LMFDB - Embedded newform 684.1.x.a.391.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [684,1,Mod(7,684)] 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("684.7"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(684, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 684 = 2^{2} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	684.x (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:	$0.341360468641$
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 4.0.467856.1
Artin image:	$\SL(2,3):C_2$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

Embedding invariants

Embedding label			391.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	684.391
Dual form			684.1.x.a.7.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +(0.500000 + 0.866025i) q^{5} -1.00000i q^{6} +(-0.866025 + 0.500000i) q^{7} -1.00000 q^{8} -1.00000 q^{9} +(-0.500000 - 0.866025i) q^{10} +(-0.866025 + 0.500000i) q^{11} +1.00000i q^{12} +(0.866025 - 0.500000i) q^{14} +(-0.866025 + 0.500000i) q^{15} +1.00000 q^{16} +(-0.500000 + 0.866025i) q^{17} +1.00000 q^{18} -1.00000i q^{19} +(0.500000 + 0.866025i) q^{20} +(-0.500000 - 0.866025i) q^{21} +(0.866025 - 0.500000i) q^{22} -1.00000i q^{24} -1.00000i q^{27} +(-0.866025 + 0.500000i) q^{28} +(0.500000 - 0.866025i) q^{29} +(0.866025 - 0.500000i) q^{30} +(0.866025 + 0.500000i) q^{31} -1.00000 q^{32} +(-0.500000 - 0.866025i) q^{33} +(0.500000 - 0.866025i) q^{34} +(-0.866025 - 0.500000i) q^{35} -1.00000 q^{36} +1.00000i q^{38} +(-0.500000 - 0.866025i) q^{40} +(0.500000 + 0.866025i) q^{41} +(0.500000 + 0.866025i) q^{42} +2.00000i q^{43} +(-0.866025 + 0.500000i) q^{44} +(-0.500000 - 0.866025i) q^{45} +(0.866025 + 0.500000i) q^{47} +1.00000i q^{48} +(-0.866025 - 0.500000i) q^{51} +(-0.500000 - 0.866025i) q^{53} +1.00000i q^{54} +(-0.866025 - 0.500000i) q^{55} +(0.866025 - 0.500000i) q^{56} +1.00000 q^{57} +(-0.500000 + 0.866025i) q^{58} +(-0.866025 + 0.500000i) q^{59} +(-0.866025 + 0.500000i) q^{60} +(-0.500000 + 0.866025i) q^{61} +(-0.866025 - 0.500000i) q^{62} +(0.866025 - 0.500000i) q^{63} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(0.866025 + 0.500000i) q^{70} +(0.866025 + 0.500000i) q^{71} +1.00000 q^{72} +(-0.500000 + 0.866025i) q^{73} -1.00000i q^{76} +(0.500000 - 0.866025i) q^{77} -2.00000i q^{79} +(0.500000 + 0.866025i) q^{80} +1.00000 q^{81} +(-0.500000 - 0.866025i) q^{82} +(0.866025 - 0.500000i) q^{83} +(-0.500000 - 0.866025i) q^{84} -1.00000 q^{85} -2.00000i 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^{93} +(-0.866025 - 0.500000i) q^{94} +(0.866025 - 0.500000i) q^{95} -1.00000i q^{96} +(0.866025 - 0.500000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{8} - 4 q^{9} - 2 q^{10} + 4 q^{16} - 2 q^{17} + 4 q^{18} + 2 q^{20} - 2 q^{21} + 2 q^{29} - 4 q^{32} - 2 q^{33} + 2 q^{34} - 4 q^{36} - 2 q^{40} + 2 q^{41} + 2 q^{42}+ \cdots - 2 q^{93}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^8 - 4 * q^9 - 2 * q^10 + 4 * q^16 - 2 * q^17 + 4 * q^18 + 2 * q^20 - 2 * q^21 + 2 * q^29 - 4 * q^32 - 2 * q^33 + 2 * q^34 - 4 * q^36 - 2 * q^40 + 2 * q^41 + 2 * q^42 - 2 * q^45 - 2 * q^53 + 4 * q^57 - 2 * q^58 - 2 * q^61 + 4 * q^64 + 2 * q^66 - 2 * q^68 + 4 * q^72 - 2 * q^73 + 2 * q^77 + 2 * q^80 + 4 * q^81 - 2 * q^82 - 2 * q^84 - 4 * q^85 + 2 * q^89 + 2 * q^90 - 2 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/684\mathbb{Z}\right)^\times$.

$n$	$325$	$343$	$533$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$-1$	$e\left(\frac{1}{3}\right)$

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 $

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

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	684.1.x.a.391.2	yes	4
3.2	odd	2		2052.1.x.a.847.1		4
4.3	odd	2	inner	684.1.x.a.391.1	yes	4
9.2	odd	6		2052.1.bm.a.1531.2		4
9.7	even	3		684.1.bm.a.619.2	yes	4
12.11	even	2		2052.1.x.a.847.2		4
19.7	even	3		684.1.bm.a.463.1	yes	4
36.7	odd	6		684.1.bm.a.619.1	yes	4
36.11	even	6		2052.1.bm.a.1531.1		4
57.26	odd	6		2052.1.bm.a.1603.1		4
76.7	odd	6		684.1.bm.a.463.2	yes	4
171.7	even	3	inner	684.1.x.a.7.2	yes	4
171.83	odd	6		2052.1.x.a.235.2		4
228.83	even	6		2052.1.bm.a.1603.2		4
684.7	odd	6	inner	684.1.x.a.7.1	✓	4
684.83	even	6		2052.1.x.a.235.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.1.x.a.7.1	✓	4	684.7	odd	6	inner
684.1.x.a.7.2	yes	4	171.7	even	3	inner
684.1.x.a.391.1	yes	4	4.3	odd	2	inner
684.1.x.a.391.2	yes	4	1.1	even	1	trivial
684.1.bm.a.463.1	yes	4	19.7	even	3
684.1.bm.a.463.2	yes	4	76.7	odd	6
684.1.bm.a.619.1	yes	4	36.7	odd	6
684.1.bm.a.619.2	yes	4	9.7	even	3
2052.1.x.a.235.1		4	684.83	even	6
2052.1.x.a.235.2		4	171.83	odd	6
2052.1.x.a.847.1		4	3.2	odd	2
2052.1.x.a.847.2		4	12.11	even	2
2052.1.bm.a.1531.1		4	36.11	even	6
2052.1.bm.a.1531.2		4	9.2	odd	6
2052.1.bm.a.1603.1		4	57.26	odd	6
2052.1.bm.a.1603.2		4	228.83	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	684.x (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000i q^{3} +1.00000 q^{4} +(0.500000 + 0.866025i) q^{5} -1.00000i q^{6} +(-0.866025 + 0.500000i) q^{7} -1.00000 q^{8} -1.00000 q^{9} +(-0.500000 - 0.866025i) q^{10} +(-0.866025 + 0.500000i) q^{11} +1.00000i q^{12} +(0.866025 - 0.500000i) q^{14} +(-0.866025 + 0.500000i) q^{15} +1.00000 q^{16} +(-0.500000 + 0.866025i) q^{17} +1.00000 q^{18} -1.00000i q^{19} +(0.500000 + 0.866025i) q^{20} +(-0.500000 - 0.866025i) q^{21} +(0.866025 - 0.500000i) q^{22} -1.00000i q^{24} -1.00000i q^{27} +(-0.866025 + 0.500000i) q^{28} +(0.500000 - 0.866025i) q^{29} +(0.866025 - 0.500000i) q^{30} +(0.866025 + 0.500000i) q^{31} -1.00000 q^{32} +(-0.500000 - 0.866025i) q^{33} +(0.500000 - 0.866025i) q^{34} +(-0.866025 - 0.500000i) q^{35} -1.00000 q^{36} +1.00000i q^{38} +(-0.500000 - 0.866025i) q^{40} +(0.500000 + 0.866025i) q^{41} +(0.500000 + 0.866025i) q^{42} +2.00000i q^{43} +(-0.866025 + 0.500000i) q^{44} +(-0.500000 - 0.866025i) q^{45} +(0.866025 + 0.500000i) q^{47} +1.00000i q^{48} +(-0.866025 - 0.500000i) q^{51} +(-0.500000 - 0.866025i) q^{53} +1.00000i q^{54} +(-0.866025 - 0.500000i) q^{55} +(0.866025 - 0.500000i) q^{56} +1.00000 q^{57} +(-0.500000 + 0.866025i) q^{58} +(-0.866025 + 0.500000i) q^{59} +(-0.866025 + 0.500000i) q^{60} +(-0.500000 + 0.866025i) q^{61} +(-0.866025 - 0.500000i) q^{62} +(0.866025 - 0.500000i) q^{63} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{66} +(-0.500000 + 0.866025i) q^{68} +(0.866025 + 0.500000i) q^{70} +(0.866025 + 0.500000i) q^{71} +1.00000 q^{72} +(-0.500000 + 0.866025i) q^{73} -1.00000i q^{76} +(0.500000 - 0.866025i) q^{77} -2.00000i q^{79} +(0.500000 + 0.866025i) q^{80} +1.00000 q^{81} +(-0.500000 - 0.866025i) q^{82} +(0.866025 - 0.500000i) q^{83} +(-0.500000 - 0.866025i) q^{84} -1.00000 q^{85} -2.00000i 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^{93} +(-0.866025 - 0.500000i) q^{94} +(0.866025 - 0.500000i) q^{95} -1.00000i q^{96} +(0.866025 - 0.500000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 2 q^{5} - 4 q^{8} - 4 q^{9} - 2 q^{10} + 4 q^{16} - 2 q^{17} + 4 q^{18} + 2 q^{20} - 2 q^{21} + 2 q^{29} - 4 q^{32} - 2 q^{33} + 2 q^{34} - 4 q^{36} - 2 q^{40} + 2 q^{41} + 2 q^{42}+ \cdots - 2 q^{93}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 + 2 * q^5 - 4 * q^8 - 4 * q^9 - 2 * q^10 + 4 * q^16 - 2 * q^17 + 4 * q^18 + 2 * q^20 - 2 * q^21 + 2 * q^29 - 4 * q^32 - 2 * q^33 + 2 * q^34 - 4 * q^36 - 2 * q^40 + 2 * q^41 + 2 * q^42 - 2 * q^45 - 2 * q^53 + 4 * q^57 - 2 * q^58 - 2 * q^61 + 4 * q^64 + 2 * q^66 - 2 * q^68 + 4 * q^72 - 2 * q^73 + 2 * q^77 + 2 * q^80 + 4 * q^81 - 2 * q^82 - 2 * q^84 - 4 * q^85 + 2 * q^89 + 2 * q^90 - 2 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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