LMFDB - Embedded newform 855.1.z.a.664.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,1,Mod(94,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("855.94");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	855.z (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:	$0.426700585801$
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 3.1.7695.1
Artin image:	$C_6\times S_3$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			664.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	855.664
Dual form			855.1.z.a.94.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 + 0.866025i) q^{6} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 + 0.866025i) q^{6} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +1.00000 q^{10} +(0.500000 - 0.866025i) q^{11} +(-0.500000 - 0.866025i) q^{13} -1.00000 q^{15} +(0.500000 - 0.866025i) q^{16} +1.00000 q^{18} +1.00000 q^{19} +(0.500000 + 0.866025i) q^{22} +(-0.500000 + 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{26} -1.00000 q^{27} +(0.500000 - 0.866025i) q^{30} +(-0.500000 - 0.866025i) q^{33} +1.00000 q^{37} +(-0.500000 + 0.866025i) q^{38} -1.00000 q^{39} +(0.500000 + 0.866025i) q^{40} +(-0.500000 + 0.866025i) q^{45} +(-0.500000 - 0.866025i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(-0.500000 - 0.866025i) q^{50} +1.00000 q^{53} +(0.500000 - 0.866025i) q^{54} -1.00000 q^{55} +(0.500000 - 0.866025i) q^{57} +(-1.00000 + 1.73205i) q^{61} +1.00000 q^{64} +(-0.500000 + 0.866025i) q^{65} +1.00000 q^{66} +(1.00000 + 1.73205i) q^{67} +(0.500000 + 0.866025i) q^{72} +(-0.500000 + 0.866025i) q^{74} +(0.500000 + 0.866025i) q^{75} +(0.500000 - 0.866025i) q^{78} -1.00000 q^{80} +(-0.500000 + 0.866025i) q^{81} +(-0.500000 + 0.866025i) q^{88} +(-0.500000 - 0.866025i) q^{90} +(-0.500000 - 0.866025i) q^{95} +(1.00000 - 1.73205i) q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{10} + q^{11} - q^{13} - 2 q^{15} + q^{16} + 2 q^{18} + 2 q^{19} + q^{22} - q^{24} - q^{25} + 2 q^{26} - 2 q^{27} + q^{30} - q^{33} + 2 q^{37} - q^{38} - 2 q^{39} + q^{40} - q^{45} - q^{48} - q^{49} - q^{50} + 2 q^{53} + q^{54} - 2 q^{55} + q^{57} - 2 q^{61} + 2 q^{64} - q^{65} + 2 q^{66} + 2 q^{67} + q^{72} - q^{74} + q^{75} + q^{78} - 2 q^{80} - q^{81} - q^{88} - q^{90} - q^{95} + 2 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) $ 2 * q - q^2 + q^3 - q^5 + q^6 - 2 * q^8 - q^9 + 2 * q^10 + q^11 - q^13 - 2 * q^15 + q^16 + 2 * q^18 + 2 * q^19 + q^22 - q^24 - q^25 + 2 * q^26 - 2 * q^27 + q^30 - q^33 + 2 * q^37 - q^38 - 2 * q^39 + q^40 - q^45 - q^48 - q^49 - q^50 + 2 * q^53 + q^54 - 2 * q^55 + q^57 - 2 * q^61 + 2 * q^64 - q^65 + 2 * q^66 + 2 * q^67 + q^72 - q^74 + q^75 + q^78 - 2 * q^80 - q^81 - q^88 - q^90 - q^95 + 2 * q^97 + 2 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$.

$n$	$172$	$191$	$496$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$-1$

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	0.500000	+	0.866025i	$0.333333\pi$
$2$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$3$	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		0.500000	−	0.866025i	$-0.333333\pi$
$7$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$8$	−1.00000			−1.00000
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$	1.00000			1.00000
$11$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$13$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$14$		0			0
$15$	−1.00000			−1.00000
$16$	0.500000	−	0.866025i	0.500000	−	0.866025i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	1.00000			1.00000
$19$	1.00000			1.00000
$19$	1.00000			1.00000
$20$		0			0
$21$		0			0
$22$	0.500000	+	0.866025i	0.500000	+	0.866025i
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$	1.00000			1.00000
$27$	−1.00000			−1.00000
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$	0.500000	−	0.866025i	0.500000	−	0.866025i
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$		0			0
$33$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	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
$39$	−1.00000			−1.00000
$40$	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	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$51$		0			0
$52$		0			0
$53$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$53$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$54$	0.500000	−	0.866025i	0.500000	−	0.866025i
$55$	−1.00000			−1.00000
$56$		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			0
$63$		0			0
$64$	1.00000			1.00000
$65$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$66$	1.00000			1.00000
$67$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$67$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.500000	+	0.866025i	0.500000	+	0.866025i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$75$	0.500000	+	0.866025i	0.500000	+	0.866025i
$76$		0			0
$77$		0			0
$78$	0.500000	−	0.866025i	0.500000	−	0.866025i
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$	−1.00000			−1.00000
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$96$		0			0
$97$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$97$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$98$	1.00000			1.00000
$99$	−1.00000			−1.00000
$100$		0			0
$101$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$101$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$103$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$104$	0.500000	+	0.866025i	0.500000	+	0.866025i
$105$		0			0
$106$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$107$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$107$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	0.500000	−	0.866025i	0.500000	−	0.866025i
$111$	0.500000	−	0.866025i	0.500000	−	0.866025i
$112$		0			0
$113$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$113$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$114$	0.500000	+	0.866025i	0.500000	+	0.866025i
$115$		0			0
$116$		0			0
$117$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$118$		0			0
$119$		0			0
$120$	1.00000			1.00000
$121$		0			0
$122$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$123$		0			0
$124$		0			0
$125$	1.00000			1.00000
$126$		0			0
$127$	−2.00000			−2.00000			−1.00000			$\pi$
$127$	−2.00000			−2.00000			−1.00000			$\pi$
$128$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$129$		0			0
$130$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$131$	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			0
$134$	−2.00000			−2.00000
$135$	0.500000	+	0.866025i	0.500000	+	0.866025i
$136$		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	$-0.666667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.00000			−1.00000
$144$	−1.00000			−1.00000
$145$		0			0
$146$		0			0
$147$	−1.00000			−1.00000
$148$		0			0
$149$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$149$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$150$	−1.00000			−1.00000
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$	−1.00000			−1.00000
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$	0.500000	−	0.866025i	0.500000	−	0.866025i
$160$		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
$165$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$166$		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
$169$		0			0
$170$		0			0
$171$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$172$		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			0
$176$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	1.00000	+	1.73205i	1.00000	+	1.73205i
$184$		0			0
$185$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	1.00000			1.00000
$191$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$192$	0.500000	−	0.866025i	0.500000	−	0.866025i
$193$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$193$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$194$	1.00000	+	1.73205i	1.00000	+	1.73205i
$195$	0.500000	+	0.866025i	0.500000	+	0.866025i
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	0.500000	−	0.866025i	0.500000	−	0.866025i
$199$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$199$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$200$	0.500000	−	0.866025i	0.500000	−	0.866025i
$201$	2.00000			2.00000
$202$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$203$		0			0
$204$		0			0
$205$		0			0
$206$	1.00000			1.00000
$207$		0			0
$208$	−1.00000			−1.00000
$209$	0.500000	−	0.866025i	0.500000	−	0.866025i
$210$		0			0
$211$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$211$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$212$		0			0
$213$		0			0
$214$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$215$		0			0
$216$	1.00000			1.00000
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$	0.500000	+	0.866025i	0.500000	+	0.866025i
$223$	−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			0
$225$	1.00000			1.00000
$226$	−2.00000			−2.00000
$227$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$227$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$228$		0			0
$229$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$229$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$239$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$240$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$	0.500000	+	0.866025i	0.500000	+	0.866025i
$244$		0			0
$245$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$246$		0			0
$247$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$248$		0			0
$249$		0			0
$250$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$251$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$251$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$252$		0			0
$253$		0			0
$254$	1.00000	−	1.73205i	1.00000	−	1.73205i
$255$		0			0
$256$		0			0
$257$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$257$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−1.00000			−1.00000
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$	0.500000	+	0.866025i	0.500000	+	0.866025i
$265$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	−1.00000			−1.00000
$271$	2.00000			2.00000			1.00000			$0$
$271$	2.00000			2.00000			1.00000			$0$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.500000	+	0.866025i	0.500000	+	0.866025i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	2.00000			2.00000
$279$		0			0
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$	−1.00000			−1.00000
$286$	0.500000	−	0.866025i	0.500000	−	0.866025i
$287$		0			0
$288$		0			0
$289$	1.00000			1.00000
$290$		0			0
$291$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$292$		0			0
$293$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$293$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$294$	0.500000	−	0.866025i	0.500000	−	0.866025i
$295$		0			0
$296$	−1.00000			−1.00000
$297$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$298$	−1.00000			−1.00000
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	1.00000	+	1.73205i	1.00000	+	1.73205i
$304$	0.500000	−	0.866025i	0.500000	−	0.866025i
$305$	2.00000			2.00000
$306$		0			0
$307$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$307$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$308$		0			0
$309$	−1.00000			−1.00000
$310$		0			0
$311$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$311$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$312$	1.00000			1.00000
$313$		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
$317$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$317$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$318$	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			0
$324$		0			0
$325$	1.00000			1.00000
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$331$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$331$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$334$	1.00000			1.00000
$335$	1.00000	−	1.73205i	1.00000	−	1.73205i
$336$		0			0
$337$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$337$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$338$		0			0
$339$	2.00000			2.00000
$340$		0			0
$341$		0			0
$342$	1.00000			1.00000
$343$		0			0
$344$		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
$349$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$349$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$350$		0			0
$351$	0.500000	+	0.866025i	0.500000	+	0.866025i
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$359$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$360$	0.500000	−	0.866025i	0.500000	−	0.866025i
$361$	1.00000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$	−2.00000			−2.00000
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$	1.00000			1.00000
$371$		0			0
$372$		0			0
$373$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$373$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$374$		0			0
$375$	0.500000	−	0.866025i	0.500000	−	0.866025i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$382$	0.500000	+	0.866025i	0.500000	+	0.866025i
$383$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$383$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$384$	0.500000	+	0.866025i	0.500000	+	0.866025i
$385$		0			0
$386$	1.00000			1.00000
$387$		0			0
$388$		0			0
$389$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$389$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$390$	−1.00000			−1.00000
$391$		0			0
$392$	0.500000	+	0.866025i	0.500000	+	0.866025i
$393$	1.00000			1.00000
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$	0.500000	−	0.866025i	0.500000	−	0.866025i
$399$		0			0
$400$	0.500000	+	0.866025i	0.500000	+	0.866025i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$403$		0			0
$404$		0			0
$405$	1.00000			1.00000
$406$		0			0
$407$	0.500000	−	0.866025i	0.500000	−	0.866025i
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−2.00000			−2.00000
$418$	0.500000	+	0.866025i	0.500000	+	0.866025i
$419$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$	−1.00000			−1.00000
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$433$	−2.00000			−2.00000			−1.00000			$\pi$
$433$	−2.00000			−2.00000			−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$	1.00000			1.00000
$441$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$442$		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
$445$		0			0
$446$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$447$	1.00000			1.00000
$448$		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			0
$454$	1.00000	+	1.73205i	1.00000	+	1.73205i
$455$		0			0
$456$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	−1.00000			−1.00000
$459$		0			0
$460$		0			0
$461$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$461$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		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			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$476$		0			0
$477$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$478$	−1.00000			−1.00000
$479$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$480$		0			0
$481$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.00000			−2.00000
$486$	−1.00000			−1.00000
$487$	−2.00000			−2.00000			−1.00000			$\pi$
$487$	−2.00000			−2.00000			−1.00000			$\pi$
$488$	1.00000	−	1.73205i	1.00000	−	1.73205i
$489$		0			0
$490$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$491$	0.500000	+	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			1.00000
$495$	0.500000	+	0.866025i	0.500000	+	0.866025i
$496$		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			0
$501$	−1.00000			−1.00000
$502$	0.500000	−	0.866025i	0.500000	−	0.866025i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$	2.00000			2.00000
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$	−1.00000			−1.00000
$514$	−2.00000			−2.00000
$515$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	0.500000	+	0.866025i	0.500000	+	0.866025i
$520$	0.500000	−	0.866025i	0.500000	−	0.866025i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$523$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−1.00000			−1.00000
$529$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$530$	1.00000			1.00000
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$536$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$537$		0			0
$538$		0			0
$539$	−1.00000			−1.00000
$540$		0			0
$541$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$541$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$542$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$547$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$548$		0			0
$549$	2.00000			2.00000
$550$	−1.00000			−1.00000
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−1.00000			−1.00000
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$563$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$564$		0			0
$565$	1.00000	−	1.73205i	1.00000	−	1.73205i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$	0.500000	−	0.866025i	0.500000	−	0.866025i
$571$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$571$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$574$		0			0
$575$		0			0
$576$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$579$	−1.00000			−1.00000
$580$		0			0
$581$		0			0
$582$	2.00000			2.00000
$583$	0.500000	−	0.866025i	0.500000	−	0.866025i
$584$		0			0
$585$	1.00000			1.00000
$586$	−2.00000			−2.00000
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.500000	−	0.866025i	0.500000	−	0.866025i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$595$		0			0
$596$		0			0
$597$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$	1.00000	−	1.73205i	1.00000	−	1.73205i
$604$		0			0
$605$		0			0
$606$	−2.00000			−2.00000
$607$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$607$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$608$		0			0
$609$		0			0
$610$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$	0.500000	−	0.866025i	0.500000	−	0.866025i
$619$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$619$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$620$		0			0
$621$		0			0
$622$	2.00000			2.00000
$623$		0			0
$624$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$631$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$		0			0
$634$	1.00000	+	1.73205i	1.00000	+	1.73205i
$635$	1.00000	+	1.73205i	1.00000	+	1.73205i
$636$		0			0
$637$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$638$		0			0
$639$		0			0
$640$	1.00000			1.00000
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$	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
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	0.500000	−	0.866025i	0.500000	−	0.866025i
$649$		0			0
$650$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$	0.500000	−	0.866025i	0.500000	−	0.866025i
$656$		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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.00000			1.00000
$667$		0			0
$668$		0			0
$669$	0.500000	+	0.866025i	0.500000	+	0.866025i
$670$	1.00000	+	1.73205i	1.00000	+	1.73205i
$671$	1.00000	+	1.73205i	1.00000	+	1.73205i
$672$		0			0
$673$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$673$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$674$	−2.00000			−2.00000
$675$	0.500000	−	0.866025i	0.500000	−	0.866025i
$676$		0			0
$677$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$677$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$678$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$679$		0			0
$680$		0			0
$681$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$682$		0			0
$683$	−2.00000			−2.00000			−1.00000			$\pi$
$683$	−2.00000			−2.00000			−1.00000			$\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	1.00000			1.00000
$688$		0			0
$689$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$690$		0			0
$691$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$691$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$696$		0			0
$697$		0			0
$698$	0.500000	+	0.866025i	0.500000	+	0.866025i
$699$		0			0
$700$		0			0
$701$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$701$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$702$	−1.00000			−1.00000
$703$	1.00000			1.00000
$704$	0.500000	−	0.866025i	0.500000	−	0.866025i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$709$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	0.500000	+	0.866025i	0.500000	+	0.866025i
$716$		0			0
$717$	1.00000			1.00000
$718$	0.500000	−	0.866025i	0.500000	−	0.866025i
$719$	2.00000			2.00000			1.00000			$0$
$719$	2.00000			2.00000			1.00000			$0$
$720$	0.500000	+	0.866025i	0.500000	+	0.866025i
$721$		0			0
$722$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$727$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$	0.500000	+	0.866025i	0.500000	+	0.866025i
$736$		0			0
$737$	2.00000			2.00000
$738$		0			0
$739$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$	−1.00000			−1.00000
$742$		0			0
$743$	−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
$745$	0.500000	−	0.866025i	0.500000	−	0.866025i
$746$	1.00000			1.00000
$747$		0			0
$748$		0			0
$749$		0			0
$750$	0.500000	+	0.866025i	0.500000	+	0.866025i
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$	0.500000	+	0.866025i	0.500000	+	0.866025i
$761$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$761$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$762$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$763$		0			0
$764$		0			0
$765$		0			0
$766$	1.00000			1.00000
$767$		0			0
$768$		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			0
$771$	2.00000			2.00000
$772$		0			0
$773$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$773$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$		0			0
$776$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$777$		0			0
$778$	0.500000	+	0.866025i	0.500000	+	0.866025i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.00000			−1.00000
$785$		0			0
$786$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$787$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$787$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	1.00000			1.00000
$793$	2.00000			2.00000
$794$		0			0
$795$	−1.00000			−1.00000
$796$		0			0
$797$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$797$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.00000	−	1.73205i	1.00000	−	1.73205i
$809$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$809$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$810$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	1.00000	−	1.73205i	1.00000	−	1.73205i
$814$	0.500000	+	0.866025i	0.500000	+	0.866025i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$821$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$	0.500000	+	0.866025i	0.500000	+	0.866025i
$825$	1.00000			1.00000
$826$		0			0
$827$	−2.00000			−2.00000			−1.00000			$\pi$
$827$	−2.00000			−2.00000			−1.00000			$\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$833$		0			0
$834$	1.00000	−	1.73205i	1.00000	−	1.73205i
$835$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$836$		0			0
$837$		0			0
$838$	2.00000			2.00000
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$	0.500000	−	0.866025i	0.500000	−	0.866025i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$856$	−1.00000			−1.00000
$857$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$857$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$858$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$859$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$859$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$860$		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
$865$	1.00000			1.00000
$866$	1.00000	−	1.73205i	1.00000	−	1.73205i
$867$	0.500000	−	0.866025i	0.500000	−	0.866025i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	1.00000	−	1.73205i	1.00000	−	1.73205i
$872$		0			0
$873$	−2.00000			−2.00000
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$877$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$	2.00000			2.00000
$880$	−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	$0.666667\pi$
$882$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		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	$0.333333\pi$
$888$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$889$		0			0
$890$		0			0
$891$	0.500000	+	0.866025i	0.500000	+	0.866025i
$892$		0			0
$893$		0			0
$894$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$905$		0			0
$906$		0			0
$907$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$907$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$908$		0			0
$909$	2.00000			2.00000
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$913$		0			0
$914$		0			0
$915$	1.00000	−	1.73205i	1.00000	−	1.73205i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$919$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$	0.500000	−	0.866025i	0.500000	−	0.866025i
$922$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$923$		0			0
$924$		0			0
$925$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$926$		0			0
$927$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$928$		0			0
$929$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$929$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$930$		0			0
$931$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$932$		0			0
$933$	−2.00000			−2.00000
$934$		0			0
$935$		0			0
$936$	0.500000	−	0.866025i	0.500000	−	0.866025i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		0			0
$950$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$951$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$952$		0			0
$953$	−2.00000			−2.00000			−1.00000			$\pi$
$953$	−2.00000			−2.00000			−1.00000			$\pi$
$954$	1.00000			1.00000
$955$	−1.00000			−1.00000
$956$		0			0
$957$		0			0
$958$	0.500000	+	0.866025i	0.500000	+	0.866025i
$959$		0			0
$960$	−1.00000			−1.00000
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$	1.00000			1.00000
$963$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$964$		0			0
$965$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$	1.00000	−	1.73205i	1.00000	−	1.73205i
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$	1.00000	−	1.73205i	1.00000	−	1.73205i
$975$	0.500000	−	0.866025i	0.500000	−	0.866025i
$976$	1.00000	+	1.73205i	1.00000	+	1.73205i
$977$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$977$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−1.00000			−1.00000
$983$	−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			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$	−1.00000			−1.00000
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	0.500000	+	0.866025i	0.500000	+	0.866025i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$	−1.00000			−1.00000
$999$	−1.00000			−1.00000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.1.z.a.664.1	yes	2
3.2	odd	2		2565.1.z.b.1234.1		2
5.4	even	2		855.1.z.b.664.1	yes	2
9.4	even	3	inner	855.1.z.a.94.1	✓	2
9.5	odd	6		2565.1.z.b.2089.1		2
15.14	odd	2		2565.1.z.a.1234.1		2
19.18	odd	2		855.1.z.b.664.1	yes	2
45.4	even	6		855.1.z.b.94.1	yes	2
45.14	odd	6		2565.1.z.a.2089.1		2
57.56	even	2		2565.1.z.a.1234.1		2
95.94	odd	2	CM	855.1.z.a.664.1	yes	2
171.94	odd	6		855.1.z.b.94.1	yes	2
171.113	even	6		2565.1.z.a.2089.1		2
285.284	even	2		2565.1.z.b.1234.1		2
855.94	odd	6	inner	855.1.z.a.94.1	✓	2
855.284	even	6		2565.1.z.b.2089.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
855.1.z.a.94.1	✓	2	9.4	even	3	inner
855.1.z.a.94.1	✓	2	855.94	odd	6	inner
855.1.z.a.664.1	yes	2	1.1	even	1	trivial
855.1.z.a.664.1	yes	2	95.94	odd	2	CM
855.1.z.b.94.1	yes	2	45.4	even	6
855.1.z.b.94.1	yes	2	171.94	odd	6
855.1.z.b.664.1	yes	2	5.4	even	2
855.1.z.b.664.1	yes	2	19.18	odd	2
2565.1.z.a.1234.1		2	15.14	odd	2
2565.1.z.a.1234.1		2	57.56	even	2
2565.1.z.a.2089.1		2	45.14	odd	6
2565.1.z.a.2089.1		2	171.113	even	6
2565.1.z.b.1234.1		2	3.2	odd	2
2565.1.z.b.1234.1		2	285.284	even	2
2565.1.z.b.2089.1		2	9.5	odd	6
2565.1.z.b.2089.1		2	855.284	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	855.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 + 0.866025i) q^{6} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{5} +(0.500000 + 0.866025i) q^{6} -1.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +1.00000 q^{10} +(0.500000 - 0.866025i) q^{11} +(-0.500000 - 0.866025i) q^{13} -1.00000 q^{15} +(0.500000 - 0.866025i) q^{16} +1.00000 q^{18} +1.00000 q^{19} +(0.500000 + 0.866025i) q^{22} +(-0.500000 + 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{26} -1.00000 q^{27} +(0.500000 - 0.866025i) q^{30} +(-0.500000 - 0.866025i) q^{33} +1.00000 q^{37} +(-0.500000 + 0.866025i) q^{38} -1.00000 q^{39} +(0.500000 + 0.866025i) q^{40} +(-0.500000 + 0.866025i) q^{45} +(-0.500000 - 0.866025i) q^{48} +(-0.500000 - 0.866025i) q^{49} +(-0.500000 - 0.866025i) q^{50} +1.00000 q^{53} +(0.500000 - 0.866025i) q^{54} -1.00000 q^{55} +(0.500000 - 0.866025i) q^{57} +(-1.00000 + 1.73205i) q^{61} +1.00000 q^{64} +(-0.500000 + 0.866025i) q^{65} +1.00000 q^{66} +(1.00000 + 1.73205i) q^{67} +(0.500000 + 0.866025i) q^{72} +(-0.500000 + 0.866025i) q^{74} +(0.500000 + 0.866025i) q^{75} +(0.500000 - 0.866025i) q^{78} -1.00000 q^{80} +(-0.500000 + 0.866025i) q^{81} +(-0.500000 + 0.866025i) q^{88} +(-0.500000 - 0.866025i) q^{90} +(-0.500000 - 0.866025i) q^{95} +(1.00000 - 1.73205i) q^{97} +1.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + q^{3} - q^{5} + q^{6} - 2 q^{8} - q^{9}+O(q^{10}) \) 2 * q - q^2 + q^3 - q^5 + q^6 - 2 * q^8 - q^9	\( 2 q - q^{2} + q^{3} - q^{5} + q^{6} - 2 q^{8} - q^{9} + 2 q^{10} + q^{11} - q^{13} - 2 q^{15} + q^{16} + 2 q^{18} + 2 q^{19} + q^{22} - q^{24} - q^{25} + 2 q^{26} - 2 q^{27} + q^{30} - q^{33} + 2 q^{37} - q^{38} - 2 q^{39} + q^{40} - q^{45} - q^{48} - q^{49} - q^{50} + 2 q^{53} + q^{54} - 2 q^{55} + q^{57} - 2 q^{61} + 2 q^{64} - q^{65} + 2 q^{66} + 2 q^{67} + q^{72} - q^{74} + q^{75} + q^{78} - 2 q^{80} - q^{81} - q^{88} - q^{90} - q^{95} + 2 q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) 2 * q - q^2 + q^3 - q^5 + q^6 - 2 * q^8 - q^9 + 2 * q^10 + q^11 - q^13 - 2 * q^15 + q^16 + 2 * q^18 + 2 * q^19 + q^22 - q^24 - q^25 + 2 * q^26 - 2 * q^27 + q^30 - q^33 + 2 * q^37 - q^38 - 2 * q^39 + q^40 - q^45 - q^48 - q^49 - q^50 + 2 * q^53 + q^54 - 2 * q^55 + q^57 - 2 * q^61 + 2 * q^64 - q^65 + 2 * q^66 + 2 * q^67 + q^72 - q^74 + q^75 + q^78 - 2 * q^80 - q^81 - q^88 - q^90 - q^95 + 2 * q^97 + 2 * q^98 - 2 * q^99

Introduction

L-functions

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