LMFDB - Embedded newform 504.1.ce.a.499.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [504,1,Mod(403,504)] 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("504.403"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 4, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	504.ce (of order $6$, degree $2$, 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:	$0.251528766367$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$A_{4}$
Projective field:	Galois closure of 4.0.254016.3

Embedding invariants

Embedding label			499.2
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	504.499
Dual form			504.1.ce.a.403.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +(0.866025 - 0.500000i) q^{5} -1.00000i q^{6} -1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +(0.500000 + 0.866025i) q^{10} +(0.500000 - 0.866025i) q^{11} +1.00000 q^{12} +(-0.866025 - 0.500000i) q^{13} +1.00000 q^{14} +(-0.866025 + 0.500000i) q^{15} +1.00000 q^{16} +(0.500000 + 0.866025i) q^{17} +1.00000i q^{18} +(-0.500000 + 0.866025i) q^{19} +(-0.866025 + 0.500000i) q^{20} +1.00000i q^{21} +(0.866025 + 0.500000i) q^{22} +(0.866025 - 0.500000i) q^{23} +1.00000i q^{24} +(0.500000 - 0.866025i) q^{26} -1.00000 q^{27} +1.00000i q^{28} +(0.866025 - 0.500000i) q^{29} +(-0.500000 - 0.866025i) q^{30} +1.00000i q^{32} +(-0.500000 + 0.866025i) q^{33} +(-0.866025 + 0.500000i) q^{34} +(-0.500000 - 0.866025i) q^{35} -1.00000 q^{36} +(-0.866025 - 0.500000i) q^{37} +(-0.866025 - 0.500000i) q^{38} +(0.866025 + 0.500000i) q^{39} +(-0.500000 - 0.866025i) q^{40} +(0.500000 - 0.866025i) q^{41} -1.00000 q^{42} +(0.500000 + 0.866025i) q^{43} +(-0.500000 + 0.866025i) q^{44} +(0.866025 - 0.500000i) q^{45} +(0.500000 + 0.866025i) q^{46} -1.00000 q^{48} -1.00000 q^{49} +(-0.500000 - 0.866025i) q^{51} +(0.866025 + 0.500000i) q^{52} +(-0.866025 + 0.500000i) q^{53} -1.00000i q^{54} -1.00000i q^{55} -1.00000 q^{56} +(0.500000 - 0.866025i) q^{57} +(0.500000 + 0.866025i) q^{58} +(0.866025 - 0.500000i) q^{60} -1.00000i q^{63} -1.00000 q^{64} -1.00000 q^{65} +(-0.866025 - 0.500000i) q^{66} +(-0.500000 - 0.866025i) q^{68} +(-0.866025 + 0.500000i) q^{69} +(0.866025 - 0.500000i) q^{70} -1.00000i q^{72} +(-0.500000 - 0.866025i) q^{73} +(0.500000 - 0.866025i) q^{74} +(0.500000 - 0.866025i) q^{76} +(-0.866025 - 0.500000i) q^{77} +(-0.500000 + 0.866025i) q^{78} +2.00000i q^{79} +(0.866025 - 0.500000i) q^{80} +1.00000 q^{81} +(0.866025 + 0.500000i) q^{82} +(0.500000 + 0.866025i) q^{83} -1.00000i q^{84} +(0.866025 + 0.500000i) q^{85} +(-0.866025 + 0.500000i) q^{86} +(-0.866025 + 0.500000i) q^{87} +(-0.866025 - 0.500000i) q^{88} +(-0.500000 + 0.866025i) q^{89} +(0.500000 + 0.866025i) q^{90} +(-0.500000 + 0.866025i) q^{91} +(-0.866025 + 0.500000i) q^{92} +1.00000i q^{95} -1.00000i q^{96} +(-0.500000 - 0.866025i) q^{97} -1.00000i q^{98} +(0.500000 - 0.866025i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + 2 q^{10} + 2 q^{11} + 4 q^{12} + 4 q^{14} + 4 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{26} - 4 q^{27} - 2 q^{30} - 2 q^{33} - 2 q^{35} - 4 q^{36} - 2 q^{40} + 2 q^{41} - 4 q^{42}+ \cdots + 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 + 2 * q^10 + 2 * q^11 + 4 * q^12 + 4 * q^14 + 4 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^26 - 4 * q^27 - 2 * q^30 - 2 * q^33 - 2 * q^35 - 4 * q^36 - 2 * q^40 + 2 * q^41 - 4 * q^42 + 2 * q^43 - 2 * q^44 + 2 * q^46 - 4 * q^48 - 4 * q^49 - 2 * q^51 - 4 * q^56 + 2 * q^57 + 2 * q^58 - 4 * q^64 - 4 * q^65 - 2 * q^68 - 2 * q^73 + 2 * q^74 + 2 * q^76 - 2 * q^78 + 4 * q^81 + 2 * q^83 - 2 * q^89 + 2 * q^90 - 2 * q^91 - 2 * q^97 + 2 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$

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 $

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

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.1.ce.a.499.2	yes	4
3.2	odd	2		1512.1.ce.a.1171.1		4
4.3	odd	2		2016.1.cm.a.751.2		4
7.2	even	3		3528.1.cg.d.3235.2		4
7.3	odd	6		3528.1.ba.d.67.1		4
7.4	even	3		504.1.ba.a.67.1	✓	4
7.5	odd	6		3528.1.cg.c.3235.2		4
7.6	odd	2		3528.1.ce.c.3019.2		4
8.3	odd	2	inner	504.1.ce.a.499.1	yes	4
8.5	even	2		2016.1.cm.a.751.1		4
9.2	odd	6		1512.1.ba.a.667.1		4
9.7	even	3		504.1.ba.a.331.2	yes	4
21.11	odd	6		1512.1.ba.a.739.2		4
24.11	even	2		1512.1.ce.a.1171.2		4
28.11	odd	6		2016.1.bi.a.1327.2		4
36.7	odd	6		2016.1.bi.a.79.2		4
56.3	even	6		3528.1.ba.d.67.2		4
56.11	odd	6		504.1.ba.a.67.2	yes	4
56.19	even	6		3528.1.cg.c.3235.1		4
56.27	even	2		3528.1.ce.c.3019.1		4
56.51	odd	6		3528.1.cg.d.3235.1		4
56.53	even	6		2016.1.bi.a.1327.1		4
63.11	odd	6		1512.1.ce.a.235.1		4
63.16	even	3		3528.1.cg.d.2059.1		4
63.25	even	3	inner	504.1.ce.a.403.2	yes	4
63.34	odd	6		3528.1.ba.d.1843.2		4
63.52	odd	6		3528.1.ce.c.2419.2		4
63.61	odd	6		3528.1.cg.c.2059.1		4
72.11	even	6		1512.1.ba.a.667.2		4
72.43	odd	6		504.1.ba.a.331.1	yes	4
72.61	even	6		2016.1.bi.a.79.1		4
168.11	even	6		1512.1.ba.a.739.1		4
252.151	odd	6		2016.1.cm.a.655.1		4
504.11	even	6		1512.1.ce.a.235.2		4
504.115	even	6		3528.1.ce.c.2419.1		4
504.187	even	6		3528.1.cg.c.2059.2		4
504.277	even	6		2016.1.cm.a.655.2		4
504.331	odd	6		3528.1.cg.d.2059.2		4
504.403	odd	6	inner	504.1.ce.a.403.1	yes	4
504.475	even	6		3528.1.ba.d.1843.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.1.ba.a.67.1	✓	4	7.4	even	3
504.1.ba.a.67.2	yes	4	56.11	odd	6
504.1.ba.a.331.1	yes	4	72.43	odd	6
504.1.ba.a.331.2	yes	4	9.7	even	3
504.1.ce.a.403.1	yes	4	504.403	odd	6	inner
504.1.ce.a.403.2	yes	4	63.25	even	3	inner
504.1.ce.a.499.1	yes	4	8.3	odd	2	inner
504.1.ce.a.499.2	yes	4	1.1	even	1	trivial
1512.1.ba.a.667.1		4	9.2	odd	6
1512.1.ba.a.667.2		4	72.11	even	6
1512.1.ba.a.739.1		4	168.11	even	6
1512.1.ba.a.739.2		4	21.11	odd	6
1512.1.ce.a.235.1		4	63.11	odd	6
1512.1.ce.a.235.2		4	504.11	even	6
1512.1.ce.a.1171.1		4	3.2	odd	2
1512.1.ce.a.1171.2		4	24.11	even	2
2016.1.bi.a.79.1		4	72.61	even	6
2016.1.bi.a.79.2		4	36.7	odd	6
2016.1.bi.a.1327.1		4	56.53	even	6
2016.1.bi.a.1327.2		4	28.11	odd	6
2016.1.cm.a.655.1		4	252.151	odd	6
2016.1.cm.a.655.2		4	504.277	even	6
2016.1.cm.a.751.1		4	8.5	even	2
2016.1.cm.a.751.2		4	4.3	odd	2
3528.1.ba.d.67.1		4	7.3	odd	6
3528.1.ba.d.67.2		4	56.3	even	6
3528.1.ba.d.1843.1		4	504.475	even	6
3528.1.ba.d.1843.2		4	63.34	odd	6
3528.1.ce.c.2419.1		4	504.115	even	6
3528.1.ce.c.2419.2		4	63.52	odd	6
3528.1.ce.c.3019.1		4	56.27	even	2
3528.1.ce.c.3019.2		4	7.6	odd	2
3528.1.cg.c.2059.1		4	63.61	odd	6
3528.1.cg.c.2059.2		4	504.187	even	6
3528.1.cg.c.3235.1		4	56.19	even	6
3528.1.cg.c.3235.2		4	7.5	odd	6
3528.1.cg.d.2059.1		4	63.16	even	3
3528.1.cg.d.2059.2		4	504.331	odd	6
3528.1.cg.d.3235.1		4	56.51	odd	6
3528.1.cg.d.3235.2		4	7.2	even	3

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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	504.ce (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +(0.866025 - 0.500000i) q^{5} -1.00000i q^{6} -1.00000i q^{7} -1.00000i q^{8} +1.00000 q^{9} +(0.500000 + 0.866025i) q^{10} +(0.500000 - 0.866025i) q^{11} +1.00000 q^{12} +(-0.866025 - 0.500000i) q^{13} +1.00000 q^{14} +(-0.866025 + 0.500000i) q^{15} +1.00000 q^{16} +(0.500000 + 0.866025i) q^{17} +1.00000i q^{18} +(-0.500000 + 0.866025i) q^{19} +(-0.866025 + 0.500000i) q^{20} +1.00000i q^{21} +(0.866025 + 0.500000i) q^{22} +(0.866025 - 0.500000i) q^{23} +1.00000i q^{24} +(0.500000 - 0.866025i) q^{26} -1.00000 q^{27} +1.00000i q^{28} +(0.866025 - 0.500000i) q^{29} +(-0.500000 - 0.866025i) q^{30} +1.00000i q^{32} +(-0.500000 + 0.866025i) q^{33} +(-0.866025 + 0.500000i) q^{34} +(-0.500000 - 0.866025i) q^{35} -1.00000 q^{36} +(-0.866025 - 0.500000i) q^{37} +(-0.866025 - 0.500000i) q^{38} +(0.866025 + 0.500000i) q^{39} +(-0.500000 - 0.866025i) q^{40} +(0.500000 - 0.866025i) q^{41} -1.00000 q^{42} +(0.500000 + 0.866025i) q^{43} +(-0.500000 + 0.866025i) q^{44} +(0.866025 - 0.500000i) q^{45} +(0.500000 + 0.866025i) q^{46} -1.00000 q^{48} -1.00000 q^{49} +(-0.500000 - 0.866025i) q^{51} +(0.866025 + 0.500000i) q^{52} +(-0.866025 + 0.500000i) q^{53} -1.00000i q^{54} -1.00000i q^{55} -1.00000 q^{56} +(0.500000 - 0.866025i) q^{57} +(0.500000 + 0.866025i) q^{58} +(0.866025 - 0.500000i) q^{60} -1.00000i q^{63} -1.00000 q^{64} -1.00000 q^{65} +(-0.866025 - 0.500000i) q^{66} +(-0.500000 - 0.866025i) q^{68} +(-0.866025 + 0.500000i) q^{69} +(0.866025 - 0.500000i) q^{70} -1.00000i q^{72} +(-0.500000 - 0.866025i) q^{73} +(0.500000 - 0.866025i) q^{74} +(0.500000 - 0.866025i) q^{76} +(-0.866025 - 0.500000i) q^{77} +(-0.500000 + 0.866025i) q^{78} +2.00000i q^{79} +(0.866025 - 0.500000i) q^{80} +1.00000 q^{81} +(0.866025 + 0.500000i) q^{82} +(0.500000 + 0.866025i) q^{83} -1.00000i q^{84} +(0.866025 + 0.500000i) q^{85} +(-0.866025 + 0.500000i) q^{86} +(-0.866025 + 0.500000i) q^{87} +(-0.866025 - 0.500000i) q^{88} +(-0.500000 + 0.866025i) q^{89} +(0.500000 + 0.866025i) q^{90} +(-0.500000 + 0.866025i) q^{91} +(-0.866025 + 0.500000i) q^{92} +1.00000i q^{95} -1.00000i q^{96} +(-0.500000 - 0.866025i) q^{97} -1.00000i q^{98} +(0.500000 - 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} + 2 q^{10} + 2 q^{11} + 4 q^{12} + 4 q^{14} + 4 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{26} - 4 q^{27} - 2 q^{30} - 2 q^{33} - 2 q^{35} - 4 q^{36} - 2 q^{40} + 2 q^{41} - 4 q^{42}+ \cdots + 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 + 2 * q^10 + 2 * q^11 + 4 * q^12 + 4 * q^14 + 4 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^26 - 4 * q^27 - 2 * q^30 - 2 * q^33 - 2 * q^35 - 4 * q^36 - 2 * q^40 + 2 * q^41 - 4 * q^42 + 2 * q^43 - 2 * q^44 + 2 * q^46 - 4 * q^48 - 4 * q^49 - 2 * q^51 - 4 * q^56 + 2 * q^57 + 2 * q^58 - 4 * q^64 - 4 * q^65 - 2 * q^68 - 2 * q^73 + 2 * q^74 + 2 * q^76 - 2 * q^78 + 4 * q^81 + 2 * q^83 - 2 * q^89 + 2 * q^90 - 2 * q^91 - 2 * q^97 + 2 * q^99

Introduction

L-functions

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