LMFDB - Embedded newform 285.1.n.b.254.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [285,1,Mod(239,285)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(285, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("285.239");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 285 = 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	285.n (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:	$0.142233528600$
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 3.1.5415.1
Artin image:	$C_3\times S_3$
Artin field:	Galois closure of 6.0.1218375.1

Embedding invariants

Embedding label			254.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	285.254
Dual form			285.1.n.b.239.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{6} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{6} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{10} +(-0.500000 + 0.866025i) q^{15} +(0.500000 + 0.866025i) q^{16} +(0.500000 + 0.866025i) q^{17} -1.00000 q^{18} +(-0.500000 - 0.866025i) q^{19} +(-1.00000 + 1.73205i) q^{23} +(-0.500000 - 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{27} -1.00000 q^{30} -1.00000 q^{31} +(-0.500000 + 0.866025i) q^{34} +(0.500000 - 0.866025i) q^{38} +(-0.500000 - 0.866025i) q^{40} +1.00000 q^{45} -2.00000 q^{46} +(0.500000 - 0.866025i) q^{47} +(0.500000 - 0.866025i) q^{48} +1.00000 q^{49} -1.00000 q^{50} +(0.500000 - 0.866025i) q^{51} +(0.500000 - 0.866025i) q^{53} +(0.500000 + 0.866025i) q^{54} +(-0.500000 + 0.866025i) q^{57} +(-1.00000 + 1.73205i) q^{61} +(-0.500000 - 0.866025i) q^{62} +1.00000 q^{64} +2.00000 q^{69} +(-0.500000 + 0.866025i) q^{72} +1.00000 q^{75} +(-1.00000 - 1.73205i) q^{79} +(0.500000 - 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} -1.00000 q^{83} +(0.500000 - 0.866025i) q^{85} +(0.500000 + 0.866025i) q^{90} +(0.500000 + 0.866025i) q^{93} +1.00000 q^{94} +(-0.500000 + 0.866025i) q^{95} +(0.500000 + 0.866025i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{3} - q^{5} + q^{6} + 2 q^{8} - q^{9}+O(q^{10}) $ 2 * q + q^2 - q^3 - q^5 + q^6 + 2 * q^8 - q^9	$ 2 q + q^{2} - q^{3} - q^{5} + q^{6} + 2 q^{8} - q^{9} + q^{10} - q^{15} + q^{16} + q^{17} - 2 q^{18} - q^{19} - 2 q^{23} - q^{24} - q^{25} + 2 q^{27} - 2 q^{30} - 2 q^{31} - q^{34} + q^{38} - q^{40} + 2 q^{45} - 4 q^{46} + q^{47} + q^{48} + 2 q^{49} - 2 q^{50} + q^{51} + q^{53} + q^{54} - q^{57} - 2 q^{61} - q^{62} + 2 q^{64} + 4 q^{69} - q^{72} + 2 q^{75} - 2 q^{79} + q^{80} - q^{81} - 2 q^{83} + q^{85} + q^{90} + q^{93} + 2 q^{94} - q^{95} + q^{98}+O(q^{100}) $ 2 * q + q^2 - q^3 - q^5 + q^6 + 2 * q^8 - q^9 + q^10 - q^15 + q^16 + q^17 - 2 * q^18 - q^19 - 2 * q^23 - q^24 - q^25 + 2 * q^27 - 2 * q^30 - 2 * q^31 - q^34 + q^38 - q^40 + 2 * q^45 - 4 * q^46 + q^47 + q^48 + 2 * q^49 - 2 * q^50 + q^51 + q^53 + q^54 - q^57 - 2 * q^61 - q^62 + 2 * q^64 + 4 * q^69 - q^72 + 2 * q^75 - 2 * q^79 + q^80 - q^81 - 2 * q^83 + q^85 + q^90 + q^93 + 2 * q^94 - q^95 + q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/285\mathbb{Z}\right)^\times$.

$n$	$172$	$191$	$211$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{3}\right)$

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	1.00000			$0$
$2$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$3$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$3$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$4$		0			0
$5$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$6$	0.500000	−	0.866025i	0.500000	−	0.866025i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$	1.00000			1.00000
$9$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$10$	0.500000	−	0.866025i	0.500000	−	0.866025i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$16$	0.500000	+	0.866025i	0.500000	+	0.866025i
$17$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$17$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$18$	−1.00000			−1.00000
$19$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$19$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$23$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$24$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$	1.00000			1.00000
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$	−1.00000			−1.00000
$31$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$31$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$32$		0			0
$33$		0			0
$34$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	0.500000	−	0.866025i	0.500000	−	0.866025i
$39$		0			0
$40$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$		0			0
$45$	1.00000			1.00000
$46$	−2.00000			−2.00000
$47$	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
$49$	1.00000			1.00000
$50$	−1.00000			−1.00000
$51$	0.500000	−	0.866025i	0.500000	−	0.866025i
$52$		0			0
$53$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$53$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$54$	0.500000	+	0.866025i	0.500000	+	0.866025i
$55$		0			0
$56$		0			0
$57$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−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	$0.666667\pi$
$62$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$63$		0			0
$64$	1.00000			1.00000
$65$		0			0
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$	2.00000			2.00000
$70$		0			0
$71$		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			0
$75$	1.00000			1.00000
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$79$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$80$	0.500000	−	0.866025i	0.500000	−	0.866025i
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$83$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$	0.500000	−	0.866025i	0.500000	−	0.866025i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$	0.500000	+	0.866025i	0.500000	+	0.866025i
$91$		0			0
$92$		0			0
$93$	0.500000	+	0.866025i	0.500000	+	0.866025i
$94$	1.00000			1.00000
$95$	−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	$0.333333\pi$
$98$	0.500000	+	0.866025i	0.500000	+	0.866025i
$99$		0			0
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$	1.00000			1.00000
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	1.00000			1.00000
$107$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$107$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$109$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$113$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$114$	−1.00000			−1.00000
$115$	2.00000			2.00000
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$121$	1.00000			1.00000
$122$	−2.00000			−2.00000
$123$		0			0
$124$		0			0
$125$	1.00000			1.00000
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$128$	0.500000	+	0.866025i	0.500000	+	0.866025i
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$136$	0.500000	+	0.866025i	0.500000	+	0.866025i
$137$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$137$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$138$	1.00000	+	1.73205i	1.00000	+	1.73205i
$139$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$139$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$140$		0			0
$141$	−1.00000			−1.00000
$142$		0			0
$143$		0			0
$144$	−1.00000			−1.00000
$145$		0			0
$146$		0			0
$147$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$	0.500000	+	0.866025i	0.500000	+	0.866025i
$151$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$151$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$152$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$153$	−1.00000			−1.00000
$154$		0			0
$155$	0.500000	+	0.866025i	0.500000	+	0.866025i
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$	1.00000	−	1.73205i	1.00000	−	1.73205i
$159$	−1.00000			−1.00000
$160$		0			0
$161$		0			0
$162$	0.500000	−	0.866025i	0.500000	−	0.866025i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$167$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$167$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$168$		0			0
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$	1.00000			1.00000
$171$	1.00000			1.00000
$172$		0			0
$173$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$173$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$181$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$182$		0			0
$183$	2.00000			2.00000
$184$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$185$		0			0
$186$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−1.00000			−1.00000
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$197$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$199$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−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			0
$206$		0			0
$207$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$208$		0			0
$209$		0			0
$210$		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			0
$213$		0			0
$214$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$215$		0			0
$216$	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
$223$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$223$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$224$		0			0
$225$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$226$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$227$	2.00000			2.00000			1.00000			$0$
$227$	2.00000			2.00000			1.00000			$0$
$228$		0			0
$229$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$229$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$230$	1.00000	+	1.73205i	1.00000	+	1.73205i
$231$		0			0
$232$		0			0
$233$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$233$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$	−1.00000			−1.00000
$236$		0			0
$237$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$	−1.00000			−1.00000
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$241$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$242$	0.500000	+	0.866025i	0.500000	+	0.866025i
$243$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$244$		0			0
$245$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$246$		0			0
$247$		0			0
$248$	−1.00000			−1.00000
$249$	0.500000	+	0.866025i	0.500000	+	0.866025i
$250$	0.500000	+	0.866025i	0.500000	+	0.866025i
$251$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$251$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−1.00000			−1.00000
$256$		0			0
$257$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$257$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$263$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$	−1.00000			−1.00000
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$	0.500000	−	0.866025i	0.500000	−	0.866025i
$271$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$271$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$272$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$273$		0			0
$274$	1.00000			1.00000
$275$		0			0
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$	1.00000			1.00000
$279$	0.500000	−	0.866025i	0.500000	−	0.866025i
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$	1.00000			1.00000
$286$		0			0
$287$		0			0
$288$		0			0
$289$		0			0
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$293$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$294$	0.500000	−	0.866025i	0.500000	−	0.866025i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$303$		0			0
$304$	0.500000	−	0.866025i	0.500000	−	0.866025i
$305$	2.00000			2.00000
$306$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$307$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$307$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$308$		0			0
$309$		0			0
$310$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$311$		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			0
$315$		0			0
$316$		0			0
$317$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$317$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$318$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$319$		0			0
$320$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$321$	0.500000	+	0.866025i	0.500000	+	0.866025i
$322$		0			0
$323$	0.500000	−	0.866025i	0.500000	−	0.866025i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	0.500000	−	0.866025i	0.500000	−	0.866025i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.00000			2.00000			1.00000			$0$
$331$	2.00000			2.00000			1.00000			$0$
$332$		0			0
$333$		0			0
$334$	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	$0.333333\pi$
$338$	0.500000	−	0.866025i	0.500000	−	0.866025i
$339$	0.500000	+	0.866025i	0.500000	+	0.866025i
$340$		0			0
$341$		0			0
$342$	0.500000	+	0.866025i	0.500000	+	0.866025i
$343$		0			0
$344$		0			0
$345$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$346$	1.00000	−	1.73205i	1.00000	−	1.73205i
$347$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$347$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$349$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.00000			2.00000			1.00000			$0$
$353$	2.00000			2.00000			1.00000			$0$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	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
$362$	1.00000			1.00000
$363$	−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			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$	−2.00000			−2.00000
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$376$	0.500000	−	0.866025i	0.500000	−	0.866025i
$377$		0			0
$378$		0			0
$379$	2.00000			2.00000			1.00000			$0$
$379$	2.00000			2.00000			1.00000			$0$
$380$		0			0
$381$		0			0
$382$		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			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$	−2.00000			−2.00000
$392$	1.00000			1.00000
$393$		0			0
$394$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$395$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$396$		0			0
$397$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$397$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$398$	−2.00000			−2.00000
$399$		0			0
$400$	−1.00000			−1.00000
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$406$		0			0
$407$		0			0
$408$	0.500000	−	0.866025i	0.500000	−	0.866025i
$409$	−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	$0.666667\pi$
$410$		0			0
$411$	−1.00000			−1.00000
$412$		0			0
$413$		0			0
$414$	1.00000	−	1.73205i	1.00000	−	1.73205i
$415$	0.500000	+	0.866025i	0.500000	+	0.866025i
$416$		0			0
$417$	−1.00000			−1.00000
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−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	$-0.666667\pi$
$422$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$423$	0.500000	+	0.866025i	0.500000	+	0.866025i
$424$	0.500000	−	0.866025i	0.500000	−	0.866025i
$425$	−1.00000			−1.00000
$426$		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	$0.666667\pi$
$432$	0.500000	+	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			0
$436$		0			0
$437$	2.00000			2.00000
$438$		0			0
$439$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$439$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$442$		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
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	0.500000	−	0.866025i	0.500000	−	0.866025i
$451$		0			0
$452$		0			0
$453$	0.500000	+	0.866025i	0.500000	+	0.866025i
$454$	1.00000	+	1.73205i	1.00000	+	1.73205i
$455$		0			0
$456$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$459$	0.500000	+	0.866025i	0.500000	+	0.866025i
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$	0.500000	−	0.866025i	0.500000	−	0.866025i
$466$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$467$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$467$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$		0			0
$470$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−2.00000			−2.00000
$475$	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
$481$		0			0
$482$	1.00000			1.00000
$483$		0			0
$484$		0			0
$485$		0			0
$486$	−1.00000			−1.00000
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$489$		0			0
$490$	0.500000	−	0.866025i	0.500000	−	0.866025i
$491$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$491$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$497$		0			0
$498$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$499$	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			0
$501$	−1.00000			−1.00000
$502$		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
$505$		0			0
$506$		0			0
$507$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$511$		0			0
$512$	1.00000			1.00000
$513$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$514$	−2.00000			−2.00000
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$527$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$528$		0			0
$529$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$530$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	0.500000	+	0.866025i	0.500000	+	0.866025i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$541$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$542$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$543$	−1.00000			−1.00000
$544$		0			0
$545$	0.500000	−	0.866025i	0.500000	−	0.866025i
$546$		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
$549$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$550$		0			0
$551$		0			0
$552$	2.00000			2.00000
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$557$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$558$	1.00000			1.00000
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−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			0
$565$	0.500000	+	0.866025i	0.500000	+	0.866025i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$	0.500000	+	0.866025i	0.500000	+	0.866025i
$571$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$576$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$587$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$587$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$	0.500000	+	0.866025i	0.500000	+	0.866025i
$590$		0			0
$591$	0.500000	+	0.866025i	0.500000	+	0.866025i
$592$		0			0
$593$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$593$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	2.00000			2.00000
$598$		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
$601$	2.00000			2.00000			1.00000			$0$
$601$	2.00000			2.00000			1.00000			$0$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$	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	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$617$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$618$		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
$621$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$631$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$632$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$633$	0.500000	−	0.866025i	0.500000	−	0.866025i
$634$	1.00000			1.00000
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.500000	−	0.866025i	0.500000	−	0.866025i
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$643$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$		0			0
$645$		0			0
$646$	1.00000			1.00000
$647$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$648$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.00000			2.00000			1.00000			$0$
$653$	2.00000			2.00000			1.00000			$0$
$654$	1.00000			1.00000
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$662$	1.00000	+	1.73205i	1.00000	+	1.73205i
$663$		0			0
$664$	−1.00000			−1.00000
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		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
$675$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$676$		0			0
$677$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$677$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$678$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$679$		0			0
$680$	0.500000	−	0.866025i	0.500000	−	0.866025i
$681$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$682$		0			0
$683$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$683$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$	−1.00000			−1.00000
$686$		0			0
$687$	0.500000	+	0.866025i	0.500000	+	0.866025i
$688$		0			0
$689$		0			0
$690$	1.00000	−	1.73205i	1.00000	−	1.73205i
$691$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$691$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$695$	−1.00000			−1.00000
$696$		0			0
$697$		0			0
$698$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$699$	0.500000	−	0.866025i	0.500000	−	0.866025i
$700$		0			0
$701$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$701$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	0.500000	+	0.866025i	0.500000	+	0.866025i
$706$	1.00000	+	1.73205i	1.00000	+	1.73205i
$707$		0			0
$708$		0			0
$709$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$709$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$710$		0			0
$711$	2.00000			2.00000
$712$		0			0
$713$	1.00000	−	1.73205i	1.00000	−	1.73205i
$714$		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	$0.333333\pi$
$720$	0.500000	+	0.866025i	0.500000	+	0.866025i
$721$		0			0
$722$	−1.00000			−1.00000
$723$	−1.00000			−1.00000
$724$		0			0
$725$		0			0
$726$	0.500000	−	0.866025i	0.500000	−	0.866025i
$727$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$727$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$739$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$743$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$744$	0.500000	+	0.866025i	0.500000	+	0.866025i
$745$		0			0
$746$		0			0
$747$	0.500000	−	0.866025i	0.500000	−	0.866025i
$748$		0			0
$749$		0			0
$750$	0.500000	−	0.866025i	0.500000	−	0.866025i
$751$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$751$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$752$	1.00000			1.00000
$753$		0			0
$754$		0			0
$755$	0.500000	+	0.866025i	0.500000	+	0.866025i
$756$		0			0
$757$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$757$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$758$	1.00000	+	1.73205i	1.00000	+	1.73205i
$759$		0			0
$760$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.500000	+	0.866025i	0.500000	+	0.866025i
$766$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$767$		0			0
$768$		0			0
$769$	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
$771$	2.00000			2.00000
$772$		0			0
$773$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$773$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	0.500000	−	0.866025i	0.500000	−	0.866025i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$783$		0			0
$784$	0.500000	+	0.866025i	0.500000	+	0.866025i
$785$		0			0
$786$		0			0
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		0			0
$789$	0.500000	−	0.866025i	0.500000	−	0.866025i
$790$	−2.00000			−2.00000
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	0.500000	+	0.866025i	0.500000	+	0.866025i
$796$		0			0
$797$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$797$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$	1.00000			1.00000
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	−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
$813$	0.500000	−	0.866025i	0.500000	−	0.866025i
$814$		0			0
$815$		0			0
$816$	1.00000			1.00000
$817$		0			0
$818$	−2.00000			−2.00000
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$827$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$828$		0			0
$829$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$829$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$830$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$831$		0			0
$832$		0			0
$833$	0.500000	+	0.866025i	0.500000	+	0.866025i
$834$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$835$	−1.00000			−1.00000
$836$		0			0
$837$	−1.00000			−1.00000
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$	1.00000	−	1.73205i	1.00000	−	1.73205i
$843$		0			0
$844$		0			0
$845$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$846$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$847$		0			0
$848$	1.00000			1.00000
$849$		0			0
$850$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$851$		0			0
$852$		0			0
$853$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$853$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$854$		0			0
$855$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$856$	−1.00000			−1.00000
$857$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$857$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$858$		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
$863$	2.00000			2.00000			1.00000			$0$
$863$	2.00000			2.00000			1.00000			$0$
$864$		0			0
$865$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.500000	+	0.866025i	0.500000	+	0.866025i
$873$		0			0
$874$	1.00000	+	1.73205i	1.00000	+	1.73205i
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$879$	0.500000	+	0.866025i	0.500000	+	0.866025i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.00000			−1.00000
$883$		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
$886$	1.00000			1.00000
$887$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$887$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.00000			−1.00000
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	1.00000			1.00000
$902$		0			0
$903$		0			0
$904$	−1.00000			−1.00000
$905$	−1.00000			−1.00000
$906$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$907$		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
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−1.00000			−1.00000
$913$		0			0
$914$		0			0
$915$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$916$		0			0
$917$		0			0
$918$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$919$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$919$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$920$	2.00000			2.00000
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$	1.00000			1.00000
$931$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$932$		0			0
$933$		0			0
$934$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$935$		0			0
$936$		0			0
$937$		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
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$947$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$948$		0			0
$949$		0			0
$950$	0.500000	+	0.866025i	0.500000	+	0.866025i
$951$	−1.00000			−1.00000
$952$		0			0
$953$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$953$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$954$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$961$		0			0
$962$		0			0
$963$	0.500000	−	0.866025i	0.500000	−	0.866025i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$	1.00000			1.00000
$969$	−1.00000			−1.00000
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−2.00000			−2.00000
$977$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$977$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.00000			−1.00000
$982$		0			0
$983$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$983$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$	0.500000	+	0.866025i	0.500000	+	0.866025i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$991$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$994$		0			0
$995$	2.00000			2.00000
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	−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	285.1.n.b.254.1	yes	2
3.2	odd	2		285.1.n.a.254.1	yes	2
5.2	odd	4		1425.1.t.b.26.1		4
5.3	odd	4		1425.1.t.b.26.2		4
5.4	even	2		285.1.n.a.254.1	yes	2
15.2	even	4		1425.1.t.b.26.2		4
15.8	even	4		1425.1.t.b.26.1		4
15.14	odd	2	CM	285.1.n.b.254.1	yes	2
19.11	even	3	inner	285.1.n.b.239.1	yes	2
57.11	odd	6		285.1.n.a.239.1	✓	2
95.49	even	6		285.1.n.a.239.1	✓	2
95.68	odd	12		1425.1.t.b.1151.1		4
95.87	odd	12		1425.1.t.b.1151.2		4
285.68	even	12		1425.1.t.b.1151.2		4
285.182	even	12		1425.1.t.b.1151.1		4
285.239	odd	6	inner	285.1.n.b.239.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.1.n.a.239.1	✓	2	57.11	odd	6
285.1.n.a.239.1	✓	2	95.49	even	6
285.1.n.a.254.1	yes	2	3.2	odd	2
285.1.n.a.254.1	yes	2	5.4	even	2
285.1.n.b.239.1	yes	2	19.11	even	3	inner
285.1.n.b.239.1	yes	2	285.239	odd	6	inner
285.1.n.b.254.1	yes	2	1.1	even	1	trivial
285.1.n.b.254.1	yes	2	15.14	odd	2	CM
1425.1.t.b.26.1		4	5.2	odd	4
1425.1.t.b.26.1		4	15.8	even	4
1425.1.t.b.26.2		4	5.3	odd	4
1425.1.t.b.26.2		4	15.2	even	4
1425.1.t.b.1151.1		4	95.68	odd	12
1425.1.t.b.1151.1		4	285.182	even	12
1425.1.t.b.1151.2		4	95.87	odd	12
1425.1.t.b.1151.2		4	285.68	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 285 = 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	285.n (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{6} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{6} +1.00000 q^{8} +(-0.500000 + 0.866025i) q^{9} +(0.500000 - 0.866025i) q^{10} +(-0.500000 + 0.866025i) q^{15} +(0.500000 + 0.866025i) q^{16} +(0.500000 + 0.866025i) q^{17} -1.00000 q^{18} +(-0.500000 - 0.866025i) q^{19} +(-1.00000 + 1.73205i) q^{23} +(-0.500000 - 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{27} -1.00000 q^{30} -1.00000 q^{31} +(-0.500000 + 0.866025i) q^{34} +(0.500000 - 0.866025i) q^{38} +(-0.500000 - 0.866025i) q^{40} +1.00000 q^{45} -2.00000 q^{46} +(0.500000 - 0.866025i) q^{47} +(0.500000 - 0.866025i) q^{48} +1.00000 q^{49} -1.00000 q^{50} +(0.500000 - 0.866025i) q^{51} +(0.500000 - 0.866025i) q^{53} +(0.500000 + 0.866025i) q^{54} +(-0.500000 + 0.866025i) q^{57} +(-1.00000 + 1.73205i) q^{61} +(-0.500000 - 0.866025i) q^{62} +1.00000 q^{64} +2.00000 q^{69} +(-0.500000 + 0.866025i) q^{72} +1.00000 q^{75} +(-1.00000 - 1.73205i) q^{79} +(0.500000 - 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} -1.00000 q^{83} +(0.500000 - 0.866025i) q^{85} +(0.500000 + 0.866025i) q^{90} +(0.500000 + 0.866025i) q^{93} +1.00000 q^{94} +(-0.500000 + 0.866025i) q^{95} +(0.500000 + 0.866025i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{3} - q^{5} + q^{6} + 2 q^{8} - q^{9}+O(q^{10}) \) 2 * q + q^2 - q^3 - q^5 + q^6 + 2 * q^8 - q^9	\( 2 q + q^{2} - q^{3} - q^{5} + q^{6} + 2 q^{8} - q^{9} + q^{10} - q^{15} + q^{16} + q^{17} - 2 q^{18} - q^{19} - 2 q^{23} - q^{24} - q^{25} + 2 q^{27} - 2 q^{30} - 2 q^{31} - q^{34} + q^{38} - q^{40} + 2 q^{45} - 4 q^{46} + q^{47} + q^{48} + 2 q^{49} - 2 q^{50} + q^{51} + q^{53} + q^{54} - q^{57} - 2 q^{61} - q^{62} + 2 q^{64} + 4 q^{69} - q^{72} + 2 q^{75} - 2 q^{79} + q^{80} - q^{81} - 2 q^{83} + q^{85} + q^{90} + q^{93} + 2 q^{94} - q^{95} + q^{98}+O(q^{100}) \) 2 * q + q^2 - q^3 - q^5 + q^6 + 2 * q^8 - q^9 + q^10 - q^15 + q^16 + q^17 - 2 * q^18 - q^19 - 2 * q^23 - q^24 - q^25 + 2 * q^27 - 2 * q^30 - 2 * q^31 - q^34 + q^38 - q^40 + 2 * q^45 - 4 * q^46 + q^47 + q^48 + 2 * q^49 - 2 * q^50 + q^51 + q^53 + q^54 - q^57 - 2 * q^61 - q^62 + 2 * q^64 + 4 * q^69 - q^72 + 2 * q^75 - 2 * q^79 + q^80 - q^81 - 2 * q^83 + q^85 + q^90 + q^93 + 2 * q^94 - q^95 + q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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