LMFDB - Embedded newform 1368.1.cj.c.1075.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,1,Mod(691,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 4, 2]))

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

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

chi := DirichletCharacter("1368.691");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1368.cj (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:	$0.682720937282$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$A_{4}$
Projective field:	Galois closure of 4.0.1871424.3

Embedding invariants

Embedding label			1075.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1368.1075
Dual form			1368.1.cj.c.691.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +(0.866025 - 0.500000i) q^{5} +1.00000i q^{6} +(-0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +(0.866025 - 0.500000i) q^{5} +1.00000i q^{6} +(-0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +1.00000 q^{9} +(-0.500000 - 0.866025i) q^{10} +(0.500000 + 0.866025i) q^{11} +1.00000 q^{12} +(0.500000 + 0.866025i) q^{14} +(-0.866025 + 0.500000i) q^{15} +1.00000 q^{16} +(-0.500000 + 0.866025i) q^{17} -1.00000i q^{18} +1.00000 q^{19} +(-0.866025 + 0.500000i) q^{20} +(0.866025 - 0.500000i) q^{21} +(0.866025 - 0.500000i) q^{22} -1.00000i q^{24} -1.00000 q^{27} +(0.866025 - 0.500000i) q^{28} +(0.866025 + 0.500000i) q^{29} +(0.500000 + 0.866025i) q^{30} +(0.866025 + 0.500000i) q^{31} -1.00000i q^{32} +(-0.500000 - 0.866025i) q^{33} +(0.866025 + 0.500000i) q^{34} +(-0.500000 + 0.866025i) q^{35} -1.00000 q^{36} -2.00000i q^{37} -1.00000i q^{38} +(0.500000 + 0.866025i) q^{40} +(0.500000 + 0.866025i) q^{41} +(-0.500000 - 0.866025i) q^{42} +(-0.500000 - 0.866025i) q^{44} +(0.866025 - 0.500000i) q^{45} +(-0.866025 - 0.500000i) q^{47} -1.00000 q^{48} +(0.500000 - 0.866025i) q^{51} +(0.866025 - 0.500000i) q^{53} +1.00000i q^{54} +(0.866025 + 0.500000i) q^{55} +(-0.500000 - 0.866025i) q^{56} -1.00000 q^{57} +(0.500000 - 0.866025i) q^{58} +(-0.500000 - 0.866025i) q^{59} +(0.866025 - 0.500000i) q^{60} +(0.866025 + 0.500000i) q^{61} +(0.500000 - 0.866025i) q^{62} +(-0.866025 + 0.500000i) q^{63} -1.00000 q^{64} +(-0.866025 + 0.500000i) q^{66} +2.00000 q^{67} +(0.500000 - 0.866025i) q^{68} +(0.866025 + 0.500000i) q^{70} +(-0.866025 - 0.500000i) q^{71} +1.00000i q^{72} +(-0.500000 + 0.866025i) q^{73} -2.00000 q^{74} -1.00000 q^{76} +(-0.866025 - 0.500000i) q^{77} +(0.866025 - 0.500000i) q^{80} +1.00000 q^{81} +(0.866025 - 0.500000i) q^{82} +(0.500000 + 0.866025i) q^{83} +(-0.866025 + 0.500000i) q^{84} +1.00000i q^{85} +(-0.866025 - 0.500000i) q^{87} +(-0.866025 + 0.500000i) q^{88} +(-0.500000 - 0.866025i) q^{89} +(-0.500000 - 0.866025i) q^{90} +(-0.866025 - 0.500000i) q^{93} +(-0.500000 + 0.866025i) q^{94} +(0.866025 - 0.500000i) q^{95} +1.00000i q^{96} +(0.500000 + 0.866025i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9	$ 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 2 q^{14} + 4 q^{16} - 2 q^{17} + 4 q^{19} - 4 q^{27} + 2 q^{30} - 2 q^{33} - 2 q^{35} - 4 q^{36} + 2 q^{40} + 2 q^{41} - 2 q^{42} - 2 q^{44} - 4 q^{48} + 2 q^{51} - 2 q^{56} - 4 q^{57} + 2 q^{58} - 2 q^{59} + 2 q^{62} - 4 q^{64} + 8 q^{67} + 2 q^{68} - 2 q^{73} - 8 q^{74} - 4 q^{76} + 4 q^{81} + 2 q^{83} - 2 q^{89} - 2 q^{90} - 2 q^{94} + 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 - 2 * q^10 + 2 * q^11 + 4 * q^12 + 2 * q^14 + 4 * q^16 - 2 * q^17 + 4 * q^19 - 4 * q^27 + 2 * q^30 - 2 * q^33 - 2 * q^35 - 4 * q^36 + 2 * q^40 + 2 * q^41 - 2 * q^42 - 2 * q^44 - 4 * q^48 + 2 * q^51 - 2 * q^56 - 4 * q^57 + 2 * q^58 - 2 * q^59 + 2 * q^62 - 4 * q^64 + 8 * q^67 + 2 * q^68 - 2 * q^73 - 8 * q^74 - 4 * q^76 + 4 * q^81 + 2 * q^83 - 2 * q^89 - 2 * q^90 - 2 * q^94 + 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{2}{3}\right)$	$e\left(\frac{1}{3}\right)$

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 $

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

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.1.cj.c.1075.1	yes	4
8.3	odd	2	inner	1368.1.cj.c.1075.2	yes	4
9.7	even	3		1368.1.ba.c.619.1	✓	4
19.7	even	3		1368.1.ba.c.1147.2	yes	4
72.43	odd	6		1368.1.ba.c.619.2	yes	4
152.83	odd	6		1368.1.ba.c.1147.1	yes	4
171.7	even	3	inner	1368.1.cj.c.691.1	yes	4
1368.691	odd	6	inner	1368.1.cj.c.691.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.1.ba.c.619.1	✓	4	9.7	even	3
1368.1.ba.c.619.2	yes	4	72.43	odd	6
1368.1.ba.c.1147.1	yes	4	152.83	odd	6
1368.1.ba.c.1147.2	yes	4	19.7	even	3
1368.1.cj.c.691.1	yes	4	171.7	even	3	inner
1368.1.cj.c.691.2	yes	4	1368.691	odd	6	inner
1368.1.cj.c.1075.1	yes	4	1.1	even	1	trivial
1368.1.cj.c.1075.2	yes	4	8.3	odd	2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1368.cj (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +(0.866025 - 0.500000i) q^{5} +1.00000i q^{6} +(-0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +(0.866025 - 0.500000i) q^{5} +1.00000i q^{6} +(-0.866025 + 0.500000i) q^{7} +1.00000i q^{8} +1.00000 q^{9} +(-0.500000 - 0.866025i) q^{10} +(0.500000 + 0.866025i) q^{11} +1.00000 q^{12} +(0.500000 + 0.866025i) q^{14} +(-0.866025 + 0.500000i) q^{15} +1.00000 q^{16} +(-0.500000 + 0.866025i) q^{17} -1.00000i q^{18} +1.00000 q^{19} +(-0.866025 + 0.500000i) q^{20} +(0.866025 - 0.500000i) q^{21} +(0.866025 - 0.500000i) q^{22} -1.00000i q^{24} -1.00000 q^{27} +(0.866025 - 0.500000i) q^{28} +(0.866025 + 0.500000i) q^{29} +(0.500000 + 0.866025i) q^{30} +(0.866025 + 0.500000i) q^{31} -1.00000i q^{32} +(-0.500000 - 0.866025i) q^{33} +(0.866025 + 0.500000i) q^{34} +(-0.500000 + 0.866025i) q^{35} -1.00000 q^{36} -2.00000i q^{37} -1.00000i q^{38} +(0.500000 + 0.866025i) q^{40} +(0.500000 + 0.866025i) q^{41} +(-0.500000 - 0.866025i) q^{42} +(-0.500000 - 0.866025i) q^{44} +(0.866025 - 0.500000i) q^{45} +(-0.866025 - 0.500000i) q^{47} -1.00000 q^{48} +(0.500000 - 0.866025i) q^{51} +(0.866025 - 0.500000i) q^{53} +1.00000i q^{54} +(0.866025 + 0.500000i) q^{55} +(-0.500000 - 0.866025i) q^{56} -1.00000 q^{57} +(0.500000 - 0.866025i) q^{58} +(-0.500000 - 0.866025i) q^{59} +(0.866025 - 0.500000i) q^{60} +(0.866025 + 0.500000i) q^{61} +(0.500000 - 0.866025i) q^{62} +(-0.866025 + 0.500000i) q^{63} -1.00000 q^{64} +(-0.866025 + 0.500000i) q^{66} +2.00000 q^{67} +(0.500000 - 0.866025i) q^{68} +(0.866025 + 0.500000i) q^{70} +(-0.866025 - 0.500000i) q^{71} +1.00000i q^{72} +(-0.500000 + 0.866025i) q^{73} -2.00000 q^{74} -1.00000 q^{76} +(-0.866025 - 0.500000i) q^{77} +(0.866025 - 0.500000i) q^{80} +1.00000 q^{81} +(0.866025 - 0.500000i) q^{82} +(0.500000 + 0.866025i) q^{83} +(-0.866025 + 0.500000i) q^{84} +1.00000i q^{85} +(-0.866025 - 0.500000i) q^{87} +(-0.866025 + 0.500000i) q^{88} +(-0.500000 - 0.866025i) q^{89} +(-0.500000 - 0.866025i) q^{90} +(-0.866025 - 0.500000i) q^{93} +(-0.500000 + 0.866025i) q^{94} +(0.866025 - 0.500000i) q^{95} +1.00000i q^{96} +(0.500000 + 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9	\( 4 q - 4 q^{3} - 4 q^{4} + 4 q^{9} - 2 q^{10} + 2 q^{11} + 4 q^{12} + 2 q^{14} + 4 q^{16} - 2 q^{17} + 4 q^{19} - 4 q^{27} + 2 q^{30} - 2 q^{33} - 2 q^{35} - 4 q^{36} + 2 q^{40} + 2 q^{41} - 2 q^{42} - 2 q^{44} - 4 q^{48} + 2 q^{51} - 2 q^{56} - 4 q^{57} + 2 q^{58} - 2 q^{59} + 2 q^{62} - 4 q^{64} + 8 q^{67} + 2 q^{68} - 2 q^{73} - 8 q^{74} - 4 q^{76} + 4 q^{81} + 2 q^{83} - 2 q^{89} - 2 q^{90} - 2 q^{94} + 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^4 + 4 * q^9 - 2 * q^10 + 2 * q^11 + 4 * q^12 + 2 * q^14 + 4 * q^16 - 2 * q^17 + 4 * q^19 - 4 * q^27 + 2 * q^30 - 2 * q^33 - 2 * q^35 - 4 * q^36 + 2 * q^40 + 2 * q^41 - 2 * q^42 - 2 * q^44 - 4 * q^48 + 2 * q^51 - 2 * q^56 - 4 * q^57 + 2 * q^58 - 2 * q^59 + 2 * q^62 - 4 * q^64 + 8 * q^67 + 2 * q^68 - 2 * q^73 - 8 * q^74 - 4 * q^76 + 4 * q^81 + 2 * q^83 - 2 * q^89 - 2 * q^90 - 2 * q^94 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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