LMFDB - Embedded newform 468.1.y.a.367.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [468,1,Mod(139,468)] 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("468.139"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(468, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 2, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 468 = 2^{2} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	468.y (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:	$0.233562425912$
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 4.0.219024.1
Artin image:	$\SL(2,3):C_2$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

Embedding invariants

Embedding label			367.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	468.367
Dual form			468.1.y.a.139.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(0.500000 + 0.866025i) q^{5} +1.00000 q^{6} +(-0.866025 + 0.500000i) q^{7} +1.00000i q^{8} -1.00000 q^{9} +(0.866025 - 0.500000i) q^{10} -1.00000i q^{12} +1.00000 q^{13} +(0.500000 + 0.866025i) q^{14} +(-0.866025 + 0.500000i) q^{15} +1.00000 q^{16} +(-0.500000 + 0.866025i) q^{17} +1.00000i q^{18} +(0.866025 + 0.500000i) q^{19} +(-0.500000 - 0.866025i) q^{20} +(-0.500000 - 0.866025i) q^{21} +(-0.866025 - 0.500000i) q^{23} -1.00000 q^{24} -1.00000i q^{26} -1.00000i q^{27} +(0.866025 - 0.500000i) q^{28} +(0.500000 + 0.866025i) q^{30} +(0.866025 - 0.500000i) q^{31} -1.00000i q^{32} +(0.866025 + 0.500000i) q^{34} +(-0.866025 - 0.500000i) q^{35} +1.00000 q^{36} +(-0.500000 - 0.866025i) q^{37} +(0.500000 - 0.866025i) q^{38} +1.00000i q^{39} +(-0.866025 + 0.500000i) q^{40} +(0.500000 - 0.866025i) q^{41} +(-0.866025 + 0.500000i) q^{42} +(0.866025 - 0.500000i) q^{43} +(-0.500000 - 0.866025i) q^{45} +(-0.500000 + 0.866025i) q^{46} +(0.866025 + 0.500000i) q^{47} +1.00000i q^{48} +(-0.866025 - 0.500000i) q^{51} -1.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +(-0.500000 - 0.866025i) q^{56} +(-0.500000 + 0.866025i) q^{57} +(0.866025 - 0.500000i) q^{60} +(-0.500000 - 0.866025i) q^{61} +(-0.500000 - 0.866025i) q^{62} +(0.866025 - 0.500000i) q^{63} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{65} +(0.866025 + 0.500000i) q^{67} +(0.500000 - 0.866025i) q^{68} +(0.500000 - 0.866025i) q^{69} +(-0.500000 + 0.866025i) q^{70} +(0.866025 + 0.500000i) q^{71} -1.00000i q^{72} +(-0.866025 + 0.500000i) q^{74} +(-0.866025 - 0.500000i) q^{76} +1.00000 q^{78} +(-0.866025 - 0.500000i) q^{79} +(0.500000 + 0.866025i) q^{80} +1.00000 q^{81} +(-0.866025 - 0.500000i) q^{82} +(-0.866025 - 0.500000i) q^{83} +(0.500000 + 0.866025i) q^{84} -1.00000 q^{85} +(-0.500000 - 0.866025i) q^{86} +(0.500000 + 0.866025i) q^{89} +(-0.866025 + 0.500000i) q^{90} +(-0.866025 + 0.500000i) q^{91} +(0.866025 + 0.500000i) q^{92} +(0.500000 + 0.866025i) q^{93} +(0.500000 - 0.866025i) q^{94} +1.00000i q^{95} +1.00000 q^{96} +(0.500000 + 0.866025i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{9} + 4 q^{13} + 2 q^{14} + 4 q^{16} - 2 q^{17} - 2 q^{20} - 2 q^{21} - 4 q^{24} + 2 q^{30} + 4 q^{36} - 2 q^{37} + 2 q^{38} + 2 q^{41} - 2 q^{45} - 2 q^{46} - 4 q^{52}+ \cdots + 2 q^{97}+O(q^{100}) $ 4 * q - 4 * q^4 + 2 * q^5 + 4 * q^6 - 4 * q^9 + 4 * q^13 + 2 * q^14 + 4 * q^16 - 2 * q^17 - 2 * q^20 - 2 * q^21 - 4 * q^24 + 2 * q^30 + 4 * q^36 - 2 * q^37 + 2 * q^38 + 2 * q^41 - 2 * q^45 - 2 * q^46 - 4 * q^52 - 8 * q^53 - 4 * q^54 - 2 * q^56 - 2 * q^57 - 2 * q^61 - 2 * q^62 - 4 * q^64 + 2 * q^65 + 2 * q^68 + 2 * q^69 - 2 * q^70 + 4 * q^78 + 2 * q^80 + 4 * q^81 + 2 * q^84 - 4 * q^85 - 2 * q^86 + 2 * q^89 + 2 * q^93 + 2 * q^94 + 4 * q^96 + 2 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/468\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$235$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{2}{3}\right)$	$-1$

Coefficient data

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

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

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

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

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	468.1.y.a.367.1	yes	4
3.2	odd	2		1404.1.y.a.991.2		4
4.3	odd	2	inner	468.1.y.a.367.2	yes	4
9.4	even	3		468.1.v.a.211.2	yes	4
9.5	odd	6		1404.1.v.a.523.1		4
12.11	even	2		1404.1.y.a.991.1		4
13.9	even	3		468.1.v.a.295.1	yes	4
36.23	even	6		1404.1.v.a.523.2		4
36.31	odd	6		468.1.v.a.211.1	✓	4
39.35	odd	6		1404.1.v.a.451.2		4
52.35	odd	6		468.1.v.a.295.2	yes	4
117.22	even	3	inner	468.1.y.a.139.1	yes	4
117.113	odd	6		1404.1.y.a.1387.2		4
156.35	even	6		1404.1.v.a.451.1		4
468.139	odd	6	inner	468.1.y.a.139.2	yes	4
468.347	even	6		1404.1.y.a.1387.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.1.v.a.211.1	✓	4	36.31	odd	6
468.1.v.a.211.2	yes	4	9.4	even	3
468.1.v.a.295.1	yes	4	13.9	even	3
468.1.v.a.295.2	yes	4	52.35	odd	6
468.1.y.a.139.1	yes	4	117.22	even	3	inner
468.1.y.a.139.2	yes	4	468.139	odd	6	inner
468.1.y.a.367.1	yes	4	1.1	even	1	trivial
468.1.y.a.367.2	yes	4	4.3	odd	2	inner
1404.1.v.a.451.1		4	156.35	even	6
1404.1.v.a.451.2		4	39.35	odd	6
1404.1.v.a.523.1		4	9.5	odd	6
1404.1.v.a.523.2		4	36.23	even	6
1404.1.y.a.991.1		4	12.11	even	2
1404.1.y.a.991.2		4	3.2	odd	2
1404.1.y.a.1387.1		4	468.347	even	6
1404.1.y.a.1387.2		4	117.113	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	468.y (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +(0.500000 + 0.866025i) q^{5} +1.00000 q^{6} +(-0.866025 + 0.500000i) q^{7} +1.00000i q^{8} -1.00000 q^{9} +(0.866025 - 0.500000i) q^{10} -1.00000i q^{12} +1.00000 q^{13} +(0.500000 + 0.866025i) q^{14} +(-0.866025 + 0.500000i) q^{15} +1.00000 q^{16} +(-0.500000 + 0.866025i) q^{17} +1.00000i q^{18} +(0.866025 + 0.500000i) q^{19} +(-0.500000 - 0.866025i) q^{20} +(-0.500000 - 0.866025i) q^{21} +(-0.866025 - 0.500000i) q^{23} -1.00000 q^{24} -1.00000i q^{26} -1.00000i q^{27} +(0.866025 - 0.500000i) q^{28} +(0.500000 + 0.866025i) q^{30} +(0.866025 - 0.500000i) q^{31} -1.00000i q^{32} +(0.866025 + 0.500000i) q^{34} +(-0.866025 - 0.500000i) q^{35} +1.00000 q^{36} +(-0.500000 - 0.866025i) q^{37} +(0.500000 - 0.866025i) q^{38} +1.00000i q^{39} +(-0.866025 + 0.500000i) q^{40} +(0.500000 - 0.866025i) q^{41} +(-0.866025 + 0.500000i) q^{42} +(0.866025 - 0.500000i) q^{43} +(-0.500000 - 0.866025i) q^{45} +(-0.500000 + 0.866025i) q^{46} +(0.866025 + 0.500000i) q^{47} +1.00000i q^{48} +(-0.866025 - 0.500000i) q^{51} -1.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +(-0.500000 - 0.866025i) q^{56} +(-0.500000 + 0.866025i) q^{57} +(0.866025 - 0.500000i) q^{60} +(-0.500000 - 0.866025i) q^{61} +(-0.500000 - 0.866025i) q^{62} +(0.866025 - 0.500000i) q^{63} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{65} +(0.866025 + 0.500000i) q^{67} +(0.500000 - 0.866025i) q^{68} +(0.500000 - 0.866025i) q^{69} +(-0.500000 + 0.866025i) q^{70} +(0.866025 + 0.500000i) q^{71} -1.00000i q^{72} +(-0.866025 + 0.500000i) q^{74} +(-0.866025 - 0.500000i) q^{76} +1.00000 q^{78} +(-0.866025 - 0.500000i) q^{79} +(0.500000 + 0.866025i) q^{80} +1.00000 q^{81} +(-0.866025 - 0.500000i) q^{82} +(-0.866025 - 0.500000i) q^{83} +(0.500000 + 0.866025i) q^{84} -1.00000 q^{85} +(-0.500000 - 0.866025i) q^{86} +(0.500000 + 0.866025i) q^{89} +(-0.866025 + 0.500000i) q^{90} +(-0.866025 + 0.500000i) q^{91} +(0.866025 + 0.500000i) q^{92} +(0.500000 + 0.866025i) q^{93} +(0.500000 - 0.866025i) q^{94} +1.00000i q^{95} +1.00000 q^{96} +(0.500000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{9} + 4 q^{13} + 2 q^{14} + 4 q^{16} - 2 q^{17} - 2 q^{20} - 2 q^{21} - 4 q^{24} + 2 q^{30} + 4 q^{36} - 2 q^{37} + 2 q^{38} + 2 q^{41} - 2 q^{45} - 2 q^{46} - 4 q^{52}+ \cdots + 2 q^{97}+O(q^{100}) \) 4 * q - 4 * q^4 + 2 * q^5 + 4 * q^6 - 4 * q^9 + 4 * q^13 + 2 * q^14 + 4 * q^16 - 2 * q^17 - 2 * q^20 - 2 * q^21 - 4 * q^24 + 2 * q^30 + 4 * q^36 - 2 * q^37 + 2 * q^38 + 2 * q^41 - 2 * q^45 - 2 * q^46 - 4 * q^52 - 8 * q^53 - 4 * q^54 - 2 * q^56 - 2 * q^57 - 2 * q^61 - 2 * q^62 - 4 * q^64 + 2 * q^65 + 2 * q^68 + 2 * q^69 - 2 * q^70 + 4 * q^78 + 2 * q^80 + 4 * q^81 + 2 * q^84 - 4 * q^85 - 2 * q^86 + 2 * q^89 + 2 * q^93 + 2 * q^94 + 4 * q^96 + 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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