LMFDB - Embedded newform 1140.1.bl.c.179.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1140,1,Mod(179,1140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1140, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 3, 1]))

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

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

chi := DirichletCharacter("1140.179");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1140.bl (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:	$0.568934114402$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.35655825600.2

Embedding invariants

Embedding label			179.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1140.179
Dual form			1140.1.bl.c.1019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} -1.00000 q^{6} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} -1.00000 q^{6} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} -1.00000 q^{10} +(-0.500000 - 0.866025i) q^{12} +(-0.500000 - 0.866025i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-0.500000 + 0.866025i) q^{19} +(-0.500000 - 0.866025i) q^{20} +(0.500000 - 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +(0.500000 - 0.866025i) q^{30} +1.00000 q^{31} +(0.500000 - 0.866025i) q^{32} -1.00000 q^{34} +1.00000 q^{36} -1.00000 q^{38} +(0.500000 - 0.866025i) q^{40} +1.00000 q^{45} +(1.50000 - 0.866025i) q^{47} +1.00000 q^{48} -1.00000 q^{49} +(0.500000 - 0.866025i) q^{50} +(-0.500000 - 0.866025i) q^{51} +(-1.50000 + 0.866025i) q^{53} +(0.500000 + 0.866025i) q^{54} +(-0.500000 - 0.866025i) q^{57} +1.00000 q^{60} +(1.00000 + 1.73205i) q^{61} +(0.500000 + 0.866025i) q^{62} +1.00000 q^{64} +(-0.500000 - 0.866025i) q^{68} +(0.500000 + 0.866025i) q^{72} +1.00000 q^{75} +(-0.500000 - 0.866025i) q^{76} +(-1.00000 + 1.73205i) q^{79} +1.00000 q^{80} +(-0.500000 + 0.866025i) q^{81} +1.73205i q^{83} +(-0.500000 - 0.866025i) q^{85} +(0.500000 + 0.866025i) q^{90} +(-0.500000 + 0.866025i) q^{93} +(1.50000 + 0.866025i) q^{94} +(-0.500000 - 0.866025i) q^{95} +(0.500000 + 0.866025i) q^{96} +(-0.500000 - 0.866025i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{3} - q^{4} - q^{5} - 2 q^{6} - 2 q^{8} - q^{9}+O(q^{10}) $ 2 * q + q^2 - q^3 - q^4 - q^5 - 2 * q^6 - 2 * q^8 - q^9	$ 2 q + q^{2} - q^{3} - q^{4} - q^{5} - 2 q^{6} - 2 q^{8} - q^{9} - 2 q^{10} - q^{12} - q^{15} - q^{16} - q^{17} + q^{18} - q^{19} - q^{20} + q^{24} - q^{25} + 2 q^{27} + q^{30} + 2 q^{31} + q^{32} - 2 q^{34} + 2 q^{36} - 2 q^{38} + q^{40} + 2 q^{45} + 3 q^{47} + 2 q^{48} - 2 q^{49} + q^{50} - q^{51} - 3 q^{53} + q^{54} - q^{57} + 2 q^{60} + 2 q^{61} + q^{62} + 2 q^{64} - q^{68} + q^{72} + 2 q^{75} - q^{76} - 2 q^{79} + 2 q^{80} - q^{81} - q^{85} + q^{90} - q^{93} + 3 q^{94} - q^{95} + q^{96} - q^{98}+O(q^{100}) $ 2 * q + q^2 - q^3 - q^4 - q^5 - 2 * q^6 - 2 * q^8 - q^9 - 2 * q^10 - q^12 - q^15 - q^16 - q^17 + q^18 - q^19 - q^20 + q^24 - q^25 + 2 * q^27 + q^30 + 2 * q^31 + q^32 - 2 * q^34 + 2 * q^36 - 2 * q^38 + q^40 + 2 * q^45 + 3 * q^47 + 2 * q^48 - 2 * q^49 + q^50 - q^51 - 3 * q^53 + q^54 - q^57 + 2 * q^60 + 2 * q^61 + q^62 + 2 * q^64 - q^68 + q^72 + 2 * q^75 - q^76 - 2 * q^79 + 2 * q^80 - q^81 - q^85 + q^90 - q^93 + 3 * q^94 - q^95 + q^96 - q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1140\mathbb{Z}\right)^\times$.

$n$	$457$	$571$	$761$	$781$
$\chi(n)$	$-1$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

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
$3$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$3$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$4$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$5$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$5$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$6$	−1.00000			−1.00000
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−1.00000			−1.00000
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$	−1.00000			−1.00000
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$13$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$13$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$14$		0			0
$15$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$16$	−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			$\pi$
$18$	0.500000	−	0.866025i	0.500000	−	0.866025i
$19$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$19$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$20$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	0.500000	−	0.866025i	0.500000	−	0.866025i
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$		0			0
$27$	1.00000			1.00000
$28$		0			0
$29$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$29$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$30$	0.500000	−	0.866025i	0.500000	−	0.866025i
$31$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$31$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$32$	0.500000	−	0.866025i	0.500000	−	0.866025i
$33$		0			0
$34$	−1.00000			−1.00000
$35$		0			0
$36$	1.00000			1.00000
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	−1.00000			−1.00000
$39$		0			0
$40$	0.500000	−	0.866025i	0.500000	−	0.866025i
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$43$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$44$		0			0
$45$	1.00000			1.00000
$46$		0			0
$47$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$47$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$48$	1.00000			1.00000
$49$	−1.00000			−1.00000
$50$	0.500000	−	0.866025i	0.500000	−	0.866025i
$51$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$52$		0			0
$53$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$53$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$54$	0.500000	+	0.866025i	0.500000	+	0.866025i
$55$		0			0
$56$		0			0
$57$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	1.00000			1.00000
$61$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$61$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$62$	0.500000	+	0.866025i	0.500000	+	0.866025i
$63$		0			0
$64$	1.00000			1.00000
$65$		0			0
$66$		0			0
$67$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$69$		0			0
$70$		0			0
$71$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$71$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$72$	0.500000	+	0.866025i	0.500000	+	0.866025i
$73$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$73$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$	1.00000			1.00000
$76$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$77$		0			0
$78$		0			0
$79$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$79$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$80$	1.00000			1.00000
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$83$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$	0.500000	+	0.866025i	0.500000	+	0.866025i
$91$		0			0
$92$		0			0
$93$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$94$	1.50000	+	0.866025i	1.50000	+	0.866025i
$95$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$96$	0.500000	+	0.866025i	0.500000	+	0.866025i
$97$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$97$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$98$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$99$		0			0
$100$	1.00000			1.00000
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$	0.500000	−	0.866025i	0.500000	−	0.866025i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$107$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$107$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$108$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$109$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$109$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$113$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$114$	0.500000	−	0.866025i	0.500000	−	0.866025i
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	0.500000	+	0.866025i	0.500000	+	0.866025i
$121$	−1.00000			−1.00000
$122$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$123$		0			0
$124$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$125$	1.00000			1.00000
$126$		0			0
$127$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$127$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$128$	0.500000	+	0.866025i	0.500000	+	0.866025i
$129$		0			0
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$136$	0.500000	−	0.866025i	0.500000	−	0.866025i
$137$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$137$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$139$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$140$		0			0
$141$			1.73205i			1.73205i
$142$		0			0
$143$		0			0
$144$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	0.500000	−	0.866025i	0.500000	−	0.866025i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	0.500000	+	0.866025i	0.500000	+	0.866025i
$151$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$151$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$152$	0.500000	−	0.866025i	0.500000	−	0.866025i
$153$	1.00000			1.00000
$154$		0			0
$155$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$	−2.00000			−2.00000
$159$		−	1.73205i		−	1.73205i
$160$	0.500000	+	0.866025i	0.500000	+	0.866025i
$161$		0			0
$162$	−1.00000			−1.00000
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$167$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$167$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$168$		0			0
$169$	0.500000	−	0.866025i	0.500000	−	0.866025i
$170$	0.500000	−	0.866025i	0.500000	−	0.866025i
$171$	1.00000			1.00000
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$181$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$181$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$182$		0			0
$183$	−2.00000			−2.00000
$184$		0			0
$185$		0			0
$186$	−1.00000			−1.00000
$187$		0			0
$188$			1.73205i			1.73205i
$189$		0			0
$190$	0.500000	−	0.866025i	0.500000	−	0.866025i
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$193$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$193$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$194$		0			0
$195$		0			0
$196$	0.500000	−	0.866025i	0.500000	−	0.866025i
$197$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$197$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	0.500000	+	0.866025i	0.500000	+	0.866025i
$201$		0			0
$202$		0			0
$203$		0			0
$204$	1.00000			1.00000
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$211$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$212$		−	1.73205i		−	1.73205i
$213$		0			0
$214$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$215$		0			0
$216$	−1.00000			−1.00000
$217$		0			0
$218$			1.73205i			1.73205i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$223$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$226$	1.50000	−	0.866025i	1.50000	−	0.866025i
$227$	−2.00000			−2.00000			−1.00000			$\pi$
$227$	−2.00000			−2.00000			−1.00000			$\pi$
$228$	1.00000			1.00000
$229$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$229$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	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
$235$			1.73205i			1.73205i
$236$		0			0
$237$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$241$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$241$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$242$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$243$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$244$	−2.00000			−2.00000
$245$	0.500000	−	0.866025i	0.500000	−	0.866025i
$246$		0			0
$247$		0			0
$248$	−1.00000			−1.00000
$249$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$250$	0.500000	+	0.866025i	0.500000	+	0.866025i
$251$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$251$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	1.00000			1.00000
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$263$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		−	1.73205i		−	1.73205i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$	−1.00000			−1.00000
$271$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$271$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$272$	1.00000			1.00000
$273$		0			0
$274$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	1.50000	+	0.866025i	1.50000	+	0.866025i
$279$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$280$		0			0
$281$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$281$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$282$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$		0			0
$285$	1.00000			1.00000
$286$		0			0
$287$		0			0
$288$	−1.00000			−1.00000
$289$		0			0
$290$		0			0
$291$		0			0
$292$		0			0
$293$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$293$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$294$	1.00000			1.00000
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$301$		0			0
$302$	0.500000	+	0.866025i	0.500000	+	0.866025i
$303$		0			0
$304$	1.00000			1.00000
$305$	−2.00000			−2.00000
$306$	0.500000	+	0.866025i	0.500000	+	0.866025i
$307$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$307$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$308$		0			0
$309$		0			0
$310$	−1.00000			−1.00000
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$313$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$317$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$317$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$318$	1.50000	−	0.866025i	1.50000	−	0.866025i
$319$		0			0
$320$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$321$	0.500000	−	0.866025i	0.500000	−	0.866025i
$322$		0			0
$323$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$324$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$325$		0			0
$326$		0			0
$327$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.00000			−2.00000			−1.00000			$\pi$
$331$	−2.00000			−2.00000			−1.00000			$\pi$
$332$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$333$		0			0
$334$	0.500000	−	0.866025i	0.500000	−	0.866025i
$335$		0			0
$336$		0			0
$337$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$337$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$338$	1.00000			1.00000
$339$	1.50000	+	0.866025i	1.50000	+	0.866025i
$340$	1.00000			1.00000
$341$		0			0
$342$	0.500000	+	0.866025i	0.500000	+	0.866025i
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$347$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$349$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.00000			2.00000			1.00000			$0$
$353$	2.00000			2.00000			1.00000			$0$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$	−1.00000			−1.00000
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$	1.50000	+	0.866025i	1.50000	+	0.866025i
$363$	0.500000	−	0.866025i	0.500000	−	0.866025i
$364$		0			0
$365$		0			0
$366$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$376$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$377$		0			0
$378$		0			0
$379$	−2.00000			−2.00000			−1.00000			$\pi$
$379$	−2.00000			−2.00000			−1.00000			$\pi$
$380$	1.00000			1.00000
$381$		0			0
$382$		0			0
$383$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$383$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$384$	−1.00000			−1.00000
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	1.00000			1.00000
$393$		0			0
$394$	0.500000	+	0.866025i	0.500000	+	0.866025i
$395$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$396$		0			0
$397$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$397$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$401$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$401$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$406$		0			0
$407$		0			0
$408$	0.500000	+	0.866025i	0.500000	+	0.866025i
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$	−1.00000			−1.00000
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$416$		0			0
$417$			1.73205i			1.73205i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$	1.00000			1.00000
$423$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$424$	1.50000	−	0.866025i	1.50000	−	0.866025i
$425$	1.00000			1.00000
$426$		0			0
$427$		0			0
$428$	0.500000	−	0.866025i	0.500000	−	0.866025i
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$433$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$433$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$434$		0			0
$435$		0			0
$436$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$437$		0			0
$438$		0			0
$439$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$439$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$440$		0			0
$441$	0.500000	+	0.866025i	0.500000	+	0.866025i
$442$		0			0
$443$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$443$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	−1.00000			−1.00000
$451$		0			0
$452$	1.50000	+	0.866025i	1.50000	+	0.866025i
$453$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$454$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$455$		0			0
$456$	0.500000	+	0.866025i	0.500000	+	0.866025i
$457$		0			0		1.00000			$0$
$457$		0			0		−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		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$466$	1.00000			1.00000
$467$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$467$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$		0			0
$470$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	1.00000	−	1.73205i	1.00000	−	1.73205i
$475$	1.00000			1.00000
$476$		0			0
$477$	1.50000	+	0.866025i	1.50000	+	0.866025i
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$	−1.00000			−1.00000
$481$		0			0
$482$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$483$		0			0
$484$	0.500000	−	0.866025i	0.500000	−	0.866025i
$485$		0			0
$486$	0.500000	−	0.866025i	0.500000	−	0.866025i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$489$		0			0
$490$	1.00000			1.00000
$491$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$491$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$497$		0			0
$498$		−	1.73205i		−	1.73205i
$499$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$499$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$500$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$501$	1.00000			1.00000
$502$		0			0
$503$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$503$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.500000	+	0.866025i	0.500000	+	0.866025i
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$	0.500000	+	0.866025i	0.500000	+	0.866025i
$511$		0			0
$512$	−1.00000			−1.00000
$513$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$523$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$			1.73205i			1.73205i
$527$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$528$		0			0
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$	1.50000	−	0.866025i	1.50000	−	0.866025i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.500000	−	0.866025i	0.500000	−	0.866025i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$541$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$541$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$542$			1.73205i			1.73205i
$543$			1.73205i			1.73205i
$544$	0.500000	+	0.866025i	0.500000	+	0.866025i
$545$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$546$		0			0
$547$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$547$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$548$	−1.00000			−1.00000
$549$	1.00000	−	1.73205i	1.00000	−	1.73205i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$			1.73205i			1.73205i
$557$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$557$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$558$	0.500000	−	0.866025i	0.500000	−	0.866025i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$563$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$564$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$565$	1.50000	+	0.866025i	1.50000	+	0.866025i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$	0.500000	+	0.866025i	0.500000	+	0.866025i
$571$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$571$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	1.50000	−	0.866025i	1.50000	−	0.866025i
$587$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$587$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$588$	0.500000	+	0.866025i	0.500000	+	0.866025i
$589$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$590$		0			0
$591$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$592$		0			0
$593$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$593$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	−1.00000			−1.00000
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$605$	0.500000	−	0.866025i	0.500000	−	0.866025i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$	0.500000	+	0.866025i	0.500000	+	0.866025i
$609$		0			0
$610$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$611$		0			0
$612$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$617$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$618$		0			0
$619$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$619$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$620$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$631$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$632$	1.00000	−	1.73205i	1.00000	−	1.73205i
$633$	0.500000	+	0.866025i	0.500000	+	0.866025i
$634$	1.50000	+	0.866025i	1.50000	+	0.866025i
$635$		0			0
$636$	1.50000	+	0.866025i	1.50000	+	0.866025i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−1.00000			−1.00000
$641$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$641$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$642$	1.00000			1.00000
$643$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$643$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$644$		0			0
$645$		0			0
$646$	0.500000	−	0.866025i	0.500000	−	0.866025i
$647$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$647$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$648$	0.500000	−	0.866025i	0.500000	−	0.866025i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.00000			−2.00000			−1.00000			$\pi$
$653$	−2.00000			−2.00000			−1.00000			$\pi$
$654$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$661$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$662$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$663$		0			0
$664$		−	1.73205i		−	1.73205i
$665$		0			0
$666$		0			0
$667$		0			0
$668$	1.00000			1.00000
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$676$	0.500000	+	0.866025i	0.500000	+	0.866025i
$677$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$677$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$678$			1.73205i			1.73205i
$679$		0			0
$680$	0.500000	+	0.866025i	0.500000	+	0.866025i
$681$	1.00000	−	1.73205i	1.00000	−	1.73205i
$682$		0			0
$683$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$683$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$684$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$685$	−1.00000			−1.00000
$686$		0			0
$687$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$688$		0			0
$689$		0			0
$690$		0			0
$691$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$691$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$			1.73205i			1.73205i
$695$			1.73205i			1.73205i
$696$		0			0
$697$		0			0
$698$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$699$	0.500000	+	0.866025i	0.500000	+	0.866025i
$700$		0			0
$701$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$701$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$706$	1.00000	+	1.73205i	1.00000	+	1.73205i
$707$		0			0
$708$		0			0
$709$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$709$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$710$		0			0
$711$	2.00000			2.00000
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$721$		0			0
$722$	0.500000	−	0.866025i	0.500000	−	0.866025i
$723$		−	1.73205i		−	1.73205i
$724$			1.73205i			1.73205i
$725$		0			0
$726$	1.00000			1.00000
$727$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$727$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$	1.00000	−	1.73205i	1.00000	−	1.73205i
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	0.500000	+	0.866025i	0.500000	+	0.866025i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$739$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$744$	0.500000	−	0.866025i	0.500000	−	0.866025i
$745$		0			0
$746$		0			0
$747$	1.50000	−	0.866025i	1.50000	−	0.866025i
$748$		0			0
$749$		0			0
$750$	−1.00000			−1.00000
$751$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$751$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$752$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$753$		0			0
$754$		0			0
$755$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$756$		0			0
$757$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$757$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$758$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$759$		0			0
$760$	0.500000	+	0.866025i	0.500000	+	0.866025i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$766$	1.00000			1.00000
$767$		0			0
$768$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$769$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$769$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.500000	+	0.866025i	0.500000	+	0.866025i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$789$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$790$	1.00000	−	1.73205i	1.00000	−	1.73205i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	1.50000	+	0.866025i	1.50000	+	0.866025i
$796$		0			0
$797$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$797$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$			1.73205i			1.73205i
$800$	−1.00000			−1.00000
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	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	$0.666667\pi$
$812$		0			0
$813$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$814$		0			0
$815$		0			0
$816$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$827$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$828$		0			0
$829$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$829$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$830$		−	1.73205i		−	1.73205i
$831$		0			0
$832$		0			0
$833$	0.500000	−	0.866025i	0.500000	−	0.866025i
$834$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$835$	1.00000			1.00000
$836$		0			0
$837$	1.00000			1.00000
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	0.500000	−	0.866025i	0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	0.500000	+	0.866025i	0.500000	+	0.866025i
$845$	0.500000	+	0.866025i	0.500000	+	0.866025i
$846$		−	1.73205i		−	1.73205i
$847$		0			0
$848$	1.50000	+	0.866025i	1.50000	+	0.866025i
$849$		0			0
$850$	0.500000	+	0.866025i	0.500000	+	0.866025i
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$856$	1.00000			1.00000
$857$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$857$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$858$		0			0
$859$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$859$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.00000			2.00000			1.00000			$0$
$863$	2.00000			2.00000			1.00000			$0$
$864$	0.500000	−	0.866025i	0.500000	−	0.866025i
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$877$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$878$	1.00000			1.00000
$879$	1.50000	+	0.866025i	1.50000	+	0.866025i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$883$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$883$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$884$		0			0
$885$		0			0
$886$	1.50000	+	0.866025i	1.50000	+	0.866025i
$887$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$887$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$			1.73205i			1.73205i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$901$		−	1.73205i		−	1.73205i
$902$		0			0
$903$		0			0
$904$			1.73205i			1.73205i
$905$			1.73205i			1.73205i
$906$	−1.00000			−1.00000
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$	1.00000	−	1.73205i	1.00000	−	1.73205i
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$913$		0			0
$914$		0			0
$915$	1.00000	−	1.73205i	1.00000	−	1.73205i
$916$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$917$		0			0
$918$	−1.00000			−1.00000
$919$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$919$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$	0.500000	−	0.866025i	0.500000	−	0.866025i
$931$	0.500000	−	0.866025i	0.500000	−	0.866025i
$932$	0.500000	+	0.866025i	0.500000	+	0.866025i
$933$		0			0
$934$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$935$		0			0
$936$		0			0
$937$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$937$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$938$		0			0
$939$		0			0
$940$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	2.00000			2.00000
$949$		0			0
$950$	0.500000	+	0.866025i	0.500000	+	0.866025i
$951$			1.73205i			1.73205i
$952$		0			0
$953$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$953$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$954$			1.73205i			1.73205i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$961$		0			0
$962$		0			0
$963$	0.500000	+	0.866025i	0.500000	+	0.866025i
$964$		−	1.73205i		−	1.73205i
$965$		0			0
$966$		0			0
$967$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$967$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$968$	1.00000			1.00000
$969$	1.00000			1.00000
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	1.00000			1.00000
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.00000	−	1.73205i	1.00000	−	1.73205i
$977$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$977$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$	0.500000	+	0.866025i	0.500000	+	0.866025i
$981$		−	1.73205i		−	1.73205i
$982$		0			0
$983$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$983$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$984$		0			0
$985$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$991$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$992$	0.500000	−	0.866025i	0.500000	−	0.866025i
$993$	1.00000	−	1.73205i	1.00000	−	1.73205i
$994$		0			0
$995$		0			0
$996$	1.50000	−	0.866025i	1.50000	−	0.866025i
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		−	1.73205i		−	1.73205i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1140.1.bl.c.179.1	yes	2
3.2	odd	2		1140.1.bl.b.179.1	yes	2
4.3	odd	2		1140.1.bl.d.179.1	yes	2
5.4	even	2		1140.1.bl.b.179.1	yes	2
12.11	even	2		1140.1.bl.a.179.1	✓	2
15.14	odd	2	CM	1140.1.bl.c.179.1	yes	2
19.12	odd	6		1140.1.bl.d.1019.1	yes	2
20.19	odd	2		1140.1.bl.a.179.1	✓	2
57.50	even	6		1140.1.bl.a.1019.1	yes	2
60.59	even	2		1140.1.bl.d.179.1	yes	2
76.31	even	6	inner	1140.1.bl.c.1019.1	yes	2
95.69	odd	6		1140.1.bl.a.1019.1	yes	2
228.107	odd	6		1140.1.bl.b.1019.1	yes	2
285.164	even	6		1140.1.bl.d.1019.1	yes	2
380.259	even	6		1140.1.bl.b.1019.1	yes	2
1140.1019	odd	6	inner	1140.1.bl.c.1019.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.1.bl.a.179.1	✓	2	12.11	even	2
1140.1.bl.a.179.1	✓	2	20.19	odd	2
1140.1.bl.a.1019.1	yes	2	57.50	even	6
1140.1.bl.a.1019.1	yes	2	95.69	odd	6
1140.1.bl.b.179.1	yes	2	3.2	odd	2
1140.1.bl.b.179.1	yes	2	5.4	even	2
1140.1.bl.b.1019.1	yes	2	228.107	odd	6
1140.1.bl.b.1019.1	yes	2	380.259	even	6
1140.1.bl.c.179.1	yes	2	1.1	even	1	trivial
1140.1.bl.c.179.1	yes	2	15.14	odd	2	CM
1140.1.bl.c.1019.1	yes	2	76.31	even	6	inner
1140.1.bl.c.1019.1	yes	2	1140.1019	odd	6	inner
1140.1.bl.d.179.1	yes	2	4.3	odd	2
1140.1.bl.d.179.1	yes	2	60.59	even	2
1140.1.bl.d.1019.1	yes	2	19.12	odd	6
1140.1.bl.d.1019.1	yes	2	285.164	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:	\( N \)	\(=\)	\( 1140 = 2^{2} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1140.bl (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} -1.00000 q^{6} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} -1.00000 q^{6} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} -1.00000 q^{10} +(-0.500000 - 0.866025i) q^{12} +(-0.500000 - 0.866025i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-0.500000 + 0.866025i) q^{19} +(-0.500000 - 0.866025i) q^{20} +(0.500000 - 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} +(0.500000 - 0.866025i) q^{30} +1.00000 q^{31} +(0.500000 - 0.866025i) q^{32} -1.00000 q^{34} +1.00000 q^{36} -1.00000 q^{38} +(0.500000 - 0.866025i) q^{40} +1.00000 q^{45} +(1.50000 - 0.866025i) q^{47} +1.00000 q^{48} -1.00000 q^{49} +(0.500000 - 0.866025i) q^{50} +(-0.500000 - 0.866025i) q^{51} +(-1.50000 + 0.866025i) q^{53} +(0.500000 + 0.866025i) q^{54} +(-0.500000 - 0.866025i) q^{57} +1.00000 q^{60} +(1.00000 + 1.73205i) q^{61} +(0.500000 + 0.866025i) q^{62} +1.00000 q^{64} +(-0.500000 - 0.866025i) q^{68} +(0.500000 + 0.866025i) q^{72} +1.00000 q^{75} +(-0.500000 - 0.866025i) q^{76} +(-1.00000 + 1.73205i) q^{79} +1.00000 q^{80} +(-0.500000 + 0.866025i) q^{81} +1.73205i q^{83} +(-0.500000 - 0.866025i) q^{85} +(0.500000 + 0.866025i) q^{90} +(-0.500000 + 0.866025i) q^{93} +(1.50000 + 0.866025i) q^{94} +(-0.500000 - 0.866025i) q^{95} +(0.500000 + 0.866025i) q^{96} +(-0.500000 - 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{3} - q^{4} - q^{5} - 2 q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \) 2 * q + q^2 - q^3 - q^4 - q^5 - 2 * q^6 - 2 * q^8 - q^9	\( 2 q + q^{2} - q^{3} - q^{4} - q^{5} - 2 q^{6} - 2 q^{8} - q^{9} - 2 q^{10} - q^{12} - q^{15} - q^{16} - q^{17} + q^{18} - q^{19} - q^{20} + q^{24} - q^{25} + 2 q^{27} + q^{30} + 2 q^{31} + q^{32} - 2 q^{34} + 2 q^{36} - 2 q^{38} + q^{40} + 2 q^{45} + 3 q^{47} + 2 q^{48} - 2 q^{49} + q^{50} - q^{51} - 3 q^{53} + q^{54} - q^{57} + 2 q^{60} + 2 q^{61} + q^{62} + 2 q^{64} - q^{68} + q^{72} + 2 q^{75} - q^{76} - 2 q^{79} + 2 q^{80} - q^{81} - q^{85} + q^{90} - q^{93} + 3 q^{94} - q^{95} + q^{96} - q^{98}+O(q^{100}) \) 2 * q + q^2 - q^3 - q^4 - q^5 - 2 * q^6 - 2 * q^8 - q^9 - 2 * q^10 - q^12 - q^15 - q^16 - q^17 + q^18 - q^19 - q^20 + q^24 - q^25 + 2 * q^27 + q^30 + 2 * q^31 + q^32 - 2 * q^34 + 2 * q^36 - 2 * q^38 + q^40 + 2 * q^45 + 3 * q^47 + 2 * q^48 - 2 * q^49 + q^50 - q^51 - 3 * q^53 + q^54 - q^57 + 2 * q^60 + 2 * q^61 + q^62 + 2 * q^64 - q^68 + q^72 + 2 * q^75 - q^76 - 2 * q^79 + 2 * q^80 - q^81 - q^85 + q^90 - q^93 + 3 * q^94 - q^95 + q^96 - q^98

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