LMFDB - Embedded newform 855.1.u.a.539.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,1,Mod(539,855)]

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([3, 3, 2]))

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

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

chi := DirichletCharacter("855.539");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	855.u (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:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.243675.1

Embedding invariants

Embedding label			539.4
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	855.539
Dual form			855.1.u.a.809.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 1.22474i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.965926 + 0.258819i) q^{5} +1.00000i q^{7} +O(q^{10})$	\(q+(0.707107 + 1.22474i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.965926 + 0.258819i) q^{5} +1.00000i q^{7} +(0.366025 + 1.36603i) q^{10} -1.41421i q^{11} +(-0.866025 - 0.500000i) q^{13} +(-1.22474 + 0.707107i) q^{14} +(0.500000 + 0.866025i) q^{16} +(-0.707107 - 1.22474i) q^{17} -1.00000 q^{19} +(-0.707107 + 0.707107i) q^{20} +(1.73205 - 1.00000i) q^{22} +(0.866025 + 0.500000i) q^{25} -1.41421i q^{26} +(-0.866025 - 0.500000i) q^{28} -1.00000 q^{31} +(-0.707107 + 1.22474i) q^{32} +(1.00000 - 1.73205i) q^{34} +(-0.258819 + 0.965926i) q^{35} +1.00000i q^{37} +(-0.707107 - 1.22474i) q^{38} +(-1.22474 + 0.707107i) q^{41} +(-0.866025 + 0.500000i) q^{43} +(1.22474 + 0.707107i) q^{44} +1.41421i q^{50} +(0.866025 - 0.500000i) q^{52} +(0.707107 - 1.22474i) q^{53} +(0.366025 - 1.36603i) q^{55} +(0.500000 - 0.866025i) q^{61} +(-0.707107 - 1.22474i) q^{62} -1.00000 q^{64} +(-0.707107 - 0.707107i) q^{65} +(-0.866025 - 0.500000i) q^{67} +1.41421 q^{68} +(-1.36603 + 0.366025i) q^{70} +(1.22474 - 0.707107i) q^{71} +(0.866025 - 0.500000i) q^{73} +(-1.22474 + 0.707107i) q^{74} +(0.500000 - 0.866025i) q^{76} +1.41421 q^{77} +(0.500000 + 0.866025i) q^{79} +(0.258819 + 0.965926i) q^{80} +(-1.73205 - 1.00000i) q^{82} +(-0.366025 - 1.36603i) q^{85} +(-1.22474 - 0.707107i) q^{86} +(1.22474 + 0.707107i) q^{89} +(0.500000 - 0.866025i) q^{91} +(-0.965926 - 0.258819i) q^{95} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{4}+O(q^{10}) $ 8 * q - 4 * q^4	$ 8 q - 4 q^{4} - 4 q^{10} + 4 q^{16} - 8 q^{19} - 8 q^{31} + 8 q^{34} - 4 q^{55} + 4 q^{61} - 8 q^{64} - 4 q^{70} + 4 q^{76} + 4 q^{79} + 4 q^{85} + 4 q^{91}+O(q^{100}) $ 8 * q - 4 * q^4 - 4 * q^10 + 4 * q^16 - 8 * q^19 - 8 * q^31 + 8 * q^34 - 4 * q^55 + 4 * q^61 - 8 * q^64 - 4 * q^70 + 4 * q^76 + 4 * q^79 + 4 * q^85 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

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

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

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 $

$2$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$2$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$5$	0.965926	+	0.258819i	0.965926	+	0.258819i
$5$	0.965926	+	0.258819i	0.965926	+	0.258819i
$6$		0			0
$7$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$7$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$8$		0			0
$9$		0			0
$10$	0.366025	+	1.36603i	0.366025	+	1.36603i
$11$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$11$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$12$		0			0
$13$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$13$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$14$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$15$		0			0
$16$	0.500000	+	0.866025i	0.500000	+	0.866025i
$17$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$17$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$18$		0			0
$19$	−1.00000			−1.00000
$19$	−1.00000			−1.00000
$20$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$21$		0			0
$22$	1.73205	−	1.00000i	1.73205	−	1.00000i
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	0.866025	+	0.500000i	0.866025	+	0.500000i
$26$		−	1.41421i		−	1.41421i
$27$		0			0
$28$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$31$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$32$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$33$		0			0
$34$	1.00000	−	1.73205i	1.00000	−	1.73205i
$35$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$36$		0			0
$37$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$37$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$38$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$39$		0			0
$40$		0			0
$41$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$41$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$42$		0			0
$43$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$43$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$44$	1.22474	+	0.707107i	1.22474	+	0.707107i
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$		0			0
$50$			1.41421i			1.41421i
$51$		0			0
$52$	0.866025	−	0.500000i	0.866025	−	0.500000i
$53$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$53$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$54$		0			0
$55$	0.366025	−	1.36603i	0.366025	−	1.36603i
$56$		0			0
$57$		0			0
$58$		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
$61$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$61$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$62$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$63$		0			0
$64$	−1.00000			−1.00000
$65$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$66$		0			0
$67$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$67$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$68$	1.41421			1.41421
$69$		0			0
$70$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$71$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$71$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$72$		0			0
$73$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$73$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$74$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$75$		0			0
$76$	0.500000	−	0.866025i	0.500000	−	0.866025i
$77$	1.41421			1.41421
$78$		0			0
$79$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$79$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$80$	0.258819	+	0.965926i	0.258819	+	0.965926i
$81$		0			0
$82$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$86$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$87$		0			0
$88$		0			0
$89$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$89$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$90$		0			0
$91$	0.500000	−	0.866025i	0.500000	−	0.866025i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$96$		0			0
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$		0			0
$99$		0			0
$100$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$103$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$	2.00000			2.00000
$107$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$107$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$108$		0			0
$109$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$109$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$110$	1.93185	−	0.517638i	1.93185	−	0.517638i
$111$		0			0
$112$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.22474	−	0.707107i	1.22474	−	0.707107i
$120$		0			0
$121$	−1.00000			−1.00000
$122$	1.41421			1.41421
$123$		0			0
$124$	0.500000	−	0.866025i	0.500000	−	0.866025i
$125$	0.707107	+	0.707107i	0.707107	+	0.707107i
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$128$		0			0
$129$		0			0
$130$	0.366025	−	1.36603i	0.366025	−	1.36603i
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		−	1.00000i		−	1.00000i
$134$		−	1.41421i		−	1.41421i
$135$		0			0
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$		0			0
$139$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$139$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$140$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$141$		0			0
$142$	1.73205	+	1.00000i	1.73205	+	1.00000i
$143$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$144$		0			0
$145$		0			0
$146$	1.22474	+	0.707107i	1.22474	+	0.707107i
$147$		0			0
$148$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$149$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$149$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$		0			0
$153$		0			0
$154$	1.00000	+	1.73205i	1.00000	+	1.73205i
$155$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$156$		0			0
$157$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$157$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$158$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$159$		0			0
$160$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$161$		0			0
$162$		0			0
$163$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$163$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$164$		−	1.41421i		−	1.41421i
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$		0			0
$170$	1.41421	−	1.41421i	1.41421	−	1.41421i
$171$		0			0
$172$		−	1.00000i		−	1.00000i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$176$	1.22474	−	0.707107i	1.22474	−	0.707107i
$177$		0			0
$178$			2.00000i			2.00000i
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$181$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$182$	1.41421			1.41421
$183$		0			0
$184$		0			0
$185$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$186$		0			0
$187$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$188$		0			0
$189$		0			0
$190$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		0			0
$193$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$193$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$199$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$206$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$207$		0			0
$208$		−	1.00000i		−	1.00000i
$209$			1.41421i			1.41421i
$210$		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	$0.666667\pi$
$212$	0.707107	+	1.22474i	0.707107	+	1.22474i
$213$		0			0
$214$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$215$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$216$		0			0
$217$		−	1.00000i		−	1.00000i
$218$		0			0
$219$		0			0
$220$	1.00000	+	1.00000i	1.00000	+	1.00000i
$221$			1.41421i			1.41421i
$222$		0			0
$223$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$223$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$224$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$225$		0			0
$226$		0			0
$227$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$227$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$228$		0			0
$229$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$229$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$233$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	1.73205	+	1.00000i	1.73205	+	1.00000i
$239$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$239$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$241$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$242$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$243$		0			0
$244$	0.500000	+	0.866025i	0.500000	+	0.866025i
$245$		0			0
$246$		0			0
$247$	0.866025	+	0.500000i	0.866025	+	0.500000i
$248$		0			0
$249$		0			0
$250$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$251$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$251$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	−1.00000			−1.00000
$260$	0.965926	−	0.258819i	0.965926	−	0.258819i
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$	1.00000	−	1.00000i	1.00000	−	1.00000i
$266$	1.22474	−	0.707107i	1.22474	−	0.707107i
$267$		0			0
$268$	0.866025	−	0.500000i	0.866025	−	0.500000i
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$	0.707107	−	1.22474i	0.707107	−	1.22474i
$273$		0			0
$274$		0			0
$275$	0.707107	−	1.22474i	0.707107	−	1.22474i
$276$		0			0
$277$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$277$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$278$	−1.41421			−1.41421
$279$		0			0
$280$		0			0
$281$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$281$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$282$		0			0
$283$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$283$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$284$			1.41421i			1.41421i
$285$		0			0
$286$	−2.00000			−2.00000
$287$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$288$		0			0
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$			1.00000i			1.00000i
$293$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$293$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	1.73205	+	1.00000i	1.73205	+	1.00000i
$299$		0			0
$300$		0			0
$301$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$302$		0			0
$303$		0			0
$304$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$305$	0.707107	−	0.707107i	0.707107	−	0.707107i
$306$		0			0
$307$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$307$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$308$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$309$		0			0
$310$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$311$		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	$0.666667\pi$
$314$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$315$		0			0
$316$	−1.00000			−1.00000
$317$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$317$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$318$		0			0
$319$		0			0
$320$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$321$		0			0
$322$		0			0
$323$	0.707107	+	1.22474i	0.707107	+	1.22474i
$324$		0			0
$325$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$326$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$331$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$336$		0			0
$337$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$337$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$338$		0			0
$339$		0			0
$340$	1.36603	+	0.366025i	1.36603	+	0.366025i
$341$			1.41421i			1.41421i
$342$		0			0
$343$			1.00000i			1.00000i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$347$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$349$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$350$	−1.41421			−1.41421
$351$		0			0
$352$	1.73205	+	1.00000i	1.73205	+	1.00000i
$353$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$353$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$354$		0			0
$355$	1.36603	−	0.366025i	1.36603	−	0.366025i
$356$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$357$		0			0
$358$		0			0
$359$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$359$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$		0			0
$363$		0			0
$364$	0.500000	+	0.866025i	0.500000	+	0.866025i
$365$	0.965926	−	0.258819i	0.965926	−	0.258819i
$366$		0			0
$367$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$367$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$368$		0			0
$369$		0			0
$370$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$371$	1.22474	+	0.707107i	1.22474	+	0.707107i
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$	−2.44949	−	1.41421i	−2.44949	−	1.41421i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$380$	0.707107	−	0.707107i	0.707107	−	0.707107i
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$	1.36603	+	0.366025i	1.36603	+	0.366025i
$386$	1.22474	+	0.707107i	1.22474	+	0.707107i
$387$		0			0
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.258819	+	0.965926i	0.258819	+	0.965926i
$396$		0			0
$397$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$397$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$398$	1.41421			1.41421
$399$		0			0
$400$			1.00000i			1.00000i
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	0.866025	+	0.500000i	0.866025	+	0.500000i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.41421			1.41421
$408$		0			0
$409$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$409$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$410$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$411$		0			0
$412$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	1.22474	−	0.707107i	1.22474	−	0.707107i
$417$		0			0
$418$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$421$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$422$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$423$		0			0
$424$		0			0
$425$		−	1.41421i		−	1.41421i
$426$		0			0
$427$	0.866025	+	0.500000i	0.866025	+	0.500000i
$428$	0.707107	−	1.22474i	0.707107	−	1.22474i
$429$		0			0
$430$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$433$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$434$	1.22474	−	0.707107i	1.22474	−	0.707107i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$439$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$443$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$443$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$444$		0			0
$445$	1.00000	+	1.00000i	1.00000	+	1.00000i
$446$	1.22474	+	0.707107i	1.22474	+	0.707107i
$447$		0			0
$448$		−	1.00000i		−	1.00000i
$449$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$449$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$450$		0			0
$451$	1.00000	+	1.73205i	1.00000	+	1.73205i
$452$		0			0
$453$		0			0
$454$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$455$	0.707107	−	0.707107i	0.707107	−	0.707107i
$456$		0			0
$457$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$457$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$458$	0.707107	+	1.22474i	0.707107	+	1.22474i
$459$		0			0
$460$		0			0
$461$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$461$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$462$		0			0
$463$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$463$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$464$		0			0
$465$		0			0
$466$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$467$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$467$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$468$		0			0
$469$	0.500000	−	0.866025i	0.500000	−	0.866025i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.707107	+	1.22474i	0.707107	+	1.22474i
$474$		0			0
$475$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$476$			1.41421i			1.41421i
$477$		0			0
$478$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	0.500000	−	0.866025i	0.500000	−	0.866025i
$482$	−1.41421			−1.41421
$483$		0			0
$484$	0.500000	−	0.866025i	0.500000	−	0.866025i
$485$		0			0
$486$		0			0
$487$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$487$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$491$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$492$		0			0
$493$		0			0
$494$			1.41421i			1.41421i
$495$		0			0
$496$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$497$	0.707107	+	1.22474i	0.707107	+	1.22474i
$498$		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	$0.666667\pi$
$500$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$501$		0			0
$502$		−	2.00000i		−	2.00000i
$503$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$503$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	0.500000	+	0.866025i	0.500000	+	0.866025i
$512$	−1.41421			−1.41421
$513$		0			0
$514$		0			0
$515$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$516$		0			0
$517$		0			0
$518$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$523$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	0.707107	+	1.22474i	0.707107	+	1.22474i
$528$		0			0
$529$	0.500000	+	0.866025i	0.500000	+	0.866025i
$530$	1.93185	+	0.517638i	1.93185	+	0.517638i
$531$		0			0
$532$	0.866025	+	0.500000i	0.866025	+	0.500000i
$533$	1.41421			1.41421
$534$		0			0
$535$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$536$		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			0
$543$		0			0
$544$	2.00000			2.00000
$545$		0			0
$546$		0			0
$547$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$547$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$548$		0			0
$549$		0			0
$550$	2.00000			2.00000
$551$		0			0
$552$		0			0
$553$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$554$	2.44949	−	1.41421i	2.44949	−	1.41421i
$555$		0			0
$556$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$557$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$557$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$558$		0			0
$559$	1.00000			1.00000
$560$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$561$		0			0
$562$		−	2.00000i		−	2.00000i
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$	2.44949	+	1.41421i	2.44949	+	1.41421i
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$571$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$572$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$573$		0			0
$574$	1.00000	−	1.73205i	1.00000	−	1.73205i
$575$		0			0
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−1.41421			−1.41421
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$584$		0			0
$585$		0			0
$586$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$587$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$587$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$588$		0			0
$589$	1.00000			1.00000
$590$		0			0
$591$		0			0
$592$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$593$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$593$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$594$		0			0
$595$	1.36603	−	0.366025i	1.36603	−	0.366025i
$596$			1.41421i			1.41421i
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$601$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$602$	0.707107	−	1.22474i	0.707107	−	1.22474i
$603$		0			0
$604$		0			0
$605$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$606$		0			0
$607$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$607$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$608$	0.707107	−	1.22474i	0.707107	−	1.22474i
$609$		0			0
$610$	1.36603	+	0.366025i	1.36603	+	0.366025i
$611$		0			0
$612$		0			0
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$	2.44949	+	1.41421i	2.44949	+	1.41421i
$615$		0			0
$616$		0			0
$617$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$617$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$618$		0			0
$619$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$619$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$620$	0.707107	−	0.707107i	0.707107	−	0.707107i
$621$		0			0
$622$		0			0
$623$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$624$		0			0
$625$	0.500000	+	0.866025i	0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$		−	1.00000i		−	1.00000i
$629$	1.22474	−	0.707107i	1.22474	−	0.707107i
$630$		0			0
$631$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$631$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	−2.00000			−2.00000
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$643$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$	0.707107	−	1.22474i	0.707107	−	1.22474i
$651$		0			0
$652$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$653$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$653$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$657$		0			0
$658$		0			0
$659$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$659$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$663$		0			0
$664$		0			0
$665$	0.258819	−	0.965926i	0.258819	−	0.965926i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	0.366025	−	1.36603i	0.366025	−	1.36603i
$671$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$672$		0			0
$673$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$673$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$674$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$675$		0			0
$676$		0			0
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$687$		0			0
$688$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$689$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$696$		0			0
$697$	1.73205	+	1.00000i	1.73205	+	1.00000i
$698$	0.707107	+	1.22474i	0.707107	+	1.22474i
$699$		0			0
$700$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$701$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$701$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$702$		0			0
$703$		−	1.00000i		−	1.00000i
$704$			1.41421i			1.41421i
$705$		0			0
$706$	−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			$0$
$710$	1.41421	+	1.41421i	1.41421	+	1.41421i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$716$		0			0
$717$		0			0
$718$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$	−1.00000			−1.00000
$722$	0.707107	+	1.22474i	0.707107	+	1.22474i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$727$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$728$		0			0
$729$		0			0
$730$	1.00000	+	1.00000i	1.00000	+	1.00000i
$731$	1.22474	+	0.707107i	1.22474	+	0.707107i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		−	1.41421i		−	1.41421i
$735$		0			0
$736$		0			0
$737$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$738$		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	$0.666667\pi$
$740$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$741$		0			0
$742$			2.00000i			2.00000i
$743$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$743$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$744$		0			0
$745$	1.36603	−	0.366025i	1.36603	−	0.366025i
$746$		0			0
$747$		0			0
$748$		−	2.00000i		−	2.00000i
$749$		−	1.41421i		−	1.41421i
$750$		0			0
$751$	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			$0$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$757$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$758$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$769$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$770$	0.517638	+	1.93185i	0.517638	+	1.93185i
$771$		0			0
$772$			1.00000i			1.00000i
$773$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$773$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$774$		0			0
$775$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.22474	−	0.707107i	1.22474	−	0.707107i
$780$		0			0
$781$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$786$		0			0
$787$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$787$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$788$		0			0
$789$		0			0
$790$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$791$		0			0
$792$		0			0
$793$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$794$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$795$		0			0
$796$	0.500000	+	0.866025i	0.500000	+	0.866025i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$801$		0			0
$802$		0			0
$803$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$804$		0			0
$805$		0			0
$806$			1.41421i			1.41421i
$807$		0			0
$808$		0			0
$809$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$809$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$810$		0			0
$811$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$811$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$812$		0			0
$813$		0			0
$814$	1.00000	+	1.73205i	1.00000	+	1.73205i
$815$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$816$		0			0
$817$	0.866025	−	0.500000i	0.866025	−	0.500000i
$818$	−2.82843			−2.82843
$819$		0			0
$820$	0.366025	−	1.36603i	0.366025	−	1.36603i
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$		0			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			0		−0.866025	−	0.500000i	$-0.833333\pi$
$827$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$828$		0			0
$829$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$829$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$	0.866025	+	0.500000i	0.866025	+	0.500000i
$833$		0			0
$834$		0			0
$835$		0			0
$836$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$837$		0			0
$838$		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
$842$		0			0
$843$		0			0
$844$	−1.00000			−1.00000
$845$		0			0
$846$		0			0
$847$		−	1.00000i		−	1.00000i
$848$	1.41421			1.41421
$849$		0			0
$850$	1.73205	−	1.00000i	1.73205	−	1.00000i
$851$		0			0
$852$		0			0
$853$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$853$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$854$			1.41421i			1.41421i
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		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			$0$
$860$	0.258819	−	0.965926i	0.258819	−	0.965926i
$861$		0			0
$862$		0			0
$863$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$863$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$864$		0			0
$865$		0			0
$866$			1.41421i			1.41421i
$867$		0			0
$868$	0.866025	+	0.500000i	0.866025	+	0.500000i
$869$	1.22474	−	0.707107i	1.22474	−	0.707107i
$870$		0			0
$871$	0.500000	+	0.866025i	0.500000	+	0.866025i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$876$		0			0
$877$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$877$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$878$	0.707107	−	1.22474i	0.707107	−	1.22474i
$879$		0			0
$880$	1.36603	−	0.366025i	1.36603	−	0.366025i
$881$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$881$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$882$		0			0
$883$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$883$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$884$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$885$		0			0
$886$	2.00000			2.00000
$887$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$887$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$888$		0			0
$889$		0			0
$890$	−0.517638	+	1.93185i	−0.517638	+	1.93185i
$891$		0			0
$892$			1.00000i			1.00000i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.73205	−	1.00000i	1.73205	−	1.00000i
$899$		0			0
$900$		0			0
$901$	−2.00000			−2.00000
$902$	−1.41421	+	2.44949i	−1.41421	+	2.44949i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$	0.707107	−	1.22474i	0.707107	−	1.22474i
$909$		0			0
$910$	1.36603	+	0.366025i	1.36603	+	0.366025i
$911$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$911$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$912$		0			0
$913$		0			0
$914$	1.22474	−	0.707107i	1.22474	−	0.707107i
$915$		0			0
$916$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$917$		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			0
$922$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$923$	−1.41421			−1.41421
$924$		0			0
$925$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$926$	1.22474	−	0.707107i	1.22474	−	0.707107i
$927$		0			0
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$	−1.41421			−1.41421
$933$		0			0
$934$	1.00000	+	1.73205i	1.00000	+	1.73205i
$935$	−1.93185	+	0.517638i	−1.93185	+	0.517638i
$936$		0			0
$937$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$937$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$938$	1.41421			1.41421
$939$		0			0
$940$		0			0
$941$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$941$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$		0			0
$949$	−1.00000			−1.00000
$950$		−	1.41421i		−	1.41421i
$951$		0			0
$952$		0			0
$953$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$953$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$954$		0			0
$955$		0			0
$956$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		0			0
$962$	1.41421			1.41421
$963$		0			0
$964$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$965$	0.965926	−	0.258819i	0.965926	−	0.258819i
$966$		0			0
$967$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$967$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		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
$973$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$974$	−2.44949	+	1.41421i	−2.44949	+	1.41421i
$975$		0			0
$976$	1.00000			1.00000
$977$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$977$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$978$		0			0
$979$	1.00000	−	1.73205i	1.00000	−	1.73205i
$980$		0			0
$981$		0			0
$982$	1.73205	+	1.00000i	1.73205	+	1.00000i
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$989$		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	$0.666667\pi$
$992$	0.707107	−	1.22474i	0.707107	−	1.22474i
$993$		0			0
$994$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$995$	0.707107	−	0.707107i	0.707107	−	0.707107i
$996$		0			0
$997$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$997$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$998$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.1.u.a.539.4	yes	8
3.2	odd	2	inner	855.1.u.a.539.1	✓	8
5.4	even	2	inner	855.1.u.a.539.2	yes	8
15.14	odd	2	inner	855.1.u.a.539.3	yes	8
19.11	even	3	inner	855.1.u.a.809.3	yes	8
57.11	odd	6	inner	855.1.u.a.809.2	yes	8
95.49	even	6	inner	855.1.u.a.809.1	yes	8
285.239	odd	6	inner	855.1.u.a.809.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
855.1.u.a.539.1	✓	8	3.2	odd	2	inner
855.1.u.a.539.2	yes	8	5.4	even	2	inner
855.1.u.a.539.3	yes	8	15.14	odd	2	inner
855.1.u.a.539.4	yes	8	1.1	even	1	trivial
855.1.u.a.809.1	yes	8	95.49	even	6	inner
855.1.u.a.809.2	yes	8	57.11	odd	6	inner
855.1.u.a.809.3	yes	8	19.11	even	3	inner
855.1.u.a.809.4	yes	8	285.239	odd	6	inner

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.u (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 1.22474i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.965926 + 0.258819i) q^{5} +1.00000i q^{7} +O(q^{10})\)	\(q+(0.707107 + 1.22474i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(0.965926 + 0.258819i) q^{5} +1.00000i q^{7} +(0.366025 + 1.36603i) q^{10} -1.41421i q^{11} +(-0.866025 - 0.500000i) q^{13} +(-1.22474 + 0.707107i) q^{14} +(0.500000 + 0.866025i) q^{16} +(-0.707107 - 1.22474i) q^{17} -1.00000 q^{19} +(-0.707107 + 0.707107i) q^{20} +(1.73205 - 1.00000i) q^{22} +(0.866025 + 0.500000i) q^{25} -1.41421i q^{26} +(-0.866025 - 0.500000i) q^{28} -1.00000 q^{31} +(-0.707107 + 1.22474i) q^{32} +(1.00000 - 1.73205i) q^{34} +(-0.258819 + 0.965926i) q^{35} +1.00000i q^{37} +(-0.707107 - 1.22474i) q^{38} +(-1.22474 + 0.707107i) q^{41} +(-0.866025 + 0.500000i) q^{43} +(1.22474 + 0.707107i) q^{44} +1.41421i q^{50} +(0.866025 - 0.500000i) q^{52} +(0.707107 - 1.22474i) q^{53} +(0.366025 - 1.36603i) q^{55} +(0.500000 - 0.866025i) q^{61} +(-0.707107 - 1.22474i) q^{62} -1.00000 q^{64} +(-0.707107 - 0.707107i) q^{65} +(-0.866025 - 0.500000i) q^{67} +1.41421 q^{68} +(-1.36603 + 0.366025i) q^{70} +(1.22474 - 0.707107i) q^{71} +(0.866025 - 0.500000i) q^{73} +(-1.22474 + 0.707107i) q^{74} +(0.500000 - 0.866025i) q^{76} +1.41421 q^{77} +(0.500000 + 0.866025i) q^{79} +(0.258819 + 0.965926i) q^{80} +(-1.73205 - 1.00000i) q^{82} +(-0.366025 - 1.36603i) q^{85} +(-1.22474 - 0.707107i) q^{86} +(1.22474 + 0.707107i) q^{89} +(0.500000 - 0.866025i) q^{91} +(-0.965926 - 0.258819i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{4}+O(q^{10}) \) 8 * q - 4 * q^4	\( 8 q - 4 q^{4} - 4 q^{10} + 4 q^{16} - 8 q^{19} - 8 q^{31} + 8 q^{34} - 4 q^{55} + 4 q^{61} - 8 q^{64} - 4 q^{70} + 4 q^{76} + 4 q^{79} + 4 q^{85} + 4 q^{91}+O(q^{100}) \) 8 * q - 4 * q^4 - 4 * q^10 + 4 * q^16 - 8 * q^19 - 8 * q^31 + 8 * q^34 - 4 * q^55 + 4 * q^61 - 8 * q^64 - 4 * q^70 + 4 * q^76 + 4 * q^79 + 4 * q^85 + 4 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more