LMFDB - Embedded newform 140.1.p.b.79.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [140,1,Mod(39,140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(140, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("140.39");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$140 = 2^{2} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	140.p (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:	$0.0698691017686$
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:	$D_{3}$
Projective field:	Galois closure of 3.1.980.1
Artin image:	$C_6\times S_3$
Artin field:	Galois closure of 12.0.153664000000.1

Embedding invariants

Embedding label			79.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	140.79
Dual form			140.1.p.b.39.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 + 0.866025i) q^{5} -1.00000 q^{6} +(0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +(0.500000 + 0.866025i) q^{10} +(-0.500000 + 0.866025i) q^{12} +1.00000 q^{14} +1.00000 q^{15} +(-0.500000 + 0.866025i) q^{16} +1.00000 q^{20} +(0.500000 - 0.866025i) q^{21} +(-0.500000 + 0.866025i) q^{23} +(0.500000 + 0.866025i) q^{24} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} +(0.500000 - 0.866025i) q^{28} -1.00000 q^{29} +(0.500000 - 0.866025i) q^{30} +(0.500000 + 0.866025i) q^{32} -1.00000 q^{35} +(0.500000 - 0.866025i) q^{40} -1.00000 q^{41} +(-0.500000 - 0.866025i) q^{42} +1.00000 q^{43} +(0.500000 + 0.866025i) q^{46} +(1.00000 - 1.73205i) q^{47} +1.00000 q^{48} +(-0.500000 + 0.866025i) q^{49} -1.00000 q^{50} +(-0.500000 + 0.866025i) q^{54} +(-0.500000 - 0.866025i) q^{56} +(-0.500000 + 0.866025i) q^{58} +(-0.500000 - 0.866025i) q^{60} +(0.500000 - 0.866025i) q^{61} +1.00000 q^{64} +(-0.500000 - 0.866025i) q^{67} +1.00000 q^{69} +(-0.500000 + 0.866025i) q^{70} +(-0.500000 + 0.866025i) q^{75} +(-0.500000 - 0.866025i) q^{80} +(0.500000 + 0.866025i) q^{81} +(-0.500000 + 0.866025i) q^{82} +1.00000 q^{83} -1.00000 q^{84} +(0.500000 - 0.866025i) q^{86} +(0.500000 + 0.866025i) q^{87} +(0.500000 - 0.866025i) q^{89} +1.00000 q^{92} +(-1.00000 - 1.73205i) q^{94} +(0.500000 - 0.866025i) q^{96} +(0.500000 + 0.866025i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{2} - q^{3} - q^{4} - q^{5} - 2 q^{6} + q^{7} - 2 q^{8} + q^{10} - q^{12} + 2 q^{14} + 2 q^{15} - q^{16} + 2 q^{20} + q^{21} - q^{23} + q^{24} - q^{25} - 2 q^{27} + q^{28} - 2 q^{29}+ \cdots + q^{98}+O(q^{100})$ 2 * q + q^2 - q^3 - q^4 - q^5 - 2 * q^6 + q^7 - 2 * q^8 + q^10 - q^12 + 2 * q^14 + 2 * q^15 - q^16 + 2 * q^20 + q^21 - q^23 + q^24 - q^25 - 2 * q^27 + q^28 - 2 * q^29 + q^30 + q^32 - 2 * q^35 + q^40 - 2 * q^41 - q^42 + 2 * q^43 + q^46 + 2 * q^47 + 2 * q^48 - q^49 - 2 * q^50 - q^54 - q^56 - q^58 - q^60 + q^61 + 2 * q^64 - q^67 + 2 * q^69 - q^70 - q^75 - q^80 + q^81 - q^82 + 2 * q^83 - 2 * q^84 + q^86 + q^87 + q^89 + 2 * q^92 - 2 * q^94 + q^96 + q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/140\mathbb{Z}\right)^\times$ .

$n$	$57$	$71$	$101$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	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	0.500000	−	0.866025i	$-0.333333\pi$
$3$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$4$	−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
$7$	0.500000	+	0.866025i	0.500000	+	0.866025i
$7$	0.500000	+	0.866025i	0.500000	+	0.866025i
$8$	−1.00000			−1.00000
$9$		0			0
$10$	0.500000	+	0.866025i	0.500000	+	0.866025i
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$	1.00000			1.00000
$15$	1.00000			1.00000
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$	1.00000			1.00000
$21$	0.500000	−	0.866025i	0.500000	−	0.866025i
$22$		0			0
$23$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$23$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$24$	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
$28$	0.500000	−	0.866025i	0.500000	−	0.866025i
$29$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$29$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$30$	0.500000	−	0.866025i	0.500000	−	0.866025i
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	0.500000	+	0.866025i	0.500000	+	0.866025i
$33$		0			0
$34$		0			0
$35$	−1.00000			−1.00000
$36$		0			0
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$		0			0
$40$	0.500000	−	0.866025i	0.500000	−	0.866025i
$41$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$41$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$42$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$43$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$43$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$44$		0			0
$45$		0			0
$46$	0.500000	+	0.866025i	0.500000	+	0.866025i
$47$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$47$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$48$	1.00000			1.00000
$49$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$50$	−1.00000			−1.00000
$51$		0			0
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$55$		0			0
$56$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$57$		0			0
$58$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$61$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$61$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$62$		0			0
$63$		0			0
$64$	1.00000			1.00000
$65$		0			0
$66$		0			0
$67$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$67$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$68$		0			0
$69$	1.00000			1.00000
$70$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$		0			0
$75$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$81$	0.500000	+	0.866025i	0.500000	+	0.866025i
$82$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$83$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$83$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$84$	−1.00000			−1.00000
$85$		0			0
$86$	0.500000	−	0.866025i	0.500000	−	0.866025i
$87$	0.500000	+	0.866025i	0.500000	+	0.866025i
$88$		0			0
$89$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$89$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$90$		0			0
$91$		0			0
$92$	1.00000			1.00000
$93$		0			0
$94$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$95$		0			0
$96$	0.500000	−	0.866025i	0.500000	−	0.866025i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	0.500000	+	0.866025i	0.500000	+	0.866025i
$99$		0			0
$100$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$101$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$101$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$103$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$104$		0			0
$105$	0.500000	+	0.866025i	0.500000	+	0.866025i
$106$		0			0
$107$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$107$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$108$	0.500000	+	0.866025i	0.500000	+	0.866025i
$109$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$109$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$	−1.00000			−1.00000
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$116$	0.500000	+	0.866025i	0.500000	+	0.866025i
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−1.00000			−1.00000
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$123$	0.500000	+	0.866025i	0.500000	+	0.866025i
$124$		0			0
$125$	1.00000			1.00000
$126$		0			0
$127$	−2.00000			−2.00000			−1.00000			$\pi$
$127$	−2.00000			−2.00000			−1.00000			$\pi$
$128$	0.500000	−	0.866025i	0.500000	−	0.866025i
$129$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$	−1.00000			−1.00000
$135$	0.500000	−	0.866025i	0.500000	−	0.866025i
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$	0.500000	−	0.866025i	0.500000	−	0.866025i
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$	0.500000	+	0.866025i	0.500000	+	0.866025i
$141$	−2.00000			−2.00000
$142$		0			0
$143$		0			0
$144$		0			0
$145$	0.500000	−	0.866025i	0.500000	−	0.866025i
$146$		0			0
$147$	1.00000			1.00000
$148$		0			0
$149$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$149$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$150$	0.500000	+	0.866025i	0.500000	+	0.866025i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$	−1.00000			−1.00000
$161$	−1.00000			−1.00000
$162$	1.00000			1.00000
$163$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$163$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$164$	0.500000	+	0.866025i	0.500000	+	0.866025i
$165$		0			0
$166$	0.500000	−	0.866025i	0.500000	−	0.866025i
$167$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$167$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$168$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$169$	1.00000			1.00000
$170$		0			0
$171$		0			0
$172$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$	1.00000			1.00000
$175$	0.500000	−	0.866025i	0.500000	−	0.866025i
$176$		0			0
$177$		0			0
$178$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$181$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$	−1.00000			−1.00000
$184$	0.500000	−	0.866025i	0.500000	−	0.866025i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−2.00000			−2.00000
$189$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$	1.00000			1.00000
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		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
$201$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$202$	1.00000			1.00000
$203$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$204$		0			0
$205$	0.500000	−	0.866025i	0.500000	−	0.866025i
$206$	0.500000	+	0.866025i	0.500000	+	0.866025i
$207$		0			0
$208$		0			0
$209$		0			0
$210$	1.00000			1.00000
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	0.500000	+	0.866025i	0.500000	+	0.866025i
$215$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$216$	1.00000			1.00000
$217$		0			0
$218$	1.00000			1.00000
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−2.00000			−2.00000			−1.00000			$\pi$
$223$	−2.00000			−2.00000			−1.00000			$\pi$
$224$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$225$		0			0
$226$		0			0
$227$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$227$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$229$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$230$	−1.00000			−1.00000
$231$		0			0
$232$	1.00000			1.00000
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$	1.00000	+	1.73205i	1.00000	+	1.73205i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$241$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$242$	0.500000	+	0.866025i	0.500000	+	0.866025i
$243$		0			0
$244$	−1.00000			−1.00000
$245$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$246$	1.00000			1.00000
$247$		0			0
$248$		0			0
$249$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$250$	0.500000	−	0.866025i	0.500000	−	0.866025i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$	−1.00000			−1.00000
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$263$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−1.00000			−1.00000
$268$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$269$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$269$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$270$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$		0			0
$280$	1.00000			1.00000
$281$	2.00000			2.00000			1.00000			$0$
$281$	2.00000			2.00000			1.00000			$0$
$282$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$283$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$283$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$288$		0			0
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	0.500000	−	0.866025i	0.500000	−	0.866025i
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$299$		0			0
$300$	1.00000			1.00000
$301$	0.500000	+	0.866025i	0.500000	+	0.866025i
$302$		0			0
$303$	0.500000	−	0.866025i	0.500000	−	0.866025i
$304$		0			0
$305$	0.500000	+	0.866025i	0.500000	+	0.866025i
$306$		0			0
$307$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$307$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$308$		0			0
$309$	1.00000			1.00000
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$321$	1.00000			1.00000
$322$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$323$		0			0
$324$	0.500000	−	0.866025i	0.500000	−	0.866025i
$325$		0			0
$326$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$327$	0.500000	−	0.866025i	0.500000	−	0.866025i
$328$	1.00000			1.00000
$329$	2.00000			2.00000
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$333$		0			0
$334$	0.500000	−	0.866025i	0.500000	−	0.866025i
$335$	1.00000			1.00000
$336$	0.500000	+	0.866025i	0.500000	+	0.866025i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	0.500000	−	0.866025i	0.500000	−	0.866025i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.00000			−1.00000
$344$	−1.00000			−1.00000
$345$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$346$		0			0
$347$	−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			$\pi$
$348$	0.500000	−	0.866025i	0.500000	−	0.866025i
$349$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$349$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$350$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$351$		0			0
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$	−1.00000			−1.00000
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$363$	1.00000			1.00000
$364$		0			0
$365$		0			0
$366$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$367$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$367$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$368$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$376$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$377$		0			0
$378$	−1.00000			−1.00000
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	1.00000	+	1.73205i	1.00000	+	1.73205i
$382$		0			0
$383$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$383$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$384$	−1.00000			−1.00000
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$389$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$390$		0			0
$391$		0			0
$392$	0.500000	−	0.866025i	0.500000	−	0.866025i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		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
$400$	1.00000			1.00000
$401$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$402$	0.500000	+	0.866025i	0.500000	+	0.866025i
$403$		0			0
$404$	0.500000	−	0.866025i	0.500000	−	0.866025i
$405$	−1.00000			−1.00000
$406$	−1.00000			−1.00000
$407$		0			0
$408$		0			0
$409$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$409$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$410$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$411$		0			0
$412$	1.00000			1.00000
$413$		0			0
$414$		0			0
$415$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$	0.500000	−	0.866025i	0.500000	−	0.866025i
$421$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$421$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.00000			1.00000
$428$	1.00000			1.00000
$429$		0			0
$430$	0.500000	+	0.866025i	0.500000	+	0.866025i
$431$		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		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$	−1.00000			−1.00000
$436$	0.500000	−	0.866025i	0.500000	−	0.866025i
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$443$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$444$		0			0
$445$	0.500000	+	0.866025i	0.500000	+	0.866025i
$446$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$447$	−1.00000			−1.00000
$448$	0.500000	+	0.866025i	0.500000	+	0.866025i
$449$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$449$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$	2.00000			2.00000
$455$		0			0
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	1.00000	+	1.73205i	1.00000	+	1.73205i
$459$		0			0
$460$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$461$	2.00000			2.00000			1.00000			$0$
$461$	2.00000			2.00000			1.00000			$0$
$462$		0			0
$463$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$463$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$464$	0.500000	−	0.866025i	0.500000	−	0.866025i
$465$		0			0
$466$		0			0
$467$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$467$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$468$		0			0
$469$	0.500000	−	0.866025i	0.500000	−	0.866025i
$470$	2.00000			2.00000
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$	0.500000	+	0.866025i	0.500000	+	0.866025i
$481$		0			0
$482$	−2.00000			−2.00000
$483$	0.500000	+	0.866025i	0.500000	+	0.866025i
$484$	1.00000			1.00000
$485$		0			0
$486$		0			0
$487$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$487$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$488$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$489$	−2.00000			−2.00000
$490$	−1.00000			−1.00000
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	0.500000	−	0.866025i	0.500000	−	0.866025i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−1.00000			−1.00000
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$501$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$502$		0			0
$503$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$503$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$	−1.00000			−1.00000
$506$		0			0
$507$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$508$	1.00000	+	1.73205i	1.00000	+	1.73205i
$509$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$509$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$		0			0
$514$		0			0
$515$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$516$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$522$		0			0
$523$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$523$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$524$		0			0
$525$	−1.00000			−1.00000
$526$	−1.00000			−1.00000
$527$		0			0
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$535$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$536$	0.500000	+	0.866025i	0.500000	+	0.866025i
$537$		0			0
$538$	1.00000			1.00000
$539$		0			0
$540$	−1.00000			−1.00000
$541$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$541$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$542$		0			0
$543$	0.500000	+	0.866025i	0.500000	+	0.866025i
$544$		0			0
$545$	−1.00000			−1.00000
$546$		0			0
$547$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$547$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−1.00000			−1.00000
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$	0.500000	−	0.866025i	0.500000	−	0.866025i
$561$		0			0
$562$	1.00000	−	1.73205i	1.00000	−	1.73205i
$563$	−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			$\pi$
$564$	1.00000	+	1.73205i	1.00000	+	1.73205i
$565$		0			0
$566$	2.00000			2.00000
$567$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$568$		0			0
$569$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$569$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$	−1.00000			−1.00000
$575$	1.00000			1.00000
$576$		0			0
$577$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$577$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$578$	0.500000	+	0.866025i	0.500000	+	0.866025i
$579$		0			0
$580$	−1.00000			−1.00000
$581$	0.500000	+	0.866025i	0.500000	+	0.866025i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2.00000			−2.00000			−1.00000			$\pi$
$587$	−2.00000			−2.00000			−1.00000			$\pi$
$588$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$	−1.00000			−1.00000
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	0.500000	−	0.866025i	0.500000	−	0.866025i
$601$	2.00000			2.00000			1.00000			$0$
$601$	2.00000			2.00000			1.00000			$0$
$602$	1.00000			1.00000
$603$		0			0
$604$		0			0
$605$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$606$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$607$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$607$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$608$		0			0
$609$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$610$	1.00000			1.00000
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	0.500000	−	0.866025i	0.500000	−	0.866025i
$615$	−1.00000			−1.00000
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	0.500000	−	0.866025i	0.500000	−	0.866025i
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$	0.500000	−	0.866025i	0.500000	−	0.866025i
$622$		0			0
$623$	1.00000			1.00000
$624$		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
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.00000	−	1.73205i	1.00000	−	1.73205i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.500000	+	0.866025i	0.500000	+	0.866025i
$641$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$641$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$642$	0.500000	−	0.866025i	0.500000	−	0.866025i
$643$	−2.00000			−2.00000			−1.00000			$\pi$
$643$	−2.00000			−2.00000			−1.00000			$\pi$
$644$	0.500000	+	0.866025i	0.500000	+	0.866025i
$645$	1.00000			1.00000
$646$		0			0
$647$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$647$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$648$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−2.00000			−2.00000
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$655$		0			0
$656$	0.500000	−	0.866025i	0.500000	−	0.866025i
$657$		0			0
$658$	1.00000	−	1.73205i	1.00000	−	1.73205i
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$661$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$	−1.00000			−1.00000
$665$		0			0
$666$		0			0
$667$	0.500000	−	0.866025i	0.500000	−	0.866025i
$668$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$669$	1.00000	+	1.73205i	1.00000	+	1.73205i
$670$	0.500000	−	0.866025i	0.500000	−	0.866025i
$671$		0			0
$672$	1.00000			1.00000
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	0.500000	+	0.866025i	0.500000	+	0.866025i
$676$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	1.00000	−	1.73205i	1.00000	−	1.73205i
$682$		0			0
$683$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$683$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$687$	2.00000			2.00000
$688$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$689$		0			0
$690$	0.500000	+	0.866025i	0.500000	+	0.866025i
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$	−1.00000			−1.00000
$695$		0			0
$696$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$697$		0			0
$698$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$699$		0			0
$700$	−1.00000			−1.00000
$701$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$701$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	1.00000	−	1.73205i	1.00000	−	1.73205i
$706$		0			0
$707$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$708$		0			0
$709$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$709$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$710$		0			0
$711$		0			0
$712$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$	−1.00000			−1.00000
$722$	−1.00000			−1.00000
$723$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$724$	0.500000	+	0.866025i	0.500000	+	0.866025i
$725$	0.500000	+	0.866025i	0.500000	+	0.866025i
$726$	0.500000	−	0.866025i	0.500000	−	0.866025i
$727$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$727$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$	0.500000	+	0.866025i	0.500000	+	0.866025i
$733$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$733$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$734$	−1.00000			−1.00000
$735$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$736$	−1.00000			−1.00000
$737$		0			0
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$743$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$	0.500000	+	0.866025i	0.500000	+	0.866025i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.00000			−1.00000
$750$	−1.00000			−1.00000
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	1.00000	+	1.73205i	1.00000	+	1.73205i
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$761$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$762$	2.00000			2.00000
$763$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$764$		0			0
$765$		0			0
$766$	0.500000	+	0.866025i	0.500000	+	0.866025i
$767$		0			0
$768$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$769$	2.00000			2.00000			1.00000			$0$
$769$	2.00000			2.00000			1.00000			$0$
$770$		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			0
$776$		0			0
$777$		0			0
$778$	−2.00000			−2.00000
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	1.00000			1.00000
$784$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$785$		0			0
$786$		0			0
$787$	−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
$789$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	0.500000	−	0.866025i	0.500000	−	0.866025i
$801$		0			0
$802$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$803$		0			0
$804$	1.00000			1.00000
$805$	0.500000	−	0.866025i	0.500000	−	0.866025i
$806$		0			0
$807$	0.500000	−	0.866025i	0.500000	−	0.866025i
$808$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$809$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$809$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$810$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$813$		0			0
$814$		0			0
$815$	1.00000	+	1.73205i	1.00000	+	1.73205i
$816$		0			0
$817$		0			0
$818$	1.00000			1.00000
$819$		0			0
$820$	−1.00000			−1.00000
$821$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$821$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$823$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$824$	0.500000	−	0.866025i	0.500000	−	0.866025i
$825$		0			0
$826$		0			0
$827$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$827$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$828$		0			0
$829$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$829$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$830$	0.500000	+	0.866025i	0.500000	+	0.866025i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$841$		0			0
$842$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$843$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$844$		0			0
$845$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$846$		0			0
$847$	−1.00000			−1.00000
$848$		0			0
$849$	1.00000	−	1.73205i	1.00000	−	1.73205i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$	0.500000	−	0.866025i	0.500000	−	0.866025i
$855$		0			0
$856$	0.500000	−	0.866025i	0.500000	−	0.866025i
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$	1.00000			1.00000
$861$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$862$		0			0
$863$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$863$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$864$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$865$		0			0
$866$		0			0
$867$	1.00000			1.00000
$868$		0			0
$869$		0			0
$870$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$871$		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			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$	−2.00000			−2.00000			−1.00000			$\pi$
$883$	−2.00000			−2.00000			−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	0.500000	+	0.866025i	0.500000	+	0.866025i
$887$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$887$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$888$		0			0
$889$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$890$	1.00000			1.00000
$891$		0			0
$892$	1.00000	+	1.73205i	1.00000	+	1.73205i
$893$		0			0
$894$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$895$		0			0
$896$	1.00000			1.00000
$897$		0			0
$898$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	0.500000	−	0.866025i	0.500000	−	0.866025i
$904$		0			0
$905$	0.500000	−	0.866025i	0.500000	−	0.866025i
$906$		0			0
$907$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$907$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$908$	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			0
$913$		0			0
$914$		0			0
$915$	0.500000	−	0.866025i	0.500000	−	0.866025i
$916$	2.00000			2.00000
$917$		0			0
$918$		0			0
$919$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$919$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$920$	0.500000	+	0.866025i	0.500000	+	0.866025i
$921$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$922$	1.00000	−	1.73205i	1.00000	−	1.73205i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	0.500000	−	0.866025i	0.500000	−	0.866025i
$927$		0			0
$928$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$929$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$929$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	0.500000	+	0.866025i	0.500000	+	0.866025i
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$939$		0			0
$940$	1.00000	−	1.73205i	1.00000	−	1.73205i
$941$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$941$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$942$		0			0
$943$	0.500000	−	0.866025i	0.500000	−	0.866025i
$944$		0			0
$945$	1.00000			1.00000
$946$		0			0
$947$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$947$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	1.00000			1.00000
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$965$		0			0
$966$	1.00000			1.00000
$967$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$967$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$968$	0.500000	−	0.866025i	0.500000	−	0.866025i
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$		0			0
$974$	2.00000			2.00000
$975$		0			0
$976$	0.500000	+	0.866025i	0.500000	+	0.866025i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$979$		0			0
$980$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$981$		0			0
$982$		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			$\pi$
$984$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$985$		0			0
$986$		0			0
$987$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$988$		0			0
$989$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	140.1.p.b.79.1	yes	2
3.2	odd	2		1260.1.ci.a.919.1		2
4.3	odd	2		140.1.p.a.79.1	yes	2
5.2	odd	4		700.1.u.a.51.2		4
5.3	odd	4		700.1.u.a.51.1		4
5.4	even	2		140.1.p.a.79.1	yes	2
7.2	even	3		980.1.f.b.99.1		1
7.3	odd	6		980.1.p.b.459.1		2
7.4	even	3	inner	140.1.p.b.39.1	yes	2
7.5	odd	6		980.1.f.a.99.1		1
7.6	odd	2		980.1.p.b.79.1		2
8.3	odd	2		2240.1.bt.a.639.1		2
8.5	even	2		2240.1.bt.b.639.1		2
12.11	even	2		1260.1.ci.b.919.1		2
15.14	odd	2		1260.1.ci.b.919.1		2
20.3	even	4		700.1.u.a.51.2		4
20.7	even	4		700.1.u.a.51.1		4
20.19	odd	2	CM	140.1.p.b.79.1	yes	2
21.11	odd	6		1260.1.ci.a.739.1		2
28.3	even	6		980.1.p.a.459.1		2
28.11	odd	6		140.1.p.a.39.1	✓	2
28.19	even	6		980.1.f.d.99.1		1
28.23	odd	6		980.1.f.c.99.1		1
28.27	even	2		980.1.p.a.79.1		2
35.4	even	6		140.1.p.a.39.1	✓	2
35.9	even	6		980.1.f.c.99.1		1
35.18	odd	12		700.1.u.a.151.2		4
35.19	odd	6		980.1.f.d.99.1		1
35.24	odd	6		980.1.p.a.459.1		2
35.32	odd	12		700.1.u.a.151.1		4
35.34	odd	2		980.1.p.a.79.1		2
40.19	odd	2		2240.1.bt.b.639.1		2
40.29	even	2		2240.1.bt.a.639.1		2
56.11	odd	6		2240.1.bt.a.319.1		2
56.53	even	6		2240.1.bt.b.319.1		2
60.59	even	2		1260.1.ci.a.919.1		2
84.11	even	6		1260.1.ci.b.739.1		2
105.74	odd	6		1260.1.ci.b.739.1		2
140.19	even	6		980.1.f.a.99.1		1
140.39	odd	6	inner	140.1.p.b.39.1	yes	2
140.59	even	6		980.1.p.b.459.1		2
140.67	even	12		700.1.u.a.151.2		4
140.79	odd	6		980.1.f.b.99.1		1
140.123	even	12		700.1.u.a.151.1		4
140.139	even	2		980.1.p.b.79.1		2
280.109	even	6		2240.1.bt.a.319.1		2
280.179	odd	6		2240.1.bt.b.319.1		2
420.179	even	6		1260.1.ci.a.739.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.1.p.a.39.1	✓	2	28.11	odd	6
140.1.p.a.39.1	✓	2	35.4	even	6
140.1.p.a.79.1	yes	2	4.3	odd	2
140.1.p.a.79.1	yes	2	5.4	even	2
140.1.p.b.39.1	yes	2	7.4	even	3	inner
140.1.p.b.39.1	yes	2	140.39	odd	6	inner
140.1.p.b.79.1	yes	2	1.1	even	1	trivial
140.1.p.b.79.1	yes	2	20.19	odd	2	CM
700.1.u.a.51.1		4	5.3	odd	4
700.1.u.a.51.1		4	20.7	even	4
700.1.u.a.51.2		4	5.2	odd	4
700.1.u.a.51.2		4	20.3	even	4
700.1.u.a.151.1		4	35.32	odd	12
700.1.u.a.151.1		4	140.123	even	12
700.1.u.a.151.2		4	35.18	odd	12
700.1.u.a.151.2		4	140.67	even	12
980.1.f.a.99.1		1	7.5	odd	6
980.1.f.a.99.1		1	140.19	even	6
980.1.f.b.99.1		1	7.2	even	3
980.1.f.b.99.1		1	140.79	odd	6
980.1.f.c.99.1		1	28.23	odd	6
980.1.f.c.99.1		1	35.9	even	6
980.1.f.d.99.1		1	28.19	even	6
980.1.f.d.99.1		1	35.19	odd	6
980.1.p.a.79.1		2	28.27	even	2
980.1.p.a.79.1		2	35.34	odd	2
980.1.p.a.459.1		2	28.3	even	6
980.1.p.a.459.1		2	35.24	odd	6
980.1.p.b.79.1		2	7.6	odd	2
980.1.p.b.79.1		2	140.139	even	2
980.1.p.b.459.1		2	7.3	odd	6
980.1.p.b.459.1		2	140.59	even	6
1260.1.ci.a.739.1		2	21.11	odd	6
1260.1.ci.a.739.1		2	420.179	even	6
1260.1.ci.a.919.1		2	3.2	odd	2
1260.1.ci.a.919.1		2	60.59	even	2
1260.1.ci.b.739.1		2	84.11	even	6
1260.1.ci.b.739.1		2	105.74	odd	6
1260.1.ci.b.919.1		2	12.11	even	2
1260.1.ci.b.919.1		2	15.14	odd	2
2240.1.bt.a.319.1		2	56.11	odd	6
2240.1.bt.a.319.1		2	280.109	even	6
2240.1.bt.a.639.1		2	8.3	odd	2
2240.1.bt.a.639.1		2	40.29	even	2
2240.1.bt.b.319.1		2	56.53	even	6
2240.1.bt.b.319.1		2	280.179	odd	6
2240.1.bt.b.639.1		2	8.5	even	2
2240.1.bt.b.639.1		2	40.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	140.1.p.b.79.1

Level	$140$
Weight	$1$
Character	140.79
Analytic conductor	$0.070$
Analytic rank	$0$
Dimension	$2$
Projective image	$D_{3}$
CM discriminant	-20
Inner twists	$4$