LMFDB - Embedded newform 760.1.bm.b.539.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [760,1,Mod(539,760)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("760.539"); S:= CuspForms(chi, 1); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(760, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 3, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]

Level:	$ N $	$=$	$ 760 = 2^{3} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	760.bm (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.379289409601$
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 3.1.14440.1
Artin image:	$C_6\times S_3$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			539.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	760.539
Dual form			760.1.bm.b.619.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +1.00000 q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.500000 + 0.866025i) q^{10} -1.00000 q^{11} +(1.00000 - 1.73205i) q^{13} +(0.500000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} -1.00000 q^{18} +(-0.500000 - 0.866025i) q^{19} -1.00000 q^{20} +(-0.500000 - 0.866025i) q^{22} +(-0.500000 + 0.866025i) q^{23} +(-0.500000 + 0.866025i) q^{25} +2.00000 q^{26} +(-0.500000 + 0.866025i) q^{28} +(0.500000 - 0.866025i) q^{32} +(0.500000 + 0.866025i) q^{35} +(-0.500000 - 0.866025i) q^{36} +1.00000 q^{37} +(0.500000 - 0.866025i) q^{38} +(-0.500000 - 0.866025i) q^{40} +(0.500000 + 0.866025i) q^{41} +(0.500000 - 0.866025i) q^{44} -1.00000 q^{45} -1.00000 q^{46} +(1.00000 - 1.73205i) q^{47} -1.00000 q^{50} +(1.00000 + 1.73205i) q^{52} +(-0.500000 + 0.866025i) q^{53} +(-0.500000 - 0.866025i) q^{55} -1.00000 q^{56} +(-1.00000 - 1.73205i) q^{59} +(-0.500000 + 0.866025i) q^{63} +1.00000 q^{64} +2.00000 q^{65} +(-0.500000 + 0.866025i) q^{70} +(0.500000 - 0.866025i) q^{72} +(0.500000 + 0.866025i) q^{74} +1.00000 q^{76} -1.00000 q^{77} +(0.500000 - 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.500000 + 0.866025i) q^{82} +1.00000 q^{88} +(0.500000 - 0.866025i) q^{89} +(-0.500000 - 0.866025i) q^{90} +(1.00000 - 1.73205i) q^{91} +(-0.500000 - 0.866025i) q^{92} +2.00000 q^{94} +(0.500000 - 0.866025i) q^{95} +(0.500000 - 0.866025i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} + q^{5} + 2 q^{7} - 2 q^{8} - q^{9} - q^{10} - 2 q^{11} + 2 q^{13} + q^{14} - q^{16} - 2 q^{18} - q^{19} - 2 q^{20} - q^{22} - q^{23} - q^{25} + 4 q^{26} - q^{28} + q^{32}+ \cdots + q^{99}+O(q^{100}) $ 2 * q + q^2 - q^4 + q^5 + 2 * q^7 - 2 * q^8 - q^9 - q^10 - 2 * q^11 + 2 * q^13 + q^14 - q^16 - 2 * q^18 - q^19 - 2 * q^20 - q^22 - q^23 - q^25 + 4 * q^26 - q^28 + q^32 + q^35 - q^36 + 2 * q^37 + q^38 - q^40 + q^41 + q^44 - 2 * q^45 - 2 * q^46 + 2 * q^47 - 2 * q^50 + 2 * q^52 - q^53 - q^55 - 2 * q^56 - 2 * q^59 - q^63 + 2 * q^64 + 4 * q^65 - q^70 + q^72 + q^74 + 2 * q^76 - 2 * q^77 + q^80 - q^81 - q^82 + 2 * q^88 + q^89 - q^90 + 2 * q^91 - q^92 + 4 * q^94 + q^95 + q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/760\mathbb{Z}\right)^\times$.

$n$	$191$	$381$	$401$	$457$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$	$-1$

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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$3$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$4$	−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
$6$		0			0
$7$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$7$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$8$	−1.00000			−1.00000
$9$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$10$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$11$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$13$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$14$	0.500000	+	0.866025i	0.500000	+	0.866025i
$15$		0			0
$16$	−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	$0.333333\pi$
$18$	−1.00000			−1.00000
$19$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$19$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$20$	−1.00000			−1.00000
$21$		0			0
$22$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$23$	−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			0
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$	2.00000			2.00000
$27$		0			0
$28$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	0.500000	−	0.866025i	0.500000	−	0.866025i
$33$		0			0
$34$		0			0
$35$	0.500000	+	0.866025i	0.500000	+	0.866025i
$36$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$37$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$37$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$38$	0.500000	−	0.866025i	0.500000	−	0.866025i
$39$		0			0
$40$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$41$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$41$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$	0.500000	−	0.866025i	0.500000	−	0.866025i
$45$	−1.00000			−1.00000
$46$	−1.00000			−1.00000
$47$	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	$-0.333333\pi$
$48$		0			0
$49$		0			0
$50$	−1.00000			−1.00000
$51$		0			0
$52$	1.00000	+	1.73205i	1.00000	+	1.73205i
$53$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$53$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$54$		0			0
$55$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$56$	−1.00000			−1.00000
$57$		0			0
$58$		0			0
$59$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$59$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$64$	1.00000			1.00000
$65$	2.00000			2.00000
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$	0.500000	−	0.866025i	0.500000	−	0.866025i
$73$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$73$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$74$	0.500000	+	0.866025i	0.500000	+	0.866025i
$75$		0			0
$76$	1.00000			1.00000
$77$	−1.00000			−1.00000
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$	0.500000	−	0.866025i	0.500000	−	0.866025i
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	1.00000			1.00000
$89$	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			$0$
$90$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$91$	1.00000	−	1.73205i	1.00000	−	1.73205i
$92$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$93$		0			0
$94$	2.00000			2.00000
$95$	0.500000	−	0.866025i	0.500000	−	0.866025i
$96$		0			0
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$		0			0
$99$	0.500000	−	0.866025i	0.500000	−	0.866025i
$100$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$103$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$104$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$105$		0			0
$106$	−1.00000			−1.00000
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$109$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$110$	0.500000	−	0.866025i	0.500000	−	0.866025i
$111$		0			0
$112$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$	−1.00000			−1.00000
$116$		0			0
$117$	1.00000	+	1.73205i	1.00000	+	1.73205i
$118$	1.00000	−	1.73205i	1.00000	−	1.73205i
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−1.00000
$126$	−1.00000			−1.00000
$127$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$127$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$128$	0.500000	+	0.866025i	0.500000	+	0.866025i
$129$		0			0
$130$	1.00000	+	1.73205i	1.00000	+	1.73205i
$131$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$131$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$139$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$140$	−1.00000			−1.00000
$141$		0			0
$142$		0			0
$143$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$144$	1.00000			1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	0.500000	+	0.866025i	0.500000	+	0.866025i
$153$		0			0
$154$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$155$		0			0
$156$		0			0
$157$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$157$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$	1.00000			1.00000
$161$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$162$	0.500000	−	0.866025i	0.500000	−	0.866025i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	−1.00000			−1.00000
$165$		0			0
$166$		0			0
$167$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$167$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$168$		0			0
$169$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$170$		0			0
$171$	1.00000			1.00000
$172$		0			0
$173$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$173$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$174$		0			0
$175$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$176$	0.500000	+	0.866025i	0.500000	+	0.866025i
$177$		0			0
$178$	1.00000			1.00000
$179$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$179$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$180$	0.500000	−	0.866025i	0.500000	−	0.866025i
$181$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$181$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$182$	2.00000			2.00000
$183$		0			0
$184$	0.500000	−	0.866025i	0.500000	−	0.866025i
$185$	0.500000	+	0.866025i	0.500000	+	0.866025i
$186$		0			0
$187$		0			0
$188$	1.00000	+	1.73205i	1.00000	+	1.73205i
$189$		0			0
$190$	1.00000			1.00000
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$		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
$196$		0			0
$197$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$197$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$198$	1.00000			1.00000
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$	0.500000	−	0.866025i	0.500000	−	0.866025i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$206$	0.500000	+	0.866025i	0.500000	+	0.866025i
$207$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$208$	−2.00000			−2.00000
$209$	0.500000	+	0.866025i	0.500000	+	0.866025i
$210$		0			0
$211$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$211$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$212$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	1.00000			1.00000
$221$		0			0
$222$		0			0
$223$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$223$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$224$	0.500000	−	0.866025i	0.500000	−	0.866025i
$225$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$235$	2.00000			2.00000
$236$	2.00000			2.00000
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−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			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2.00000			−2.00000
$248$		0			0
$249$		0			0
$250$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$251$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$251$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$252$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$253$	0.500000	−	0.866025i	0.500000	−	0.866025i
$254$	−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			0
$259$	1.00000			1.00000
$260$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$261$		0			0
$262$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$263$	−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
$265$	−1.00000			−1.00000
$266$	0.500000	−	0.866025i	0.500000	−	0.866025i
$267$		0			0
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.500000	−	0.866025i	0.500000	−	0.866025i
$276$		0			0
$277$	−2.00000			−2.00000			−1.00000			$\pi$
$277$	−2.00000			−2.00000			−1.00000			$\pi$
$278$	−2.00000			−2.00000
$279$		0			0
$280$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$281$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$281$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$	−2.00000			−2.00000
$287$	0.500000	+	0.866025i	0.500000	+	0.866025i
$288$	0.500000	+	0.866025i	0.500000	+	0.866025i
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$293$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$	1.00000	−	1.73205i	1.00000	−	1.73205i
$296$	−1.00000			−1.00000
$297$		0			0
$298$		0			0
$299$	1.00000	+	1.73205i	1.00000	+	1.73205i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$305$		0			0
$306$		0			0
$307$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$307$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$308$	0.500000	−	0.866025i	0.500000	−	0.866025i
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	0.500000	−	0.866025i	0.500000	−	0.866025i
$315$	−1.00000			−1.00000
$316$		0			0
$317$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$317$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$	0.500000	+	0.866025i	0.500000	+	0.866025i
$321$		0			0
$322$	−1.00000			−1.00000
$323$		0			0
$324$	1.00000			1.00000
$325$	1.00000	+	1.73205i	1.00000	+	1.73205i
$326$		0			0
$327$		0			0
$328$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$329$	1.00000	−	1.73205i	1.00000	−	1.73205i
$330$		0			0
$331$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$331$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$334$	−1.00000			−1.00000
$335$		0			0
$336$		0			0
$337$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$337$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$338$	1.50000	−	2.59808i	1.50000	−	2.59808i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	0.500000	+	0.866025i	0.500000	+	0.866025i
$343$	−1.00000			−1.00000
$344$		0			0
$345$		0			0
$346$	0.500000	−	0.866025i	0.500000	−	0.866025i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−1.00000			−1.00000
$351$		0			0
$352$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$353$		0			0		1.00000			$0$
$353$		0			0		−1.00000			$\pi$
$354$		0			0
$355$		0			0
$356$	0.500000	+	0.866025i	0.500000	+	0.866025i
$357$		0			0
$358$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$	1.00000			1.00000
$361$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$362$		0			0
$363$		0			0
$364$	1.00000	+	1.73205i	1.00000	+	1.73205i
$365$		0			0
$366$		0			0
$367$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$367$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$368$	1.00000			1.00000
$369$	−1.00000			−1.00000
$370$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$371$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$372$		0			0
$373$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$373$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$377$		0			0
$378$		0			0
$379$	2.00000			2.00000			1.00000			$0$
$379$	2.00000			2.00000			1.00000			$0$
$380$	0.500000	+	0.866025i	0.500000	+	0.866025i
$381$		0			0
$382$		0			0
$383$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$383$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	0.500000	+	0.866025i	0.500000	+	0.866025i
$395$		0			0
$396$	0.500000	+	0.866025i	0.500000	+	0.866025i
$397$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$397$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			1.00000
$401$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	0.500000	−	0.866025i	0.500000	−	0.866025i
$406$		0			0
$407$	−1.00000			−1.00000
$408$		0			0
$409$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$409$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$410$	−1.00000			−1.00000
$411$		0			0
$412$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$413$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$414$	0.500000	−	0.866025i	0.500000	−	0.866025i
$415$		0			0
$416$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$417$		0			0
$418$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$419$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$423$	1.00000	+	1.73205i	1.00000	+	1.73205i
$424$	0.500000	−	0.866025i	0.500000	−	0.866025i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$433$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.00000			1.00000
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$	0.500000	+	0.866025i	0.500000	+	0.866025i
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$	1.00000			1.00000
$446$	0.500000	−	0.866025i	0.500000	−	0.866025i
$447$		0			0
$448$	1.00000			1.00000
$449$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$449$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$450$	0.500000	−	0.866025i	0.500000	−	0.866025i
$451$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.00000			2.00000
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$	0.500000	−	0.866025i	0.500000	−	0.866025i
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		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			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	−2.00000			−2.00000
$469$		0			0
$470$	1.00000	+	1.73205i	1.00000	+	1.73205i
$471$		0			0
$472$	1.00000	+	1.73205i	1.00000	+	1.73205i
$473$		0			0
$474$		0			0
$475$	1.00000			1.00000
$476$		0			0
$477$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	1.00000	−	1.73205i	1.00000	−	1.73205i
$482$	−2.00000			−2.00000
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$487$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$491$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$495$	1.00000			1.00000
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$499$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$500$	0.500000	−	0.866025i	0.500000	−	0.866025i
$501$		0			0
$502$	−2.00000			−2.00000
$503$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$503$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$504$	0.500000	−	0.866025i	0.500000	−	0.866025i
$505$		0			0
$506$	1.00000			1.00000
$507$		0			0
$508$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		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			0
$517$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$518$	0.500000	+	0.866025i	0.500000	+	0.866025i
$519$		0			0
$520$	−2.00000			−2.00000
$521$	2.00000			2.00000			1.00000			$0$
$521$	2.00000			2.00000			1.00000			$0$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$	−1.00000			−1.00000
$525$		0			0
$526$	0.500000	−	0.866025i	0.500000	−	0.866025i
$527$		0			0
$528$		0			0
$529$		0			0
$530$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$531$	2.00000			2.00000
$532$	1.00000			1.00000
$533$	2.00000			2.00000
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$		0			0
$549$		0			0
$550$	1.00000			1.00000
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$555$		0			0
$556$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$557$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$557$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$	0.500000	−	0.866025i	0.500000	−	0.866025i
$561$		0			0
$562$	1.00000			1.00000
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$568$		0			0
$569$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$569$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	2.00000			2.00000			1.00000			$0$
$571$	2.00000			2.00000			1.00000			$0$
$572$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$573$		0			0
$574$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$575$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$576$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−1.00000			−1.00000
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.500000	−	0.866025i	0.500000	−	0.866025i
$584$		0			0
$585$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$586$	0.500000	+	0.866025i	0.500000	+	0.866025i
$587$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$587$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$		0			0
$590$	2.00000			2.00000
$591$		0			0
$592$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$593$		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
$596$		0			0
$597$		0			0
$598$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$601$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$607$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$608$	−1.00000			−1.00000
$609$		0			0
$610$		0			0
$611$	−2.00000	−	3.46410i	−2.00000	−	3.46410i
$612$		0			0
$613$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$613$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$	1.00000			1.00000
$617$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$619$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	0.500000	−	0.866025i	0.500000	−	0.866025i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	1.00000			1.00000
$629$		0			0
$630$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$631$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$631$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$		0			0
$634$	−1.00000			−1.00000
$635$	−1.00000			−1.00000
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$641$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$642$		0			0
$643$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$645$		0			0
$646$		0			0
$647$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$647$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$648$	0.500000	+	0.866025i	0.500000	+	0.866025i
$649$	1.00000	+	1.73205i	1.00000	+	1.73205i
$650$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$651$		0			0
$652$		0			0
$653$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$653$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$656$	0.500000	−	0.866025i	0.500000	−	0.866025i
$657$		0			0
$658$	2.00000			2.00000
$659$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$659$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$663$		0			0
$664$		0			0
$665$	0.500000	−	0.866025i	0.500000	−	0.866025i
$666$	−1.00000			−1.00000
$667$		0			0
$668$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$	3.00000			3.00000
$677$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$677$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$685$		0			0
$686$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$687$		0			0
$688$		0			0
$689$	1.00000	+	1.73205i	1.00000	+	1.73205i
$690$		0			0
$691$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$691$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$692$	1.00000			1.00000
$693$	0.500000	−	0.866025i	0.500000	−	0.866025i
$694$		0			0
$695$	−2.00000			−2.00000
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$701$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$701$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$702$		0			0
$703$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$704$	−1.00000			−1.00000
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$713$		0			0
$714$		0			0
$715$	−2.00000			−2.00000
$716$	0.500000	−	0.866025i	0.500000	−	0.866025i
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$	0.500000	+	0.866025i	0.500000	+	0.866025i
$721$	1.00000			1.00000
$722$	−1.00000			−1.00000
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$727$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$728$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$733$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$734$	2.00000			2.00000
$735$		0			0
$736$	0.500000	+	0.866025i	0.500000	+	0.866025i
$737$		0			0
$738$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$739$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$739$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$740$	−1.00000			−1.00000
$741$		0			0
$742$	−1.00000			−1.00000
$743$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$743$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$	0.500000	+	0.866025i	0.500000	+	0.866025i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	−2.00000			−2.00000
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$757$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$758$	1.00000	+	1.73205i	1.00000	+	1.73205i
$759$		0			0
$760$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$761$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$761$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$767$	−4.00000			−4.00000
$768$		0			0
$769$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$769$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$770$	0.500000	−	0.866025i	0.500000	−	0.866025i
$771$		0			0
$772$		0			0
$773$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$773$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.500000	−	0.866025i	0.500000	−	0.866025i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.500000	−	0.866025i	0.500000	−	0.866025i
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$793$		0			0
$794$	0.500000	−	0.866025i	0.500000	−	0.866025i
$795$		0			0
$796$		0			0
$797$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$797$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$	0.500000	+	0.866025i	0.500000	+	0.866025i
$801$	0.500000	+	0.866025i	0.500000	+	0.866025i
$802$	1.00000	−	1.73205i	1.00000	−	1.73205i
$803$		0			0
$804$		0			0
$805$	−1.00000			−1.00000
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.00000			2.00000			1.00000			$0$
$809$	2.00000			2.00000			1.00000			$0$
$810$	1.00000			1.00000
$811$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$811$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$812$		0			0
$813$		0			0
$814$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.00000			1.00000
$819$	1.00000	+	1.73205i	1.00000	+	1.73205i
$820$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−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			$\pi$
$824$	−1.00000			−1.00000
$825$		0			0
$826$	1.00000	−	1.73205i	1.00000	−	1.73205i
$827$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$827$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$828$	1.00000			1.00000
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	1.00000	−	1.73205i	1.00000	−	1.73205i
$833$		0			0
$834$		0			0
$835$	−1.00000			−1.00000
$836$	−1.00000			−1.00000
$837$		0			0
$838$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	−1.00000			−1.00000
$845$	1.50000	−	2.59808i	1.50000	−	2.59808i
$846$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$847$		0			0
$848$	1.00000			1.00000
$849$		0			0
$850$		0			0
$851$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$852$		0			0
$853$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$853$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$	0.500000	+	0.866025i	0.500000	+	0.866025i
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$859$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$863$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$	0.500000	−	0.866025i	0.500000	−	0.866025i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	0.500000	+	0.866025i	0.500000	+	0.866025i
$875$	−1.00000			−1.00000
$876$		0			0
$877$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$877$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$881$	−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
$883$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$883$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$887$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$888$		0			0
$889$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$890$	0.500000	+	0.866025i	0.500000	+	0.866025i
$891$	0.500000	+	0.866025i	0.500000	+	0.866025i
$892$	1.00000			1.00000
$893$	−2.00000			−2.00000
$894$		0			0
$895$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$896$	0.500000	+	0.866025i	0.500000	+	0.866025i
$897$		0			0
$898$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$899$		0			0
$900$	1.00000			1.00000
$901$		0			0
$902$	0.500000	−	0.866025i	0.500000	−	0.866025i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$		0			0
$909$		0			0
$910$	1.00000	+	1.73205i	1.00000	+	1.73205i
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	0.500000	+	0.866025i	0.500000	+	0.866025i
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$	1.00000			1.00000
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$926$	0.500000	+	0.866025i	0.500000	+	0.866025i
$927$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$928$		0			0
$929$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$929$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$937$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$937$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$938$		0			0
$939$		0			0
$940$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$	−1.00000			−1.00000
$944$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$		0			0
$950$	0.500000	+	0.866025i	0.500000	+	0.866025i
$951$		0			0
$952$		0			0
$953$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$953$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$954$	0.500000	−	0.866025i	0.500000	−	0.866025i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.00000			1.00000
$962$	2.00000			2.00000
$963$		0			0
$964$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$965$		0			0
$966$		0			0
$967$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$967$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$971$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$972$		0			0
$973$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$974$	0.500000	+	0.866025i	0.500000	+	0.866025i
$975$		0			0
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$980$		0			0
$981$		0			0
$982$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$983$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$983$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$984$		0			0
$985$	0.500000	+	0.866025i	0.500000	+	0.866025i
$986$		0			0
$987$		0			0
$988$	1.00000	−	1.73205i	1.00000	−	1.73205i
$989$		0			0
$990$	0.500000	+	0.866025i	0.500000	+	0.866025i
$991$		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			0
$997$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$997$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$998$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	760.1.bm.b.539.1	yes	2
4.3	odd	2		3040.1.cc.b.1679.1		2
5.2	odd	4		3800.1.bd.f.1451.1		4
5.3	odd	4		3800.1.bd.f.1451.2		4
5.4	even	2		760.1.bm.a.539.1	✓	2
8.3	odd	2		760.1.bm.a.539.1	✓	2
8.5	even	2		3040.1.cc.a.1679.1		2
19.11	even	3	inner	760.1.bm.b.619.1	yes	2
20.19	odd	2		3040.1.cc.a.1679.1		2
40.3	even	4		3800.1.bd.f.1451.1		4
40.19	odd	2	CM	760.1.bm.b.539.1	yes	2
40.27	even	4		3800.1.bd.f.1451.2		4
40.29	even	2		3040.1.cc.b.1679.1		2
76.11	odd	6		3040.1.cc.b.239.1		2
95.49	even	6		760.1.bm.a.619.1	yes	2
95.68	odd	12		3800.1.bd.f.3051.1		4
95.87	odd	12		3800.1.bd.f.3051.2		4
152.11	odd	6		760.1.bm.a.619.1	yes	2
152.125	even	6		3040.1.cc.a.239.1		2
380.239	odd	6		3040.1.cc.a.239.1		2
760.163	even	12		3800.1.bd.f.3051.2		4
760.429	even	6		3040.1.cc.b.239.1		2
760.467	even	12		3800.1.bd.f.3051.1		4
760.619	odd	6	inner	760.1.bm.b.619.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.1.bm.a.539.1	✓	2	5.4	even	2
760.1.bm.a.539.1	✓	2	8.3	odd	2
760.1.bm.a.619.1	yes	2	95.49	even	6
760.1.bm.a.619.1	yes	2	152.11	odd	6
760.1.bm.b.539.1	yes	2	1.1	even	1	trivial
760.1.bm.b.539.1	yes	2	40.19	odd	2	CM
760.1.bm.b.619.1	yes	2	19.11	even	3	inner
760.1.bm.b.619.1	yes	2	760.619	odd	6	inner
3040.1.cc.a.239.1		2	152.125	even	6
3040.1.cc.a.239.1		2	380.239	odd	6
3040.1.cc.a.1679.1		2	8.5	even	2
3040.1.cc.a.1679.1		2	20.19	odd	2
3040.1.cc.b.239.1		2	76.11	odd	6
3040.1.cc.b.239.1		2	760.429	even	6
3040.1.cc.b.1679.1		2	4.3	odd	2
3040.1.cc.b.1679.1		2	40.29	even	2
3800.1.bd.f.1451.1		4	5.2	odd	4
3800.1.bd.f.1451.1		4	40.3	even	4
3800.1.bd.f.1451.2		4	5.3	odd	4
3800.1.bd.f.1451.2		4	40.27	even	4
3800.1.bd.f.3051.1		4	95.68	odd	12
3800.1.bd.f.3051.1		4	760.467	even	12
3800.1.bd.f.3051.2		4	95.87	odd	12
3800.1.bd.f.3051.2		4	760.163	even	12

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

Level:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	760.bm (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.500000 + 0.866025i) q^{5} +1.00000 q^{7} -1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(-0.500000 + 0.866025i) q^{10} -1.00000 q^{11} +(1.00000 - 1.73205i) q^{13} +(0.500000 + 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} -1.00000 q^{18} +(-0.500000 - 0.866025i) q^{19} -1.00000 q^{20} +(-0.500000 - 0.866025i) q^{22} +(-0.500000 + 0.866025i) q^{23} +(-0.500000 + 0.866025i) q^{25} +2.00000 q^{26} +(-0.500000 + 0.866025i) q^{28} +(0.500000 - 0.866025i) q^{32} +(0.500000 + 0.866025i) q^{35} +(-0.500000 - 0.866025i) q^{36} +1.00000 q^{37} +(0.500000 - 0.866025i) q^{38} +(-0.500000 - 0.866025i) q^{40} +(0.500000 + 0.866025i) q^{41} +(0.500000 - 0.866025i) q^{44} -1.00000 q^{45} -1.00000 q^{46} +(1.00000 - 1.73205i) q^{47} -1.00000 q^{50} +(1.00000 + 1.73205i) q^{52} +(-0.500000 + 0.866025i) q^{53} +(-0.500000 - 0.866025i) q^{55} -1.00000 q^{56} +(-1.00000 - 1.73205i) q^{59} +(-0.500000 + 0.866025i) q^{63} +1.00000 q^{64} +2.00000 q^{65} +(-0.500000 + 0.866025i) q^{70} +(0.500000 - 0.866025i) q^{72} +(0.500000 + 0.866025i) q^{74} +1.00000 q^{76} -1.00000 q^{77} +(0.500000 - 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.500000 + 0.866025i) q^{82} +1.00000 q^{88} +(0.500000 - 0.866025i) q^{89} +(-0.500000 - 0.866025i) q^{90} +(1.00000 - 1.73205i) q^{91} +(-0.500000 - 0.866025i) q^{92} +2.00000 q^{94} +(0.500000 - 0.866025i) q^{95} +(0.500000 - 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} + q^{5} + 2 q^{7} - 2 q^{8} - q^{9} - q^{10} - 2 q^{11} + 2 q^{13} + q^{14} - q^{16} - 2 q^{18} - q^{19} - 2 q^{20} - q^{22} - q^{23} - q^{25} + 4 q^{26} - q^{28} + q^{32}+ \cdots + q^{99}+O(q^{100}) \) 2 * q + q^2 - q^4 + q^5 + 2 * q^7 - 2 * q^8 - q^9 - q^10 - 2 * q^11 + 2 * q^13 + q^14 - q^16 - 2 * q^18 - q^19 - 2 * q^20 - q^22 - q^23 - q^25 + 4 * q^26 - q^28 + q^32 + q^35 - q^36 + 2 * q^37 + q^38 - q^40 + q^41 + q^44 - 2 * q^45 - 2 * q^46 + 2 * q^47 - 2 * q^50 + 2 * q^52 - q^53 - q^55 - 2 * q^56 - 2 * q^59 - q^63 + 2 * q^64 + 4 * q^65 - q^70 + q^72 + q^74 + 2 * q^76 - 2 * q^77 + q^80 - q^81 - q^82 + 2 * q^88 + q^89 - q^90 + 2 * q^91 - q^92 + 4 * q^94 + q^95 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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