LMFDB - Embedded newform 777.1.be.a.137.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [777,1,Mod(137,777)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(777, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4, 2])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("777.137"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$ N $	$=$	$ 777 = 3 \cdot 7 \cdot 37 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	777.be (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:	$0.387773514816$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$A_{4}$
Projective field:	Galois closure of 4.0.603729.1

Embedding invariants

Embedding label			137.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	777.137
Dual form			777.1.be.a.380.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.866025 - 0.500000i) q^{3} +(-0.500000 - 0.866025i) q^{4} -1.00000i q^{5} +(0.500000 + 0.866025i) q^{7} +(0.500000 - 0.866025i) q^{9} +(0.866025 + 0.500000i) q^{11} +(-0.866025 - 0.500000i) q^{12} +(-0.500000 + 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{15} +(-0.500000 + 0.866025i) q^{16} -1.00000i q^{17} -1.00000 q^{19} +(-0.866025 + 0.500000i) q^{20} +(0.866025 + 0.500000i) q^{21} +(-0.866025 + 0.500000i) q^{23} -1.00000i q^{27} +(0.500000 - 0.866025i) q^{28} -2.00000i q^{29} +(-0.500000 + 0.866025i) q^{31} +1.00000 q^{33} +(0.866025 - 0.500000i) q^{35} -1.00000 q^{36} +(-0.500000 + 0.866025i) q^{37} +1.00000i q^{39} +(0.866025 + 0.500000i) q^{41} -1.00000i q^{44} +(-0.866025 - 0.500000i) q^{45} +(-0.866025 + 0.500000i) q^{47} +1.00000i q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.500000 - 0.866025i) q^{51} +1.00000 q^{52} +1.00000i q^{53} +(0.500000 - 0.866025i) q^{55} +(-0.866025 + 0.500000i) q^{57} +1.00000i q^{59} +(-0.500000 + 0.866025i) q^{60} +1.00000 q^{61} +1.00000 q^{63} +1.00000 q^{64} +(0.866025 + 0.500000i) q^{65} +(-0.866025 + 0.500000i) q^{68} +(-0.500000 + 0.866025i) q^{69} +(0.866025 + 0.500000i) q^{71} +(0.500000 - 0.866025i) q^{73} +(0.500000 + 0.866025i) q^{76} +1.00000i q^{77} +1.00000 q^{79} +(0.866025 + 0.500000i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.866025 + 0.500000i) q^{83} -1.00000i q^{84} -1.00000 q^{85} +(-1.00000 - 1.73205i) q^{87} -1.00000 q^{91} +(0.866025 + 0.500000i) q^{92} +1.00000i q^{93} +1.00000i q^{95} +(0.866025 - 0.500000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{4} + 2 q^{7} + 2 q^{9} - 2 q^{13} - 2 q^{15} - 2 q^{16} - 4 q^{19} + 2 q^{28} - 2 q^{31} + 4 q^{33} - 4 q^{36} - 2 q^{37} - 2 q^{49} - 2 q^{51} + 4 q^{52} + 2 q^{55} - 2 q^{60} + 4 q^{61} + 4 q^{63}+ \cdots - 4 q^{91}+O(q^{100}) $ 4 * q - 2 * q^4 + 2 * q^7 + 2 * q^9 - 2 * q^13 - 2 * q^15 - 2 * q^16 - 4 * q^19 + 2 * q^28 - 2 * q^31 + 4 * q^33 - 4 * q^36 - 2 * q^37 - 2 * q^49 - 2 * q^51 + 4 * q^52 + 2 * q^55 - 2 * q^60 + 4 * q^61 + 4 * q^63 + 4 * q^64 - 2 * q^69 + 2 * q^73 + 2 * q^76 + 4 * q^79 - 2 * q^81 - 4 * q^85 - 4 * q^87 - 4 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/777\mathbb{Z}\right)^\times$.

$n$	$260$	$556$	$631$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$

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 $

$2$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	0.866025	−	0.500000i	0.866025	−	0.500000i
$3$	0.866025	−	0.500000i	0.866025	−	0.500000i
$4$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$5$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$6$		0			0
$7$	0.500000	+	0.866025i	0.500000	+	0.866025i
$7$	0.500000	+	0.866025i	0.500000	+	0.866025i
$8$		0			0
$9$	0.500000	−	0.866025i	0.500000	−	0.866025i
$10$		0			0
$11$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$11$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$12$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$13$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$13$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$14$		0			0
$15$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$17$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$18$		0			0
$19$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$19$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$20$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$21$	0.866025	+	0.500000i	0.866025	+	0.500000i
$22$		0			0
$23$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$23$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		−	1.00000i		−	1.00000i
$28$	0.500000	−	0.866025i	0.500000	−	0.866025i
$29$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$29$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$30$		0			0
$31$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$31$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$32$		0			0
$33$	1.00000			1.00000
$34$		0			0
$35$	0.866025	−	0.500000i	0.866025	−	0.500000i
$36$	−1.00000			−1.00000
$37$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$37$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$38$		0			0
$39$			1.00000i			1.00000i
$40$		0			0
$41$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$41$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		−	1.00000i		−	1.00000i
$45$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$46$		0			0
$47$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$47$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$48$			1.00000i			1.00000i
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$		0			0
$51$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$52$	1.00000			1.00000
$53$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$53$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$	0.500000	−	0.866025i	0.500000	−	0.866025i
$56$		0			0
$57$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$58$		0			0
$59$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$59$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$60$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$61$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$61$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$	1.00000			1.00000
$64$	1.00000			1.00000
$65$	0.866025	+	0.500000i	0.866025	+	0.500000i
$66$		0			0
$67$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$67$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$68$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$69$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$70$		0			0
$71$	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	$0.5\pi$
$72$		0			0
$73$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$73$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$74$		0			0
$75$		0			0
$76$	0.500000	+	0.866025i	0.500000	+	0.866025i
$77$			1.00000i			1.00000i
$78$		0			0
$79$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$79$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$80$	0.866025	+	0.500000i	0.866025	+	0.500000i
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$83$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$84$		−	1.00000i		−	1.00000i
$85$	−1.00000			−1.00000
$86$		0			0
$87$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	−1.00000			−1.00000
$92$	0.866025	+	0.500000i	0.866025	+	0.500000i
$93$			1.00000i			1.00000i
$94$		0			0
$95$			1.00000i			1.00000i
$96$		0			0
$97$		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
$101$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$101$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$103$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$104$		0			0
$105$	0.500000	−	0.866025i	0.500000	−	0.866025i
$106$		0			0
$107$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$107$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$108$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$109$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$109$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$			1.00000i			1.00000i
$112$	−1.00000			−1.00000
$113$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$113$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$114$		0			0
$115$	0.500000	+	0.866025i	0.500000	+	0.866025i
$116$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$117$	0.500000	+	0.866025i	0.500000	+	0.866025i
$118$		0			0
$119$	0.866025	−	0.500000i	0.866025	−	0.500000i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	1.00000			1.00000
$124$	1.00000			1.00000
$125$		−	1.00000i		−	1.00000i
$126$		0			0
$127$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$127$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$133$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$134$		0			0
$135$	−1.00000			−1.00000
$136$		0			0
$137$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$137$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$139$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$140$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$141$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$142$		0			0
$143$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$144$	0.500000	+	0.866025i	0.500000	+	0.866025i
$145$	−2.00000			−2.00000
$146$		0			0
$147$			1.00000i			1.00000i
$148$	1.00000			1.00000
$149$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$149$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$150$		0			0
$151$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$151$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$152$		0			0
$153$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$154$		0			0
$155$	0.866025	+	0.500000i	0.866025	+	0.500000i
$156$	0.866025	−	0.500000i	0.866025	−	0.500000i
$157$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$157$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$	0.500000	+	0.866025i	0.500000	+	0.866025i
$160$		0			0
$161$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$162$		0			0
$163$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$163$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$164$		−	1.00000i		−	1.00000i
$165$		−	1.00000i		−	1.00000i
$166$		0			0
$167$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$167$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$177$	0.500000	+	0.866025i	0.500000	+	0.866025i
$178$		0			0
$179$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$179$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$180$			1.00000i			1.00000i
$181$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$181$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$182$		0			0
$183$	0.866025	−	0.500000i	0.866025	−	0.500000i
$184$		0			0
$185$	0.866025	+	0.500000i	0.866025	+	0.500000i
$186$		0			0
$187$	0.500000	−	0.866025i	0.500000	−	0.866025i
$188$	0.866025	+	0.500000i	0.866025	+	0.500000i
$189$	0.866025	−	0.500000i	0.866025	−	0.500000i
$190$		0			0
$191$	−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	$0.5\pi$
$192$	0.866025	−	0.500000i	0.866025	−	0.500000i
$193$	−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			0
$195$	1.00000			1.00000
$196$	1.00000			1.00000
$197$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$197$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$198$		0			0
$199$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$199$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.73205	−	1.00000i	1.73205	−	1.00000i
$204$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$205$	0.500000	−	0.866025i	0.500000	−	0.866025i
$206$		0			0
$207$			1.00000i			1.00000i
$208$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$209$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$210$		0			0
$211$		0			0			−	1.00000i	$-0.5\pi$
$211$		0			0				1.00000i	$0.5\pi$
$212$	0.866025	−	0.500000i	0.866025	−	0.500000i
$213$	1.00000			1.00000
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.00000			−1.00000
$218$		0			0
$219$		−	1.00000i		−	1.00000i
$220$	−1.00000			−1.00000
$221$	0.866025	+	0.500000i	0.866025	+	0.500000i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	0.866025	+	0.500000i	0.866025	+	0.500000i
$229$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$229$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$	0.500000	+	0.866025i	0.500000	+	0.866025i
$232$		0			0
$233$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$233$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$234$		0			0
$235$	0.500000	+	0.866025i	0.500000	+	0.866025i
$236$	0.866025	−	0.500000i	0.866025	−	0.500000i
$237$	0.866025	−	0.500000i	0.866025	−	0.500000i
$238$		0			0
$239$	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	$0.166667\pi$
$240$	1.00000			1.00000
$241$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$241$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$242$		0			0
$243$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$244$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$245$	0.866025	+	0.500000i	0.866025	+	0.500000i
$246$		0			0
$247$	0.500000	−	0.866025i	0.500000	−	0.866025i
$248$		0			0
$249$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$253$	−1.00000			−1.00000
$254$		0			0
$255$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	−1.00000			−1.00000
$260$		−	1.00000i		−	1.00000i
$261$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$262$		0			0
$263$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$263$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$264$		0			0
$265$	1.00000			1.00000
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$269$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$	0.866025	+	0.500000i	0.866025	+	0.500000i
$273$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$274$		0			0
$275$		0			0
$276$	1.00000			1.00000
$277$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$277$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$278$		0			0
$279$	0.500000	+	0.866025i	0.500000	+	0.866025i
$280$		0			0
$281$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$281$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$		−	1.00000i		−	1.00000i
$285$	0.500000	+	0.866025i	0.500000	+	0.866025i
$286$		0			0
$287$			1.00000i			1.00000i
$288$		0			0
$289$		0			0
$290$		0			0
$291$		0			0
$292$	−1.00000			−1.00000
$293$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$293$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$	1.00000			1.00000
$296$		0			0
$297$	0.500000	−	0.866025i	0.500000	−	0.866025i
$298$		0			0
$299$		−	1.00000i		−	1.00000i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−1.00000			−1.00000
$304$	0.500000	−	0.866025i	0.500000	−	0.866025i
$305$		−	1.00000i		−	1.00000i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	0.866025	−	0.500000i	0.866025	−	0.500000i
$309$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$310$		0			0
$311$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$311$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$312$		0			0
$313$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$313$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$314$		0			0
$315$		−	1.00000i		−	1.00000i
$316$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$	1.00000	−	1.73205i	1.00000	−	1.73205i
$320$		−	1.00000i		−	1.00000i
$321$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$322$		0			0
$323$			1.00000i			1.00000i
$324$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$325$		0			0
$326$		0			0
$327$	0.866025	−	0.500000i	0.866025	−	0.500000i
$328$		0			0
$329$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$330$		0			0
$331$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$331$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$332$	0.866025	+	0.500000i	0.866025	+	0.500000i
$333$	0.500000	+	0.866025i	0.500000	+	0.866025i
$334$		0			0
$335$		0			0
$336$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$337$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$337$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$338$		0			0
$339$	0.500000	−	0.866025i	0.500000	−	0.866025i
$340$	0.500000	+	0.866025i	0.500000	+	0.866025i
$341$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$342$		0			0
$343$	−1.00000			−1.00000
$344$		0			0
$345$	0.866025	+	0.500000i	0.866025	+	0.500000i
$346$		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	$0.833333\pi$
$348$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$349$	−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
$351$	0.866025	+	0.500000i	0.866025	+	0.500000i
$352$		0			0
$353$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$353$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$	0.500000	−	0.866025i	0.500000	−	0.866025i
$356$		0			0
$357$	0.500000	−	0.866025i	0.500000	−	0.866025i
$358$		0			0
$359$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$359$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$	0.500000	+	0.866025i	0.500000	+	0.866025i
$365$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$366$		0			0
$367$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$367$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$368$		−	1.00000i		−	1.00000i
$369$	0.866025	−	0.500000i	0.866025	−	0.500000i
$370$		0			0
$371$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$372$	0.866025	−	0.500000i	0.866025	−	0.500000i
$373$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$373$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$376$		0			0
$377$	1.73205	+	1.00000i	1.73205	+	1.00000i
$378$		0			0
$379$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$379$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$380$	0.866025	−	0.500000i	0.866025	−	0.500000i
$381$	0.866025	+	0.500000i	0.866025	+	0.500000i
$382$		0			0
$383$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$383$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$	1.00000			1.00000
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$	0.500000	+	0.866025i	0.500000	+	0.866025i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	1.00000i		−	1.00000i
$396$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$397$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$397$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$400$		0			0
$401$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$401$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$404$			1.00000i			1.00000i
$405$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$406$		0			0
$407$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$408$		0			0
$409$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$409$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$	−1.00000			−1.00000
$412$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$413$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$414$		0			0
$415$	0.500000	+	0.866025i	0.500000	+	0.866025i
$416$		0			0
$417$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$418$		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	$0.166667\pi$
$420$	−1.00000			−1.00000
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$			1.00000i			1.00000i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.500000	+	0.866025i	0.500000	+	0.866025i
$428$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$429$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	0.866025	+	0.500000i	0.866025	+	0.500000i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$436$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$437$	0.866025	−	0.500000i	0.866025	−	0.500000i
$438$		0			0
$439$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$439$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	0.500000	+	0.866025i	0.500000	+	0.866025i
$442$		0			0
$443$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$443$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$444$	0.866025	−	0.500000i	0.866025	−	0.500000i
$445$		0			0
$446$		0			0
$447$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$448$	0.500000	+	0.866025i	0.500000	+	0.866025i
$449$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$449$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$450$		0			0
$451$	0.500000	+	0.866025i	0.500000	+	0.866025i
$452$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$453$		0			0
$454$		0			0
$455$			1.00000i			1.00000i
$456$		0			0
$457$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$457$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$458$		0			0
$459$	−1.00000			−1.00000
$460$	0.500000	−	0.866025i	0.500000	−	0.866025i
$461$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$461$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$463$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$464$	1.73205	+	1.00000i	1.73205	+	1.00000i
$465$	1.00000			1.00000
$466$		0			0
$467$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$467$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$468$	0.500000	−	0.866025i	0.500000	−	0.866025i
$469$		0			0
$470$		0			0
$471$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$477$	0.866025	+	0.500000i	0.866025	+	0.500000i
$478$		0			0
$479$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$479$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$482$		0			0
$483$	−1.00000			−1.00000
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$487$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$493$	−2.00000			−2.00000
$494$		0			0
$495$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$496$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$497$			1.00000i			1.00000i
$498$		0			0
$499$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$499$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$500$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$501$	−1.00000			−1.00000
$502$		0			0
$503$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$503$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$504$		0			0
$505$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$506$		0			0
$507$		0			0
$508$	0.500000	−	0.866025i	0.500000	−	0.866025i
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	1.00000			1.00000
$512$		0			0
$513$			1.00000i			1.00000i
$514$		0			0
$515$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$516$		0			0
$517$	−1.00000			−1.00000
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$521$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$522$		0			0
$523$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$523$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.866025	+	0.500000i	0.866025	+	0.500000i
$528$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$529$		0			0
$530$		0			0
$531$	0.866025	+	0.500000i	0.866025	+	0.500000i
$532$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$533$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$534$		0			0
$535$	−1.00000			−1.00000
$536$		0			0
$537$	1.00000			1.00000
$538$		0			0
$539$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$540$	0.500000	+	0.866025i	0.500000	+	0.866025i
$541$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$541$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$542$		0			0
$543$		−	1.00000i		−	1.00000i
$544$		0			0
$545$		−	1.00000i		−	1.00000i
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$			1.00000i			1.00000i
$549$	0.500000	−	0.866025i	0.500000	−	0.866025i
$550$		0			0
$551$			2.00000i			2.00000i
$552$		0			0
$553$	0.500000	+	0.866025i	0.500000	+	0.866025i
$554$		0			0
$555$	1.00000			1.00000
$556$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$557$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$557$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		0			0
$560$			1.00000i			1.00000i
$561$		−	1.00000i		−	1.00000i
$562$		0			0
$563$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$563$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$564$	1.00000			1.00000
$565$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$566$		0			0
$567$	0.500000	−	0.866025i	0.500000	−	0.866025i
$568$		0			0
$569$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$569$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$570$		0			0
$571$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$571$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$572$	0.866025	+	0.500000i	0.866025	+	0.500000i
$573$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$574$		0			0
$575$		0			0
$576$	0.500000	−	0.866025i	0.500000	−	0.866025i
$577$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$577$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$578$		0			0
$579$			1.00000i			1.00000i
$580$	1.00000	+	1.73205i	1.00000	+	1.73205i
$581$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$582$		0			0
$583$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$584$		0			0
$585$	0.866025	−	0.500000i	0.866025	−	0.500000i
$586$		0			0
$587$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$587$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$588$	0.866025	−	0.500000i	0.866025	−	0.500000i
$589$	0.500000	−	0.866025i	0.500000	−	0.866025i
$590$		0			0
$591$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$592$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$593$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$593$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$596$	0.866025	+	0.500000i	0.866025	+	0.500000i
$597$			1.00000i			1.00000i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$601$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$	1.00000	−	1.73205i	1.00000	−	1.73205i
$610$		0			0
$611$		−	1.00000i		−	1.00000i
$612$			1.00000i			1.00000i
$613$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$613$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		−	1.00000i		−	1.00000i
$616$		0			0
$617$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$617$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$618$		0			0
$619$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$619$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$620$		−	1.00000i		−	1.00000i
$621$	0.500000	+	0.866025i	0.500000	+	0.866025i
$622$		0			0
$623$		0			0
$624$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$625$	−1.00000			−1.00000
$626$		0			0
$627$	−1.00000			−1.00000
$628$	0.500000	+	0.866025i	0.500000	+	0.866025i
$629$	0.866025	+	0.500000i	0.866025	+	0.500000i
$630$		0			0
$631$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$631$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.866025	−	0.500000i	0.866025	−	0.500000i
$636$	0.500000	−	0.866025i	0.500000	−	0.866025i
$637$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$638$		0			0
$639$	0.866025	−	0.500000i	0.866025	−	0.500000i
$640$		0			0
$641$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$641$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$642$		0			0
$643$	2.00000			2.00000			1.00000			$0$
$643$	2.00000			2.00000			1.00000			$0$
$644$			1.00000i			1.00000i
$645$		0			0
$646$		0			0
$647$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$647$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$650$		0			0
$651$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$652$		0			0
$653$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$653$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$657$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$658$		0			0
$659$	−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	$0.833333\pi$
$660$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$661$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$662$		0			0
$663$	1.00000			1.00000
$664$		0			0
$665$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$666$		0			0
$667$	1.00000	+	1.73205i	1.00000	+	1.73205i
$668$			1.00000i			1.00000i
$669$		0			0
$670$		0			0
$671$	0.866025	+	0.500000i	0.866025	+	0.500000i
$672$		0			0
$673$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$673$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$677$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$683$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$684$	1.00000			1.00000
$685$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$686$		0			0
$687$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$688$		0			0
$689$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$690$		0			0
$691$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$691$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$692$		0			0
$693$	0.866025	+	0.500000i	0.866025	+	0.500000i
$694$		0			0
$695$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$696$		0			0
$697$	0.500000	−	0.866025i	0.500000	−	0.866025i
$698$		0			0
$699$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$700$		0			0
$701$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$701$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$702$		0			0
$703$	0.500000	−	0.866025i	0.500000	−	0.866025i
$704$	0.866025	+	0.500000i	0.866025	+	0.500000i
$705$	0.866025	+	0.500000i	0.866025	+	0.500000i
$706$		0			0
$707$		−	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	$-0.333333\pi$
$709$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$710$		0			0
$711$	0.500000	−	0.866025i	0.500000	−	0.866025i
$712$		0			0
$713$		−	1.00000i		−	1.00000i
$714$		0			0
$715$	0.500000	+	0.866025i	0.500000	+	0.866025i
$716$		−	1.00000i		−	1.00000i
$717$	0.500000	−	0.866025i	0.500000	−	0.866025i
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$	0.866025	−	0.500000i	0.866025	−	0.500000i
$721$	0.500000	−	0.866025i	0.500000	−	0.866025i
$722$		0			0
$723$		0			0
$724$	−1.00000			−1.00000
$725$		0			0
$726$		0			0
$727$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$727$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$728$		0			0
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$733$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$733$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$	1.00000			1.00000
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$739$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$740$		−	1.00000i		−	1.00000i
$741$		−	1.00000i		−	1.00000i
$742$		0			0
$743$	−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			0
$745$	0.500000	+	0.866025i	0.500000	+	0.866025i
$746$		0			0
$747$			1.00000i			1.00000i
$748$	−1.00000			−1.00000
$749$	0.866025	−	0.500000i	0.866025	−	0.500000i
$750$		0			0
$751$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$751$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$752$		−	1.00000i		−	1.00000i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$757$	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			0
$759$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	0.500000	+	0.866025i	0.500000	+	0.866025i
$764$	0.866025	+	0.500000i	0.866025	+	0.500000i
$765$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$766$		0			0
$767$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$768$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$		0			0
$772$	1.00000			1.00000
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$778$		0			0
$779$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$780$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$781$	0.500000	+	0.866025i	0.500000	+	0.866025i
$782$		0			0
$783$	−2.00000			−2.00000
$784$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$785$			1.00000i			1.00000i
$786$		0			0
$787$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$787$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$788$	0.866025	+	0.500000i	0.866025	+	0.500000i
$789$	1.00000	−	1.73205i	1.00000	−	1.73205i
$790$		0			0
$791$	0.866025	+	0.500000i	0.866025	+	0.500000i
$792$		0			0
$793$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$794$		0			0
$795$	0.866025	−	0.500000i	0.866025	−	0.500000i
$796$	1.00000			1.00000
$797$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$797$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$798$		0			0
$799$	0.500000	+	0.866025i	0.500000	+	0.866025i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.866025	−	0.500000i	0.866025	−	0.500000i
$804$		0			0
$805$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$806$		0			0
$807$	−1.00000			−1.00000
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$811$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$812$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$813$		0			0
$814$		0			0
$815$		0			0
$816$	1.00000			1.00000
$817$		0			0
$818$		0			0
$819$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$820$	−1.00000			−1.00000
$821$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$821$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$822$		0			0
$823$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$823$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$827$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$828$	0.866025	−	0.500000i	0.866025	−	0.500000i
$829$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$829$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$833$	0.866025	+	0.500000i	0.866025	+	0.500000i
$834$		0			0
$835$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$836$			1.00000i			1.00000i
$837$	0.866025	+	0.500000i	0.866025	+	0.500000i
$838$		0			0
$839$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$839$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	−3.00000			−3.00000
$842$		0			0
$843$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$849$		0			0
$850$		0			0
$851$		−	1.00000i		−	1.00000i
$852$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$853$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$853$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$854$		0			0
$855$	0.866025	+	0.500000i	0.866025	+	0.500000i
$856$		0			0
$857$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$857$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$858$		0			0
$859$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$859$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$	0.500000	+	0.866025i	0.500000	+	0.866025i
$862$		0			0
$863$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$863$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	0.500000	+	0.866025i	0.500000	+	0.866025i
$869$	0.866025	+	0.500000i	0.866025	+	0.500000i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		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
$879$	−1.00000			−1.00000
$880$	0.500000	+	0.866025i	0.500000	+	0.866025i
$881$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$881$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$882$		0			0
$883$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$883$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$884$		−	1.00000i		−	1.00000i
$885$	0.866025	−	0.500000i	0.866025	−	0.500000i
$886$		0			0
$887$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$887$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$890$		0			0
$891$		−	1.00000i		−	1.00000i
$892$		0			0
$893$	0.866025	−	0.500000i	0.866025	−	0.500000i
$894$		0			0
$895$	0.500000	−	0.866025i	0.500000	−	0.866025i
$896$		0			0
$897$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$898$		0			0
$899$	1.73205	+	1.00000i	1.73205	+	1.00000i
$900$		0			0
$901$	1.00000			1.00000
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$906$		0			0
$907$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$907$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$908$		0			0
$909$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		−	1.00000i		−	1.00000i
$913$	−1.00000			−1.00000
$914$		0			0
$915$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$916$	0.500000	+	0.866025i	0.500000	+	0.866025i
$917$		0			0
$918$		0			0
$919$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$919$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$924$	0.500000	−	0.866025i	0.500000	−	0.866025i
$925$		0			0
$926$		0			0
$927$	−1.00000			−1.00000
$928$		0			0
$929$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$929$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	0.500000	−	0.866025i	0.500000	−	0.866025i
$932$	0.866025	+	0.500000i	0.866025	+	0.500000i
$933$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$934$		0			0
$935$	−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			0
$940$	0.500000	−	0.866025i	0.500000	−	0.866025i
$941$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$941$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$	−1.00000			−1.00000
$944$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$945$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$949$	0.500000	+	0.866025i	0.500000	+	0.866025i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$953$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$954$		0			0
$955$	0.500000	+	0.866025i	0.500000	+	0.866025i
$956$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$957$		−	2.00000i		−	2.00000i
$958$		0			0
$959$		−	1.00000i		−	1.00000i
$960$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$961$		0			0
$962$		0			0
$963$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$964$		0			0
$965$	0.866025	+	0.500000i	0.866025	+	0.500000i
$966$		0			0
$967$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$967$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$968$		0			0
$969$	0.500000	+	0.866025i	0.500000	+	0.866025i
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$			1.00000i			1.00000i
$973$	0.500000	−	0.866025i	0.500000	−	0.866025i
$974$		0			0
$975$		0			0
$976$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$977$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$977$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$978$		0			0
$979$		0			0
$980$		−	1.00000i		−	1.00000i
$981$	0.500000	−	0.866025i	0.500000	−	0.866025i
$982$		0			0
$983$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$983$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$984$		0			0
$985$	0.500000	+	0.866025i	0.500000	+	0.866025i
$986$		0			0
$987$	−1.00000			−1.00000
$988$	−1.00000			−1.00000
$989$		0			0
$990$		0			0
$991$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$991$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.866025	+	0.500000i	0.866025	+	0.500000i
$996$	1.00000			1.00000
$997$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$997$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$	0.866025	+	0.500000i	0.866025	+	0.500000i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	777.1.be.a.137.2	yes	4
3.2	odd	2	inner	777.1.be.a.137.1	yes	4
7.2	even	3		777.1.t.a.359.2	yes	4
21.2	odd	6		777.1.t.a.359.1	yes	4
37.10	even	3		777.1.t.a.158.2	yes	4
111.47	odd	6		777.1.t.a.158.1	✓	4
259.121	even	3	inner	777.1.be.a.380.1	yes	4
777.380	odd	6	inner	777.1.be.a.380.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
777.1.t.a.158.1	✓	4	111.47	odd	6
777.1.t.a.158.2	yes	4	37.10	even	3
777.1.t.a.359.1	yes	4	21.2	odd	6
777.1.t.a.359.2	yes	4	7.2	even	3
777.1.be.a.137.1	yes	4	3.2	odd	2	inner
777.1.be.a.137.2	yes	4	1.1	even	1	trivial
777.1.be.a.380.1	yes	4	259.121	even	3	inner
777.1.be.a.380.2	yes	4	777.380	odd	6	inner

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:	\( N \)	\(=\)	\( 777 = 3 \cdot 7 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	777.be (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{3} +(-0.500000 - 0.866025i) q^{4} -1.00000i q^{5} +(0.500000 + 0.866025i) q^{7} +(0.500000 - 0.866025i) q^{9} +(0.866025 + 0.500000i) q^{11} +(-0.866025 - 0.500000i) q^{12} +(-0.500000 + 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{15} +(-0.500000 + 0.866025i) q^{16} -1.00000i q^{17} -1.00000 q^{19} +(-0.866025 + 0.500000i) q^{20} +(0.866025 + 0.500000i) q^{21} +(-0.866025 + 0.500000i) q^{23} -1.00000i q^{27} +(0.500000 - 0.866025i) q^{28} -2.00000i q^{29} +(-0.500000 + 0.866025i) q^{31} +1.00000 q^{33} +(0.866025 - 0.500000i) q^{35} -1.00000 q^{36} +(-0.500000 + 0.866025i) q^{37} +1.00000i q^{39} +(0.866025 + 0.500000i) q^{41} -1.00000i q^{44} +(-0.866025 - 0.500000i) q^{45} +(-0.866025 + 0.500000i) q^{47} +1.00000i q^{48} +(-0.500000 + 0.866025i) q^{49} +(-0.500000 - 0.866025i) q^{51} +1.00000 q^{52} +1.00000i q^{53} +(0.500000 - 0.866025i) q^{55} +(-0.866025 + 0.500000i) q^{57} +1.00000i q^{59} +(-0.500000 + 0.866025i) q^{60} +1.00000 q^{61} +1.00000 q^{63} +1.00000 q^{64} +(0.866025 + 0.500000i) q^{65} +(-0.866025 + 0.500000i) q^{68} +(-0.500000 + 0.866025i) q^{69} +(0.866025 + 0.500000i) q^{71} +(0.500000 - 0.866025i) q^{73} +(0.500000 + 0.866025i) q^{76} +1.00000i q^{77} +1.00000 q^{79} +(0.866025 + 0.500000i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.866025 + 0.500000i) q^{83} -1.00000i q^{84} -1.00000 q^{85} +(-1.00000 - 1.73205i) q^{87} -1.00000 q^{91} +(0.866025 + 0.500000i) q^{92} +1.00000i q^{93} +1.00000i q^{95} +(0.866025 - 0.500000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{4} + 2 q^{7} + 2 q^{9} - 2 q^{13} - 2 q^{15} - 2 q^{16} - 4 q^{19} + 2 q^{28} - 2 q^{31} + 4 q^{33} - 4 q^{36} - 2 q^{37} - 2 q^{49} - 2 q^{51} + 4 q^{52} + 2 q^{55} - 2 q^{60} + 4 q^{61} + 4 q^{63}+ \cdots - 4 q^{91}+O(q^{100}) \) 4 * q - 2 * q^4 + 2 * q^7 + 2 * q^9 - 2 * q^13 - 2 * q^15 - 2 * q^16 - 4 * q^19 + 2 * q^28 - 2 * q^31 + 4 * q^33 - 4 * q^36 - 2 * q^37 - 2 * q^49 - 2 * q^51 + 4 * q^52 + 2 * q^55 - 2 * q^60 + 4 * q^61 + 4 * q^63 + 4 * q^64 - 2 * q^69 + 2 * q^73 + 2 * q^76 + 4 * q^79 - 2 * q^81 - 4 * q^85 - 4 * q^87 - 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more