LMFDB - Embedded newform 3447.1.f.a.382.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3447,1,Mod(382,3447)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3447, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("3447.382");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3447 = 3^{2} \cdot 383 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3447.f (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:	$1.72027709855$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.31023.1
Artin image:	$C_3\times S_3$
Artin field:	Galois closure of 6.0.4550732847.1

Embedding invariants

Embedding label			382.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3447.382
Dual form			3447.1.f.a.2680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-1.50000 + 2.59808i) q^{4} +2.00000 q^{6} +(0.500000 + 0.866025i) q^{7} +4.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-1.00000 - 1.73205i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-1.50000 + 2.59808i) q^{4} +2.00000 q^{6} +(0.500000 + 0.866025i) q^{7} +4.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-1.50000 - 2.59808i) q^{12} +(1.00000 - 1.73205i) q^{14} +(-2.50000 - 4.33013i) q^{16} +2.00000 q^{17} +(-1.00000 + 1.73205i) q^{18} -1.00000 q^{19} -1.00000 q^{21} +(0.500000 - 0.866025i) q^{23} +(-2.00000 + 3.46410i) q^{24} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} -3.00000 q^{28} +(-1.00000 - 1.73205i) q^{29} +(0.500000 - 0.866025i) q^{31} +(-3.00000 + 5.19615i) q^{32} +(-2.00000 - 3.46410i) q^{34} +3.00000 q^{36} +(1.00000 + 1.73205i) q^{38} +(1.00000 + 1.73205i) q^{42} +(0.500000 + 0.866025i) q^{43} -2.00000 q^{46} +5.00000 q^{48} +(-1.00000 + 1.73205i) q^{50} +(-1.00000 + 1.73205i) q^{51} +(-1.00000 - 1.73205i) q^{54} +(2.00000 + 3.46410i) q^{56} +(0.500000 - 0.866025i) q^{57} +(-2.00000 + 3.46410i) q^{58} -2.00000 q^{62} +(0.500000 - 0.866025i) q^{63} +7.00000 q^{64} +(0.500000 - 0.866025i) q^{67} +(-3.00000 + 5.19615i) q^{68} +(0.500000 + 0.866025i) q^{69} +2.00000 q^{71} +(-2.00000 - 3.46410i) q^{72} -1.00000 q^{73} +1.00000 q^{75} +(1.50000 - 2.59808i) q^{76} +(-0.500000 + 0.866025i) q^{81} +(1.50000 - 2.59808i) q^{84} +(1.00000 - 1.73205i) q^{86} +2.00000 q^{87} +(1.50000 + 2.59808i) q^{92} +(0.500000 + 0.866025i) q^{93} +(-3.00000 - 5.19615i) q^{96} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - q^{3} - 3 q^{4} + 4 q^{6} + q^{7} + 8 q^{8} - q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - q^3 - 3 * q^4 + 4 * q^6 + q^7 + 8 * q^8 - q^9	$ 2 q - 2 q^{2} - q^{3} - 3 q^{4} + 4 q^{6} + q^{7} + 8 q^{8} - q^{9} - 3 q^{12} + 2 q^{14} - 5 q^{16} + 4 q^{17} - 2 q^{18} - 2 q^{19} - 2 q^{21} + q^{23} - 4 q^{24} - q^{25} + 2 q^{27} - 6 q^{28} - 2 q^{29} + q^{31} - 6 q^{32} - 4 q^{34} + 6 q^{36} + 2 q^{38} + 2 q^{42} + q^{43} - 4 q^{46} + 10 q^{48} - 2 q^{50} - 2 q^{51} - 2 q^{54} + 4 q^{56} + q^{57} - 4 q^{58} - 4 q^{62} + q^{63} + 14 q^{64} + q^{67} - 6 q^{68} + q^{69} + 4 q^{71} - 4 q^{72} - 2 q^{73} + 2 q^{75} + 3 q^{76} - q^{81} + 3 q^{84} + 2 q^{86} + 4 q^{87} + 3 q^{92} + q^{93} - 6 q^{96}+O(q^{100}) $ 2 * q - 2 * q^2 - q^3 - 3 * q^4 + 4 * q^6 + q^7 + 8 * q^8 - q^9 - 3 * q^12 + 2 * q^14 - 5 * q^16 + 4 * q^17 - 2 * q^18 - 2 * q^19 - 2 * q^21 + q^23 - 4 * q^24 - q^25 + 2 * q^27 - 6 * q^28 - 2 * q^29 + q^31 - 6 * q^32 - 4 * q^34 + 6 * q^36 + 2 * q^38 + 2 * q^42 + q^43 - 4 * q^46 + 10 * q^48 - 2 * q^50 - 2 * q^51 - 2 * q^54 + 4 * q^56 + q^57 - 4 * q^58 - 4 * q^62 + q^63 + 14 * q^64 + q^67 - 6 * q^68 + q^69 + 4 * q^71 - 4 * q^72 - 2 * q^73 + 2 * q^75 + 3 * q^76 - q^81 + 3 * q^84 + 2 * q^86 + 4 * q^87 + 3 * q^92 + q^93 - 6 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3447\mathbb{Z}\right)^\times$.

$n$	$388$	$767$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$3$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$4$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$	2.00000			2.00000
$7$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$7$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$8$	4.00000			4.00000
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$13$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$13$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$14$	1.00000	−	1.73205i	1.00000	−	1.73205i
$15$		0			0
$16$	−2.50000	−	4.33013i	−2.50000	−	4.33013i
$17$	2.00000			2.00000			1.00000			$0$
$17$	2.00000			2.00000			1.00000			$0$
$18$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$19$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$19$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$20$		0			0
$21$	−1.00000			−1.00000
$22$		0			0
$23$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$23$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$24$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$		0			0
$27$	1.00000			1.00000
$28$	−3.00000			−3.00000
$29$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$29$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$30$		0			0
$31$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$31$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$32$	−3.00000	+	5.19615i	−3.00000	+	5.19615i
$33$		0			0
$34$	−2.00000	−	3.46410i	−2.00000	−	3.46410i
$35$		0			0
$36$	3.00000			3.00000
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	1.00000	+	1.73205i	1.00000	+	1.73205i
$39$		0			0
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$	1.00000	+	1.73205i	1.00000	+	1.73205i
$43$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$43$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$		0			0
$46$	−2.00000			−2.00000
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	5.00000			5.00000
$49$		0			0
$50$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$51$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$55$		0			0
$56$	2.00000	+	3.46410i	2.00000	+	3.46410i
$57$	0.500000	−	0.866025i	0.500000	−	0.866025i
$58$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$	−2.00000			−2.00000
$63$	0.500000	−	0.866025i	0.500000	−	0.866025i
$64$	7.00000			7.00000
$65$		0			0
$66$		0			0
$67$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$67$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$68$	−3.00000	+	5.19615i	−3.00000	+	5.19615i
$69$	0.500000	+	0.866025i	0.500000	+	0.866025i
$70$		0			0
$71$	2.00000			2.00000			1.00000			$0$
$71$	2.00000			2.00000			1.00000			$0$
$72$	−2.00000	−	3.46410i	−2.00000	−	3.46410i
$73$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$73$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$	1.00000			1.00000
$76$	1.50000	−	2.59808i	1.50000	−	2.59808i
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$		0			0
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$	1.50000	−	2.59808i	1.50000	−	2.59808i
$85$		0			0
$86$	1.00000	−	1.73205i	1.00000	−	1.73205i
$87$	2.00000			2.00000
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	1.50000	+	2.59808i	1.50000	+	2.59808i
$93$	0.500000	+	0.866025i	0.500000	+	0.866025i
$94$		0			0
$95$		0			0
$96$	−3.00000	−	5.19615i	−3.00000	−	5.19615i
$97$		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
$100$	3.00000			3.00000
$101$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$101$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$102$	4.00000			4.00000
$103$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$103$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	2.50000	−	4.33013i	2.50000	−	4.33013i
$113$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$113$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$114$	−2.00000			−2.00000
$115$		0			0
$116$	6.00000			6.00000
$117$		0			0
$118$		0			0
$119$	1.00000	+	1.73205i	1.00000	+	1.73205i
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	1.50000	+	2.59808i	1.50000	+	2.59808i
$125$		0			0
$126$	−2.00000			−2.00000
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$	−4.00000	−	6.92820i	−4.00000	−	6.92820i
$129$	−1.00000			−1.00000
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$134$	−2.00000			−2.00000
$135$		0			0
$136$	8.00000			8.00000
$137$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$137$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$138$	1.00000	−	1.73205i	1.00000	−	1.73205i
$139$	−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
$142$	−2.00000	−	3.46410i	−2.00000	−	3.46410i
$143$		0			0
$144$	−2.50000	+	4.33013i	−2.50000	+	4.33013i
$145$		0			0
$146$	1.00000	+	1.73205i	1.00000	+	1.73205i
$147$		0			0
$148$		0			0
$149$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$149$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$150$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$151$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$151$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$152$	−4.00000			−4.00000
$153$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$157$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.00000			1.00000
$162$	2.00000			2.00000
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$	−4.00000			−4.00000
$169$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$170$		0			0
$171$	0.500000	+	0.866025i	0.500000	+	0.866025i
$172$	−3.00000			−3.00000
$173$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$173$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$174$	−2.00000	−	3.46410i	−2.00000	−	3.46410i
$175$	0.500000	−	0.866025i	0.500000	−	0.866025i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	2.00000	−	3.46410i	2.00000	−	3.46410i
$185$		0			0
$186$	1.00000	−	1.73205i	1.00000	−	1.73205i
$187$		0			0
$188$		0			0
$189$	0.500000	+	0.866025i	0.500000	+	0.866025i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−3.50000	+	6.06218i	−3.50000	+	6.06218i
$193$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$193$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−2.00000	−	3.46410i	−2.00000	−	3.46410i
$201$	0.500000	+	0.866025i	0.500000	+	0.866025i
$202$	1.00000	−	1.73205i	1.00000	−	1.73205i
$203$	1.00000	−	1.73205i	1.00000	−	1.73205i
$204$	−3.00000	−	5.19615i	−3.00000	−	5.19615i
$205$		0			0
$206$	−2.00000			−2.00000
$207$	−1.00000			−1.00000
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$		0			0
$213$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$214$		0			0
$215$		0			0
$216$	4.00000			4.00000
$217$	1.00000			1.00000
$218$		0			0
$219$	0.500000	−	0.866025i	0.500000	−	0.866025i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$223$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$224$	−6.00000			−6.00000
$225$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$226$	−2.00000			−2.00000
$227$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$227$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$228$	1.50000	+	2.59808i	1.50000	+	2.59808i
$229$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$229$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$	−4.00000	−	6.92820i	−4.00000	−	6.92820i
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$	2.00000	−	3.46410i	2.00000	−	3.46410i
$239$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$239$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	2.00000			2.00000
$243$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	2.00000	−	3.46410i	2.00000	−	3.46410i
$249$		0			0
$250$		0			0
$251$	2.00000			2.00000			1.00000			$0$
$251$	2.00000			2.00000			1.00000			$0$
$252$	1.50000	+	2.59808i	1.50000	+	2.59808i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−4.50000	+	7.79423i	−4.50000	+	7.79423i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$	1.00000	+	1.73205i	1.00000	+	1.73205i
$259$		0			0
$260$		0			0
$261$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$262$		0			0
$263$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$263$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$267$		0			0
$268$	1.50000	+	2.59808i	1.50000	+	2.59808i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−5.00000	−	8.66025i	−5.00000	−	8.66025i
$273$		0			0
$274$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$275$		0			0
$276$	−3.00000			−3.00000
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$277$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$278$	4.00000			4.00000
$279$	−1.00000			−1.00000
$280$		0			0
$281$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$281$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$	−3.00000	+	5.19615i	−3.00000	+	5.19615i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	6.00000			6.00000
$289$	3.00000			3.00000
$290$		0			0
$291$		0			0
$292$	1.50000	−	2.59808i	1.50000	−	2.59808i
$293$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$293$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$	−2.00000			−2.00000
$299$		0			0
$300$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$301$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$302$		0			0
$303$	−1.00000			−1.00000
$304$	2.50000	+	4.33013i	2.50000	+	4.33013i
$305$		0			0
$306$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$	0.500000	+	0.866025i	0.500000	+	0.866025i
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$313$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$317$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$323$	−2.00000			−2.00000
$324$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$331$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$	2.50000	+	4.33013i	2.50000	+	4.33013i
$337$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$337$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$338$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$339$	0.500000	+	0.866025i	0.500000	+	0.866025i
$340$		0			0
$341$		0			0
$342$	1.00000	−	1.73205i	1.00000	−	1.73205i
$343$	1.00000			1.00000
$344$	2.00000	+	3.46410i	2.00000	+	3.46410i
$345$		0			0
$346$	1.00000	−	1.73205i	1.00000	−	1.73205i
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$	−3.00000	+	5.19615i	−3.00000	+	5.19615i
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$	−2.00000			−2.00000
$351$		0			0
$352$		0			0
$353$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$353$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−2.00000			−2.00000
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$367$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$368$	−5.00000			−5.00000
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−3.00000			−3.00000
$373$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$373$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$	1.00000	−	1.73205i	1.00000	−	1.73205i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$383$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$384$	8.00000			8.00000
$385$		0			0
$386$	−2.00000			−2.00000
$387$	0.500000	−	0.866025i	0.500000	−	0.866025i
$388$		0			0
$389$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$389$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$390$		0			0
$391$	1.00000	−	1.73205i	1.00000	−	1.73205i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$397$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$	1.00000			1.00000
$400$	−2.50000	+	4.33013i	−2.50000	+	4.33013i
$401$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$401$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$402$	1.00000	−	1.73205i	1.00000	−	1.73205i
$403$		0			0
$404$	−3.00000			−3.00000
$405$		0			0
$406$	−4.00000			−4.00000
$407$		0			0
$408$	−4.00000	+	6.92820i	−4.00000	+	6.92820i
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$	2.00000			2.00000
$412$	1.50000	+	2.59808i	1.50000	+	2.59808i
$413$		0			0
$414$	1.00000	+	1.73205i	1.00000	+	1.73205i
$415$		0			0
$416$		0			0
$417$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$418$		0			0
$419$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$420$		0			0
$421$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$421$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$426$	4.00000			4.00000
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$431$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$432$	−2.50000	−	4.33013i	−2.50000	−	4.33013i
$433$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$433$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$434$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$435$		0			0
$436$		0			0
$437$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$438$	−2.00000			−2.00000
$439$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$439$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$	1.00000	−	1.73205i	1.00000	−	1.73205i
$447$	0.500000	+	0.866025i	0.500000	+	0.866025i
$448$	3.50000	+	6.06218i	3.50000	+	6.06218i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$	2.00000			2.00000
$451$		0			0
$452$	1.50000	+	2.59808i	1.50000	+	2.59808i
$453$		0			0
$454$	1.00000	−	1.73205i	1.00000	−	1.73205i
$455$		0			0
$456$	2.00000	−	3.46410i	2.00000	−	3.46410i
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$	4.00000			4.00000
$459$	2.00000			2.00000
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$	−5.00000	+	8.66025i	−5.00000	+	8.66025i
$465$		0			0
$466$		0			0
$467$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$467$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$	1.00000			1.00000
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.500000	+	0.866025i	0.500000	+	0.866025i
$476$	−6.00000			−6.00000
$477$		0			0
$478$		0			0
$479$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$479$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$484$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$485$		0			0
$486$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$491$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$492$		0			0
$493$	−2.00000	−	3.46410i	−2.00000	−	3.46410i
$494$		0			0
$495$		0			0
$496$	−5.00000			−5.00000
$497$	1.00000	+	1.73205i	1.00000	+	1.73205i
$498$		0			0
$499$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$499$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$500$		0			0
$501$		0			0
$502$	−2.00000	−	3.46410i	−2.00000	−	3.46410i
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$	2.00000	−	3.46410i	2.00000	−	3.46410i
$505$		0			0
$506$		0			0
$507$	1.00000			1.00000
$508$		0			0
$509$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$509$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$510$		0			0
$511$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$512$	10.0000			10.0000
$513$	−1.00000			−1.00000
$514$		0			0
$515$		0			0
$516$	1.50000	−	2.59808i	1.50000	−	2.59808i
$517$		0			0
$518$		0			0
$519$	−1.00000			−1.00000
$520$		0			0
$521$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$522$	4.00000			4.00000
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$	0.500000	+	0.866025i	0.500000	+	0.866025i
$526$	1.00000	−	1.73205i	1.00000	−	1.73205i
$527$	1.00000	−	1.73205i	1.00000	−	1.73205i
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$	3.00000			3.00000
$533$		0			0
$534$		0			0
$535$		0			0
$536$	2.00000	−	3.46410i	2.00000	−	3.46410i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	−6.00000	+	10.3923i	−6.00000	+	10.3923i
$545$		0			0
$546$		0			0
$547$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$547$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$548$	6.00000			6.00000
$549$		0			0
$550$		0			0
$551$	1.00000	+	1.73205i	1.00000	+	1.73205i
$552$	2.00000	+	3.46410i	2.00000	+	3.46410i
$553$		0			0
$554$	1.00000	−	1.73205i	1.00000	−	1.73205i
$555$		0			0
$556$	−3.00000	−	5.19615i	−3.00000	−	5.19615i
$557$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$557$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$558$	1.00000	+	1.73205i	1.00000	+	1.73205i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.00000			−1.00000
$568$	8.00000			8.00000
$569$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$569$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.00000			−1.00000
$576$	−3.50000	−	6.06218i	−3.50000	−	6.06218i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−3.00000	−	5.19615i	−3.00000	−	5.19615i
$579$	0.500000	+	0.866025i	0.500000	+	0.866025i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−4.00000			−4.00000
$585$		0			0
$586$	−2.00000			−2.00000
$587$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$587$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$	1.50000	+	2.59808i	1.50000	+	2.59808i
$597$		0			0
$598$		0			0
$599$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$599$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$600$	4.00000			4.00000
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$	2.00000			2.00000
$603$	−1.00000			−1.00000
$604$		0			0
$605$		0			0
$606$	1.00000	+	1.73205i	1.00000	+	1.73205i
$607$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$607$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$608$	3.00000	−	5.19615i	3.00000	−	5.19615i
$609$	1.00000	+	1.73205i	1.00000	+	1.73205i
$610$		0			0
$611$		0			0
$612$	6.00000			6.00000
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$	1.00000	−	1.73205i	1.00000	−	1.73205i
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$	0.500000	−	0.866025i	0.500000	−	0.866025i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$	1.00000	−	1.73205i	1.00000	−	1.73205i
$627$		0			0
$628$		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
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$640$		0			0
$641$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$641$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$643$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$644$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$645$		0			0
$646$	2.00000	+	3.46410i	2.00000	+	3.46410i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$649$		0			0
$650$		0			0
$651$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.500000	+	0.866025i	0.500000	+	0.866025i
$658$		0			0
$659$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$659$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$660$		0			0
$661$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$661$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$662$	1.00000	−	1.73205i	1.00000	−	1.73205i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−2.00000			−2.00000
$668$		0			0
$669$	−1.00000			−1.00000
$670$		0			0
$671$		0			0
$672$	3.00000	−	5.19615i	3.00000	−	5.19615i
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$		0			0
$675$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$676$	3.00000			3.00000
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$678$	1.00000	−	1.73205i	1.00000	−	1.73205i
$679$		0			0
$680$		0			0
$681$	−1.00000			−1.00000
$682$		0			0
$683$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$683$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$684$	−3.00000			−3.00000
$685$		0			0
$686$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$687$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$688$	2.50000	−	4.33013i	2.50000	−	4.33013i
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	−3.00000			−3.00000
$693$		0			0
$694$		0			0
$695$		0			0
$696$	8.00000			8.00000
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.50000	+	2.59808i	1.50000	+	2.59808i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$707$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$708$		0			0
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$714$	2.00000	+	3.46410i	2.00000	+	3.46410i
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	2.00000			2.00000			1.00000			$0$
$719$	2.00000			2.00000			1.00000			$0$
$720$		0			0
$721$	1.00000			1.00000
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$726$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$727$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$727$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$	1.00000	+	1.73205i	1.00000	+	1.73205i
$732$		0			0
$733$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$733$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$734$		0			0
$735$		0			0
$736$	3.00000	+	5.19615i	3.00000	+	5.19615i
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$	2.00000	+	3.46410i	2.00000	+	3.46410i
$745$		0			0
$746$	−2.00000			−2.00000
$747$		0			0
$748$		0			0
$749$		0			0
$750$		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
$753$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$754$		0			0
$755$		0			0
$756$	−3.00000			−3.00000
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$761$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	2.00000			2.00000
$767$		0			0
$768$	−4.50000	−	7.79423i	−4.50000	−	7.79423i
$769$	−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
$771$		0			0
$772$	1.50000	+	2.59808i	1.50000	+	2.59808i
$773$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$773$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$774$	−2.00000			−2.00000
$775$	−1.00000			−1.00000
$776$		0			0
$777$		0			0
$778$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−4.00000			−4.00000
$783$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$787$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$788$		0			0
$789$	−1.00000			−1.00000
$790$		0			0
$791$	1.00000			1.00000
$792$		0			0
$793$		0			0
$794$	1.00000	+	1.73205i	1.00000	+	1.73205i
$795$		0			0
$796$		0			0
$797$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$797$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$798$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$799$		0			0
$800$	6.00000			6.00000
$801$		0			0
$802$	−2.00000			−2.00000
$803$		0			0
$804$	−3.00000			−3.00000
$805$		0			0
$806$		0			0
$807$		0			0
$808$	2.00000	+	3.46410i	2.00000	+	3.46410i
$809$	−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			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	3.00000	+	5.19615i	3.00000	+	5.19615i
$813$		0			0
$814$		0			0
$815$		0			0
$816$	10.0000			10.0000
$817$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$821$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$822$	−2.00000	−	3.46410i	−2.00000	−	3.46410i
$823$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$823$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$824$	2.00000	−	3.46410i	2.00000	−	3.46410i
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	1.50000	−	2.59808i	1.50000	−	2.59808i
$829$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$829$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$	−1.00000			−1.00000
$832$		0			0
$833$		0			0
$834$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$835$		0			0
$836$		0			0
$837$	0.500000	−	0.866025i	0.500000	−	0.866025i
$838$	−2.00000			−2.00000
$839$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$839$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$842$	1.00000	−	1.73205i	1.00000	−	1.73205i
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−1.00000			−1.00000
$848$		0			0
$849$		0			0
$850$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$851$		0			0
$852$	−3.00000	−	5.19615i	−3.00000	−	5.19615i
$853$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$853$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$859$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$860$		0			0
$861$		0			0
$862$	1.00000	+	1.73205i	1.00000	+	1.73205i
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$	−3.00000	+	5.19615i	−3.00000	+	5.19615i
$865$		0			0
$866$	1.00000	+	1.73205i	1.00000	+	1.73205i
$867$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$868$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	2.00000			2.00000
$875$		0			0
$876$	1.50000	+	2.59808i	1.50000	+	2.59808i
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$	1.00000	−	1.73205i	1.00000	−	1.73205i
$879$	0.500000	+	0.866025i	0.500000	+	0.866025i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$887$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−3.00000			−3.00000
$893$		0			0
$894$	1.00000	−	1.73205i	1.00000	−	1.73205i
$895$		0			0
$896$	4.00000	−	6.92820i	4.00000	−	6.92820i
$897$		0			0
$898$		0			0
$899$	−2.00000			−2.00000
$900$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$901$		0			0
$902$		0			0
$903$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$904$	2.00000	−	3.46410i	2.00000	−	3.46410i
$905$		0			0
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$	−3.00000			−3.00000
$909$	0.500000	−	0.866025i	0.500000	−	0.866025i
$910$		0			0
$911$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$911$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$912$	−5.00000			−5.00000
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−3.00000	−	5.19615i	−3.00000	−	5.19615i
$917$		0			0
$918$	−2.00000	−	3.46410i	−2.00000	−	3.46410i
$919$	−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
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1.00000			−1.00000
$928$	12.0000			12.0000
$929$		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
$932$		0			0
$933$		0			0
$934$	1.00000	+	1.73205i	1.00000	+	1.73205i
$935$		0			0
$936$		0			0
$937$	2.00000			2.00000			1.00000			$0$
$937$	2.00000			2.00000			1.00000			$0$
$938$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$939$	−1.00000			−1.00000
$940$		0			0
$941$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$941$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$		0			0
$950$	1.00000	−	1.73205i	1.00000	−	1.73205i
$951$	−1.00000			−1.00000
$952$	4.00000	+	6.92820i	4.00000	+	6.92820i
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$	1.00000	−	1.73205i	1.00000	−	1.73205i
$959$	1.00000	−	1.73205i	1.00000	−	1.73205i
$960$		0			0
$961$		0			0
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$	2.00000			2.00000
$967$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$967$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$968$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$969$	1.00000	−	1.73205i	1.00000	−	1.73205i
$970$		0			0
$971$	2.00000			2.00000			1.00000			$0$
$971$	2.00000			2.00000			1.00000			$0$
$972$	3.00000			3.00000
$973$	−2.00000			−2.00000
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−2.00000			−2.00000
$983$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$983$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$	−4.00000	+	6.92820i	−4.00000	+	6.92820i
$987$		0			0
$988$		0			0
$989$	1.00000			1.00000
$990$		0			0
$991$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$992$	3.00000	+	5.19615i	3.00000	+	5.19615i
$993$	−1.00000			−1.00000
$994$	2.00000	−	3.46410i	2.00000	−	3.46410i
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$	4.00000			4.00000
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3447.1.f.a.382.1	✓	2
9.7	even	3	inner	3447.1.f.a.2680.1	yes	2
383.382	odd	2	CM	3447.1.f.a.382.1	✓	2
3447.2680	odd	6	inner	3447.1.f.a.2680.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3447.1.f.a.382.1	✓	2	1.1	even	1	trivial
3447.1.f.a.382.1	✓	2	383.382	odd	2	CM
3447.1.f.a.2680.1	yes	2	9.7	even	3	inner
3447.1.f.a.2680.1	yes	2	3447.2680	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:	\( N \)	\(=\)	\( 3447 = 3^{2} \cdot 383 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3447.f (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-1.50000 + 2.59808i) q^{4} +2.00000 q^{6} +(0.500000 + 0.866025i) q^{7} +4.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-0.500000 + 0.866025i) q^{3} +(-1.50000 + 2.59808i) q^{4} +2.00000 q^{6} +(0.500000 + 0.866025i) q^{7} +4.00000 q^{8} +(-0.500000 - 0.866025i) q^{9} +(-1.50000 - 2.59808i) q^{12} +(1.00000 - 1.73205i) q^{14} +(-2.50000 - 4.33013i) q^{16} +2.00000 q^{17} +(-1.00000 + 1.73205i) q^{18} -1.00000 q^{19} -1.00000 q^{21} +(0.500000 - 0.866025i) q^{23} +(-2.00000 + 3.46410i) q^{24} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{27} -3.00000 q^{28} +(-1.00000 - 1.73205i) q^{29} +(0.500000 - 0.866025i) q^{31} +(-3.00000 + 5.19615i) q^{32} +(-2.00000 - 3.46410i) q^{34} +3.00000 q^{36} +(1.00000 + 1.73205i) q^{38} +(1.00000 + 1.73205i) q^{42} +(0.500000 + 0.866025i) q^{43} -2.00000 q^{46} +5.00000 q^{48} +(-1.00000 + 1.73205i) q^{50} +(-1.00000 + 1.73205i) q^{51} +(-1.00000 - 1.73205i) q^{54} +(2.00000 + 3.46410i) q^{56} +(0.500000 - 0.866025i) q^{57} +(-2.00000 + 3.46410i) q^{58} -2.00000 q^{62} +(0.500000 - 0.866025i) q^{63} +7.00000 q^{64} +(0.500000 - 0.866025i) q^{67} +(-3.00000 + 5.19615i) q^{68} +(0.500000 + 0.866025i) q^{69} +2.00000 q^{71} +(-2.00000 - 3.46410i) q^{72} -1.00000 q^{73} +1.00000 q^{75} +(1.50000 - 2.59808i) q^{76} +(-0.500000 + 0.866025i) q^{81} +(1.50000 - 2.59808i) q^{84} +(1.00000 - 1.73205i) q^{86} +2.00000 q^{87} +(1.50000 + 2.59808i) q^{92} +(0.500000 + 0.866025i) q^{93} +(-3.00000 - 5.19615i) q^{96} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - q^{3} - 3 q^{4} + 4 q^{6} + q^{7} + 8 q^{8} - q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - q^3 - 3 * q^4 + 4 * q^6 + q^7 + 8 * q^8 - q^9	\( 2 q - 2 q^{2} - q^{3} - 3 q^{4} + 4 q^{6} + q^{7} + 8 q^{8} - q^{9} - 3 q^{12} + 2 q^{14} - 5 q^{16} + 4 q^{17} - 2 q^{18} - 2 q^{19} - 2 q^{21} + q^{23} - 4 q^{24} - q^{25} + 2 q^{27} - 6 q^{28} - 2 q^{29} + q^{31} - 6 q^{32} - 4 q^{34} + 6 q^{36} + 2 q^{38} + 2 q^{42} + q^{43} - 4 q^{46} + 10 q^{48} - 2 q^{50} - 2 q^{51} - 2 q^{54} + 4 q^{56} + q^{57} - 4 q^{58} - 4 q^{62} + q^{63} + 14 q^{64} + q^{67} - 6 q^{68} + q^{69} + 4 q^{71} - 4 q^{72} - 2 q^{73} + 2 q^{75} + 3 q^{76} - q^{81} + 3 q^{84} + 2 q^{86} + 4 q^{87} + 3 q^{92} + q^{93} - 6 q^{96}+O(q^{100}) \) 2 * q - 2 * q^2 - q^3 - 3 * q^4 + 4 * q^6 + q^7 + 8 * q^8 - q^9 - 3 * q^12 + 2 * q^14 - 5 * q^16 + 4 * q^17 - 2 * q^18 - 2 * q^19 - 2 * q^21 + q^23 - 4 * q^24 - q^25 + 2 * q^27 - 6 * q^28 - 2 * q^29 + q^31 - 6 * q^32 - 4 * q^34 + 6 * q^36 + 2 * q^38 + 2 * q^42 + q^43 - 4 * q^46 + 10 * q^48 - 2 * q^50 - 2 * q^51 - 2 * q^54 + 4 * q^56 + q^57 - 4 * q^58 - 4 * q^62 + q^63 + 14 * q^64 + q^67 - 6 * q^68 + q^69 + 4 * q^71 - 4 * q^72 - 2 * q^73 + 2 * q^75 + 3 * q^76 - q^81 + 3 * q^84 + 2 * q^86 + 4 * q^87 + 3 * q^92 + q^93 - 6 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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