LMFDB - Embedded newform 684.1.bm.a.619.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [684,1,Mod(463,684)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(684, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 2, 2]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("684.463");

S:= CuspForms(chi, 1);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$A_{4}$
Projective field:	Galois closure of 4.0.467856.1

Embedding invariants

Embedding label			619.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	684.619
Dual form			684.1.bm.a.463.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(0.866025 - 0.500000i) q^{3} +(-0.500000 + 0.866025i) q^{4} -1.00000 q^{5} +(0.866025 + 0.500000i) q^{6} +(0.866025 + 0.500000i) q^{7} -1.00000 q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(0.866025 - 0.500000i) q^{3} +(-0.500000 + 0.866025i) q^{4} -1.00000 q^{5} +(0.866025 + 0.500000i) q^{6} +(0.866025 + 0.500000i) q^{7} -1.00000 q^{8} +(0.500000 - 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{10} +(0.866025 + 0.500000i) q^{11} +1.00000i q^{12} +1.00000i q^{14} +(-0.866025 + 0.500000i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +1.00000 q^{18} -1.00000i q^{19} +(0.500000 - 0.866025i) q^{20} +1.00000 q^{21} +1.00000i q^{22} +(-0.866025 + 0.500000i) q^{24} -1.00000i q^{27} +(-0.866025 + 0.500000i) q^{28} -1.00000 q^{29} +(-0.866025 - 0.500000i) q^{30} +(-0.866025 + 0.500000i) q^{31} +(0.500000 - 0.866025i) q^{32} +1.00000 q^{33} -1.00000 q^{34} +(-0.866025 - 0.500000i) q^{35} +(0.500000 + 0.866025i) q^{36} +(0.866025 - 0.500000i) q^{38} +1.00000 q^{40} -1.00000 q^{41} +(0.500000 + 0.866025i) q^{42} +(1.73205 - 1.00000i) q^{43} +(-0.866025 + 0.500000i) q^{44} +(-0.500000 + 0.866025i) q^{45} -1.00000i q^{47} +(-0.866025 - 0.500000i) q^{48} +1.00000i q^{51} +(-0.500000 - 0.866025i) q^{53} +(0.866025 - 0.500000i) q^{54} +(-0.866025 - 0.500000i) q^{55} +(-0.866025 - 0.500000i) q^{56} +(-0.500000 - 0.866025i) q^{57} +(-0.500000 - 0.866025i) q^{58} -1.00000i q^{59} -1.00000i q^{60} +1.00000 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} -1.00000i q^{70} +(0.866025 + 0.500000i) q^{71} +(-0.500000 + 0.866025i) q^{72} +(-0.500000 + 0.866025i) q^{73} +(0.866025 + 0.500000i) q^{76} +(0.500000 + 0.866025i) q^{77} +(-1.73205 + 1.00000i) q^{79} +(0.500000 + 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.500000 - 0.866025i) q^{82} +(-0.866025 - 0.500000i) q^{83} +(-0.500000 + 0.866025i) q^{84} +(0.500000 - 0.866025i) q^{85} +(1.73205 + 1.00000i) q^{86} +(-0.866025 + 0.500000i) q^{87} +(-0.866025 - 0.500000i) q^{88} +(0.500000 + 0.866025i) q^{89} -1.00000 q^{90} +(-0.500000 + 0.866025i) q^{93} +(0.866025 - 0.500000i) q^{94} +1.00000i q^{95} -1.00000i q^{96} +(0.866025 - 0.500000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{4} - 4 q^{5} - 4 q^{8} + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^5 - 4 * q^8 + 2 * q^9	$ 4 q + 2 q^{2} - 2 q^{4} - 4 q^{5} - 4 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{16} - 2 q^{17} + 4 q^{18} + 2 q^{20} + 4 q^{21} - 4 q^{29} + 2 q^{32} + 4 q^{33} - 4 q^{34} + 2 q^{36} + 4 q^{40} - 4 q^{41} + 2 q^{42} - 2 q^{45} - 2 q^{53} - 2 q^{57} - 2 q^{58} + 4 q^{61} + 4 q^{64} + 2 q^{66} - 2 q^{68} - 2 q^{72} - 2 q^{73} + 2 q^{77} + 2 q^{80} - 2 q^{81} - 2 q^{82} - 2 q^{84} + 2 q^{85} + 2 q^{89} - 4 q^{90} - 2 q^{93}+O(q^{100}) $ 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^5 - 4 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^16 - 2 * q^17 + 4 * q^18 + 2 * q^20 + 4 * q^21 - 4 * q^29 + 2 * q^32 + 4 * q^33 - 4 * q^34 + 2 * q^36 + 4 * q^40 - 4 * q^41 + 2 * q^42 - 2 * q^45 - 2 * q^53 - 2 * q^57 - 2 * q^58 + 4 * q^61 + 4 * q^64 + 2 * q^66 - 2 * q^68 - 2 * q^72 - 2 * q^73 + 2 * q^77 + 2 * q^80 - 2 * q^81 - 2 * q^82 - 2 * q^84 + 2 * q^85 + 2 * q^89 - 4 * q^90 - 2 * q^93

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	0.500000	+	0.866025i	0.500000	+	0.866025i
$2$	0.500000	+	0.866025i	0.500000	+	0.866025i
$3$	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
$5$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$6$	0.866025	+	0.500000i	0.866025	+	0.500000i
$7$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$7$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$8$	−1.00000			−1.00000
$9$	0.500000	−	0.866025i	0.500000	−	0.866025i
$10$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$11$	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	$0.5\pi$
$12$			1.00000i			1.00000i
$13$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$13$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$14$			1.00000i			1.00000i
$15$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$	−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
$21$	1.00000			1.00000
$22$			1.00000i			1.00000i
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$25$		0			0
$26$		0			0
$27$		−	1.00000i		−	1.00000i
$28$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$29$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$29$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$30$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$31$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$31$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$32$	0.500000	−	0.866025i	0.500000	−	0.866025i
$33$	1.00000			1.00000
$34$	−1.00000			−1.00000
$35$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$36$	0.500000	+	0.866025i	0.500000	+	0.866025i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	0.866025	−	0.500000i	0.866025	−	0.500000i
$39$		0			0
$40$	1.00000			1.00000
$41$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$41$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$42$	0.500000	+	0.866025i	0.500000	+	0.866025i
$43$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$43$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$44$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$45$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$46$		0			0
$47$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$47$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$48$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$49$		0			0
$50$		0			0
$51$			1.00000i			1.00000i
$52$		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			$\pi$
$54$	0.866025	−	0.500000i	0.866025	−	0.500000i
$55$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$56$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$57$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$58$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$59$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$59$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$60$		−	1.00000i		−	1.00000i
$61$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$61$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$62$	−0.866025	−	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		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$69$		0			0
$70$		−	1.00000i		−	1.00000i
$71$	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	$0.5\pi$
$72$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$73$	−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
$76$	0.866025	+	0.500000i	0.866025	+	0.500000i
$77$	0.500000	+	0.866025i	0.500000	+	0.866025i
$78$		0			0
$79$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$79$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$80$	0.500000	+	0.866025i	0.500000	+	0.866025i
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$	−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	$0.833333\pi$
$84$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$85$	0.500000	−	0.866025i	0.500000	−	0.866025i
$86$	1.73205	+	1.00000i	1.73205	+	1.00000i
$87$	−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	$0.666667\pi$
$90$	−1.00000			−1.00000
$91$		0			0
$92$		0			0
$93$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$94$	0.866025	−	0.500000i	0.866025	−	0.500000i
$95$			1.00000i			1.00000i
$96$		−	1.00000i		−	1.00000i
$97$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$97$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$98$		0			0
$99$	0.866025	−	0.500000i	0.866025	−	0.500000i
$100$		0			0
$101$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$101$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$102$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$103$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$103$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$104$		0			0
$105$	−1.00000			−1.00000
$106$	0.500000	−	0.866025i	0.500000	−	0.866025i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	0.866025	+	0.500000i	0.866025	+	0.500000i
$109$	−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			$\pi$
$110$		−	1.00000i		−	1.00000i
$111$		0			0
$112$		−	1.00000i		−	1.00000i
$113$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$113$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$114$	0.500000	−	0.866025i	0.500000	−	0.866025i
$115$		0			0
$116$	0.500000	−	0.866025i	0.500000	−	0.866025i
$117$		0			0
$118$	0.866025	−	0.500000i	0.866025	−	0.500000i
$119$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$120$	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
$124$		−	1.00000i		−	1.00000i
$125$	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	$0.166667\pi$
$128$	0.500000	+	0.866025i	0.500000	+	0.866025i
$129$	1.00000	−	1.73205i	1.00000	−	1.73205i
$130$		0			0
$131$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$131$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$132$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$133$	0.500000	−	0.866025i	0.500000	−	0.866025i
$134$		0			0
$135$			1.00000i			1.00000i
$136$	0.500000	−	0.866025i	0.500000	−	0.866025i
$137$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$137$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$	0.866025	−	0.500000i	0.866025	−	0.500000i
$141$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$142$			1.00000i			1.00000i
$143$		0			0
$144$	−1.00000			−1.00000
$145$	1.00000			1.00000
$146$	−1.00000			−1.00000
$147$		0			0
$148$		0			0
$149$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$149$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$151$	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
$155$	0.866025	−	0.500000i	0.866025	−	0.500000i
$156$		0			0
$157$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$157$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$158$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$159$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$160$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$161$		0			0
$162$	0.500000	−	0.866025i	0.500000	−	0.866025i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	0.500000	−	0.866025i	0.500000	−	0.866025i
$165$	−1.00000			−1.00000
$166$		−	1.00000i		−	1.00000i
$167$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$167$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$168$	−1.00000			−1.00000
$169$	0.500000	+	0.866025i	0.500000	+	0.866025i
$170$	1.00000			1.00000
$171$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$172$			2.00000i			2.00000i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$175$		0			0
$176$		−	1.00000i		−	1.00000i
$177$	−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
$186$	−1.00000			−1.00000
$187$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$188$	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	$0.833333\pi$
$192$	0.866025	−	0.500000i	0.866025	−	0.500000i
$193$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$193$	−1.00000			−1.00000			−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
$203$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$204$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$205$	1.00000			1.00000
$206$	−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
$211$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$211$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$212$	1.00000			1.00000
$213$	1.00000			1.00000
$214$		0			0
$215$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$216$			1.00000i			1.00000i
$217$	−1.00000			−1.00000
$218$	−1.00000			−1.00000
$219$			1.00000i			1.00000i
$220$	0.866025	−	0.500000i	0.866025	−	0.500000i
$221$		0			0
$222$		0			0
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$	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	$-0.166667\pi$
$227$	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	1.00000			$0$
$229$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$	0.866025	+	0.500000i	0.866025	+	0.500000i
$232$	1.00000			1.00000
$233$	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
$237$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$238$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$239$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$239$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$240$	0.866025	+	0.500000i	0.866025	+	0.500000i
$241$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$241$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$244$	−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
$249$	−1.00000			−1.00000
$250$	0.500000	+	0.866025i	0.500000	+	0.866025i
$251$	−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	$0.5\pi$
$252$			1.00000i			1.00000i
$253$		0			0
$254$	0.866025	+	0.500000i	0.866025	+	0.500000i
$255$		−	1.00000i		−	1.00000i
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$	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		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$	−1.00000			−1.00000
$265$	0.500000	+	0.866025i	0.500000	+	0.866025i
$266$	1.00000			1.00000
$267$	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	$0.5\pi$
$272$	1.00000			1.00000
$273$		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
$279$			1.00000i			1.00000i
$280$	0.866025	+	0.500000i	0.866025	+	0.500000i
$281$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$281$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$282$	0.500000	−	0.866025i	0.500000	−	0.866025i
$283$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$283$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$284$	−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
$288$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$289$		0			0
$290$	0.500000	+	0.866025i	0.500000	+	0.866025i
$291$		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
$295$			1.00000i			1.00000i
$296$		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
$301$	2.00000			2.00000
$302$			1.00000i			1.00000i
$303$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$304$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$305$	−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	$0.833333\pi$
$308$	−1.00000			−1.00000
$309$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$310$	0.866025	+	0.500000i	0.866025	+	0.500000i
$311$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$311$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$312$		0			0
$313$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$313$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$314$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$315$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$316$		−	2.00000i		−	2.00000i
$317$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$317$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$318$		−	1.00000i		−	1.00000i
$319$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$320$	−1.00000			−1.00000
$321$		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
$327$			1.00000i			1.00000i
$328$	1.00000			1.00000
$329$	0.500000	−	0.866025i	0.500000	−	0.866025i
$330$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$331$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$331$	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
$337$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$337$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$338$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$339$	0.866025	+	0.500000i	0.866025	+	0.500000i
$340$	0.500000	+	0.866025i	0.500000	+	0.866025i
$341$	−1.00000			−1.00000
$342$		−	1.00000i		−	1.00000i
$343$		−	1.00000i		−	1.00000i
$344$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$345$		0			0
$346$		0			0
$347$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$347$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$348$		−	1.00000i		−	1.00000i
$349$	−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
$355$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$356$	−1.00000			−1.00000
$357$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$358$		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
$365$	0.500000	−	0.866025i	0.500000	−	0.866025i
$366$	0.866025	+	0.500000i	0.866025	+	0.500000i
$367$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$367$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$368$		0			0
$369$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$370$		0			0
$371$		−	1.00000i		−	1.00000i
$372$	−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			$\pi$
$374$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$375$	0.866025	−	0.500000i	0.866025	−	0.500000i
$376$			1.00000i			1.00000i
$377$		0			0
$378$	1.00000			1.00000
$379$		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
$382$		−	1.00000i		−	1.00000i
$383$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$383$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$384$	0.866025	+	0.500000i	0.866025	+	0.500000i
$385$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$386$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$387$		−	2.00000i		−	2.00000i
$388$		0			0
$389$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$389$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$390$		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
$396$			1.00000i			1.00000i
$397$	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
$399$		−	1.00000i		−	1.00000i
$400$		0			0
$401$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.00000			−1.00000			−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
$408$		−	1.00000i		−	1.00000i
$409$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$409$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$410$	0.500000	+	0.866025i	0.500000	+	0.866025i
$411$	0.866025	−	0.500000i	0.866025	−	0.500000i
$412$		−	1.00000i		−	1.00000i
$413$	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
$418$	1.00000			1.00000
$419$	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	$0.166667\pi$
$420$	0.500000	−	0.866025i	0.500000	−	0.866025i
$421$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$421$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$422$	−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
$427$	0.866025	+	0.500000i	0.866025	+	0.500000i
$428$		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	$0.5\pi$
$432$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$433$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$433$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$434$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$435$	0.866025	−	0.500000i	0.866025	−	0.500000i
$436$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$437$		0			0
$438$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$	0.866025	+	0.500000i	0.866025	+	0.500000i
$441$		0			0
$442$		0			0
$443$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$443$			1.00000i			1.00000i	−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
$452$	−1.00000			−1.00000
$453$	1.00000			1.00000
$454$			1.00000i			1.00000i
$455$		0			0
$456$	0.500000	+	0.866025i	0.500000	+	0.866025i
$457$	−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		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$			1.00000i			1.00000i
$463$	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	$0.166667\pi$
$464$	0.500000	+	0.866025i	0.500000	+	0.866025i
$465$	0.500000	−	0.866025i	0.500000	−	0.866025i
$466$	1.00000			1.00000
$467$		−	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
$470$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$471$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$472$			1.00000i			1.00000i
$473$	2.00000			2.00000
$474$	−2.00000			−2.00000
$475$		0			0
$476$		−	1.00000i		−	1.00000i
$477$	−1.00000			−1.00000
$478$	0.866025	+	0.500000i	0.866025	+	0.500000i
$479$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$479$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$480$			1.00000i			1.00000i
$481$		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			$\pi$
$488$	−1.00000			−1.00000
$489$		0			0
$490$		0			0
$491$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$491$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$492$		−	1.00000i		−	1.00000i
$493$	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
$497$	0.500000	+	0.866025i	0.500000	+	0.866025i
$498$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$499$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$499$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$500$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$501$	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
$507$	0.866025	+	0.500000i	0.866025	+	0.500000i
$508$			1.00000i			1.00000i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$	0.866025	−	0.500000i	0.866025	−	0.500000i
$511$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$512$	−1.00000			−1.00000
$513$	−1.00000			−1.00000
$514$		0			0
$515$	0.866025	−	0.500000i	0.866025	−	0.500000i
$516$	1.00000	+	1.73205i	1.00000	+	1.73205i
$517$	0.500000	−	0.866025i	0.500000	−	0.866025i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	−1.00000			−1.00000
$523$	−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
$527$		−	1.00000i		−	1.00000i
$528$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$	−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
$534$			1.00000i			1.00000i
$535$		0			0
$536$		0			0
$537$		0			0
$538$	1.00000			1.00000
$539$		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	$0.666667\pi$
$542$			1.00000i			1.00000i
$543$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$544$	0.500000	+	0.866025i	0.500000	+	0.866025i
$545$	0.500000	−	0.866025i	0.500000	−	0.866025i
$546$		0			0
$547$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$547$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$548$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$549$	0.500000	−	0.866025i	0.500000	−	0.866025i
$550$		0			0
$551$			1.00000i			1.00000i
$552$		0			0
$553$	−2.00000			−2.00000
$554$	−1.00000			−1.00000
$555$		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
$560$			1.00000i			1.00000i
$561$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$562$	0.500000	+	0.866025i	0.500000	+	0.866025i
$563$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$563$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$564$	1.00000			1.00000
$565$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$566$	0.866025	−	0.500000i	0.866025	−	0.500000i
$567$		−	1.00000i		−	1.00000i
$568$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$569$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$569$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$570$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$571$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$571$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$572$		0			0
$573$	−1.00000			−1.00000
$574$		−	1.00000i		−	1.00000i
$575$		0			0
$576$	0.500000	−	0.866025i	0.500000	−	0.866025i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$		0			0
$579$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$580$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$581$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$582$		0			0
$583$		−	1.00000i		−	1.00000i
$584$	0.500000	−	0.866025i	0.500000	−	0.866025i
$585$		0			0
$586$	0.500000	−	0.866025i	0.500000	−	0.866025i
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		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	$0.666667\pi$
$594$	1.00000			1.00000
$595$	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
$599$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$599$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$600$		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	$0.166667\pi$
$608$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$609$	−1.00000			−1.00000
$610$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$611$		0			0
$612$	−1.00000			−1.00000
$613$	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			$0$
$614$		−	1.00000i		−	1.00000i
$615$	0.866025	−	0.500000i	0.866025	−	0.500000i
$616$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$617$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$617$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$618$	−1.00000			−1.00000
$619$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$619$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$620$			1.00000i			1.00000i
$621$		0			0
$622$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$623$			1.00000i			1.00000i
$624$		0			0
$625$	−1.00000			−1.00000
$626$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$627$		−	1.00000i		−	1.00000i
$628$	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	$0.5\pi$
$632$	1.73205	−	1.00000i	1.73205	−	1.00000i
$633$	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
$641$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$641$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$643$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$644$		0			0
$645$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$646$			1.00000i			1.00000i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	0.500000	+	0.866025i	0.500000	+	0.866025i
$649$	0.500000	−	0.866025i	0.500000	−	0.866025i
$650$		0			0
$651$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$652$		0			0
$653$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$653$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$654$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$655$		−	1.00000i		−	1.00000i
$656$	0.500000	+	0.866025i	0.500000	+	0.866025i
$657$	0.500000	+	0.866025i	0.500000	+	0.866025i
$658$	1.00000			1.00000
$659$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$659$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$660$	0.500000	−	0.866025i	0.500000	−	0.866025i
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$			1.00000i			1.00000i
$663$		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
$668$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$669$		0			0
$670$		0			0
$671$	0.866025	+	0.500000i	0.866025	+	0.500000i
$672$	0.500000	−	0.866025i	0.500000	−	0.866025i
$673$	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			$0$
$674$	0.500000	+	0.866025i	0.500000	+	0.866025i
$675$		0			0
$676$	−1.00000			−1.00000
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$678$			1.00000i			1.00000i
$679$		0			0
$680$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$681$	1.00000			1.00000
$682$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	0.866025	−	0.500000i	0.866025	−	0.500000i
$685$	−1.00000			−1.00000
$686$	0.866025	−	0.500000i	0.866025	−	0.500000i
$687$	0.866025	+	0.500000i	0.866025	+	0.500000i
$688$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$689$		0			0
$690$		0			0
$691$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$691$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$692$		0			0
$693$	1.00000			1.00000
$694$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$695$		0			0
$696$	0.866025	−	0.500000i	0.866025	−	0.500000i
$697$	0.500000	−	0.866025i	0.500000	−	0.866025i
$698$	−1.00000			−1.00000
$699$		−	1.00000i		−	1.00000i
$700$		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
$705$	0.500000	+	0.866025i	0.500000	+	0.866025i
$706$	−1.00000			−1.00000
$707$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$708$	1.00000			1.00000
$709$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$709$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$710$		−	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
$718$			1.00000i			1.00000i
$719$	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	$0.5\pi$
$720$	1.00000			1.00000
$721$	−1.00000			−1.00000
$722$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$723$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$724$	1.00000			1.00000
$725$		0			0
$726$		0			0
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	−1.00000			−1.00000
$730$	1.00000			1.00000
$731$			2.00000i			2.00000i
$732$			1.00000i			1.00000i
$733$	−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
$738$	−1.00000			−1.00000
$739$	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
$743$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$743$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$744$	0.500000	−	0.866025i	0.500000	−	0.866025i
$745$	−1.00000			−1.00000
$746$	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
$750$	0.866025	+	0.500000i	0.866025	+	0.500000i
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	−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
$760$		−	1.00000i		−	1.00000i
$761$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$761$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$762$	1.00000			1.00000
$763$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$764$	0.866025	−	0.500000i	0.866025	−	0.500000i
$765$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$766$	0.866025	−	0.500000i	0.866025	−	0.500000i
$767$		0			0
$768$			1.00000i			1.00000i
$769$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$769$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$770$	0.500000	−	0.866025i	0.500000	−	0.866025i
$771$		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			$\pi$
$774$	1.73205	−	1.00000i	1.73205	−	1.00000i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.500000	+	0.866025i	0.500000	+	0.866025i
$779$			1.00000i			1.00000i
$780$		0			0
$781$	0.500000	+	0.866025i	0.500000	+	0.866025i
$782$		0			0
$783$			1.00000i			1.00000i
$784$		0			0
$785$	1.00000			1.00000
$786$	−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	$0.166667\pi$
$788$		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
$794$	1.00000			1.00000
$795$	0.866025	+	0.500000i	0.866025	+	0.500000i
$796$			1.00000i			1.00000i
$797$	−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
$801$	1.00000			1.00000
$802$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$803$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	1.00000i		−	1.00000i
$808$	1.00000			1.00000
$809$		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	$0.166667\pi$
$812$	0.866025	−	0.500000i	0.866025	−	0.500000i
$813$	1.00000			1.00000
$814$		0			0
$815$		0			0
$816$	0.866025	−	0.500000i	0.866025	−	0.500000i
$817$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$818$	−2.00000			−2.00000
$819$		0			0
$820$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$821$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$821$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$822$	0.866025	+	0.500000i	0.866025	+	0.500000i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$	0.866025	−	0.500000i	0.866025	−	0.500000i
$825$		0			0
$826$	1.00000			1.00000
$827$	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	$0.5\pi$
$830$			1.00000i			1.00000i
$831$			1.00000i			1.00000i
$832$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$	1.00000			1.00000
$841$		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
$846$		−	1.00000i		−	1.00000i
$847$		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		0.866025	−	0.500000i	$-0.166667\pi$
$853$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$854$			1.00000i			1.00000i
$855$	0.866025	+	0.500000i	0.866025	+	0.500000i
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		−	2.00000i		−	2.00000i
$861$	−1.00000			−1.00000
$862$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$865$		0			0
$866$	0.500000	−	0.866025i	0.500000	−	0.866025i
$867$		0			0
$868$	0.500000	−	0.866025i	0.500000	−	0.866025i
$869$	−2.00000			−2.00000
$870$	0.866025	+	0.500000i	0.866025	+	0.500000i
$871$		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
$877$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$877$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$880$			1.00000i			1.00000i
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		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		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	1.00000			1.00000
$890$	0.500000	−	0.866025i	0.500000	−	0.866025i
$891$		−	1.00000i		−	1.00000i
$892$		0			0
$893$	−1.00000			−1.00000
$894$	0.866025	+	0.500000i	0.866025	+	0.500000i
$895$		0			0
$896$			1.00000i			1.00000i
$897$		0			0
$898$		0			0
$899$	0.866025	−	0.500000i	0.866025	−	0.500000i
$900$		0			0
$901$	1.00000			1.00000
$902$		−	1.00000i		−	1.00000i
$903$	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
$907$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$907$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$908$	−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	$0.833333\pi$
$912$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$913$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$914$	−1.00000			−1.00000
$915$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$916$	−1.00000			−1.00000
$917$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$918$			1.00000i			1.00000i
$919$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$919$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$920$		0			0
$921$	−1.00000			−1.00000
$922$		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
$927$			1.00000i			1.00000i
$928$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$929$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$929$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$930$	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
$934$	1.73205	−	1.00000i	1.73205	−	1.00000i
$935$	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
$940$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$	−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		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$949$		0			0
$950$		0			0
$951$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$952$	0.866025	−	0.500000i	0.866025	−	0.500000i
$953$	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
$956$			1.00000i			1.00000i
$957$	−1.00000			−1.00000
$958$	0.866025	−	0.500000i	0.866025	−	0.500000i
$959$	0.866025	+	0.500000i	0.866025	+	0.500000i
$960$	−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
$965$	1.00000			1.00000
$966$		0			0
$967$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$967$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$968$		0			0
$969$	1.00000			1.00000
$970$		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	$0.5\pi$
$972$	0.866025	−	0.500000i	0.866025	−	0.500000i
$973$		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
$979$			1.00000i			1.00000i
$980$		0			0
$981$	0.500000	+	0.866025i	0.500000	+	0.866025i
$982$	0.866025	−	0.500000i	0.866025	−	0.500000i
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$	0.866025	−	0.500000i	0.866025	−	0.500000i
$985$		0			0
$986$	1.00000			1.00000
$987$		−	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	$0.833333\pi$
$992$			1.00000i			1.00000i
$993$	1.00000			1.00000
$994$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$995$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$996$	0.500000	−	0.866025i	0.500000	−	0.866025i
$997$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$997$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$998$	0.866025	−	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.bm.a.619.2	yes	4
3.2	odd	2		2052.1.bm.a.1531.2		4
4.3	odd	2	inner	684.1.bm.a.619.1	yes	4
9.4	even	3		684.1.x.a.391.2	yes	4
9.5	odd	6		2052.1.x.a.847.1		4
12.11	even	2		2052.1.bm.a.1531.1		4
19.7	even	3		684.1.x.a.7.2	yes	4
36.23	even	6		2052.1.x.a.847.2		4
36.31	odd	6		684.1.x.a.391.1	yes	4
57.26	odd	6		2052.1.x.a.235.2		4
76.7	odd	6		684.1.x.a.7.1	✓	4
171.121	even	3	inner	684.1.bm.a.463.1	yes	4
171.140	odd	6		2052.1.bm.a.1603.1		4
228.83	even	6		2052.1.x.a.235.1		4
684.311	even	6		2052.1.bm.a.1603.2		4
684.463	odd	6	inner	684.1.bm.a.463.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
684.1.x.a.7.1	✓	4	76.7	odd	6
684.1.x.a.7.2	yes	4	19.7	even	3
684.1.x.a.391.1	yes	4	36.31	odd	6
684.1.x.a.391.2	yes	4	9.4	even	3
684.1.bm.a.463.1	yes	4	171.121	even	3	inner
684.1.bm.a.463.2	yes	4	684.463	odd	6	inner
684.1.bm.a.619.1	yes	4	4.3	odd	2	inner
684.1.bm.a.619.2	yes	4	1.1	even	1	trivial
2052.1.x.a.235.1		4	228.83	even	6
2052.1.x.a.235.2		4	57.26	odd	6
2052.1.x.a.847.1		4	9.5	odd	6
2052.1.x.a.847.2		4	36.23	even	6
2052.1.bm.a.1531.1		4	12.11	even	2
2052.1.bm.a.1531.2		4	3.2	odd	2
2052.1.bm.a.1603.1		4	171.140	odd	6
2052.1.bm.a.1603.2		4	684.311	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(0.866025 - 0.500000i) q^{3} +(-0.500000 + 0.866025i) q^{4} -1.00000 q^{5} +(0.866025 + 0.500000i) q^{6} +(0.866025 + 0.500000i) q^{7} -1.00000 q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(0.866025 - 0.500000i) q^{3} +(-0.500000 + 0.866025i) q^{4} -1.00000 q^{5} +(0.866025 + 0.500000i) q^{6} +(0.866025 + 0.500000i) q^{7} -1.00000 q^{8} +(0.500000 - 0.866025i) q^{9} +(-0.500000 - 0.866025i) q^{10} +(0.866025 + 0.500000i) q^{11} +1.00000i q^{12} +1.00000i q^{14} +(-0.866025 + 0.500000i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +1.00000 q^{18} -1.00000i q^{19} +(0.500000 - 0.866025i) q^{20} +1.00000 q^{21} +1.00000i q^{22} +(-0.866025 + 0.500000i) q^{24} -1.00000i q^{27} +(-0.866025 + 0.500000i) q^{28} -1.00000 q^{29} +(-0.866025 - 0.500000i) q^{30} +(-0.866025 + 0.500000i) q^{31} +(0.500000 - 0.866025i) q^{32} +1.00000 q^{33} -1.00000 q^{34} +(-0.866025 - 0.500000i) q^{35} +(0.500000 + 0.866025i) q^{36} +(0.866025 - 0.500000i) q^{38} +1.00000 q^{40} -1.00000 q^{41} +(0.500000 + 0.866025i) q^{42} +(1.73205 - 1.00000i) q^{43} +(-0.866025 + 0.500000i) q^{44} +(-0.500000 + 0.866025i) q^{45} -1.00000i q^{47} +(-0.866025 - 0.500000i) q^{48} +1.00000i q^{51} +(-0.500000 - 0.866025i) q^{53} +(0.866025 - 0.500000i) q^{54} +(-0.866025 - 0.500000i) q^{55} +(-0.866025 - 0.500000i) q^{56} +(-0.500000 - 0.866025i) q^{57} +(-0.500000 - 0.866025i) q^{58} -1.00000i q^{59} -1.00000i q^{60} +1.00000 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} -1.00000i q^{70} +(0.866025 + 0.500000i) q^{71} +(-0.500000 + 0.866025i) q^{72} +(-0.500000 + 0.866025i) q^{73} +(0.866025 + 0.500000i) q^{76} +(0.500000 + 0.866025i) q^{77} +(-1.73205 + 1.00000i) q^{79} +(0.500000 + 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.500000 - 0.866025i) q^{82} +(-0.866025 - 0.500000i) q^{83} +(-0.500000 + 0.866025i) q^{84} +(0.500000 - 0.866025i) q^{85} +(1.73205 + 1.00000i) q^{86} +(-0.866025 + 0.500000i) q^{87} +(-0.866025 - 0.500000i) q^{88} +(0.500000 + 0.866025i) q^{89} -1.00000 q^{90} +(-0.500000 + 0.866025i) q^{93} +(0.866025 - 0.500000i) q^{94} +1.00000i q^{95} -1.00000i q^{96} +(0.866025 - 0.500000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{5} - 4 q^{8} + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^5 - 4 * q^8 + 2 * q^9	\( 4 q + 2 q^{2} - 2 q^{4} - 4 q^{5} - 4 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{16} - 2 q^{17} + 4 q^{18} + 2 q^{20} + 4 q^{21} - 4 q^{29} + 2 q^{32} + 4 q^{33} - 4 q^{34} + 2 q^{36} + 4 q^{40} - 4 q^{41} + 2 q^{42} - 2 q^{45} - 2 q^{53} - 2 q^{57} - 2 q^{58} + 4 q^{61} + 4 q^{64} + 2 q^{66} - 2 q^{68} - 2 q^{72} - 2 q^{73} + 2 q^{77} + 2 q^{80} - 2 q^{81} - 2 q^{82} - 2 q^{84} + 2 q^{85} + 2 q^{89} - 4 q^{90} - 2 q^{93}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^4 - 4 * q^5 - 4 * q^8 + 2 * q^9 - 2 * q^10 - 2 * q^16 - 2 * q^17 + 4 * q^18 + 2 * q^20 + 4 * q^21 - 4 * q^29 + 2 * q^32 + 4 * q^33 - 4 * q^34 + 2 * q^36 + 4 * q^40 - 4 * q^41 + 2 * q^42 - 2 * q^45 - 2 * q^53 - 2 * q^57 - 2 * q^58 + 4 * q^61 + 4 * q^64 + 2 * q^66 - 2 * q^68 - 2 * q^72 - 2 * q^73 + 2 * q^77 + 2 * q^80 - 2 * q^81 - 2 * q^82 - 2 * q^84 + 2 * q^85 + 2 * q^89 - 4 * q^90 - 2 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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