LMFDB - Embedded newform 3267.1.h.a.1693.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3267,1,Mod(604,3267)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3267, 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("3267.604");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3267 = 3^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3267.h (of order $6$, degree $2$, not 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:	$1.63044539627$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1089)
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.107811.1

Embedding invariants

Embedding label			1693.1
Root			$-1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3267.1693
Dual form			3267.1.h.a.604.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.22474 - 0.707107i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(-1.22474 - 0.707107i) q^{7} +O(q^{10})$	$q+(-1.22474 - 0.707107i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(-1.22474 - 0.707107i) q^{7} +1.41421i q^{10} +(1.00000 + 1.73205i) q^{14} +(0.500000 - 0.866025i) q^{16} +(0.500000 - 0.866025i) q^{20} -1.41421i q^{28} +(-1.22474 - 0.707107i) q^{29} +(-0.500000 - 0.866025i) q^{31} +(-1.22474 + 0.707107i) q^{32} +1.41421i q^{35} -1.00000 q^{37} +(-0.500000 + 0.866025i) q^{47} +(0.500000 + 0.866025i) q^{49} -1.00000 q^{53} +(1.00000 + 1.73205i) q^{58} +(0.500000 + 0.866025i) q^{59} +(-1.22474 - 0.707107i) q^{61} +1.41421i q^{62} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{67} +(1.00000 - 1.73205i) q^{70} +1.00000 q^{71} +1.41421i q^{73} +(1.22474 + 0.707107i) q^{74} -1.00000 q^{80} +(1.22474 + 0.707107i) q^{83} +(1.22474 - 0.707107i) q^{94} +(-0.500000 + 0.866025i) q^{97} -1.41421i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{5}+O(q^{10}) $ 4 * q + 2 * q^4 - 2 * q^5	$ 4 q + 2 q^{4} - 2 q^{5} + 4 q^{14} + 2 q^{16} + 2 q^{20} - 2 q^{31} - 4 q^{37} - 2 q^{47} + 2 q^{49} - 4 q^{53} + 4 q^{58} + 2 q^{59} + 4 q^{64} + 2 q^{67} + 4 q^{70} + 4 q^{71} - 4 q^{80} - 2 q^{97}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^5 + 4 * q^14 + 2 * q^16 + 2 * q^20 - 2 * q^31 - 4 * q^37 - 2 * q^47 + 2 * q^49 - 4 * q^53 + 4 * q^58 + 2 * q^59 + 4 * q^64 + 2 * q^67 + 4 * q^70 + 4 * q^71 - 4 * q^80 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times$.

$n$	$244$	$3026$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$2$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.500000	+	0.866025i
$5$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$5$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$6$		0			0
$7$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$7$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$8$		0			0
$9$		0			0
$10$			1.41421i			1.41421i
$11$		0			0
$11$		0			0
$12$		0			0
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$	1.00000	+	1.73205i	1.00000	+	1.73205i
$15$		0			0
$16$	0.500000	−	0.866025i	0.500000	−	0.866025i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$	0.500000	−	0.866025i	0.500000	−	0.866025i
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		−	1.41421i		−	1.41421i
$29$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$29$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$30$		0			0
$31$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$31$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$32$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$33$		0			0
$34$		0			0
$35$			1.41421i			1.41421i
$36$		0			0
$37$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$37$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$41$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$47$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$48$		0			0
$49$	0.500000	+	0.866025i	0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$53$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.00000	+	1.73205i	1.00000	+	1.73205i
$59$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$59$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$61$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$62$			1.41421i			1.41421i
$63$		0			0
$64$	1.00000			1.00000
$65$		0			0
$66$		0			0
$67$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$67$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$68$		0			0
$69$		0			0
$70$	1.00000	−	1.73205i	1.00000	−	1.73205i
$71$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$71$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$72$		0			0
$73$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$73$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$74$	1.22474	+	0.707107i	1.22474	+	0.707107i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$	−1.00000			−1.00000
$81$		0			0
$82$		0			0
$83$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$83$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	1.22474	−	0.707107i	1.22474	−	0.707107i
$95$		0			0
$96$		0			0
$97$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$97$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$98$		−	1.41421i		−	1.41421i
$99$		0			0
$100$		0			0
$101$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$101$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$102$		0			0
$103$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$103$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$	1.22474	+	0.707107i	1.22474	+	0.707107i
$107$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$107$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$113$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$113$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		−	1.41421i		−	1.41421i
$117$		0			0
$118$		−	1.41421i		−	1.41421i
$119$		0			0
$120$		0			0
$121$		0			0
$122$	1.00000	+	1.73205i	1.00000	+	1.73205i
$123$		0			0
$124$	0.500000	−	0.866025i	0.500000	−	0.866025i
$125$	−1.00000			−1.00000
$126$		0			0
$127$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$127$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$		−	1.41421i		−	1.41421i
$135$		0			0
$136$		0			0
$137$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$137$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$138$		0			0
$139$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$139$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$140$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$141$		0			0
$142$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$143$		0			0
$144$		0			0
$145$			1.41421i			1.41421i
$146$	1.00000	−	1.73205i	1.00000	−	1.73205i
$147$		0			0
$148$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$156$		0			0
$157$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$157$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$	1.22474	+	0.707107i	1.22474	+	0.707107i
$161$		0			0
$162$		0			0
$163$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$163$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$		0			0
$166$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$179$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$181$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.500000	+	0.866025i	0.500000	+	0.866025i
$186$		0			0
$187$		0			0
$188$	−1.00000			−1.00000
$189$		0			0
$190$		0			0
$191$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$191$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$	1.22474	−	0.707107i	1.22474	−	0.707107i
$195$		0			0
$196$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$197$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$197$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$198$		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
$202$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$203$	1.00000	+	1.73205i	1.00000	+	1.73205i
$204$		0			0
$205$		0			0
$206$		−	1.41421i		−	1.41421i
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$211$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$212$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$213$		0			0
$214$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$215$		0			0
$216$		0			0
$217$			1.41421i			1.41421i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$223$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$224$	2.00000			2.00000
$225$		0			0
$226$			1.41421i			1.41421i
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$229$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$233$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$234$		0			0
$235$	1.00000			1.00000
$236$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$237$		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		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$		0			0
$244$		−	1.41421i		−	1.41421i
$245$	0.500000	−	0.866025i	0.500000	−	0.866025i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	1.22474	+	0.707107i	1.22474	+	0.707107i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$	1.00000	−	1.73205i	1.00000	−	1.73205i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$257$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$258$		0			0
$259$	1.22474	+	0.707107i	1.22474	+	0.707107i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$	0.500000	+	0.866025i	0.500000	+	0.866025i
$266$		0			0
$267$		0			0
$268$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$269$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$269$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$		0			0
$274$	1.22474	−	0.707107i	1.22474	−	0.707107i
$275$		0			0
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$	−2.00000			−2.00000
$279$		0			0
$280$		0			0
$281$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$281$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$282$		0			0
$283$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$283$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$284$	0.500000	+	0.866025i	0.500000	+	0.866025i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1.00000			1.00000
$290$	1.00000	−	1.73205i	1.00000	−	1.73205i
$291$		0			0
$292$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$293$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$293$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$294$		0			0
$295$	0.500000	−	0.866025i	0.500000	−	0.866025i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$			1.41421i			1.41421i
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	1.22474	−	0.707107i	1.22474	−	0.707107i
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$311$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$313$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$314$			1.41421i			1.41421i
$315$		0			0
$316$		0			0
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$		0			0
$326$	1.22474	+	0.707107i	1.22474	+	0.707107i
$327$		0			0
$328$		0			0
$329$	1.22474	−	0.707107i	1.22474	−	0.707107i
$330$		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			$\pi$
$332$			1.41421i			1.41421i
$333$		0			0
$334$		0			0
$335$	0.500000	−	0.866025i	0.500000	−	0.866025i
$336$		0			0
$337$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$337$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$338$	1.22474	−	0.707107i	1.22474	−	0.707107i
$339$		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			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$356$		0			0
$357$		0			0
$358$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$359$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$359$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$	1.22474	+	0.707107i	1.22474	+	0.707107i
$363$		0			0
$364$		0			0
$365$	1.22474	−	0.707107i	1.22474	−	0.707107i
$366$		0			0
$367$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$367$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$368$		0			0
$369$		0			0
$370$		−	1.41421i		−	1.41421i
$371$	1.22474	+	0.707107i	1.22474	+	0.707107i
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0			−	1.00000i	$-0.5\pi$
$379$		0			0				1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$	1.22474	−	0.707107i	1.22474	−	0.707107i
$383$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$383$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−1.00000			−1.00000
$389$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$389$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	1.00000	−	1.73205i	1.00000	−	1.73205i
$395$		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	$0.666667\pi$
$398$	1.22474	+	0.707107i	1.22474	+	0.707107i
$399$		0			0
$400$		0			0
$401$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$401$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$			1.41421i			1.41421i
$405$		0			0
$406$		−	2.82843i		−	2.82843i
$407$		0			0
$408$		0			0
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		0			0
$412$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$413$		−	1.41421i		−	1.41421i
$414$		0			0
$415$		−	1.41421i		−	1.41421i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$419$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$420$		0			0
$421$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$421$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$422$	−2.00000			−2.00000
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.00000	+	1.73205i	1.00000	+	1.73205i
$428$	1.22474	−	0.707107i	1.22474	−	0.707107i
$429$		0			0
$430$		0			0
$431$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$431$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$	1.00000	−	1.73205i	1.00000	−	1.73205i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$439$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$443$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$449$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$449$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$450$		0			0
$451$		0			0
$452$	0.500000	−	0.866025i	0.500000	−	0.866025i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$457$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$461$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$462$		0			0
$463$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$463$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$464$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$465$		0			0
$466$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$467$	−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
$469$		−	1.41421i		−	1.41421i
$470$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	2.00000			2.00000
$479$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$479$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.00000			1.00000
$486$		0			0
$487$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$487$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$488$		0			0
$489$		0			0
$490$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$491$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$491$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.00000			−1.00000
$497$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$498$		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			$\pi$
$500$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$501$		0			0
$502$		0			0
$503$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$503$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$504$		0			0
$505$		−	1.41421i		−	1.41421i
$506$		0			0
$507$		0			0
$508$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	1.00000	−	1.73205i	1.00000	−	1.73205i
$512$			1.41421i			1.41421i
$513$		0			0
$514$			2.82843i			2.82843i
$515$	0.500000	−	0.866025i	0.500000	−	0.866025i
$516$		0			0
$517$		0			0
$518$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$519$		0			0
$520$		0			0
$521$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$521$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.500000	−	0.866025i	0.500000	−	0.866025i
$530$		−	1.41421i		−	1.41421i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$536$		0			0
$537$		0			0
$538$	1.22474	+	0.707107i	1.22474	+	0.707107i
$539$		0			0
$540$		0			0
$541$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$541$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$547$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$548$	−1.00000			−1.00000
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	1.22474	+	0.707107i	1.22474	+	0.707107i
$557$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$557$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$		0			0
$560$	1.22474	+	0.707107i	1.22474	+	0.707107i
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$566$	2.00000			2.00000
$567$		0			0
$568$		0			0
$569$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$569$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$570$		0			0
$571$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$571$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$577$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$578$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$579$		0			0
$580$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$581$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−2.00000			−2.00000
$587$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$587$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$588$		0			0
$589$		0			0
$590$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$591$		0			0
$592$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$593$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$593$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$601$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$	1.00000	−	1.73205i	1.00000	−	1.73205i
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$617$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$618$		0			0
$619$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$619$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$620$	−1.00000			−1.00000
$621$		0			0
$622$		−	1.41421i		−	1.41421i
$623$		0			0
$624$		0			0
$625$	0.500000	+	0.866025i	0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	0.500000	−	0.866025i	0.500000	−	0.866025i
$629$		0			0
$630$		0			0
$631$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$631$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.22474	−	0.707107i	1.22474	−	0.707107i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$641$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$643$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.00000			2.00000			1.00000			$0$
$647$	2.00000			2.00000			1.00000			$0$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$653$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$653$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$	−2.00000			−2.00000
$659$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$659$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$660$		0			0
$661$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$661$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$662$	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
$670$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$671$		0			0
$672$		0			0
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$	2.00000			2.00000
$675$		0			0
$676$	−1.00000			−1.00000
$677$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$677$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$678$		0			0
$679$	1.22474	−	0.707107i	1.22474	−	0.707107i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$683$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$	1.00000			1.00000
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$691$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$692$		0			0
$693$		0			0
$694$	2.00000			2.00000
$695$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$708$		0			0
$709$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$709$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$710$			1.41421i			1.41421i
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.500000	+	0.866025i	0.500000	+	0.866025i
$717$		0			0
$718$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$719$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$719$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		−	1.41421i		−	1.41421i
$722$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$723$		0			0
$724$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$725$		0			0
$726$		0			0
$727$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$727$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$728$		0			0
$729$		0			0
$730$	−2.00000			−2.00000
$731$		0			0
$732$		0			0
$733$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$733$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$734$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$741$		0			0
$742$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$743$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$743$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$750$		0			0
$751$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$751$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$752$	0.500000	+	0.866025i	0.500000	+	0.866025i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$757$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$761$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$762$		0			0
$763$		0			0
$764$	−1.00000			−1.00000
$765$		0			0
$766$		−	1.41421i		−	1.41421i
$767$		0			0
$768$		0			0
$769$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$769$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$	1.22474	−	0.707107i	1.22474	−	0.707107i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.00000			1.00000
$785$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$786$		0			0
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$789$		0			0
$790$		0			0
$791$			1.41421i			1.41421i
$792$		0			0
$793$		0			0
$794$	1.22474	+	0.707107i	1.22474	+	0.707107i
$795$		0			0
$796$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$797$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$797$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$			1.41421i			1.41421i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$809$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$810$		0			0
$811$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$811$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$812$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$813$		0			0
$814$		0			0
$815$	0.500000	+	0.866025i	0.500000	+	0.866025i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$		0			0
$826$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$829$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$830$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$			1.41421i			1.41421i
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	0.500000	+	0.866025i	0.500000	+	0.866025i
$842$	1.22474	−	0.707107i	1.22474	−	0.707107i
$843$		0			0
$844$	1.22474	+	0.707107i	1.22474	+	0.707107i
$845$	1.00000			1.00000
$846$		0			0
$847$		0			0
$848$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$853$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$854$		−	2.82843i		−	2.82843i
$855$		0			0
$856$		0			0
$857$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$857$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$858$		0			0
$859$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$859$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$	1.00000	−	1.73205i	1.00000	−	1.73205i
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.22474	+	0.707107i	1.22474	+	0.707107i
$876$		0			0
$877$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$877$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$878$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$879$		0			0
$880$		0			0
$881$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$881$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$883$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$884$		0			0
$885$		0			0
$886$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	1.00000	−	1.73205i	1.00000	−	1.73205i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$896$		0			0
$897$		0			0
$898$	1.22474	+	0.707107i	1.22474	+	0.707107i
$899$			1.41421i			1.41421i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.500000	+	0.866025i	0.500000	+	0.866025i
$906$		0			0
$907$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$907$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$911$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$	1.00000	+	1.73205i	1.00000	+	1.73205i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$	1.00000	+	1.73205i	1.00000	+	1.73205i
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	2.00000			2.00000
$929$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$929$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$	1.22474	−	0.707107i	1.22474	−	0.707107i
$933$		0			0
$934$	1.22474	+	0.707107i	1.22474	+	0.707107i
$935$		0			0
$936$		0			0
$937$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$937$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$938$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$939$		0			0
$940$	0.500000	+	0.866025i	0.500000	+	0.866025i
$941$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$941$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$942$		0			0
$943$		0			0
$944$	1.00000			1.00000
$945$		0			0
$946$		0			0
$947$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$947$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$953$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$	1.00000			1.00000
$956$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$957$		0			0
$958$	1.00000	+	1.73205i	1.00000	+	1.73205i
$959$	1.22474	−	0.707107i	1.22474	−	0.707107i
$960$		0			0
$961$		0			0
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$967$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$968$		0			0
$969$		0			0
$970$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		0			0
$973$	−2.00000			−2.00000
$974$	1.22474	+	0.707107i	1.22474	+	0.707107i
$975$		0			0
$976$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$977$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$977$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$978$		0			0
$979$		0			0
$980$	1.00000			1.00000
$981$		0			0
$982$	2.00000			2.00000
$983$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$983$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$984$		0			0
$985$	1.22474	−	0.707107i	1.22474	−	0.707107i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$	1.22474	+	0.707107i	1.22474	+	0.707107i
$993$		0			0
$994$	1.00000	+	1.73205i	1.00000	+	1.73205i
$995$	0.500000	+	0.866025i	0.500000	+	0.866025i
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$			1.41421i			1.41421i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3267.1.h.a.1693.1		4
3.2	odd	2		1089.1.h.a.241.2	yes	4
9.4	even	3	inner	3267.1.h.a.604.2		4
9.5	odd	6		1089.1.h.a.967.1	yes	4
11.2	odd	10		3267.1.w.b.1855.1		16
11.3	even	5		3267.1.w.b.3016.1		16
11.4	even	5		3267.1.w.b.1909.2		16
11.5	even	5		3267.1.w.b.1207.2		16
11.6	odd	10		3267.1.w.b.1207.1		16
11.7	odd	10		3267.1.w.b.1909.1		16
11.8	odd	10		3267.1.w.b.3016.2		16
11.9	even	5		3267.1.w.b.1855.2		16
11.10	odd	2	inner	3267.1.h.a.1693.2		4
33.2	even	10		1089.1.s.b.403.2		16
33.5	odd	10		1089.1.s.b.844.1		16
33.8	even	10		1089.1.s.b.475.1		16
33.14	odd	10		1089.1.s.b.475.2		16
33.17	even	10		1089.1.s.b.844.2		16
33.20	odd	10		1089.1.s.b.403.1		16
33.26	odd	10		1089.1.s.b.457.1		16
33.29	even	10		1089.1.s.b.457.2		16
33.32	even	2		1089.1.h.a.241.1	✓	4
99.4	even	15		3267.1.w.b.820.2		16
99.5	odd	30		1089.1.s.b.481.2		16
99.13	odd	30		3267.1.w.b.766.2		16
99.14	odd	30		1089.1.s.b.112.2		16
99.31	even	15		3267.1.w.b.766.1		16
99.32	even	6		1089.1.h.a.967.2	yes	4
99.40	odd	30		3267.1.w.b.820.1		16
99.41	even	30		1089.1.s.b.112.1		16
99.49	even	15		3267.1.w.b.118.1		16
99.50	even	30		1089.1.s.b.481.1		16
99.58	even	15		3267.1.w.b.1927.1		16
99.59	odd	30		1089.1.s.b.94.1		16
99.68	even	30		1089.1.s.b.40.1		16
99.76	odd	6	inner	3267.1.h.a.604.1		4
99.85	odd	30		3267.1.w.b.1927.2		16
99.86	odd	30		1089.1.s.b.40.2		16
99.94	odd	30		3267.1.w.b.118.2		16
99.95	even	30		1089.1.s.b.94.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1089.1.h.a.241.1	✓	4	33.32	even	2
1089.1.h.a.241.2	yes	4	3.2	odd	2
1089.1.h.a.967.1	yes	4	9.5	odd	6
1089.1.h.a.967.2	yes	4	99.32	even	6
1089.1.s.b.40.1		16	99.68	even	30
1089.1.s.b.40.2		16	99.86	odd	30
1089.1.s.b.94.1		16	99.59	odd	30
1089.1.s.b.94.2		16	99.95	even	30
1089.1.s.b.112.1		16	99.41	even	30
1089.1.s.b.112.2		16	99.14	odd	30
1089.1.s.b.403.1		16	33.20	odd	10
1089.1.s.b.403.2		16	33.2	even	10
1089.1.s.b.457.1		16	33.26	odd	10
1089.1.s.b.457.2		16	33.29	even	10
1089.1.s.b.475.1		16	33.8	even	10
1089.1.s.b.475.2		16	33.14	odd	10
1089.1.s.b.481.1		16	99.50	even	30
1089.1.s.b.481.2		16	99.5	odd	30
1089.1.s.b.844.1		16	33.5	odd	10
1089.1.s.b.844.2		16	33.17	even	10
3267.1.h.a.604.1		4	99.76	odd	6	inner
3267.1.h.a.604.2		4	9.4	even	3	inner
3267.1.h.a.1693.1		4	1.1	even	1	trivial
3267.1.h.a.1693.2		4	11.10	odd	2	inner
3267.1.w.b.118.1		16	99.49	even	15
3267.1.w.b.118.2		16	99.94	odd	30
3267.1.w.b.766.1		16	99.31	even	15
3267.1.w.b.766.2		16	99.13	odd	30
3267.1.w.b.820.1		16	99.40	odd	30
3267.1.w.b.820.2		16	99.4	even	15
3267.1.w.b.1207.1		16	11.6	odd	10
3267.1.w.b.1207.2		16	11.5	even	5
3267.1.w.b.1855.1		16	11.2	odd	10
3267.1.w.b.1855.2		16	11.9	even	5
3267.1.w.b.1909.1		16	11.7	odd	10
3267.1.w.b.1909.2		16	11.4	even	5
3267.1.w.b.1927.1		16	99.58	even	15
3267.1.w.b.1927.2		16	99.85	odd	30
3267.1.w.b.3016.1		16	11.3	even	5
3267.1.w.b.3016.2		16	11.8	odd	10

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:	\( N \)	\(=\)	\( 3267 = 3^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3267.h (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.22474 - 0.707107i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(-1.22474 - 0.707107i) q^{7} +O(q^{10})\)	\(q+(-1.22474 - 0.707107i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(-1.22474 - 0.707107i) q^{7} +1.41421i q^{10} +(1.00000 + 1.73205i) q^{14} +(0.500000 - 0.866025i) q^{16} +(0.500000 - 0.866025i) q^{20} -1.41421i q^{28} +(-1.22474 - 0.707107i) q^{29} +(-0.500000 - 0.866025i) q^{31} +(-1.22474 + 0.707107i) q^{32} +1.41421i q^{35} -1.00000 q^{37} +(-0.500000 + 0.866025i) q^{47} +(0.500000 + 0.866025i) q^{49} -1.00000 q^{53} +(1.00000 + 1.73205i) q^{58} +(0.500000 + 0.866025i) q^{59} +(-1.22474 - 0.707107i) q^{61} +1.41421i q^{62} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{67} +(1.00000 - 1.73205i) q^{70} +1.00000 q^{71} +1.41421i q^{73} +(1.22474 + 0.707107i) q^{74} -1.00000 q^{80} +(1.22474 + 0.707107i) q^{83} +(1.22474 - 0.707107i) q^{94} +(-0.500000 + 0.866025i) q^{97} -1.41421i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{5}+O(q^{10}) \) 4 * q + 2 * q^4 - 2 * q^5	\( 4 q + 2 q^{4} - 2 q^{5} + 4 q^{14} + 2 q^{16} + 2 q^{20} - 2 q^{31} - 4 q^{37} - 2 q^{47} + 2 q^{49} - 4 q^{53} + 4 q^{58} + 2 q^{59} + 4 q^{64} + 2 q^{67} + 4 q^{70} + 4 q^{71} - 4 q^{80} - 2 q^{97}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^5 + 4 * q^14 + 2 * q^16 + 2 * q^20 - 2 * q^31 - 4 * q^37 - 2 * q^47 + 2 * q^49 - 4 * q^53 + 4 * q^58 + 2 * q^59 + 4 * q^64 + 2 * q^67 + 4 * q^70 + 4 * q^71 - 4 * q^80 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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