LMFDB - Embedded newform 1161.1.i.b.1025.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1161,1,Mod(350,1161)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1161, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1161.350");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1161 = 3^{3} \cdot 43 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1161.i (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:	$0.579414479667$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.49923.1

Embedding invariants

Embedding label			1025.1
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1161.1025
Dual form			1161.1.i.b.350.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} -1.00000 q^{4} +(1.22474 - 0.707107i) q^{5} +O(q^{10})$	\(q-1.41421i q^{2} -1.00000 q^{4} +(1.22474 - 0.707107i) q^{5} +(-1.00000 - 1.73205i) q^{10} +(-0.500000 + 0.866025i) q^{13} -1.00000 q^{16} +(-0.500000 - 0.866025i) q^{19} +(-1.22474 + 0.707107i) q^{20} +(1.22474 - 0.707107i) q^{23} +(0.500000 - 0.866025i) q^{25} +(1.22474 + 0.707107i) q^{26} +(0.500000 + 0.866025i) q^{31} +1.41421i q^{32} +(-1.22474 + 0.707107i) q^{38} +1.41421i q^{41} -1.00000 q^{43} +(-1.00000 - 1.73205i) q^{46} -1.41421i q^{47} +(0.500000 + 0.866025i) q^{49} +(-1.22474 - 0.707107i) q^{50} +(0.500000 - 0.866025i) q^{52} +(-1.22474 + 0.707107i) q^{53} +(0.500000 - 0.866025i) q^{61} +(1.22474 - 0.707107i) q^{62} +1.00000 q^{64} +1.41421i q^{65} +(-1.22474 - 0.707107i) q^{71} +(-1.00000 + 1.73205i) q^{73} +(0.500000 + 0.866025i) q^{76} +(-0.500000 + 0.866025i) q^{79} +(-1.22474 + 0.707107i) q^{80} +2.00000 q^{82} +1.41421i q^{86} +(1.22474 - 0.707107i) q^{89} +(-1.22474 + 0.707107i) q^{92} -2.00000 q^{94} +(-1.22474 - 0.707107i) q^{95} +1.00000 q^{97} +(1.22474 - 0.707107i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} - 4 q^{10} - 2 q^{13} - 4 q^{16} - 2 q^{19} + 2 q^{25} + 2 q^{31} - 4 q^{43} - 4 q^{46} + 2 q^{49} + 2 q^{52} + 2 q^{61} + 4 q^{64} - 4 q^{73} + 2 q^{76} - 2 q^{79} + 8 q^{82} - 8 q^{94} + 4 q^{97}+O(q^{100}) $ 4 * q - 4 * q^4 - 4 * q^10 - 2 * q^13 - 4 * q^16 - 2 * q^19 + 2 * q^25 + 2 * q^31 - 4 * q^43 - 4 * q^46 + 2 * q^49 + 2 * q^52 + 2 * q^61 + 4 * q^64 - 4 * q^73 + 2 * q^76 - 2 * q^79 + 8 * q^82 - 8 * q^94 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1161\mathbb{Z}\right)^\times$.

$n$	$433$	$947$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$2$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000			−1.00000
$5$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$5$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$6$		0			0
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$		0			0
$9$		0			0
$10$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$13$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	−1.00000			−1.00000
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$19$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$20$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$21$		0			0
$22$		0			0
$23$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$23$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$24$		0			0
$25$	0.500000	−	0.866025i	0.500000	−	0.866025i
$26$	1.22474	+	0.707107i	1.22474	+	0.707107i
$27$		0			0
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$31$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$32$			1.41421i			1.41421i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$37$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$38$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$39$		0			0
$40$		0			0
$41$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$41$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$42$		0			0
$43$	−1.00000			−1.00000
$43$	−1.00000			−1.00000
$44$		0			0
$45$		0			0
$46$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$47$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$47$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$48$		0			0
$49$	0.500000	+	0.866025i	0.500000	+	0.866025i
$50$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$51$		0			0
$52$	0.500000	−	0.866025i	0.500000	−	0.866025i
$53$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$53$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		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			$0$
$62$	1.22474	−	0.707107i	1.22474	−	0.707107i
$63$		0			0
$64$	1.00000			1.00000
$65$			1.41421i			1.41421i
$66$		0			0
$67$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$67$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$71$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$72$		0			0
$73$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$73$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$		0			0
$76$	0.500000	+	0.866025i	0.500000	+	0.866025i
$77$		0			0
$78$		0			0
$79$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$79$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$80$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$81$		0			0
$82$	2.00000			2.00000
$83$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$83$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$84$		0			0
$85$		0			0
$86$			1.41421i			1.41421i
$87$		0			0
$88$		0			0
$89$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$89$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$90$		0			0
$91$		0			0
$92$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$93$		0			0
$94$	−2.00000			−2.00000
$95$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$96$		0			0
$97$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$97$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$98$	1.22474	−	0.707107i	1.22474	−	0.707107i
$99$		0			0
$100$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$103$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$	1.00000	+	1.73205i	1.00000	+	1.73205i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$109$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$113$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$114$		0			0
$115$	1.00000	−	1.73205i	1.00000	−	1.73205i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000			1.00000
$122$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$123$		0			0
$124$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$125$		0			0
$126$		0			0
$127$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$127$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$128$		0			0
$129$		0			0
$130$	2.00000			2.00000
$131$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$131$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		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			0
$141$		0			0
$142$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$143$		0			0
$144$		0			0
$145$		0			0
$146$	2.44949	+	1.41421i	2.44949	+	1.41421i
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$151$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.22474	+	0.707107i	1.22474	+	0.707107i
$156$		0			0
$157$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$157$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$158$	1.22474	+	0.707107i	1.22474	+	0.707107i
$159$		0			0
$160$	1.00000	+	1.73205i	1.00000	+	1.73205i
$161$		0			0
$162$		0			0
$163$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$163$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$164$		−	1.41421i		−	1.41421i
$165$		0			0
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$	1.00000			1.00000
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$179$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$179$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$180$		0			0
$181$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$181$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$			1.41421i			1.41421i
$189$		0			0
$190$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$191$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$191$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		−	1.41421i		−	1.41421i
$195$		0			0
$196$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$197$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$197$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$198$		0			0
$199$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$199$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	1.00000	+	1.73205i	1.00000	+	1.73205i
$206$	1.22474	+	0.707107i	1.22474	+	0.707107i
$207$		0			0
$208$	0.500000	−	0.866025i	0.500000	−	0.866025i
$209$		0			0
$210$		0			0
$211$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$211$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$212$	1.22474	−	0.707107i	1.22474	−	0.707107i
$213$		0			0
$214$		0			0
$215$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$216$		0			0
$217$		0			0
$218$	2.44949	−	1.41421i	2.44949	−	1.41421i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$223$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$		0			0
$226$	−2.00000			−2.00000
$227$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$227$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$228$		0			0
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$	−2.44949	−	1.41421i	−2.44949	−	1.41421i
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$239$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$240$		0			0
$241$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$241$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$242$		−	1.41421i		−	1.41421i
$243$		0			0
$244$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$245$	1.22474	+	0.707107i	1.22474	+	0.707107i
$246$		0			0
$247$	1.00000			1.00000
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$251$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$252$		0			0
$253$		0			0
$254$		−	1.41421i		−	1.41421i
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		−	1.41421i		−	1.41421i
$261$		0			0
$262$	2.00000			2.00000
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$269$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$270$		0			0
$271$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$271$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$277$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$278$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$279$		0			0
$280$		0			0
$281$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$281$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$282$		0			0
$283$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$283$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$284$	1.22474	+	0.707107i	1.22474	+	0.707107i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$	1.00000	−	1.73205i	1.00000	−	1.73205i
$293$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$293$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$			1.41421i			1.41421i
$300$		0			0
$301$		0			0
$302$		−	1.41421i		−	1.41421i
$303$		0			0
$304$	0.500000	+	0.866025i	0.500000	+	0.866025i
$305$		−	1.41421i		−	1.41421i
$306$		0			0
$307$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$307$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$308$		0			0
$309$		0			0
$310$	1.00000	−	1.73205i	1.00000	−	1.73205i
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$313$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$314$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$315$		0			0
$316$	0.500000	−	0.866025i	0.500000	−	0.866025i
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$	1.22474	−	0.707107i	1.22474	−	0.707107i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		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
$331$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$331$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$337$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$347$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$348$		0			0
$349$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$349$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$353$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$354$		0			0
$355$	−2.00000			−2.00000
$356$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$357$		0			0
$358$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$359$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$359$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$360$		0			0
$361$		0			0
$362$	1.22474	+	0.707107i	1.22474	+	0.707107i
$363$		0			0
$364$		0			0
$365$			2.82843i			2.82843i
$366$		0			0
$367$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$367$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$368$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	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
$378$		0			0
$379$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$380$	1.22474	+	0.707107i	1.22474	+	0.707107i
$381$		0			0
$382$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$383$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$383$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−1.00000			−1.00000
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$395$			1.41421i			1.41421i
$396$		0			0
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$			1.41421i			1.41421i
$399$		0			0
$400$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$401$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$401$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$402$		0			0
$403$	−1.00000			−1.00000
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$	2.44949	−	1.41421i	2.44949	−	1.41421i
$411$		0			0
$412$	0.500000	−	0.866025i	0.500000	−	0.866025i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$417$		0			0
$418$		0			0
$419$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$419$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$421$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$422$			1.41421i			1.41421i
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	1.00000	+	1.73205i	1.00000	+	1.73205i
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$433$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$434$		0			0
$435$		0			0
$436$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$437$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$438$		0			0
$439$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$439$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$443$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$444$		0			0
$445$	1.00000	−	1.73205i	1.00000	−	1.73205i
$446$			1.41421i			1.41421i
$447$		0			0
$448$		0			0
$449$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$449$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$			1.41421i			1.41421i
$453$		0			0
$454$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$		0			0
$460$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$463$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$		0			0
$470$	−2.44949	+	1.41421i	−2.44949	+	1.41421i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.00000			−1.00000
$476$		0			0
$477$		0			0
$478$	1.00000	+	1.73205i	1.00000	+	1.73205i
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.00000			−1.00000
$485$	1.22474	−	0.707107i	1.22474	−	0.707107i
$486$		0			0
$487$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$487$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$488$		0			0
$489$		0			0
$490$	1.00000	−	1.73205i	1.00000	−	1.73205i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		−	1.41421i		−	1.41421i
$495$		0			0
$496$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$497$		0			0
$498$		0			0
$499$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$499$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$500$		0			0
$501$		0			0
$502$	1.00000	−	1.73205i	1.00000	−	1.73205i
$503$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$503$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−1.00000			−1.00000
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$		−	1.41421i		−	1.41421i
$513$		0			0
$514$		0			0
$515$			1.41421i			1.41421i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$521$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$522$		0			0
$523$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$523$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$524$		−	1.41421i		−	1.41421i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.500000	−	0.866025i	0.500000	−	0.866025i
$530$	2.44949	+	1.41421i	2.44949	+	1.41421i
$531$		0			0
$532$		0			0
$533$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	2.00000			2.00000
$539$		0			0
$540$		0			0
$541$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$541$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$542$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$543$		0			0
$544$		0			0
$545$	2.44949	+	1.41421i	2.44949	+	1.41421i
$546$		0			0
$547$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$547$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$555$		0			0
$556$	0.500000	+	0.866025i	0.500000	+	0.866025i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$	0.500000	−	0.866025i	0.500000	−	0.866025i
$560$		0			0
$561$		0			0
$562$	1.00000	+	1.73205i	1.00000	+	1.73205i
$563$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$563$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$564$		0			0
$565$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$569$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$570$		0			0
$571$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$571$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	1.41421i		−	1.41421i
$576$		0			0
$577$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$577$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$578$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−2.00000			−2.00000
$587$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$587$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$588$		0			0
$589$	0.500000	−	0.866025i	0.500000	−	0.866025i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$593$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	2.00000			2.00000
$599$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$599$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$600$		0			0
$601$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$601$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$	−1.00000			−1.00000
$605$	1.22474	−	0.707107i	1.22474	−	0.707107i
$606$		0			0
$607$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$607$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$608$	1.22474	−	0.707107i	1.22474	−	0.707107i
$609$		0			0
$610$	−2.00000			−2.00000
$611$	1.22474	+	0.707107i	1.22474	+	0.707107i
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	1.22474	−	0.707107i	1.22474	−	0.707107i
$615$		0			0
$616$		0			0
$617$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.500000	+	0.866025i	0.500000	+	0.866025i
$626$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$627$		0			0
$628$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$629$		0			0
$630$		0			0
$631$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$631$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.22474	−	0.707107i	1.22474	−	0.707107i
$636$		0			0
$637$	−1.00000			−1.00000
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$	1.22474	−	0.707107i	1.22474	−	0.707107i
$651$		0			0
$652$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$653$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$653$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$654$		0			0
$655$	1.00000	+	1.73205i	1.00000	+	1.73205i
$656$		−	1.41421i		−	1.41421i
$657$		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
$661$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$661$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$662$	1.22474	+	0.707107i	1.22474	+	0.707107i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$673$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$677$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	1.00000			1.00000
$689$		−	1.41421i		−	1.41421i
$690$		0			0
$691$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$691$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$	1.00000	+	1.73205i	1.00000	+	1.73205i
$695$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$696$		0			0
$697$		0			0
$698$	1.22474	−	0.707107i	1.22474	−	0.707107i
$699$		0			0
$700$		0			0
$701$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$701$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$707$		0			0
$708$		0			0
$709$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$709$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$710$			2.82843i			2.82843i
$711$		0			0
$712$		0			0
$713$	1.22474	+	0.707107i	1.22474	+	0.707107i
$714$		0			0
$715$		0			0
$716$	1.22474	+	0.707107i	1.22474	+	0.707107i
$717$		0			0
$718$	1.00000	−	1.73205i	1.00000	−	1.73205i
$719$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$719$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$	0.500000	−	0.866025i	0.500000	−	0.866025i
$725$		0			0
$726$		0			0
$727$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$727$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$728$		0			0
$729$		0			0
$730$	4.00000			4.00000
$731$		0			0
$732$		0			0
$733$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$733$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$734$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$735$		0			0
$736$	1.00000	+	1.73205i	1.00000	+	1.73205i
$737$		0			0
$738$		0			0
$739$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		0			0
$745$		0			0
$746$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$			1.41421i			1.41421i
$753$		0			0
$754$		0			0
$755$	1.22474	−	0.707107i	1.22474	−	0.707107i
$756$		0			0
$757$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$757$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$758$			1.41421i			1.41421i
$759$		0			0
$760$		0			0
$761$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$761$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$762$		0			0
$763$		0			0
$764$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$765$		0			0
$766$	2.00000			2.00000
$767$		0			0
$768$		0			0
$769$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$769$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$773$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$774$		0			0
$775$	1.00000			1.00000
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.22474	−	0.707107i	1.22474	−	0.707107i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$785$		−	1.41421i		−	1.41421i
$786$		0			0
$787$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$787$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$788$	1.22474	+	0.707107i	1.22474	+	0.707107i
$789$		0			0
$790$	2.00000			2.00000
$791$		0			0
$792$		0			0
$793$	0.500000	+	0.866025i	0.500000	+	0.866025i
$794$		0			0
$795$		0			0
$796$	1.00000			1.00000
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$	1.22474	+	0.707107i	1.22474	+	0.707107i
$801$		0			0
$802$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$803$		0			0
$804$		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
$811$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$811$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		−	1.41421i		−	1.41421i
$816$		0			0
$817$	0.500000	+	0.866025i	0.500000	+	0.866025i
$818$		0			0
$819$		0			0
$820$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$821$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$821$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$822$		0			0
$823$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$823$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$827$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$828$		0			0
$829$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$829$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$	2.00000			2.00000
$839$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$839$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$840$		0			0
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$843$		0			0
$844$	1.00000			1.00000
$845$		0			0
$846$		0			0
$847$		0			0
$848$	1.22474	−	0.707107i	1.22474	−	0.707107i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$853$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$857$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$858$		0			0
$859$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$859$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$860$	1.22474	−	0.707107i	1.22474	−	0.707107i
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$875$		0			0
$876$		0			0
$877$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$877$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$878$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$879$		0			0
$880$		0			0
$881$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$881$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$882$		0			0
$883$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$883$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$887$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$887$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$	−2.44949	−	1.41421i	−2.44949	−	1.41421i
$891$		0			0
$892$	1.00000			1.00000
$893$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$894$		0			0
$895$	−2.00000			−2.00000
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$			1.41421i			1.41421i
$906$		0			0
$907$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$907$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$908$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$909$		0			0
$910$		0			0
$911$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$911$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$919$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.22474	−	0.707107i	1.22474	−	0.707107i
$924$		0			0
$925$		0			0
$926$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$927$		0			0
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	0.500000	−	0.866025i	0.500000	−	0.866025i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$937$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$938$		0			0
$939$		0			0
$940$	1.00000	+	1.73205i	1.00000	+	1.73205i
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$	1.00000	+	1.73205i	1.00000	+	1.73205i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$		0			0
$949$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$950$			1.41421i			1.41421i
$951$		0			0
$952$		0			0
$953$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$953$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$954$		0			0
$955$	1.00000	−	1.73205i	1.00000	−	1.73205i
$956$	1.22474	−	0.707107i	1.22474	−	0.707107i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		0			0
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$967$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$971$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$971$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$977$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$977$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$978$		0			0
$979$		0			0
$980$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$981$		0			0
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$	−2.00000			−2.00000
$986$		0			0
$987$		0			0
$988$	−1.00000			−1.00000
$989$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$990$		0			0
$991$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$992$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$993$		0			0
$994$		0			0
$995$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$996$		0			0
$997$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$997$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$998$	1.22474	+	0.707107i	1.22474	+	0.707107i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1161.1.i.b.1025.1	yes	4
3.2	odd	2	inner	1161.1.i.b.1025.2	yes	4
9.2	odd	6		3483.1.o.b.1025.2		4
9.4	even	3		3483.1.p.b.3347.2		4
9.5	odd	6		3483.1.p.b.3347.1		4
9.7	even	3		3483.1.o.b.1025.1		4
43.6	even	3	inner	1161.1.i.b.350.1	✓	4
129.92	odd	6	inner	1161.1.i.b.350.2	yes	4
387.49	even	3		3483.1.o.b.350.2		4
387.92	odd	6		3483.1.p.b.1511.2		4
387.178	even	3		3483.1.p.b.1511.1		4
387.221	odd	6		3483.1.o.b.350.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1161.1.i.b.350.1	✓	4	43.6	even	3	inner
1161.1.i.b.350.2	yes	4	129.92	odd	6	inner
1161.1.i.b.1025.1	yes	4	1.1	even	1	trivial
1161.1.i.b.1025.2	yes	4	3.2	odd	2	inner
3483.1.o.b.350.1		4	387.221	odd	6
3483.1.o.b.350.2		4	387.49	even	3
3483.1.o.b.1025.1		4	9.7	even	3
3483.1.o.b.1025.2		4	9.2	odd	6
3483.1.p.b.1511.1		4	387.178	even	3
3483.1.p.b.1511.2		4	387.92	odd	6
3483.1.p.b.3347.1		4	9.5	odd	6
3483.1.p.b.3347.2		4	9.4	even	3

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:	\( N \)	\(=\)	\( 1161 = 3^{3} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1161.i (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -1.00000 q^{4} +(1.22474 - 0.707107i) q^{5} +O(q^{10})\)	\(q-1.41421i q^{2} -1.00000 q^{4} +(1.22474 - 0.707107i) q^{5} +(-1.00000 - 1.73205i) q^{10} +(-0.500000 + 0.866025i) q^{13} -1.00000 q^{16} +(-0.500000 - 0.866025i) q^{19} +(-1.22474 + 0.707107i) q^{20} +(1.22474 - 0.707107i) q^{23} +(0.500000 - 0.866025i) q^{25} +(1.22474 + 0.707107i) q^{26} +(0.500000 + 0.866025i) q^{31} +1.41421i q^{32} +(-1.22474 + 0.707107i) q^{38} +1.41421i q^{41} -1.00000 q^{43} +(-1.00000 - 1.73205i) q^{46} -1.41421i q^{47} +(0.500000 + 0.866025i) q^{49} +(-1.22474 - 0.707107i) q^{50} +(0.500000 - 0.866025i) q^{52} +(-1.22474 + 0.707107i) q^{53} +(0.500000 - 0.866025i) q^{61} +(1.22474 - 0.707107i) q^{62} +1.00000 q^{64} +1.41421i q^{65} +(-1.22474 - 0.707107i) q^{71} +(-1.00000 + 1.73205i) q^{73} +(0.500000 + 0.866025i) q^{76} +(-0.500000 + 0.866025i) q^{79} +(-1.22474 + 0.707107i) q^{80} +2.00000 q^{82} +1.41421i q^{86} +(1.22474 - 0.707107i) q^{89} +(-1.22474 + 0.707107i) q^{92} -2.00000 q^{94} +(-1.22474 - 0.707107i) q^{95} +1.00000 q^{97} +(1.22474 - 0.707107i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} - 4 q^{10} - 2 q^{13} - 4 q^{16} - 2 q^{19} + 2 q^{25} + 2 q^{31} - 4 q^{43} - 4 q^{46} + 2 q^{49} + 2 q^{52} + 2 q^{61} + 4 q^{64} - 4 q^{73} + 2 q^{76} - 2 q^{79} + 8 q^{82} - 8 q^{94} + 4 q^{97}+O(q^{100}) \) 4 * q - 4 * q^4 - 4 * q^10 - 2 * q^13 - 4 * q^16 - 2 * q^19 + 2 * q^25 + 2 * q^31 - 4 * q^43 - 4 * q^46 + 2 * q^49 + 2 * q^52 + 2 * q^61 + 4 * q^64 - 4 * q^73 + 2 * q^76 - 2 * q^79 + 8 * q^82 - 8 * q^94 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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