LMFDB - Embedded newform 3120.1.iq.a.1907.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3120,1,Mod(1667,3120)] 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("3120.1667"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3120, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 9, 6, 3, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3120.iq (of order $12$, degree $4$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$1.55708283941$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.129792000.3

Embedding invariants

Embedding label			1907.2
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	3120.1907
Dual form			3120.1.iq.a.2603.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{3} +(0.866025 - 0.500000i) q^{4} +(0.707107 - 0.707107i) q^{5} +(0.965926 + 0.258819i) q^{6} +(-0.366025 + 1.36603i) q^{7} +(0.707107 - 0.707107i) q^{8} +(0.500000 + 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{10} +(-0.258819 - 0.965926i) q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.41421i q^{14} +(0.965926 - 0.258819i) q^{15} +(0.500000 - 0.866025i) q^{16} +(0.707107 + 0.707107i) q^{18} +(0.258819 - 0.965926i) q^{20} +(-1.00000 + 1.00000i) q^{21} +(-0.500000 - 0.866025i) q^{22} +(-0.965926 + 0.258819i) q^{23} +(0.965926 - 0.258819i) q^{24} -1.00000i q^{25} +(-0.965926 + 0.258819i) q^{26} +1.00000i q^{27} +(0.366025 + 1.36603i) q^{28} +(0.965926 - 0.258819i) q^{29} +(0.866025 - 0.500000i) q^{30} -1.00000i q^{31} +(0.258819 - 0.965926i) q^{32} +(0.258819 - 0.965926i) q^{33} +(0.707107 + 1.22474i) q^{35} +(0.866025 + 0.500000i) q^{36} +(-0.500000 + 0.866025i) q^{37} +(-0.866025 - 0.500000i) q^{39} -1.00000i q^{40} +(-0.707107 + 1.22474i) q^{42} +(0.500000 + 0.866025i) q^{43} +(-0.707107 - 0.707107i) q^{44} +(0.965926 + 0.258819i) q^{45} +(-0.866025 + 0.500000i) q^{46} +(0.707107 + 0.707107i) q^{47} +(0.866025 - 0.500000i) q^{48} +(-0.866025 - 0.500000i) q^{49} +(-0.258819 - 0.965926i) q^{50} +(-0.866025 + 0.500000i) q^{52} -1.41421 q^{53} +(0.258819 + 0.965926i) q^{54} +(-0.866025 - 0.500000i) q^{55} +(0.707107 + 1.22474i) q^{56} +(0.866025 - 0.500000i) q^{58} +(-0.258819 + 0.965926i) q^{59} +(0.707107 - 0.707107i) q^{60} +(-0.258819 - 0.965926i) q^{62} +(-1.36603 + 0.366025i) q^{63} -1.00000i q^{64} +(-0.707107 + 0.707107i) q^{65} -1.00000i q^{66} +(-0.965926 - 0.258819i) q^{69} +(1.00000 + 1.00000i) q^{70} +(-1.22474 + 0.707107i) q^{71} +(0.965926 + 0.258819i) q^{72} +(-1.00000 - 1.00000i) q^{73} +(-0.258819 + 0.965926i) q^{74} +(0.500000 - 0.866025i) q^{75} +1.41421 q^{77} +(-0.965926 - 0.258819i) q^{78} -1.00000 q^{79} +(-0.258819 - 0.965926i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-0.366025 + 1.36603i) q^{84} +(0.707107 + 0.707107i) q^{86} +(0.965926 + 0.258819i) q^{87} +(-0.866025 - 0.500000i) q^{88} +(-1.22474 - 0.707107i) q^{89} +1.00000 q^{90} +(0.366025 - 1.36603i) q^{91} +(-0.707107 + 0.707107i) q^{92} +(0.500000 - 0.866025i) q^{93} +(0.866025 + 0.500000i) q^{94} +(0.707107 - 0.707107i) q^{96} +(-0.965926 - 0.258819i) q^{98} +(0.707107 - 0.707107i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7} + 4 q^{9} + 4 q^{10} + 8 q^{12} - 8 q^{13} + 4 q^{16} - 8 q^{21} - 4 q^{22} - 4 q^{28} - 4 q^{37} + 4 q^{43} - 4 q^{63} + 8 q^{70} - 8 q^{73} + 4 q^{75} - 8 q^{79} - 4 q^{81} + 4 q^{84} + 8 q^{90}+ \cdots + 4 q^{93}+O(q^{100}) $ 8 * q + 4 * q^7 + 4 * q^9 + 4 * q^10 + 8 * q^12 - 8 * q^13 + 4 * q^16 - 8 * q^21 - 4 * q^22 - 4 * q^28 - 4 * q^37 + 4 * q^43 - 4 * q^63 + 8 * q^70 - 8 * q^73 + 4 * q^75 - 8 * q^79 - 4 * q^81 + 4 * q^84 + 8 * q^90 - 4 * q^91 + 4 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times$.

$n$	$1951$	$2081$	$2341$	$2497$	$2641$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{3}{4}\right)$	$e\left(\frac{1}{4}\right)$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	0.965926	−	0.258819i	0.965926	−	0.258819i
$2$	0.965926	−	0.258819i	0.965926	−	0.258819i
$3$	0.866025	+	0.500000i	0.866025	+	0.500000i
$3$	0.866025	+	0.500000i	0.866025	+	0.500000i
$4$	0.866025	−	0.500000i	0.866025	−	0.500000i
$5$	0.707107	−	0.707107i	0.707107	−	0.707107i
$5$	0.707107	−	0.707107i	0.707107	−	0.707107i
$6$	0.965926	+	0.258819i	0.965926	+	0.258819i
$7$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$7$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$8$	0.707107	−	0.707107i	0.707107	−	0.707107i
$9$	0.500000	+	0.866025i	0.500000	+	0.866025i
$10$	0.500000	−	0.866025i	0.500000	−	0.866025i
$11$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$11$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$12$	1.00000			1.00000
$13$	−1.00000			−1.00000
$13$	−1.00000			−1.00000
$14$			1.41421i			1.41421i
$15$	0.965926	−	0.258819i	0.965926	−	0.258819i
$16$	0.500000	−	0.866025i	0.500000	−	0.866025i
$17$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$17$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$18$	0.707107	+	0.707107i	0.707107	+	0.707107i
$19$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$19$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$20$	0.258819	−	0.965926i	0.258819	−	0.965926i
$21$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$22$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$23$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$23$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$24$	0.965926	−	0.258819i	0.965926	−	0.258819i
$25$		−	1.00000i		−	1.00000i
$26$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$27$			1.00000i			1.00000i
$28$	0.366025	+	1.36603i	0.366025	+	1.36603i
$29$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$29$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$30$	0.866025	−	0.500000i	0.866025	−	0.500000i
$31$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$31$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$32$	0.258819	−	0.965926i	0.258819	−	0.965926i
$33$	0.258819	−	0.965926i	0.258819	−	0.965926i
$34$		0			0
$35$	0.707107	+	1.22474i	0.707107	+	1.22474i
$36$	0.866025	+	0.500000i	0.866025	+	0.500000i
$37$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$37$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$38$		0			0
$39$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$40$		−	1.00000i		−	1.00000i
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$43$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$43$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$44$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$45$	0.965926	+	0.258819i	0.965926	+	0.258819i
$46$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$47$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$47$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$48$	0.866025	−	0.500000i	0.866025	−	0.500000i
$49$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$50$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$51$		0			0
$52$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$53$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$53$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$54$	0.258819	+	0.965926i	0.258819	+	0.965926i
$55$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$56$	0.707107	+	1.22474i	0.707107	+	1.22474i
$57$		0			0
$58$	0.866025	−	0.500000i	0.866025	−	0.500000i
$59$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$59$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$60$	0.707107	−	0.707107i	0.707107	−	0.707107i
$61$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$61$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$62$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$63$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$64$		−	1.00000i		−	1.00000i
$65$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$66$		−	1.00000i		−	1.00000i
$67$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$67$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$68$		0			0
$69$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$70$	1.00000	+	1.00000i	1.00000	+	1.00000i
$71$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$71$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$72$	0.965926	+	0.258819i	0.965926	+	0.258819i
$73$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$73$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$74$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$75$	0.500000	−	0.866025i	0.500000	−	0.866025i
$76$		0			0
$77$	1.41421			1.41421
$78$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$79$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$79$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$80$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$85$		0			0
$86$	0.707107	+	0.707107i	0.707107	+	0.707107i
$87$	0.965926	+	0.258819i	0.965926	+	0.258819i
$88$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$89$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$89$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$90$	1.00000			1.00000
$91$	0.366025	−	1.36603i	0.366025	−	1.36603i
$92$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$93$	0.500000	−	0.866025i	0.500000	−	0.866025i
$94$	0.866025	+	0.500000i	0.866025	+	0.500000i
$95$		0			0
$96$	0.707107	−	0.707107i	0.707107	−	0.707107i
$97$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$97$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$98$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$99$	0.707107	−	0.707107i	0.707107	−	0.707107i
$100$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$101$	1.93185	−	0.517638i	1.93185	−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$101$	1.93185	−	0.517638i	1.93185	−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$102$		0			0
$103$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$103$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$104$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$105$			1.41421i			1.41421i
$106$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$107$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$107$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$108$	0.500000	+	0.866025i	0.500000	+	0.866025i
$109$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$109$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$110$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$111$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$112$	1.00000	+	1.00000i	1.00000	+	1.00000i
$113$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$113$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$114$		0			0
$115$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$116$	0.707107	−	0.707107i	0.707107	−	0.707107i
$117$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$118$			1.00000i			1.00000i
$119$		0			0
$120$	0.500000	−	0.866025i	0.500000	−	0.866025i
$121$		0			0
$122$		0			0
$123$		0			0
$124$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$125$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$126$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$127$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$127$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$128$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$129$			1.00000i			1.00000i
$130$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$131$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$131$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$132$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$133$		0			0
$134$		0			0
$135$	0.707107	+	0.707107i	0.707107	+	0.707107i
$136$		0			0
$137$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$137$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$138$	−1.00000			−1.00000
$139$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$139$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$140$	1.22474	+	0.707107i	1.22474	+	0.707107i
$141$	0.258819	+	0.965926i	0.258819	+	0.965926i
$142$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$143$	0.258819	+	0.965926i	0.258819	+	0.965926i
$144$	1.00000			1.00000
$145$	0.500000	−	0.866025i	0.500000	−	0.866025i
$146$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$147$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$148$			1.00000i			1.00000i
$149$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$149$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$150$	0.258819	−	0.965926i	0.258819	−	0.965926i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$	1.36603	−	0.366025i	1.36603	−	0.366025i
$155$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$156$	−1.00000			−1.00000
$157$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$157$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$158$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$159$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$160$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$161$		−	1.41421i		−	1.41421i
$162$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$163$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$163$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$164$		0			0
$165$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$166$		0			0
$167$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$167$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$168$			1.41421i			1.41421i
$169$	1.00000			1.00000
$170$		0			0
$171$		0			0
$172$	0.866025	+	0.500000i	0.866025	+	0.500000i
$173$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$173$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$174$	1.00000			1.00000
$175$	1.36603	+	0.366025i	1.36603	+	0.366025i
$176$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$177$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$178$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$179$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$179$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$180$	0.965926	−	0.258819i	0.965926	−	0.258819i
$181$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$181$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$182$		−	1.41421i		−	1.41421i
$183$		0			0
$184$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$185$	0.258819	+	0.965926i	0.258819	+	0.965926i
$186$	0.258819	−	0.965926i	0.258819	−	0.965926i
$187$		0			0
$188$	0.965926	+	0.258819i	0.965926	+	0.258819i
$189$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$	0.500000	−	0.866025i	0.500000	−	0.866025i
$193$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$193$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$194$		0			0
$195$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$196$	−1.00000			−1.00000
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$	0.500000	−	0.866025i	0.500000	−	0.866025i
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$201$		0			0
$202$	1.73205	−	1.00000i	1.73205	−	1.00000i
$203$			1.41421i			1.41421i
$204$		0			0
$205$		0			0
$206$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$207$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$208$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$209$		0			0
$210$	0.366025	+	1.36603i	0.366025	+	1.36603i
$211$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$211$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$212$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$213$	−1.41421			−1.41421
$214$		0			0
$215$	0.965926	+	0.258819i	0.965926	+	0.258819i
$216$	0.707107	+	0.707107i	0.707107	+	0.707107i
$217$	1.36603	+	0.366025i	1.36603	+	0.366025i
$218$	0.707107	−	1.22474i	0.707107	−	1.22474i
$219$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$220$	−1.00000			−1.00000
$221$		0			0
$222$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$223$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$223$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$224$	1.22474	+	0.707107i	1.22474	+	0.707107i
$225$	0.866025	−	0.500000i	0.866025	−	0.500000i
$226$		−	1.00000i		−	1.00000i
$227$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$227$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$228$		0			0
$229$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$229$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$230$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$231$	1.22474	+	0.707107i	1.22474	+	0.707107i
$232$	0.500000	−	0.866025i	0.500000	−	0.866025i
$233$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$233$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$234$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$235$	1.00000			1.00000
$236$	0.258819	+	0.965926i	0.258819	+	0.965926i
$237$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	0.258819	−	0.965926i	0.258819	−	0.965926i
$241$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$241$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$244$		0			0
$245$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$246$		0			0
$247$		0			0
$248$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$249$		0			0
$250$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$251$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$251$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$252$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$253$	0.500000	+	0.866025i	0.500000	+	0.866025i
$254$		0			0
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$257$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$258$	0.258819	+	0.965926i	0.258819	+	0.965926i
$259$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$260$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$261$	0.707107	+	0.707107i	0.707107	+	0.707107i
$262$	0.866025	+	0.500000i	0.866025	+	0.500000i
$263$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$263$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$264$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$265$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$266$		0			0
$267$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$268$		0			0
$269$	0.517638	−	1.93185i	0.517638	−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$269$	0.517638	−	1.93185i	0.517638	−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$270$	0.866025	+	0.500000i	0.866025	+	0.500000i
$271$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$271$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$	1.00000	−	1.00000i	1.00000	−	1.00000i
$274$		−	1.00000i		−	1.00000i
$275$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$276$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$278$	1.41421			1.41421
$279$	0.866025	−	0.500000i	0.866025	−	0.500000i
$280$	1.36603	+	0.366025i	1.36603	+	0.366025i
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	0.500000	+	0.866025i	0.500000	+	0.866025i
$283$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$283$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$284$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$285$		0			0
$286$	0.500000	+	0.866025i	0.500000	+	0.866025i
$287$		0			0
$288$	0.965926	−	0.258819i	0.965926	−	0.258819i
$289$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$290$	0.258819	−	0.965926i	0.258819	−	0.965926i
$291$		0			0
$292$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$295$	0.500000	+	0.866025i	0.500000	+	0.866025i
$296$	0.258819	+	0.965926i	0.258819	+	0.965926i
$297$	0.965926	−	0.258819i	0.965926	−	0.258819i
$298$	−1.00000			−1.00000
$299$	0.965926	−	0.258819i	0.965926	−	0.258819i
$300$		−	1.00000i		−	1.00000i
$301$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$302$		0			0
$303$	1.93185	+	0.517638i	1.93185	+	0.517638i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	1.22474	−	0.707107i	1.22474	−	0.707107i
$309$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$310$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$311$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$311$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$312$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$313$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$313$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$314$	0.258819	+	0.965926i	0.258819	+	0.965926i
$315$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$316$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$319$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$320$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$321$		0			0
$322$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$323$		0			0
$324$			1.00000i			1.00000i
$325$			1.00000i			1.00000i
$326$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$327$	1.36603	−	0.366025i	1.36603	−	0.366025i
$328$		0			0
$329$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$330$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$331$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$331$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$332$		0			0
$333$	−1.00000			−1.00000
$334$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$335$		0			0
$336$	0.366025	+	1.36603i	0.366025	+	1.36603i
$337$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$337$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$338$	0.965926	−	0.258819i	0.965926	−	0.258819i
$339$	0.707107	−	0.707107i	0.707107	−	0.707107i
$340$		0			0
$341$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$342$		0			0
$343$		0			0
$344$	0.965926	+	0.258819i	0.965926	+	0.258819i
$345$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$346$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$	0.965926	−	0.258819i	0.965926	−	0.258819i
$349$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$349$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$350$	1.41421			1.41421
$351$		−	1.00000i		−	1.00000i
$352$	−1.00000			−1.00000
$353$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$353$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$354$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$355$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$356$	−1.41421			−1.41421
$357$		0			0
$358$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$359$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$359$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$360$	0.866025	−	0.500000i	0.866025	−	0.500000i
$361$	0.866025	+	0.500000i	0.866025	+	0.500000i
$362$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$363$		0			0
$364$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$365$	−1.41421			−1.41421
$366$		0			0
$367$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$367$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$368$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$369$		0			0
$370$	0.500000	+	0.866025i	0.500000	+	0.866025i
$371$	0.517638	−	1.93185i	0.517638	−	1.93185i
$372$		−	1.00000i		−	1.00000i
$373$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$373$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$374$		0			0
$375$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$376$	1.00000			1.00000
$377$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$378$	−1.41421			−1.41421
$379$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$379$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$383$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$384$	0.258819	−	0.965926i	0.258819	−	0.965926i
$385$	1.00000	−	1.00000i	1.00000	−	1.00000i
$386$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$387$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$388$		0			0
$389$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$389$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$390$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$391$		0			0
$392$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$393$	0.258819	+	0.965926i	0.258819	+	0.965926i
$394$		0			0
$395$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$396$	0.258819	−	0.965926i	0.258819	−	0.965926i
$397$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$397$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$			1.00000i			1.00000i
$404$	1.41421	−	1.41421i	1.41421	−	1.41421i
$405$	0.258819	+	0.965926i	0.258819	+	0.965926i
$406$	0.366025	+	1.36603i	0.366025	+	1.36603i
$407$	0.965926	+	0.258819i	0.965926	+	0.258819i
$408$		0			0
$409$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$409$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$410$		0			0
$411$	0.707107	−	0.707107i	0.707107	−	0.707107i
$412$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$413$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$414$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$415$		0			0
$416$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$417$	1.00000	+	1.00000i	1.00000	+	1.00000i
$418$		0			0
$419$	0.517638	+	1.93185i	0.517638	+	1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$419$	0.517638	+	1.93185i	0.517638	+	1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$420$	0.707107	+	1.22474i	0.707107	+	1.22474i
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$		0			0
$423$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$424$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$425$		0			0
$426$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$427$		0			0
$428$		0			0
$429$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$430$	1.00000			1.00000
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$	0.866025	+	0.500000i	0.866025	+	0.500000i
$433$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$433$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$434$	1.41421			1.41421
$435$	0.866025	−	0.500000i	0.866025	−	0.500000i
$436$	0.366025	−	1.36603i	0.366025	−	1.36603i
$437$		0			0
$438$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$439$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$439$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$440$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$441$		−	1.00000i		−	1.00000i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$445$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$446$	1.22474	−	0.707107i	1.22474	−	0.707107i
$447$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$448$	1.36603	+	0.366025i	1.36603	+	0.366025i
$449$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$449$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$450$	0.707107	−	0.707107i	0.707107	−	0.707107i
$451$		0			0
$452$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$453$		0			0
$454$	1.00000	−	1.00000i	1.00000	−	1.00000i
$455$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$456$		0			0
$457$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$457$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$458$	1.22474	+	0.707107i	1.22474	+	0.707107i
$459$		0			0
$460$			1.00000i			1.00000i
$461$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$461$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$462$	1.36603	+	0.366025i	1.36603	+	0.366025i
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	0.258819	−	0.965926i	0.258819	−	0.965926i
$465$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$466$	0.500000	−	0.866025i	0.500000	−	0.866025i
$467$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$467$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$468$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$469$		0			0
$470$	0.965926	−	0.258819i	0.965926	−	0.258819i
$471$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$472$	0.500000	+	0.866025i	0.500000	+	0.866025i
$473$	0.707107	−	0.707107i	0.707107	−	0.707107i
$474$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$475$		0			0
$476$		0			0
$477$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		−	1.00000i		−	1.00000i
$481$	0.500000	−	0.866025i	0.500000	−	0.866025i
$482$	0.707107	+	0.707107i	0.707107	+	0.707107i
$483$	0.707107	−	1.22474i	0.707107	−	1.22474i
$484$		0			0
$485$		0			0
$486$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$487$	0.366025	−	1.36603i	0.366025	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$487$	0.366025	−	1.36603i	0.366025	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$488$		0			0
$489$	−1.00000			−1.00000
$490$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$491$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$491$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		−	1.00000i		−	1.00000i
$496$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$497$	−0.517638	−	1.93185i	−0.517638	−	1.93185i
$498$		0			0
$499$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$499$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$500$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$501$	0.258819	−	0.965926i	0.258819	−	0.965926i
$502$	1.00000			1.00000
$503$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$503$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$504$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$505$	1.00000	−	1.73205i	1.00000	−	1.73205i
$506$	0.707107	+	0.707107i	0.707107	+	0.707107i
$507$	0.866025	+	0.500000i	0.866025	+	0.500000i
$508$		0			0
$509$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$509$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$510$		0			0
$511$	1.73205	−	1.00000i	1.73205	−	1.00000i
$512$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$513$		0			0
$514$	0.500000	+	0.866025i	0.500000	+	0.866025i
$515$			1.41421i			1.41421i
$516$	0.500000	+	0.866025i	0.500000	+	0.866025i
$517$	0.500000	−	0.866025i	0.500000	−	0.866025i
$518$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$519$	−1.41421			−1.41421
$520$			1.00000i			1.00000i
$521$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$521$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$522$	0.866025	+	0.500000i	0.866025	+	0.500000i
$523$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$523$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$524$	0.965926	+	0.258819i	0.965926	+	0.258819i
$525$	1.00000	+	1.00000i	1.00000	+	1.00000i
$526$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$527$		0			0
$528$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$529$		0			0
$530$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$531$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$532$		0			0
$533$		0			0
$534$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$535$		0			0
$536$		0			0
$537$	0.258819	−	0.965926i	0.258819	−	0.965926i
$538$		−	2.00000i		−	2.00000i
$539$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$540$	0.965926	+	0.258819i	0.965926	+	0.258819i
$541$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$541$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$542$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$543$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$544$		0			0
$545$		−	1.41421i		−	1.41421i
$546$	0.707107	−	1.22474i	0.707107	−	1.22474i
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$549$		0			0
$550$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$551$		0			0
$552$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$553$	0.366025	−	1.36603i	0.366025	−	1.36603i
$554$	0.707107	+	0.707107i	0.707107	+	0.707107i
$555$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$556$	1.36603	−	0.366025i	1.36603	−	0.366025i
$557$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$557$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$558$	0.707107	−	0.707107i	0.707107	−	0.707107i
$559$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$560$	1.41421			1.41421
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$	0.707107	+	0.707107i	0.707107	+	0.707107i
$565$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$566$	0.258819	−	0.965926i	0.258819	−	0.965926i
$567$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$568$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$569$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$569$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$570$		0			0
$571$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$571$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$572$	0.707107	+	0.707107i	0.707107	+	0.707107i
$573$		0			0
$574$		0			0
$575$	0.258819	+	0.965926i	0.258819	+	0.965926i
$576$	0.866025	−	0.500000i	0.866025	−	0.500000i
$577$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$577$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$578$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$579$	0.366025	−	1.36603i	0.366025	−	1.36603i
$580$		−	1.00000i		−	1.00000i
$581$		0			0
$582$		0			0
$583$	0.366025	+	1.36603i	0.366025	+	1.36603i
$584$	−1.41421			−1.41421
$585$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$586$		0			0
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$589$		0			0
$590$	0.707107	+	0.707107i	0.707107	+	0.707107i
$591$		0			0
$592$	0.500000	+	0.866025i	0.500000	+	0.866025i
$593$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$593$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$594$	0.866025	−	0.500000i	0.866025	−	0.500000i
$595$		0			0
$596$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$597$		0			0
$598$	0.866025	−	0.500000i	0.866025	−	0.500000i
$599$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$599$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$600$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$601$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$601$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$602$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$603$		0			0
$604$		0			0
$605$		0			0
$606$	2.00000			2.00000
$607$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$607$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$608$		0			0
$609$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$610$		0			0
$611$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$612$		0			0
$613$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$613$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$	1.00000	−	1.00000i	1.00000	−	1.00000i
$617$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$617$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$618$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$619$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$619$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$620$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$621$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$622$	0.366025	+	1.36603i	0.366025	+	1.36603i
$623$	1.41421	−	1.41421i	1.41421	−	1.41421i
$624$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$625$	−1.00000			−1.00000
$626$		0			0
$627$		0			0
$628$	0.500000	+	0.866025i	0.500000	+	0.866025i
$629$		0			0
$630$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$631$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$631$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$632$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−1.41421			−1.41421
$637$	0.866025	+	0.500000i	0.866025	+	0.500000i
$638$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$639$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$640$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$641$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$641$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$642$		0			0
$643$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$645$	0.707107	+	0.707107i	0.707107	+	0.707107i
$646$		0			0
$647$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$647$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$648$	0.258819	+	0.965926i	0.258819	+	0.965926i
$649$	1.00000			1.00000
$650$	0.258819	+	0.965926i	0.258819	+	0.965926i
$651$	1.00000	+	1.00000i	1.00000	+	1.00000i
$652$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$653$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$653$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$654$	1.22474	−	0.707107i	1.22474	−	0.707107i
$655$	1.00000			1.00000
$656$		0			0
$657$	0.366025	−	1.36603i	0.366025	−	1.36603i
$658$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$659$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$659$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$660$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$661$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$661$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$667$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$668$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$669$	1.36603	+	0.366025i	1.36603	+	0.366025i
$670$		0			0
$671$		0			0
$672$	0.707107	+	1.22474i	0.707107	+	1.22474i
$673$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$673$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$674$	1.22474	+	0.707107i	1.22474	+	0.707107i
$675$	1.00000			1.00000
$676$	0.866025	−	0.500000i	0.866025	−	0.500000i
$677$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$677$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$678$	0.500000	−	0.866025i	0.500000	−	0.866025i
$679$		0			0
$680$		0			0
$681$	1.41421			1.41421
$682$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$		0			0
$685$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$686$		0			0
$687$	0.366025	+	1.36603i	0.366025	+	1.36603i
$688$	1.00000			1.00000
$689$	1.41421			1.41421
$690$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$691$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$691$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$692$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$693$	0.707107	+	1.22474i	0.707107	+	1.22474i
$694$		0			0
$695$	1.22474	−	0.707107i	1.22474	−	0.707107i
$696$	0.866025	−	0.500000i	0.866025	−	0.500000i
$697$		0			0
$698$	0.707107	+	1.22474i	0.707107	+	1.22474i
$699$	0.965926	−	0.258819i	0.965926	−	0.258819i
$700$	1.36603	−	0.366025i	1.36603	−	0.366025i
$701$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$701$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$702$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$703$		0			0
$704$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$705$	0.866025	+	0.500000i	0.866025	+	0.500000i
$706$		0			0
$707$			2.82843i			2.82843i
$708$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$709$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$709$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$710$			1.41421i			1.41421i
$711$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$712$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$713$	0.258819	+	0.965926i	0.258819	+	0.965926i
$714$		0			0
$715$	0.866025	+	0.500000i	0.866025	+	0.500000i
$716$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$717$		0			0
$718$	1.36603	−	0.366025i	1.36603	−	0.366025i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$	0.707107	−	0.707107i	0.707107	−	0.707107i
$721$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$722$	0.965926	+	0.258819i	0.965926	+	0.258819i
$723$			1.00000i			1.00000i
$724$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$725$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$729$	−1.00000			−1.00000
$730$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$735$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$736$			1.00000i			1.00000i
$737$		0			0
$738$		0			0
$739$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$739$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$740$	0.707107	+	0.707107i	0.707107	+	0.707107i
$741$		0			0
$742$		−	2.00000i		−	2.00000i
$743$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$743$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$744$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$745$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$746$	0.707107	−	0.707107i	0.707107	−	0.707107i
$747$		0			0
$748$		0			0
$749$		0			0
$750$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$751$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$751$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$752$	0.965926	−	0.258819i	0.965926	−	0.258819i
$753$	0.707107	+	0.707107i	0.707107	+	0.707107i
$754$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$755$		0			0
$756$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$757$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$757$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$758$		0			0
$759$			1.00000i			1.00000i
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	1.00000	+	1.73205i	1.00000	+	1.73205i
$764$		0			0
$765$		0			0
$766$	−1.00000			−1.00000
$767$	0.258819	−	0.965926i	0.258819	−	0.965926i
$768$		−	1.00000i		−	1.00000i
$769$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$769$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$770$	0.707107	−	1.22474i	0.707107	−	1.22474i
$771$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$772$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$775$	−1.00000			−1.00000
$776$		0			0
$777$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$778$	0.500000	−	0.866025i	0.500000	−	0.866025i
$779$		0			0
$780$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$781$	1.00000	+	1.00000i	1.00000	+	1.00000i
$782$		0			0
$783$	0.258819	+	0.965926i	0.258819	+	0.965926i
$784$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$785$	0.707107	+	0.707107i	0.707107	+	0.707107i
$786$	0.500000	+	0.866025i	0.500000	+	0.866025i
$787$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$787$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$788$		0			0
$789$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$790$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$791$	1.22474	+	0.707107i	1.22474	+	0.707107i
$792$		−	1.00000i		−	1.00000i
$793$		0			0
$794$	0.707107	−	0.707107i	0.707107	−	0.707107i
$795$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$796$		0			0
$797$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$797$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$798$		0			0
$799$		0			0
$800$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$801$		−	1.41421i		−	1.41421i
$802$		0			0
$803$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$804$		0			0
$805$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$806$	0.258819	+	0.965926i	0.258819	+	0.965926i
$807$	1.41421	−	1.41421i	1.41421	−	1.41421i
$808$	1.00000	−	1.73205i	1.00000	−	1.73205i
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$	0.500000	+	0.866025i	0.500000	+	0.866025i
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$	0.707107	+	1.22474i	0.707107	+	1.22474i
$813$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$814$	1.00000			1.00000
$815$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$816$		0			0
$817$		0			0
$818$		0			0
$819$	1.36603	−	0.366025i	1.36603	−	0.366025i
$820$		0			0
$821$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$821$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$822$	0.500000	−	0.866025i	0.500000	−	0.866025i
$823$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$823$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$824$			1.41421i			1.41421i
$825$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$826$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$829$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$829$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$830$		0			0
$831$			1.00000i			1.00000i
$832$			1.00000i			1.00000i
$833$		0			0
$834$	1.22474	+	0.707107i	1.22474	+	0.707107i
$835$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$836$		0			0
$837$	1.00000			1.00000
$838$	1.00000	+	1.73205i	1.00000	+	1.73205i
$839$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$839$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$840$	1.00000	+	1.00000i	1.00000	+	1.00000i
$841$		0			0
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.707107	−	0.707107i	0.707107	−	0.707107i
$846$			1.00000i			1.00000i
$847$		0			0
$848$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$849$	0.866025	−	0.500000i	0.866025	−	0.500000i
$850$		0			0
$851$	0.258819	−	0.965926i	0.258819	−	0.965926i
$852$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$853$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$853$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$857$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$858$			1.00000i			1.00000i
$859$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$859$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$860$	0.965926	−	0.258819i	0.965926	−	0.258819i
$861$		0			0
$862$		0			0
$863$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$863$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$864$	0.965926	+	0.258819i	0.965926	+	0.258819i
$865$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$866$		0			0
$867$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$868$	1.36603	−	0.366025i	1.36603	−	0.366025i
$869$	0.258819	+	0.965926i	0.258819	+	0.965926i
$870$	0.707107	−	0.707107i	0.707107	−	0.707107i
$871$		0			0
$872$		−	1.41421i		−	1.41421i
$873$		0			0
$874$		0			0
$875$	1.22474	−	0.707107i	1.22474	−	0.707107i
$876$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$877$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$877$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$878$	1.93185	+	0.517638i	1.93185	+	0.517638i
$879$		0			0
$880$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$881$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$881$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$882$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$883$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$883$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$884$		0			0
$885$			1.00000i			1.00000i
$886$		0			0
$887$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$887$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$888$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$889$		0			0
$890$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$891$	0.965926	+	0.258819i	0.965926	+	0.258819i
$892$	1.00000	−	1.00000i	1.00000	−	1.00000i
$893$		0			0
$894$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$895$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$896$	1.41421			1.41421
$897$	0.965926	+	0.258819i	0.965926	+	0.258819i
$898$		0			0
$899$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$900$	0.500000	−	0.866025i	0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$904$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$905$			1.41421i			1.41421i
$906$		0			0
$907$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$907$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$908$	0.707107	−	1.22474i	0.707107	−	1.22474i
$909$	1.41421	+	1.41421i	1.41421	+	1.41421i
$910$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.36603	+	0.366025i	1.36603	+	0.366025i
$917$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$918$		0			0
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$	0.258819	+	0.965926i	0.258819	+	0.965926i
$921$		0			0
$922$	−1.00000			−1.00000
$923$	1.22474	−	0.707107i	1.22474	−	0.707107i
$924$	1.41421			1.41421
$925$	0.866025	+	0.500000i	0.866025	+	0.500000i
$926$		0			0
$927$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$928$		−	1.00000i		−	1.00000i
$929$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$929$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$930$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$931$		0			0
$932$	0.258819	−	0.965926i	0.258819	−	0.965926i
$933$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$934$	0.366025	+	1.36603i	0.366025	+	1.36603i
$935$		0			0
$936$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$937$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$937$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$938$		0			0
$939$		0			0
$940$	0.866025	−	0.500000i	0.866025	−	0.500000i
$941$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$941$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$942$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$943$		0			0
$944$	0.707107	+	0.707107i	0.707107	+	0.707107i
$945$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$946$	0.500000	−	0.866025i	0.500000	−	0.866025i
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	−1.00000			−1.00000
$949$	1.00000	+	1.00000i	1.00000	+	1.00000i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$953$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$954$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$955$		0			0
$956$		0			0
$957$		−	1.00000i		−	1.00000i
$958$		0			0
$959$	1.22474	+	0.707107i	1.22474	+	0.707107i
$960$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$961$		0			0
$962$	0.258819	−	0.965926i	0.258819	−	0.965926i
$963$		0			0
$964$	0.866025	+	0.500000i	0.866025	+	0.500000i
$965$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$966$	0.366025	−	1.36603i	0.366025	−	1.36603i
$967$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$967$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$971$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$972$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$973$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$974$		−	1.41421i		−	1.41421i
$975$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$976$		0			0
$977$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$977$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$978$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$979$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$980$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$981$	1.36603	+	0.366025i	1.36603	+	0.366025i
$982$		0			0
$983$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$983$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	−1.41421			−1.41421
$988$		0			0
$989$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$990$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$991$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$991$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$992$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$993$		0			0
$994$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$995$		0			0
$996$		0			0
$997$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$997$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$998$		0			0
$999$	−0.866025	−	0.500000i	−0.866025	−	0.500000i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.1.iq.a.1907.2	yes	8
3.2	odd	2	inner	3120.1.iq.a.1907.1	yes	8
5.3	odd	4		3120.1.iv.a.1283.1	yes	8
13.3	even	3	inner	3120.1.iq.a.1667.1	✓	8
15.8	even	4		3120.1.iv.a.1283.2	yes	8
16.11	odd	4		3120.1.iv.a.347.1	yes	8
39.29	odd	6	inner	3120.1.iq.a.1667.2	yes	8
48.11	even	4		3120.1.iv.a.347.2	yes	8
65.3	odd	12		3120.1.iv.a.1043.2	yes	8
80.43	even	4	inner	3120.1.iq.a.2843.2	yes	8
195.68	even	12		3120.1.iv.a.1043.1	yes	8
208.107	odd	12		3120.1.iv.a.107.2	yes	8
240.203	odd	4	inner	3120.1.iq.a.2843.1	yes	8
624.107	even	12		3120.1.iv.a.107.1	yes	8
1040.523	even	12	inner	3120.1.iq.a.2603.1	yes	8
3120.2603	odd	12	inner	3120.1.iq.a.2603.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3120.1.iq.a.1667.1	✓	8	13.3	even	3	inner
3120.1.iq.a.1667.2	yes	8	39.29	odd	6	inner
3120.1.iq.a.1907.1	yes	8	3.2	odd	2	inner
3120.1.iq.a.1907.2	yes	8	1.1	even	1	trivial
3120.1.iq.a.2603.1	yes	8	1040.523	even	12	inner
3120.1.iq.a.2603.2	yes	8	3120.2603	odd	12	inner
3120.1.iq.a.2843.1	yes	8	240.203	odd	4	inner
3120.1.iq.a.2843.2	yes	8	80.43	even	4	inner
3120.1.iv.a.107.1	yes	8	624.107	even	12
3120.1.iv.a.107.2	yes	8	208.107	odd	12
3120.1.iv.a.347.1	yes	8	16.11	odd	4
3120.1.iv.a.347.2	yes	8	48.11	even	4
3120.1.iv.a.1043.1	yes	8	195.68	even	12
3120.1.iv.a.1043.2	yes	8	65.3	odd	12
3120.1.iv.a.1283.1	yes	8	5.3	odd	4
3120.1.iv.a.1283.2	yes	8	15.8	even	4

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:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3120.iq (of order \(12\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{3} +(0.866025 - 0.500000i) q^{4} +(0.707107 - 0.707107i) q^{5} +(0.965926 + 0.258819i) q^{6} +(-0.366025 + 1.36603i) q^{7} +(0.707107 - 0.707107i) q^{8} +(0.500000 + 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{10} +(-0.258819 - 0.965926i) q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.41421i q^{14} +(0.965926 - 0.258819i) q^{15} +(0.500000 - 0.866025i) q^{16} +(0.707107 + 0.707107i) q^{18} +(0.258819 - 0.965926i) q^{20} +(-1.00000 + 1.00000i) q^{21} +(-0.500000 - 0.866025i) q^{22} +(-0.965926 + 0.258819i) q^{23} +(0.965926 - 0.258819i) q^{24} -1.00000i q^{25} +(-0.965926 + 0.258819i) q^{26} +1.00000i q^{27} +(0.366025 + 1.36603i) q^{28} +(0.965926 - 0.258819i) q^{29} +(0.866025 - 0.500000i) q^{30} -1.00000i q^{31} +(0.258819 - 0.965926i) q^{32} +(0.258819 - 0.965926i) q^{33} +(0.707107 + 1.22474i) q^{35} +(0.866025 + 0.500000i) q^{36} +(-0.500000 + 0.866025i) q^{37} +(-0.866025 - 0.500000i) q^{39} -1.00000i q^{40} +(-0.707107 + 1.22474i) q^{42} +(0.500000 + 0.866025i) q^{43} +(-0.707107 - 0.707107i) q^{44} +(0.965926 + 0.258819i) q^{45} +(-0.866025 + 0.500000i) q^{46} +(0.707107 + 0.707107i) q^{47} +(0.866025 - 0.500000i) q^{48} +(-0.866025 - 0.500000i) q^{49} +(-0.258819 - 0.965926i) q^{50} +(-0.866025 + 0.500000i) q^{52} -1.41421 q^{53} +(0.258819 + 0.965926i) q^{54} +(-0.866025 - 0.500000i) q^{55} +(0.707107 + 1.22474i) q^{56} +(0.866025 - 0.500000i) q^{58} +(-0.258819 + 0.965926i) q^{59} +(0.707107 - 0.707107i) q^{60} +(-0.258819 - 0.965926i) q^{62} +(-1.36603 + 0.366025i) q^{63} -1.00000i q^{64} +(-0.707107 + 0.707107i) q^{65} -1.00000i q^{66} +(-0.965926 - 0.258819i) q^{69} +(1.00000 + 1.00000i) q^{70} +(-1.22474 + 0.707107i) q^{71} +(0.965926 + 0.258819i) q^{72} +(-1.00000 - 1.00000i) q^{73} +(-0.258819 + 0.965926i) q^{74} +(0.500000 - 0.866025i) q^{75} +1.41421 q^{77} +(-0.965926 - 0.258819i) q^{78} -1.00000 q^{79} +(-0.258819 - 0.965926i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(-0.366025 + 1.36603i) q^{84} +(0.707107 + 0.707107i) q^{86} +(0.965926 + 0.258819i) q^{87} +(-0.866025 - 0.500000i) q^{88} +(-1.22474 - 0.707107i) q^{89} +1.00000 q^{90} +(0.366025 - 1.36603i) q^{91} +(-0.707107 + 0.707107i) q^{92} +(0.500000 - 0.866025i) q^{93} +(0.866025 + 0.500000i) q^{94} +(0.707107 - 0.707107i) q^{96} +(-0.965926 - 0.258819i) q^{98} +(0.707107 - 0.707107i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7} + 4 q^{9} + 4 q^{10} + 8 q^{12} - 8 q^{13} + 4 q^{16} - 8 q^{21} - 4 q^{22} - 4 q^{28} - 4 q^{37} + 4 q^{43} - 4 q^{63} + 8 q^{70} - 8 q^{73} + 4 q^{75} - 8 q^{79} - 4 q^{81} + 4 q^{84} + 8 q^{90}+ \cdots + 4 q^{93}+O(q^{100}) \) 8 * q + 4 * q^7 + 4 * q^9 + 4 * q^10 + 8 * q^12 - 8 * q^13 + 4 * q^16 - 8 * q^21 - 4 * q^22 - 4 * q^28 - 4 * q^37 + 4 * q^43 - 4 * q^63 + 8 * q^70 - 8 * q^73 + 4 * q^75 - 8 * q^79 - 4 * q^81 + 4 * q^84 + 8 * q^90 - 4 * q^91 + 4 * q^93

Introduction

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