LMFDB - Embedded newform 1340.1.t.a.439.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1340,1,Mod(439,1340)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1340, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1340.439");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1340 = 2^{2} \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1340.t (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:	$0.668747116928$
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 3.1.89780.1
Artin image:	$C_3\times S_3$
Artin field:	Galois closure of 6.0.35912000.2

Embedding invariants

Embedding label			439.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1340.439
Dual form			1340.1.t.a.699.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 - 0.866025i) q^{2} -1.00000 q^{3} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{5} +(0.500000 + 0.866025i) q^{6} +(0.500000 - 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+(-0.500000 - 0.866025i) q^{2} -1.00000 q^{3} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{5} +(0.500000 + 0.866025i) q^{6} +(0.500000 - 0.866025i) q^{7} +1.00000 q^{8} +(-0.500000 - 0.866025i) q^{10} +(0.500000 - 0.866025i) q^{12} -1.00000 q^{14} -1.00000 q^{15} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{20} +(-0.500000 + 0.866025i) q^{21} +(0.500000 + 0.866025i) q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +(0.500000 + 0.866025i) q^{28} +(0.500000 - 0.866025i) q^{29} +(0.500000 + 0.866025i) q^{30} +(-0.500000 + 0.866025i) q^{32} +(0.500000 - 0.866025i) q^{35} +1.00000 q^{40} +(0.500000 - 0.866025i) q^{41} +1.00000 q^{42} -1.00000 q^{43} +(0.500000 - 0.866025i) q^{46} +(0.500000 - 0.866025i) q^{47} +(0.500000 + 0.866025i) q^{48} +(-0.500000 - 0.866025i) q^{50} +(-0.500000 - 0.866025i) q^{54} +(0.500000 - 0.866025i) q^{56} -1.00000 q^{58} +(0.500000 - 0.866025i) q^{60} +(-1.00000 - 1.73205i) q^{61} +1.00000 q^{64} +(-0.500000 - 0.866025i) q^{67} +(-0.500000 - 0.866025i) q^{69} -1.00000 q^{70} -1.00000 q^{75} +(-0.500000 - 0.866025i) q^{80} -1.00000 q^{81} -1.00000 q^{82} +(0.500000 + 0.866025i) q^{83} +(-0.500000 - 0.866025i) q^{84} +(0.500000 + 0.866025i) q^{86} +(-0.500000 + 0.866025i) q^{87} +2.00000 q^{89} -1.00000 q^{92} -1.00000 q^{94} +(0.500000 - 0.866025i) q^{96} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q - q^2 - 2 * q^3 - q^4 + 2 * q^5 + q^6 + q^7 + 2 * q^8	$ 2 q - q^{2} - 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{8} - q^{10} + q^{12} - 2 q^{14} - 2 q^{15} - q^{16} - q^{20} - q^{21} + q^{23} - 2 q^{24} + 2 q^{25} + 2 q^{27} + q^{28} + q^{29} + q^{30} - q^{32} + q^{35} + 2 q^{40} + q^{41} + 2 q^{42} - 2 q^{43} + q^{46} + q^{47} + q^{48} - q^{50} - q^{54} + q^{56} - 2 q^{58} + q^{60} - 2 q^{61} + 2 q^{64} - q^{67} - q^{69} - 2 q^{70} - 2 q^{75} - q^{80} - 2 q^{81} - 2 q^{82} + q^{83} - q^{84} + q^{86} - q^{87} + 4 q^{89} - 2 q^{92} - 2 q^{94} + q^{96}+O(q^{100}) $ 2 * q - q^2 - 2 * q^3 - q^4 + 2 * q^5 + q^6 + q^7 + 2 * q^8 - q^10 + q^12 - 2 * q^14 - 2 * q^15 - q^16 - q^20 - q^21 + q^23 - 2 * q^24 + 2 * q^25 + 2 * q^27 + q^28 + q^29 + q^30 - q^32 + q^35 + 2 * q^40 + q^41 + 2 * q^42 - 2 * q^43 + q^46 + q^47 + q^48 - q^50 - q^54 + q^56 - 2 * q^58 + q^60 - 2 * q^61 + 2 * q^64 - q^67 - q^69 - 2 * q^70 - 2 * q^75 - q^80 - 2 * q^81 - 2 * q^82 + q^83 - q^84 + q^86 - q^87 + 4 * q^89 - 2 * q^92 - 2 * q^94 + q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1340\mathbb{Z}\right)^\times$.

$n$	$537$	$671$	$1141$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$

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
$3$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$4$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$5$	1.00000			1.00000
$5$	1.00000			1.00000
$6$	0.500000	+	0.866025i	0.500000	+	0.866025i
$7$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$7$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$8$	1.00000			1.00000
$9$		0			0
$10$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	0.500000	−	0.866025i	0.500000	−	0.866025i
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$	−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	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$21$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$22$		0			0
$23$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$23$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$24$	−1.00000			−1.00000
$25$	1.00000			1.00000
$26$		0			0
$27$	1.00000			1.00000
$28$	0.500000	+	0.866025i	0.500000	+	0.866025i
$29$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$29$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$30$	0.500000	+	0.866025i	0.500000	+	0.866025i
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$33$		0			0
$34$		0			0
$35$	0.500000	−	0.866025i	0.500000	−	0.866025i
$36$		0			0
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$		0			0
$40$	1.00000			1.00000
$41$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$41$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$42$	1.00000			1.00000
$43$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$43$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$		0			0
$46$	0.500000	−	0.866025i	0.500000	−	0.866025i
$47$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$47$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$48$	0.500000	+	0.866025i	0.500000	+	0.866025i
$49$		0			0
$50$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$55$		0			0
$56$	0.500000	−	0.866025i	0.500000	−	0.866025i
$57$		0			0
$58$	−1.00000			−1.00000
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$	0.500000	−	0.866025i	0.500000	−	0.866025i
$61$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$61$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$62$		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
$67$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$68$		0			0
$69$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$70$	−1.00000			−1.00000
$71$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$71$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$72$		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
$75$	−1.00000			−1.00000
$76$		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
$81$	−1.00000			−1.00000
$82$	−1.00000			−1.00000
$83$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$83$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$84$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$85$		0			0
$86$	0.500000	+	0.866025i	0.500000	+	0.866025i
$87$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$88$		0			0
$89$	2.00000			2.00000			1.00000			$0$
$89$	2.00000			2.00000			1.00000			$0$
$90$		0			0
$91$		0			0
$92$	−1.00000			−1.00000
$93$		0			0
$94$	−1.00000			−1.00000
$95$		0			0
$96$	0.500000	−	0.866025i	0.500000	−	0.866025i
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$		0			0
$99$		0			0
$100$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$101$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$101$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$103$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$104$		0			0
$105$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$106$		0			0
$107$	2.00000			2.00000			1.00000			$0$
$107$	2.00000			2.00000			1.00000			$0$
$108$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$109$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$109$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$	−1.00000			−1.00000
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		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
$122$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$123$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$124$		0			0
$125$	1.00000			1.00000
$126$		0			0
$127$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$127$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$128$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$129$	1.00000			1.00000
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$135$	1.00000			1.00000
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	−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
$141$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	0.500000	−	0.866025i	0.500000	−	0.866025i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$149$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$150$	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
$160$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$161$	1.00000			1.00000
$162$	0.500000	+	0.866025i	0.500000	+	0.866025i
$163$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$163$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$164$	0.500000	+	0.866025i	0.500000	+	0.866025i
$165$		0			0
$166$	0.500000	−	0.866025i	0.500000	−	0.866025i
$167$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$167$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$168$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$	0.500000	−	0.866025i	0.500000	−	0.866025i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$	1.00000			1.00000
$175$	0.500000	−	0.866025i	0.500000	−	0.866025i
$176$		0			0
$177$		0			0
$178$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$181$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$	1.00000	+	1.73205i	1.00000	+	1.73205i
$184$	0.500000	+	0.866025i	0.500000	+	0.866025i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	0.500000	+	0.866025i	0.500000	+	0.866025i
$189$	0.500000	−	0.866025i	0.500000	−	0.866025i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−1.00000			−1.00000
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$	1.00000			1.00000
$201$	0.500000	+	0.866025i	0.500000	+	0.866025i
$202$	2.00000			2.00000
$203$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$204$		0			0
$205$	0.500000	−	0.866025i	0.500000	−	0.866025i
$206$	−1.00000			−1.00000
$207$		0			0
$208$		0			0
$209$		0			0
$210$	1.00000			1.00000
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$		0			0
$214$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$215$	−1.00000			−1.00000
$216$	1.00000			1.00000
$217$		0			0
$218$	0.500000	+	0.866025i	0.500000	+	0.866025i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	2.00000			2.00000			1.00000			$0$
$223$	2.00000			2.00000			1.00000			$0$
$224$	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	$0.666667\pi$
$227$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$229$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$230$	0.500000	−	0.866025i	0.500000	−	0.866025i
$231$		0			0
$232$	0.500000	−	0.866025i	0.500000	−	0.866025i
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$	0.500000	−	0.866025i	0.500000	−	0.866025i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$	0.500000	+	0.866025i	0.500000	+	0.866025i
$241$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$241$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$242$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$243$		0			0
$244$	2.00000			2.00000
$245$		0			0
$246$	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		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$		0			0
$253$		0			0
$254$	−1.00000			−1.00000
$255$		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	$0.666667\pi$
$258$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$263$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−2.00000			−2.00000
$268$	1.00000			1.00000
$269$	2.00000			2.00000			1.00000			$0$
$269$	2.00000			2.00000			1.00000			$0$
$270$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	1.00000			1.00000
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$	0.500000	−	0.866025i	0.500000	−	0.866025i
$281$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$281$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$282$	1.00000			1.00000
$283$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$283$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$284$		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
$290$	−1.00000			−1.00000
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	0.500000	+	0.866025i	0.500000	+	0.866025i
$299$		0			0
$300$	0.500000	−	0.866025i	0.500000	−	0.866025i
$301$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$302$		0			0
$303$	1.00000	−	1.73205i	1.00000	−	1.73205i
$304$		0			0
$305$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$306$		0			0
$307$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$307$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$308$		0			0
$309$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$	1.00000			1.00000
$321$	−2.00000			−2.00000
$322$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$323$		0			0
$324$	0.500000	−	0.866025i	0.500000	−	0.866025i
$325$		0			0
$326$	2.00000			2.00000
$327$	1.00000			1.00000
$328$	0.500000	−	0.866025i	0.500000	−	0.866025i
$329$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$330$		0			0
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$	−1.00000			−1.00000
$333$		0			0
$334$	2.00000			2.00000
$335$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$336$	1.00000			1.00000
$337$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$337$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$338$	1.00000			1.00000
$339$		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
$347$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$347$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$348$	−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	$0.666667\pi$
$350$	−1.00000			−1.00000
$351$		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	+	1.73205i	−1.00000	+	1.73205i
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$362$	2.00000			2.00000
$363$	0.500000	+	0.866025i	0.500000	+	0.866025i
$364$		0			0
$365$		0			0
$366$	1.00000	−	1.73205i	1.00000	−	1.73205i
$367$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$367$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$368$	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
$375$	−1.00000			−1.00000
$376$	0.500000	−	0.866025i	0.500000	−	0.866025i
$377$		0			0
$378$	−1.00000			−1.00000
$379$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$379$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$380$		0			0
$381$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$382$		0			0
$383$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$383$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$384$	0.500000	+	0.866025i	0.500000	+	0.866025i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$389$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$401$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$402$	0.500000	−	0.866025i	0.500000	−	0.866025i
$403$		0			0
$404$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$405$	−1.00000			−1.00000
$406$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$407$		0			0
$408$		0			0
$409$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$409$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$410$	−1.00000			−1.00000
$411$		0			0
$412$	0.500000	+	0.866025i	0.500000	+	0.866025i
$413$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$421$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.00000			−2.00000
$428$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$429$		0			0
$430$	0.500000	+	0.866025i	0.500000	+	0.866025i
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$433$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$433$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$436$	0.500000	−	0.866025i	0.500000	−	0.866025i
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$443$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$444$		0			0
$445$	2.00000			2.00000
$446$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$447$	1.00000			1.00000
$448$	0.500000	−	0.866025i	0.500000	−	0.866025i
$449$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$449$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$450$		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	$0.666667\pi$
$458$	0.500000	−	0.866025i	0.500000	−	0.866025i
$459$		0			0
$460$	−1.00000			−1.00000
$461$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$461$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$463$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$464$	−1.00000			−1.00000
$465$		0			0
$466$		0			0
$467$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$467$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$	−1.00000			−1.00000
$470$	−1.00000			−1.00000
$471$		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
$482$	0.500000	+	0.866025i	0.500000	+	0.866025i
$483$	−1.00000			−1.00000
$484$	1.00000			1.00000
$485$		0			0
$486$		0			0
$487$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$487$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$488$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$489$	1.00000	−	1.73205i	1.00000	−	1.73205i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$501$	1.00000	−	1.73205i	1.00000	−	1.73205i
$502$		0			0
$503$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$503$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$504$		0			0
$505$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$506$		0			0
$507$	0.500000	−	0.866025i	0.500000	−	0.866025i
$508$	0.500000	+	0.866025i	0.500000	+	0.866025i
$509$	2.00000			2.00000			1.00000			$0$
$509$	2.00000			2.00000			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			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$523$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$526$	0.500000	+	0.866025i	0.500000	+	0.866025i
$527$		0			0
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	1.00000	+	1.73205i	1.00000	+	1.73205i
$535$	2.00000			2.00000
$536$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$537$		0			0
$538$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$539$		0			0
$540$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$541$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$541$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$	1.00000	−	1.73205i	1.00000	−	1.73205i
$544$		0			0
$545$	−1.00000			−1.00000
$546$		0			0
$547$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$547$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		0			0
$560$	−1.00000			−1.00000
$561$		0			0
$562$	0.500000	−	0.866025i	0.500000	−	0.866025i
$563$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$563$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$564$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$565$		0			0
$566$	0.500000	+	0.866025i	0.500000	+	0.866025i
$567$	−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	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$575$	0.500000	+	0.866025i	0.500000	+	0.866025i
$576$		0			0
$577$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$577$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$578$	1.00000			1.00000
$579$		0			0
$580$	0.500000	+	0.866025i	0.500000	+	0.866025i
$581$	1.00000			1.00000
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$587$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$	0.500000	−	0.866025i	0.500000	−	0.866025i
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$	−1.00000			−1.00000
$601$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$601$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$602$	1.00000			1.00000
$603$		0			0
$604$		0			0
$605$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$606$	−2.00000			−2.00000
$607$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$607$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$608$		0			0
$609$	0.500000	+	0.866025i	0.500000	+	0.866025i
$610$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$615$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$	1.00000			1.00000
$619$		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	−	1.73205i	1.00000	−	1.73205i
$624$		0			0
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$631$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	0.500000	−	0.866025i	0.500000	−	0.866025i
$636$		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	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$642$	1.00000	+	1.73205i	1.00000	+	1.73205i
$643$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$643$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$644$	−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	1.00000			$0$
$647$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$648$	−1.00000			−1.00000
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$655$		0			0
$656$	−1.00000			−1.00000
$657$		0			0
$658$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	2.00000			2.00000			1.00000			$0$
$661$	2.00000			2.00000			1.00000			$0$
$662$		0			0
$663$		0			0
$664$	0.500000	+	0.866025i	0.500000	+	0.866025i
$665$		0			0
$666$		0			0
$667$	1.00000			1.00000
$668$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$669$	−2.00000			−2.00000
$670$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$671$		0			0
$672$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	1.00000			1.00000
$676$	−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
$683$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$683$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$684$		0			0
$685$		0			0
$686$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$687$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$688$	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
$694$	2.00000			2.00000
$695$		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
$700$	0.500000	+	0.866025i	0.500000	+	0.866025i
$701$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$701$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$706$		0			0
$707$	1.00000	+	1.73205i	1.00000	+	1.73205i
$708$		0			0
$709$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$709$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	2.00000			2.00000
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$722$	1.00000			1.00000
$723$	1.00000			1.00000
$724$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$725$	0.500000	−	0.866025i	0.500000	−	0.866025i
$726$	0.500000	−	0.866025i	0.500000	−	0.866025i
$727$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$727$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$	−2.00000			−2.00000
$733$		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			0
$736$	−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	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$744$		0			0
$745$	−1.00000			−1.00000
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.00000	−	1.73205i	1.00000	−	1.73205i
$750$	0.500000	+	0.866025i	0.500000	+	0.866025i
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$	−1.00000			−1.00000
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.500000	+	0.866025i	0.500000	+	0.866025i
$757$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$757$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	2.00000			2.00000			1.00000			$0$
$761$	2.00000			2.00000			1.00000			$0$
$762$	1.00000			1.00000
$763$	−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
$769$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$769$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	0.500000	−	0.866025i	0.500000	−	0.866025i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	0.500000	−	0.866025i	0.500000	−	0.866025i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$787$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$788$		0			0
$789$	1.00000			1.00000
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$797$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$798$		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
$805$	1.00000			1.00000
$806$		0			0
$807$	−2.00000			−2.00000
$808$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$809$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$809$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$810$	0.500000	+	0.866025i	0.500000	+	0.866025i
$811$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$811$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$812$	1.00000			1.00000
$813$		0			0
$814$		0			0
$815$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$816$		0			0
$817$		0			0
$818$	2.00000			2.00000
$819$		0			0
$820$	0.500000	+	0.866025i	0.500000	+	0.866025i
$821$	−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	1.00000			$0$
$823$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$824$	0.500000	−	0.866025i	0.500000	−	0.866025i
$825$		0			0
$826$		0			0
$827$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$827$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$828$		0			0
$829$	2.00000			2.00000			1.00000			$0$
$829$	2.00000			2.00000			1.00000			$0$
$830$	0.500000	−	0.866025i	0.500000	−	0.866025i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$841$		0			0
$842$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$843$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$844$		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			1.00000
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$	1.00000	+	1.73205i	1.00000	+	1.73205i
$855$		0			0
$856$	2.00000			2.00000
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$	0.500000	−	0.866025i	0.500000	−	0.866025i
$861$	0.500000	+	0.866025i	0.500000	+	0.866025i
$862$		0			0
$863$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$863$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$864$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$865$		0			0
$866$		0			0
$867$	0.500000	−	0.866025i	0.500000	−	0.866025i
$868$		0			0
$869$		0			0
$870$	1.00000			1.00000
$871$		0			0
$872$	−1.00000			−1.00000
$873$		0			0
$874$		0			0
$875$	0.500000	−	0.866025i	0.500000	−	0.866025i
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$881$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$883$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$884$		0			0
$885$		0			0
$886$	−1.00000			−1.00000
$887$	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
$889$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$890$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$891$		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
$898$	−1.00000			−1.00000
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	0.500000	−	0.866025i	0.500000	−	0.866025i
$904$		0			0
$905$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$906$		0			0
$907$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$907$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−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			0
$913$		0			0
$914$		0			0
$915$	1.00000	+	1.73205i	1.00000	+	1.73205i
$916$	−1.00000			−1.00000
$917$		0			0
$918$		0			0
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$	0.500000	+	0.866025i	0.500000	+	0.866025i
$921$	1.00000	+	1.73205i	1.00000	+	1.73205i
$922$	0.500000	+	0.866025i	0.500000	+	0.866025i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$927$		0			0
$928$	0.500000	+	0.866025i	0.500000	+	0.866025i
$929$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$929$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$930$		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
$940$	0.500000	+	0.866025i	0.500000	+	0.866025i
$941$	2.00000			2.00000			1.00000			$0$
$941$	2.00000			2.00000			1.00000			$0$
$942$		0			0
$943$	1.00000			1.00000
$944$		0			0
$945$	0.500000	−	0.866025i	0.500000	−	0.866025i
$946$		0			0
$947$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$947$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$948$		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
$964$	0.500000	−	0.866025i	0.500000	−	0.866025i
$965$		0			0
$966$	0.500000	+	0.866025i	0.500000	+	0.866025i
$967$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$967$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$968$	−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
$974$	0.500000	−	0.866025i	0.500000	−	0.866025i
$975$		0			0
$976$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$	−2.00000			−2.00000
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	2.00000			2.00000			1.00000			$0$
$983$	2.00000			2.00000			1.00000			$0$
$984$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$985$		0			0
$986$		0			0
$987$	0.500000	+	0.866025i	0.500000	+	0.866025i
$988$		0			0
$989$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$	1.00000			1.00000
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1340.1.t.a.439.1	✓	2
4.3	odd	2		1340.1.t.b.439.1	yes	2
5.4	even	2		1340.1.t.b.439.1	yes	2
20.19	odd	2	CM	1340.1.t.a.439.1	✓	2
67.29	even	3	inner	1340.1.t.a.699.1	yes	2
268.163	odd	6		1340.1.t.b.699.1	yes	2
335.29	even	6		1340.1.t.b.699.1	yes	2
1340.699	odd	6	inner	1340.1.t.a.699.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1340.1.t.a.439.1	✓	2	1.1	even	1	trivial
1340.1.t.a.439.1	✓	2	20.19	odd	2	CM
1340.1.t.a.699.1	yes	2	67.29	even	3	inner
1340.1.t.a.699.1	yes	2	1340.699	odd	6	inner
1340.1.t.b.439.1	yes	2	4.3	odd	2
1340.1.t.b.439.1	yes	2	5.4	even	2
1340.1.t.b.699.1	yes	2	268.163	odd	6
1340.1.t.b.699.1	yes	2	335.29	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:	\( N \)	\(=\)	\( 1340 = 2^{2} \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1340.t (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{2} -1.00000 q^{3} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{5} +(0.500000 + 0.866025i) q^{6} +(0.500000 - 0.866025i) q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+(-0.500000 - 0.866025i) q^{2} -1.00000 q^{3} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{5} +(0.500000 + 0.866025i) q^{6} +(0.500000 - 0.866025i) q^{7} +1.00000 q^{8} +(-0.500000 - 0.866025i) q^{10} +(0.500000 - 0.866025i) q^{12} -1.00000 q^{14} -1.00000 q^{15} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{20} +(-0.500000 + 0.866025i) q^{21} +(0.500000 + 0.866025i) q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +(0.500000 + 0.866025i) q^{28} +(0.500000 - 0.866025i) q^{29} +(0.500000 + 0.866025i) q^{30} +(-0.500000 + 0.866025i) q^{32} +(0.500000 - 0.866025i) q^{35} +1.00000 q^{40} +(0.500000 - 0.866025i) q^{41} +1.00000 q^{42} -1.00000 q^{43} +(0.500000 - 0.866025i) q^{46} +(0.500000 - 0.866025i) q^{47} +(0.500000 + 0.866025i) q^{48} +(-0.500000 - 0.866025i) q^{50} +(-0.500000 - 0.866025i) q^{54} +(0.500000 - 0.866025i) q^{56} -1.00000 q^{58} +(0.500000 - 0.866025i) q^{60} +(-1.00000 - 1.73205i) q^{61} +1.00000 q^{64} +(-0.500000 - 0.866025i) q^{67} +(-0.500000 - 0.866025i) q^{69} -1.00000 q^{70} -1.00000 q^{75} +(-0.500000 - 0.866025i) q^{80} -1.00000 q^{81} -1.00000 q^{82} +(0.500000 + 0.866025i) q^{83} +(-0.500000 - 0.866025i) q^{84} +(0.500000 + 0.866025i) q^{86} +(-0.500000 + 0.866025i) q^{87} +2.00000 q^{89} -1.00000 q^{92} -1.00000 q^{94} +(0.500000 - 0.866025i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q - q^2 - 2 * q^3 - q^4 + 2 * q^5 + q^6 + q^7 + 2 * q^8	\( 2 q - q^{2} - 2 q^{3} - q^{4} + 2 q^{5} + q^{6} + q^{7} + 2 q^{8} - q^{10} + q^{12} - 2 q^{14} - 2 q^{15} - q^{16} - q^{20} - q^{21} + q^{23} - 2 q^{24} + 2 q^{25} + 2 q^{27} + q^{28} + q^{29} + q^{30} - q^{32} + q^{35} + 2 q^{40} + q^{41} + 2 q^{42} - 2 q^{43} + q^{46} + q^{47} + q^{48} - q^{50} - q^{54} + q^{56} - 2 q^{58} + q^{60} - 2 q^{61} + 2 q^{64} - q^{67} - q^{69} - 2 q^{70} - 2 q^{75} - q^{80} - 2 q^{81} - 2 q^{82} + q^{83} - q^{84} + q^{86} - q^{87} + 4 q^{89} - 2 q^{92} - 2 q^{94} + q^{96}+O(q^{100}) \) 2 * q - q^2 - 2 * q^3 - q^4 + 2 * q^5 + q^6 + q^7 + 2 * q^8 - q^10 + q^12 - 2 * q^14 - 2 * q^15 - q^16 - q^20 - q^21 + q^23 - 2 * q^24 + 2 * q^25 + 2 * q^27 + q^28 + q^29 + q^30 - q^32 + q^35 + 2 * q^40 + q^41 + 2 * q^42 - 2 * q^43 + q^46 + q^47 + q^48 - q^50 - q^54 + q^56 - 2 * q^58 + q^60 - 2 * q^61 + 2 * q^64 - q^67 - q^69 - 2 * q^70 - 2 * q^75 - q^80 - 2 * q^81 - 2 * q^82 + q^83 - q^84 + q^86 - q^87 + 4 * q^89 - 2 * q^92 - 2 * q^94 + q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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