965926i	0.258819	−	0.965926i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	−0.739288	−	1.78480i	−0.739288	−	1.78480i
$199$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$199$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$200$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$201$	0.707107	+	1.70711i	0.707107	+	1.70711i
$202$	−1.12197	+	0.860919i	−1.12197	+	0.860919i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−0.0340742	−	0.258819i	−0.0340742	−	0.258819i
$207$		0			0
$208$	−1.05441	+	0.608761i	−1.05441	+	0.608761i
$209$	1.67303	+	0.965926i	1.67303	+	0.965926i
$210$		0			0
$211$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$211$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$212$	1.40211	−	1.40211i	1.40211	−	1.40211i
$213$		0			0
$214$	0.465926	−	1.12484i	0.465926	−	1.12484i
$215$		0			0
$216$		−	1.00000i		−	1.00000i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−1.36603	+	1.36603i	−1.36603	+	1.36603i
$221$		0			0
$222$	−1.37413	−	0.793353i	−1.37413	−	0.793353i
$223$	1.71723	+	0.991445i	1.71723	+	0.991445i	0.923880	+	0.382683i	$0.125000\pi$
$223$	1.71723	+	0.991445i	1.71723	+	0.991445i	0.793353	+	0.608761i	$0.208333\pi$
$224$		0			0
$225$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$226$	0.241181	+	1.83195i	0.241181	+	1.83195i
$227$	0.382683	−	0.662827i	0.382683	−	0.662827i	−0.608761	−	0.793353i	$-0.708333\pi$
$227$	0.382683	−	0.662827i	0.382683	−	0.662827i	0.991445	+	0.130526i	$0.0416667\pi$
$228$	0.608761	+	0.793353i	0.608761	+	0.793353i
$229$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$229$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	−1.20711	−	0.158919i	−1.20711	−	0.158919i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$239$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$240$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−2.70868	+	0.356604i	−2.70868	+	0.356604i
$243$	0.608761	−	0.793353i	0.608761	−	0.793353i
$244$	1.36603	−	0.366025i	1.36603	−	0.366025i
$245$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$246$		0			0
$247$	1.05441	−	0.608761i	1.05441	−	0.608761i
$248$		0			0
$249$		0			0
$250$	0.923880	+	0.382683i	0.923880	+	0.382683i
$251$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$251$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$		0			0
$254$	−0.707107	−	0.292893i	−0.707107	−	0.292893i
$255$		0			0
$256$	−1.00000			−1.00000
$257$	0.662827	−	0.382683i	0.662827	−	0.382683i	−0.130526	−	0.991445i	$-0.541667\pi$
$257$	0.662827	−	0.382683i	0.662827	−	0.382683i	0.793353	+	0.608761i	$0.208333\pi$
$258$		0			0
$259$		0			0
$260$	0.315118	+	1.17604i	0.315118	+	1.17604i
$261$		0			0
$262$	0.991445	−	0.130526i	0.991445	−	0.130526i
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$265$	−0.991445	−	1.71723i	−0.991445	−	1.71723i
$266$		0			0
$267$		0			0
$268$	1.78480	+	0.478235i	1.78480	+	0.478235i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$271$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$271$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.965926	+	1.67303i	0.965926	+	1.67303i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	0.184592	+	1.40211i	0.184592	+	1.40211i
$279$		0			0
$280$		0			0
$281$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$281$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$282$		0			0
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$		0			0
$285$	0.923880	−	0.382683i	0.923880	−	0.382683i
$286$	−0.900100	+	2.17303i	−0.900100	+	2.17303i
$287$		0			0
$288$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$289$	−1.00000			−1.00000
$290$		0			0
$291$	1.46593	+	1.12484i	1.46593	+	1.12484i
$292$		0			0
$293$	−0.662827	+	0.382683i	−0.662827	+	0.382683i	−0.793353	−	0.608761i	$-0.791667\pi$
$293$	−0.662827	+	0.382683i	−0.662827	+	0.382683i	0.130526	+	0.991445i	$0.458333\pi$
$294$		−	1.00000i		−	1.00000i
$295$		0			0
$296$	−1.46593	+	0.607206i	−1.46593	+	0.607206i
$297$	−1.17604	−	1.53264i	−1.17604	−	1.53264i
$298$	−0.252157	−	1.91532i	−0.252157	−	1.91532i
$299$		0			0
$300$	0.130526	+	0.991445i	0.130526	+	0.991445i
$301$		0			0
$302$		0			0
$303$	−0.860919	+	1.12197i	−0.860919	+	1.12197i
$304$	1.00000			1.00000
$305$		−	1.41421i		−	1.41421i
$306$		0			0
$307$		−	0.261052i		−	0.261052i	−0.991445	−	0.130526i	$-0.958333\pi$
$307$		−	0.261052i		−	0.261052i	0.991445	−	0.130526i	$-0.0416667\pi$
$308$		0			0
$309$	−0.0999004	−	0.241181i	−0.0999004	−	0.241181i
$310$		0			0
$311$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$311$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$312$	−0.860919	+	0.860919i	−0.860919	+	0.860919i
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.60021	+	0.923880i	1.60021	+	0.923880i	0.991445	+	0.130526i	$0.0416667\pi$
$317$	1.60021	+	0.923880i	1.60021	+	0.923880i	0.608761	+	0.793353i	$0.291667\pi$
$318$	0.991445	−	1.71723i	0.991445	−	1.71723i
$319$		0			0
$320$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$321$	0.158919	−	1.20711i	0.158919	−	1.20711i
$322$		0			0
$323$		0			0
$324$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$325$	1.21752			1.21752
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$331$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$331$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$332$		0			0
$333$	−1.53264	−	0.410670i	−1.53264	−	0.410670i
$334$	−0.258819	+	0.0340742i	−0.258819	+	0.0340742i
$335$	0.923880	−	1.60021i	0.923880	−	1.60021i
$336$		0			0
$337$	0.382683	+	0.662827i	0.382683	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$337$	0.382683	+	0.662827i	0.382683	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$338$	0.293643	+	0.382683i	0.293643	+	0.382683i
$339$	0.707107	+	1.70711i	0.707107	+	1.70711i
$340$		0			0
$341$		0			0
$342$	0.793353	+	0.608761i	0.793353	+	0.608761i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	1.25882	−	0.965926i	1.25882	−	0.965926i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$349$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$350$		0			0
$351$	−1.20711	+	0.158919i	−1.20711	+	0.158919i
$352$	−1.53264	+	1.17604i	−1.53264	+	1.17604i
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$359$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$360$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$361$	−1.00000			−1.00000
$362$		0			0
$363$	−2.52409	+	1.04551i	−2.52409	+	1.04551i
$364$		0			0
$365$		0			0
$366$	1.22474	−	0.707107i	1.22474	−	0.707107i
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$		0			0
$370$	0.207107	+	1.57313i	0.207107	+	1.57313i
$371$		0			0
$372$		0			0
$373$	−0.130526	−	0.226078i	−0.130526	−	0.226078i	0.793353	−	0.608761i	$-0.208333\pi$
$373$	−0.130526	−	0.226078i	−0.130526	−	0.226078i	−0.923880	+	0.382683i	$0.875000\pi$
$374$		0			0
$375$	0.991445	+	0.130526i	0.991445	+	0.130526i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$	0.258819	−	0.965926i	0.258819	−	0.965926i
$381$	−0.758819	−	0.0999004i	−0.758819	−	0.0999004i
$382$	1.05441	+	1.37413i	1.05441	+	1.37413i
$383$	−0.608761	−	1.05441i	−0.608761	−	1.05441i	−0.991445	−	0.130526i	$-0.958333\pi$
$383$	−0.608761	−	1.05441i	−0.608761	−	1.05441i	0.382683	−	0.923880i	$-0.375000\pi$
$384$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$385$		0			0
$386$	0.258819	−	0.0340742i	0.258819	−	0.0340742i
$387$		0			0
$388$	1.78480	−	0.478235i	1.78480	−	0.478235i
$389$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$389$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$390$	0.608761	+	1.05441i	0.608761	+	1.05441i
$391$		0			0
$392$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$393$	0.923880	−	0.382683i	0.923880	−	0.382683i
$394$		0			0
$395$		0			0
$396$	−1.93185			−1.93185
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$399$		0			0
$400$	0.866025	+	0.500000i	0.866025	+	0.500000i
$401$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$401$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$402$	1.84776			1.84776
$403$		0			0
$404$	0.366025	+	1.36603i	0.366025	+	1.36603i
$405$	−1.00000			−1.00000
$406$		0			0
$407$	−1.53264	+	2.65461i	−1.53264	+	2.65461i
$408$		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
$412$	−0.252157	−	0.0675653i	−0.252157	−	0.0675653i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.158919	+	1.20711i	0.158919	+	1.20711i
$417$	0.541196	+	1.30656i	0.541196	+	1.30656i
$418$	1.53264	−	1.17604i	1.53264	−	1.17604i
$419$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$420$		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
$424$	−0.758819	−	1.83195i	−0.758819	−	1.83195i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.860919	−	0.860919i	−0.860919	−	0.860919i
$429$	−0.307007	+	2.33195i	−0.307007	+	2.33195i
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$433$	0.765367			0.765367			0.382683	−	0.923880i	$-0.375000\pi$
$433$	0.765367			0.765367			0.382683	+	0.923880i	$0.375000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$	0.739288	+	1.78480i	0.739288	+	1.78480i
$441$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−1.25882	+	0.965926i	−1.25882	+	0.965926i
$445$		0			0
$446$	1.57313	−	1.20711i	1.57313	−	1.20711i
$447$	−0.739288	−	1.78480i	−0.739288	−	1.78480i
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	0.382683	+	0.923880i	0.382683	+	0.923880i
$451$		0			0
$452$	1.78480	+	0.478235i	1.78480	+	0.478235i
$453$		0			0
$454$	−0.465926	−	0.607206i	−0.465926	−	0.607206i
$455$		0			0
$456$	0.965926	−	0.258819i	0.965926	−	0.258819i
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	0.513210	−	0.0675653i	0.513210	−	0.0675653i
$459$		0			0
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$463$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	−0.608761	+	1.05441i	−0.608761	+	1.05441i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$476$		0			0
$477$	0.513210	−	1.91532i	0.513210	−	1.91532i
$478$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$479$	0.258819	−	0.448288i	0.258819	−	0.448288i	−0.707107	−	0.707107i	$-0.750000\pi$
$479$	0.258819	−	0.448288i	0.258819	−	0.448288i	0.965926	+	0.258819i	$0.0833333\pi$
$480$			1.00000i			1.00000i
$481$	0.965926	+	1.67303i	0.965926	+	1.67303i
$482$		0			0
$483$		0			0
$484$	−0.707107	+	2.63896i	−0.707107	+	2.63896i
$485$		−	1.84776i		−	1.84776i
$486$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$487$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$487$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$488$	0.184592	−	1.40211i	0.184592	−	1.40211i
$489$		0			0
$490$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$491$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$491$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$	−0.158919	−	1.20711i	−0.158919	−	1.20711i
$495$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.448288	−	0.258819i	0.448288	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$499$	0.448288	−	0.258819i	0.448288	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$500$	0.707107	−	0.707107i	0.707107	−	0.707107i
$501$	−0.241181	+	0.0999004i	−0.241181	+	0.0999004i
$502$	−0.662827	+	1.60021i	−0.662827	+	1.60021i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	1.41421			1.41421
$506$		0			0
$507$	0.382683	+	0.293643i	0.382683	+	0.293643i
$508$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$		0			0
$512$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$513$	0.923880	+	0.382683i	0.923880	+	0.382683i
$514$	−0.0999004	−	0.758819i	−0.0999004	−	0.758819i
$515$	−0.130526	+	0.226078i	−0.130526	+	0.226078i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	0.965926	−	1.25882i	0.965926	−	1.25882i
$520$	1.20711	+	0.158919i	1.20711	+	0.158919i
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$			1.98289i			1.98289i	0.130526	+	0.991445i	$0.458333\pi$
$523$			1.98289i			1.98289i	−0.130526	+	0.991445i	$0.541667\pi$
$524$	0.258819	−	0.965926i	0.258819	−	0.965926i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−1.17604	+	1.53264i	−1.17604	+	1.53264i
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$	−1.96593	+	0.258819i	−1.96593	+	0.258819i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.05441	+	0.608761i	−1.05441	+	0.608761i
$536$	1.12484	−	1.46593i	1.12484	−	1.46593i
$537$		0			0
$538$		0			0
$539$	−1.93185			−1.93185
$540$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$541$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$541$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$542$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.05441	−	0.608761i	−1.05441	−	0.608761i	−0.130526	−	0.991445i	$-0.541667\pi$
$547$	−1.05441	−	0.608761i	−1.05441	−	0.608761i	−0.923880	+	0.382683i	$0.875000\pi$
$548$		0			0
$549$	1.00000	−	1.00000i	1.00000	−	1.00000i
$550$	1.91532	−	0.252157i	1.91532	−	0.252157i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.607206	+	1.46593i	0.607206	+	1.46593i
$556$	1.36603	+	0.366025i	1.36603	+	0.366025i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.793353	+	1.37413i	0.793353	+	1.37413i	0.923880	+	0.382683i	$0.125000\pi$
$563$	0.793353	+	1.37413i	0.793353	+	1.37413i	−0.130526	+	0.991445i	$0.541667\pi$
$564$		0			0
$565$	0.923880	−	1.60021i	0.923880	−	1.60021i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$569$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$570$		−	1.00000i		−	1.00000i
$571$	−1.67303	+	0.965926i	−1.67303	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$571$	−1.67303	+	0.965926i	−1.67303	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$572$	1.66317	+	1.66317i	1.66317	+	1.66317i
$573$	1.37413	+	1.05441i	1.37413	+	1.05441i
$574$		0			0
$575$		0			0
$576$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$579$	0.241181	−	0.0999004i	0.241181	−	0.0999004i
$580$		0			0
$581$		0			0
$582$	1.60021	−	0.923880i	1.60021	−	0.923880i
$583$	−3.31744	−	1.91532i	−3.31744	−	1.91532i
$584$		0			0
$585$	0.860919	+	0.860919i	0.860919	+	0.860919i
$586$	0.0999004	+	0.758819i	0.0999004	+	0.758819i
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$589$		0			0
$590$		0			0
$591$		0			0
$592$			1.58671i			1.58671i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$	−1.86603	+	0.500000i	−1.86603	+	0.500000i
$595$		0			0
$596$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$597$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	0.965926	+	0.258819i	0.965926	+	0.258819i
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$	1.78480	−	0.478235i	1.78480	−	0.478235i
$604$		0			0
$605$	2.36603	+	1.36603i	2.36603	+	1.36603i
$606$	0.707107	+	1.22474i	0.707107	+	1.22474i
$607$	1.37413	−	0.793353i	1.37413	−	0.793353i	0.382683	−	0.923880i	$-0.375000\pi$
$607$	1.37413	−	0.793353i	1.37413	−	0.793353i	0.991445	+	0.130526i	$0.0416667\pi$
$608$	0.382683	−	0.923880i	0.382683	−	0.923880i
$609$		0			0
$610$	−1.30656	−	0.541196i	−1.30656	−	0.541196i
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−0.241181	−	0.0999004i	−0.241181	−	0.0999004i
$615$		0			0
$616$		0			0
$617$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$617$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$618$	−0.261052			−0.261052
$619$	0.448288	+	0.258819i	0.448288	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$619$	0.448288	+	0.258819i	0.448288	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$620$		0			0
$621$		0			0
$622$	−1.40211	+	0.184592i	−1.40211	+	0.184592i
$623$		0			0
$624$	0.465926	+	1.12484i	0.465926	+	1.12484i
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$	1.17604	−	1.53264i	1.17604	−	1.53264i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$631$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$632$		0			0
$633$		0			0
$634$	1.46593	−	1.12484i	1.46593	−	1.12484i
$635$	0.382683	+	0.662827i	0.382683	+	0.662827i
$636$	−1.20711	−	1.57313i	−1.20711	−	1.57313i
$637$	−0.608761	+	1.05441i	−0.608761	+	1.05441i
$638$		0			0
$639$		0			0
$640$	0.793353	+	0.608761i	0.793353	+	0.608761i
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$	−1.05441	−	0.608761i	−1.05441	−	0.608761i
$643$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$643$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$649$		0			0
$650$	0.465926	−	1.12484i	0.465926	−	1.12484i
$651$		0			0
$652$		0			0
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$	1.17604	+	1.53264i	1.17604	+	1.53264i
$661$		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
$666$	−0.965926	+	1.25882i	−0.965926	+	1.25882i
$667$		0			0
$668$	−0.0675653	+	0.252157i	−0.0675653	+	0.252157i
$669$	1.20711	−	1.57313i	1.20711	−	1.57313i
$670$	−1.12484	−	1.46593i	−1.12484	−	1.46593i
$671$	−1.36603	−	2.36603i	−1.36603	−	2.36603i
$672$		0			0
$673$	0.991445	−	1.71723i	0.991445	−	1.71723i	0.382683	−	0.923880i	$-0.375000\pi$
$673$	0.991445	−	1.71723i	0.991445	−	1.71723i	0.608761	−	0.793353i	$-0.291667\pi$
$674$	0.758819	−	0.0999004i	0.758819	−	0.0999004i
$675$	0.608761	+	0.793353i	0.608761	+	0.793353i
$676$	0.465926	−	0.124844i	0.465926	−	0.124844i
$677$	1.05441	+	0.608761i	1.05441	+	0.608761i	0.923880	−	0.382683i	$-0.125000\pi$
$677$	1.05441	+	0.608761i	1.05441	+	0.608761i	0.130526	+	0.991445i	$0.458333\pi$
$678$	1.84776			1.84776
$679$		0			0
$680$		0			0
$681$	−0.607206	−	0.465926i	−0.607206	−	0.465926i
$682$		0			0
$683$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$683$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$684$	0.866025	−	0.500000i	0.866025	−	0.500000i
$685$		0			0
$686$		0			0
$687$	0.478235	−	0.198092i	0.478235	−	0.198092i
$688$		0			0
$689$	−2.09077	+	1.20711i	−2.09077	+	1.20711i
$690$		0			0
$691$	1.67303	+	0.965926i	1.67303	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$691$	1.67303	+	0.965926i	1.67303	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$692$	−0.410670	−	1.53264i	−0.410670	−	1.53264i
$693$		0			0
$694$		0			0
$695$	0.707107	−	1.22474i	0.707107	−	1.22474i
$696$		0			0
$697$		0			0
$698$	−1.05441	−	1.37413i	−1.05441	−	1.37413i
$699$		0			0
$700$		0			0
$701$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$701$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$702$	−0.315118	+	1.17604i	−0.315118	+	1.17604i
$703$		−	1.58671i		−	1.58671i
$704$	0.500000	+	1.86603i	0.500000	+	1.86603i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$709$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	2.03696	−	1.17604i	2.03696	−	1.17604i
$716$		0			0
$717$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$718$	0.198092	−	0.478235i	0.198092	−	0.478235i
$719$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$719$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$720$	0.258819	+	0.965926i	0.258819	+	0.965926i
$721$		0			0
$722$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$723$		0			0
$724$		0			0
$725$		0			0
$726$			2.73205i			2.73205i
$727$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$727$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$728$		0			0
$729$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$730$		0			0
$731$		0			0
$732$	−0.184592	−	1.40211i	−0.184592	−	1.40211i
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$736$		0			0
$737$		−	3.56960i		−	3.56960i
$738$		0			0
$739$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$739$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$740$	1.53264	+	0.410670i	1.53264	+	0.410670i
$741$	−0.465926	−	1.12484i	−0.465926	−	1.12484i
$742$		0			0
$743$	0.130526	+	0.226078i	0.130526	+	0.226078i	0.923880	−	0.382683i	$-0.125000\pi$
$743$	0.130526	+	0.226078i	0.130526	+	0.226078i	−0.793353	+	0.608761i	$0.791667\pi$
$744$		0			0
$745$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$746$	−0.258819	+	0.0340742i	−0.258819	+	0.0340742i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	0.500000	−	0.866025i	0.500000	−	0.866025i
$751$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$	−0.226078	+	1.71723i	−0.226078	+	1.71723i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$761$	−0.448288	+	0.258819i	−0.448288	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$761$	−0.448288	+	0.258819i	−0.448288	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$762$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$763$		0			0
$764$	1.67303	−	0.448288i	1.67303	−	0.448288i
$765$		0			0
$766$	−1.20711	+	0.158919i	−1.20711	+	0.158919i
$767$		0			0
$768$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$769$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$769$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$770$		0			0
$771$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$772$	0.0675653	−	0.252157i	0.0675653	−	0.252157i
$773$		−	1.58671i		−	1.58671i	−0.608761	−	0.793353i	$-0.708333\pi$
$773$		−	1.58671i		−	1.58671i	0.608761	−	0.793353i	$-0.291667\pi$
$774$		0			0
$775$		0			0
$776$	0.241181	−	1.83195i	0.241181	−	1.83195i
$777$		0			0
$778$	−1.37413	+	1.05441i	−1.37413	+	1.05441i
$779$		0			0
$780$	1.20711	−	0.158919i	1.20711	−	0.158919i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$785$		0			0
$786$		−	1.00000i		−	1.00000i
$787$	−0.662827	+	0.382683i	−0.662827	+	0.382683i	−0.793353	−	0.608761i	$-0.791667\pi$
$787$	−0.662827	+	0.382683i	−0.662827	+	0.382683i	0.130526	+	0.991445i	$0.458333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.739288	+	1.78480i	−0.739288	+	1.78480i
$793$	−1.72184			−1.72184
$794$		0			0
$795$	−1.83195	+	0.758819i	−1.83195	+	0.758819i
$796$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$797$	1.05441	−	0.608761i	1.05441	−	0.608761i	0.130526	−	0.991445i	$-0.458333\pi$
$797$	1.05441	−	0.608761i	1.05441	−	0.608761i	0.923880	+	0.382683i	$0.125000\pi$
$798$		0			0
$799$		0			0
$800$	0.793353	−	0.608761i	0.793353	−	0.608761i
$801$		0			0
$802$		0			0
$803$		0			0
$804$	0.707107	−	1.70711i	0.707107	−	1.70711i
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.40211	+	0.184592i	1.40211	+	0.184592i
$809$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$809$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$810$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$		0			0
$813$	−1.40211	−	0.184592i	−1.40211	−	0.184592i
$814$	1.86603	+	2.43185i	1.86603	+	2.43185i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$821$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$	−0.158919	+	0.207107i	−0.158919	+	0.207107i
$825$	1.78480	−	0.739288i	1.78480	−	0.739288i
$826$		0			0
$827$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$827$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	1.17604	+	0.315118i	1.17604	+	0.315118i
$833$		0			0
$834$	1.41421			1.41421
$835$	0.226078	+	0.130526i	0.226078	+	0.130526i
$836$	−0.500000	−	1.86603i	−0.500000	−	1.86603i
$837$		0			0
$838$	−1.98289	+	0.261052i	−1.98289	+	0.261052i
$839$		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
$845$		−	0.482362i		−	0.482362i
$846$		0			0
$847$		0			0
$848$	−1.98289			−1.98289
$849$		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
$855$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$856$	−1.12484	+	0.465926i	−1.12484	+	0.465926i
$857$	−1.71723	−	0.991445i	−1.71723	−	0.991445i	−0.923880	−	0.382683i	$-0.875000\pi$
$857$	−1.71723	−	0.991445i	−1.71723	−	0.991445i	−0.793353	−	0.608761i	$-0.791667\pi$
$858$	2.03696	+	1.17604i	2.03696	+	1.17604i
$859$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$859$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.58671			1.58671			0.793353	−	0.608761i	$-0.208333\pi$
$863$	1.58671			1.58671			0.793353	+	0.608761i	$0.208333\pi$
$864$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$865$	−1.58671			−1.58671
$866$	0.292893	−	0.707107i	0.292893	−	0.707107i
$867$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−1.94829	−	1.12484i	−1.94829	−	1.12484i
$872$		0			0
$873$	1.30656	−	1.30656i	1.30656	−	1.30656i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.793353	−	0.608761i	$-0.791667\pi$
$877$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.130526	−	0.991445i	$-0.541667\pi$
$878$		0			0
$879$	0.292893	+	0.707107i	0.292893	+	0.707107i
$880$	1.93185			1.93185
$881$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$881$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$882$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.382683	+	0.662827i	0.382683	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$887$	0.382683	+	0.662827i	0.382683	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$888$	0.410670	+	1.53264i	0.410670	+	1.53264i
$889$		0			0
$890$		0			0
$891$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$892$	−0.513210	−	1.91532i	−0.513210	−	1.91532i
$893$		0			0
$894$	−1.93185			−1.93185
$895$		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
$904$	1.12484	−	1.46593i	1.12484	−	1.46593i
$905$		0			0
$906$		0			0
$907$	1.37413	+	0.793353i	1.37413	+	0.793353i	0.991445	−	0.130526i	$-0.0416667\pi$
$907$	1.37413	+	0.793353i	1.37413	+	0.793353i	0.382683	+	0.923880i	$0.375000\pi$
$908$	−0.739288	+	0.198092i	−0.739288	+	0.198092i
$909$	1.00000	+	1.00000i	1.00000	+	1.00000i
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$	0.130526	−	0.991445i	0.130526	−	0.991445i
$913$		0			0
$914$		0			0
$915$	−1.40211	−	0.184592i	−1.40211	−	0.184592i
$916$	0.133975	−	0.500000i	0.133975	−	0.500000i
$917$		0			0
$918$		0			0
$919$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$919$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$	−0.258819	−	0.0340742i	−0.258819	−	0.0340742i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.793353	−	1.37413i	0.793353	−	1.37413i
$926$		0			0
$927$	−0.252157	+	0.0675653i	−0.252157	+	0.0675653i
$928$		0			0
$929$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$929$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	0.866025	−	0.500000i	0.866025	−	0.500000i
$932$		0			0
$933$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$934$		0			0
$935$		0			0
$936$	0.741181	+	0.965926i	0.741181	+	0.965926i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		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
$950$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$951$	1.12484	−	1.46593i	1.12484	−	1.46593i
$952$		0			0
$953$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$953$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$954$	−1.57313	−	1.20711i	−1.57313	−	1.20711i
$955$		−	1.73205i		−	1.73205i
$956$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$957$		0			0
$958$	−0.315118	−	0.410670i	−0.315118	−	0.410670i
$959$		0			0
$960$	0.923880	+	0.382683i	0.923880	+	0.382683i
$961$	0.500000	−	0.866025i	0.500000	−	0.866025i
$962$	1.91532	−	0.252157i	1.91532	−	0.252157i
$963$	−1.17604	−	0.315118i	−1.17604	−	0.315118i
$964$		0			0
$965$	−0.226078	−	0.130526i	−0.226078	−	0.130526i
$966$		0			0
$967$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$967$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$968$	2.16748	+	1.66317i	2.16748	+	1.66317i
$969$		0			0
$970$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$973$		0			0
$974$	1.70711	+	0.707107i	1.70711	+	0.707107i
$975$	0.158919	−	1.20711i	0.158919	−	1.20711i
$976$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$977$	−0.226078	+	0.130526i	−0.226078	+	0.130526i	−0.608761	−	0.793353i	$-0.708333\pi$
$977$	−0.226078	+	0.130526i	−0.226078	+	0.130526i	0.382683	+	0.923880i	$0.375000\pi$
$978$		0			0
$979$		0			0
$980$	0.258819	+	0.965926i	0.258819	+	0.965926i
$981$		0			0
$982$	1.71723	−	0.226078i	1.71723	−	0.226078i
$983$	0.991445	−	1.71723i	0.991445	−	1.71723i	0.382683	−	0.923880i	$-0.375000\pi$
$983$	0.991445	−	1.71723i	0.991445	−	1.71723i	0.608761	−	0.793353i	$-0.291667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.17604	−	0.315118i	−1.17604	−	0.315118i
$989$		0			0
$990$	1.53264	+	1.17604i	1.53264	+	1.17604i
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.500000	+	0.866025i	0.500000	+	0.866025i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	−0.0675653	−	0.513210i	−0.0675653	−	0.513210i
$999$	−0.607206	+	1.46593i	−0.607206	+	1.46593i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.1.bn.b.2279.5	yes	16
4.3	odd	2		3420.1.bn.a.2279.3	yes	16
5.4	even	2	inner	3420.1.bn.b.2279.4	yes	16
9.5	odd	6		3420.1.bn.a.1139.3	✓	16
19.18	odd	2	inner	3420.1.bn.b.2279.4	yes	16
20.19	odd	2		3420.1.bn.a.2279.6	yes	16
36.23	even	6	inner	3420.1.bn.b.1139.6	yes	16
45.14	odd	6		3420.1.bn.a.1139.6	yes	16
76.75	even	2		3420.1.bn.a.2279.6	yes	16
95.94	odd	2	CM	3420.1.bn.b.2279.5	yes	16
171.113	even	6		3420.1.bn.a.1139.6	yes	16
180.59	even	6	inner	3420.1.bn.b.1139.3	yes	16
380.379	even	2		3420.1.bn.a.2279.3	yes	16
684.455	odd	6	inner	3420.1.bn.b.1139.3	yes	16
855.284	even	6		3420.1.bn.a.1139.3	✓	16
3420.1139	odd	6	inner	3420.1.bn.b.1139.6	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3420.1.bn.a.1139.3	✓	16	9.5	odd	6
3420.1.bn.a.1139.3	✓	16	855.284	even	6
3420.1.bn.a.1139.6	yes	16	45.14	odd	6
3420.1.bn.a.1139.6	yes	16	171.113	even	6
3420.1.bn.a.2279.3	yes	16	4.3	odd	2
3420.1.bn.a.2279.3	yes	16	380.379	even	2
3420.1.bn.a.2279.6	yes	16	20.19	odd	2
3420.1.bn.a.2279.6	yes	16	76.75	even	2
3420.1.bn.b.1139.3	yes	16	180.59	even	6	inner
3420.1.bn.b.1139.3	yes	16	684.455	odd	6	inner
3420.1.bn.b.1139.6	yes	16	36.23	even	6	inner
3420.1.bn.b.1139.6	yes	16	3420.1139	odd	6	inner
3420.1.bn.b.2279.4	yes	16	5.4	even	2	inner
3420.1.bn.b.2279.4	yes	16	19.18	odd	2	inner
3420.1.bn.b.2279.5	yes	16	1.1	even	1	trivial
3420.1.bn.b.2279.5	yes	16	95.94	odd	2	CM

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

Level:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3420.bn (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.382683 - 0.923880i) q^{2} +(0.130526 - 0.991445i) q^{3} +(-0.707107 - 0.707107i) q^{4} +(-0.866025 + 0.500000i) q^{5} +(-0.866025 - 0.500000i) q^{6} +(-0.923880 + 0.382683i) q^{8} +(-0.965926 - 0.258819i) q^{9} +(0.130526 + 0.991445i) q^{10} +(-0.965926 + 1.67303i) q^{11} +(-0.793353 + 0.608761i) q^{12} +(0.608761 + 1.05441i) q^{13} +(0.382683 + 0.923880i) q^{15} +1.00000i q^{16} +(-0.608761 + 0.793353i) q^{18} -1.00000i q^{19} +(0.965926 + 0.258819i) q^{20} +(1.17604 + 1.53264i) q^{22} +(0.258819 + 0.965926i) q^{24} +(0.500000 - 0.866025i) q^{25} +(1.20711 - 0.158919i) q^{26} +(-0.382683 + 0.923880i) q^{27} +1.00000 q^{30} +(0.923880 + 0.382683i) q^{32} +(1.53264 + 1.17604i) q^{33} +(0.500000 + 0.866025i) q^{36} +1.58671 q^{37} +(-0.923880 - 0.382683i) q^{38} +(1.12484 - 0.465926i) q^{39} +(0.608761 - 0.793353i) q^{40} +(1.86603 - 0.500000i) q^{44} +(0.965926 - 0.258819i) q^{45} +(0.991445 + 0.130526i) q^{48} +(0.500000 + 0.866025i) q^{49} +(-0.608761 - 0.793353i) q^{50} +(0.315118 - 1.17604i) q^{52} +1.98289i q^{53} +(0.707107 + 0.707107i) q^{54} -1.93185i q^{55} +(-0.991445 - 0.130526i) q^{57} +(0.382683 - 0.923880i) q^{60} +(-0.707107 + 1.22474i) q^{61} +(0.707107 - 0.707107i) q^{64} +(-1.05441 - 0.608761i) q^{65} +(1.67303 - 0.965926i) q^{66} +(-1.60021 + 0.923880i) q^{67} +(0.991445 - 0.130526i) q^{72} +(0.607206 - 1.46593i) q^{74} +(-0.793353 - 0.608761i) q^{75} +(-0.707107 + 0.707107i) q^{76} -1.21752i q^{78} +(-0.500000 - 0.866025i) q^{80} +(0.866025 + 0.500000i) q^{81} +(0.252157 - 1.91532i) q^{88} +(0.130526 - 0.991445i) q^{90} +(0.500000 + 0.866025i) q^{95} +(0.500000 - 0.866025i) q^{96} +(-0.923880 + 1.60021i) q^{97} +(0.991445 - 0.130526i) q^{98} +(1.36603 - 1.36603i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{25} + 8 q^{26} + 16 q^{30} + 8 q^{36} + 16 q^{44} + 8 q^{49} - 8 q^{80} + 8 q^{95} + 8 q^{96} + 8 q^{99}+O(q^{100}) \) 16 * q + 8 * q^25 + 8 * q^26 + 16 * q^30 + 8 * q^36 + 16 * q^44 + 8 * q^49 - 8 * q^80 + 8 * q^95 + 8 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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