LMFDB - Embedded newform 2205.1.bn.a.754.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2205,1,Mod(754,2205)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2205, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 3, 5]))

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

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

chi := DirichletCharacter("2205.754");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2205 = 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2205.bn (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:	$1.10043835286$
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:	no (minimal twist has level 315)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.2835.1
Artin image:	$C_3\times S_3$
Artin field:	Galois closure of 6.0.170170875.1

Embedding invariants

Embedding label			754.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	2205.754
Dual form			2205.1.bn.a.1354.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{5} +(-0.500000 + 0.866025i) q^{9} -1.00000 q^{11} +1.00000 q^{12} +(0.500000 + 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(0.500000 + 0.866025i) q^{17} +(-0.500000 + 0.866025i) q^{20} +1.00000 q^{25} +1.00000 q^{27} +(-1.00000 + 1.73205i) q^{29} +(0.500000 + 0.866025i) q^{33} +(-0.500000 - 0.866025i) q^{36} +(0.500000 - 0.866025i) q^{39} +(0.500000 - 0.866025i) q^{44} +(-0.500000 + 0.866025i) q^{45} +(0.500000 + 0.866025i) q^{47} +(-0.500000 + 0.866025i) q^{48} +(0.500000 - 0.866025i) q^{51} -1.00000 q^{52} -1.00000 q^{55} +1.00000 q^{60} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{65} -1.00000 q^{68} -1.00000 q^{71} +(0.500000 + 0.866025i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(0.500000 - 0.866025i) q^{83} +(0.500000 + 0.866025i) q^{85} +2.00000 q^{87} +(0.500000 - 0.866025i) q^{97} +(0.500000 - 0.866025i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{4} + 2 q^{5} - q^{9} - 2 q^{11} + 2 q^{12} + q^{13} - q^{15} - q^{16} + q^{17} - q^{20} + 2 q^{25} + 2 q^{27} - 2 q^{29} + q^{33} - q^{36} + q^{39} + q^{44} - q^{45} + q^{47}+ \cdots + q^{99}+O(q^{100}) $ 2 * q - q^3 - q^4 + 2 * q^5 - q^9 - 2 * q^11 + 2 * q^12 + q^13 - q^15 - q^16 + q^17 - q^20 + 2 * q^25 + 2 * q^27 - 2 * q^29 + q^33 - q^36 + q^39 + q^44 - q^45 + q^47 - q^48 + q^51 - 2 * q^52 - 2 * q^55 + 2 * q^60 + 2 * q^64 + q^65 - 2 * q^68 - 2 * q^71 + q^73 - q^75 + q^79 - q^80 - q^81 + q^83 + q^85 + 4 * q^87 + q^97 + q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times$.

$n$	$442$	$1081$	$1226$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{2}{3}\right)$

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 $

$2$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$3$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$4$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$5$	1.00000			1.00000
$5$	1.00000			1.00000
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$10$		0			0
$11$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$12$	1.00000			1.00000
$13$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$13$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$14$		0			0
$15$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$17$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$19$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$20$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	1.00000			1.00000
$26$		0			0
$27$	1.00000			1.00000
$28$		0			0
$29$	−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			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$		0			0
$33$	0.500000	+	0.866025i	0.500000	+	0.866025i
$34$		0			0
$35$		0			0
$36$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$	0.500000	−	0.866025i	0.500000	−	0.866025i
$40$		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	$0.666667\pi$
$44$	0.500000	−	0.866025i	0.500000	−	0.866025i
$45$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$46$		0			0
$47$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$47$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$48$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$49$		0			0
$50$		0			0
$51$	0.500000	−	0.866025i	0.500000	−	0.866025i
$52$	−1.00000			−1.00000
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$	−1.00000			−1.00000
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	1.00000			1.00000
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	1.00000			1.00000
$65$	0.500000	+	0.866025i	0.500000	+	0.866025i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	−1.00000			−1.00000
$69$		0			0
$70$		0			0
$71$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$71$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$72$		0			0
$73$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$73$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$79$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$80$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$83$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$84$		0			0
$85$	0.500000	+	0.866025i	0.500000	+	0.866025i
$86$		0			0
$87$	2.00000			2.00000
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$97$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$98$		0			0
$99$	0.500000	−	0.866025i	0.500000	−	0.866025i
$100$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$	2.00000			2.00000			1.00000			$0$
$103$	2.00000			2.00000			1.00000			$0$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$109$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$109$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$		0			0
$116$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$117$	−1.00000			−1.00000
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			1.00000
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	−1.00000			−1.00000
$133$		0			0
$134$		0			0
$135$	1.00000			1.00000
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$139$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$140$		0			0
$141$	0.500000	−	0.866025i	0.500000	−	0.866025i
$142$		0			0
$143$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$144$	1.00000			1.00000
$145$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$149$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$151$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$	−1.00000			−1.00000
$154$		0			0
$155$		0			0
$156$	0.500000	+	0.866025i	0.500000	+	0.866025i
$157$	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			$0$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$163$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$164$		0			0
$165$	0.500000	+	0.866025i	0.500000	+	0.866025i
$166$		0			0
$167$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$167$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$173$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$174$		0			0
$175$		0			0
$176$	0.500000	+	0.866025i	0.500000	+	0.866025i
$177$		0			0
$178$		0			0
$179$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$179$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$180$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$188$	−1.00000			−1.00000
$189$		0			0
$190$		0			0
$191$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$191$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$192$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$	0.500000	−	0.866025i	0.500000	−	0.866025i
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$	0.500000	+	0.866025i	0.500000	+	0.866025i
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.500000	−	0.866025i	0.500000	−	0.866025i
$209$		0			0
$210$		0			0
$211$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$211$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$212$		0			0
$213$	0.500000	+	0.866025i	0.500000	+	0.866025i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	0.500000	−	0.866025i	0.500000	−	0.866025i
$220$	0.500000	−	0.866025i	0.500000	−	0.866025i
$221$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$222$		0			0
$223$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$223$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$224$		0			0
$225$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$226$		0			0
$227$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$227$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		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
$235$	0.500000	+	0.866025i	0.500000	+	0.866025i
$236$		0			0
$237$	0.500000	−	0.866025i	0.500000	−	0.866025i
$238$		0			0
$239$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$239$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$240$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−1.00000			−1.00000
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	0.500000	−	0.866025i	0.500000	−	0.866025i
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$257$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$	−1.00000			−1.00000
$261$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$	0.500000	−	0.866025i	0.500000	−	0.866025i
$273$		0			0
$274$		0			0
$275$	−1.00000			−1.00000
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$281$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$282$		0			0
$283$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$283$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$284$	0.500000	−	0.866025i	0.500000	−	0.866025i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$		0			0
$290$		0			0
$291$	−1.00000			−1.00000
$292$	−1.00000			−1.00000
$293$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$293$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.00000			−1.00000
$298$		0			0
$299$		0			0
$300$	1.00000			1.00000
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$307$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$308$		0			0
$309$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$310$		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
$313$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$313$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$314$		0			0
$315$		0			0
$316$	−1.00000			−1.00000
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$	1.00000	−	1.73205i	1.00000	−	1.73205i
$320$	1.00000			1.00000
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.00000			1.00000
$325$	0.500000	+	0.866025i	0.500000	+	0.866025i
$326$		0			0
$327$	0.500000	−	0.866025i	0.500000	−	0.866025i
$328$		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	$0.666667\pi$
$332$	0.500000	+	0.866025i	0.500000	+	0.866025i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$337$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$338$		0			0
$339$		0			0
$340$	−1.00000			−1.00000
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$	0.500000	+	0.866025i	0.500000	+	0.866025i
$352$		0			0
$353$	2.00000			2.00000			1.00000			$0$
$353$	2.00000			2.00000			1.00000			$0$
$354$		0			0
$355$	−1.00000			−1.00000
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$359$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.500000	+	0.866025i	0.500000	+	0.866025i
$366$		0			0
$367$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$367$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$376$		0			0
$377$	−2.00000			−2.00000
$378$		0			0
$379$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$383$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.500000	+	0.866025i	0.500000	+	0.866025i
$389$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$389$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.500000	+	0.866025i	0.500000	+	0.866025i
$396$	0.500000	+	0.866025i	0.500000	+	0.866025i
$397$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$397$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$401$	2.00000			2.00000			1.00000			$0$
$401$	2.00000			2.00000			1.00000			$0$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$413$		0			0
$414$		0			0
$415$	0.500000	−	0.866025i	0.500000	−	0.866025i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$419$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$420$		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			$0$
$422$		0			0
$423$	−1.00000			−1.00000
$424$		0			0
$425$	0.500000	+	0.866025i	0.500000	+	0.866025i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$430$		0			0
$431$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$431$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$432$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$433$	2.00000			2.00000			1.00000			$0$
$433$	2.00000			2.00000			1.00000			$0$
$434$		0			0
$435$	2.00000			2.00000
$436$	−1.00000			−1.00000
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		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
$446$		0			0
$447$	0.500000	+	0.866025i	0.500000	+	0.866025i
$448$		0			0
$449$	2.00000			2.00000			1.00000			$0$
$449$	2.00000			2.00000			1.00000			$0$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	0.500000	+	0.866025i	0.500000	+	0.866025i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$		0			0
$459$	0.500000	+	0.866025i	0.500000	+	0.866025i
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$463$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$464$	2.00000			2.00000
$465$		0			0
$466$		0			0
$467$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$467$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$468$	0.500000	−	0.866025i	0.500000	−	0.866025i
$469$		0			0
$470$		0			0
$471$	−1.00000			−1.00000
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		1.00000			$0$
$479$		0			0		−1.00000			$\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	0.500000	−	0.866025i	0.500000	−	0.866025i
$486$		0			0
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$491$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$	−2.00000			−2.00000
$494$		0			0
$495$	0.500000	−	0.866025i	0.500000	−	0.866025i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$499$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$500$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$501$	0.500000	−	0.866025i	0.500000	−	0.866025i
$502$		0			0
$503$	2.00000			2.00000			1.00000			$0$
$503$	2.00000			2.00000			1.00000			$0$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	2.00000			2.00000
$516$		0			0
$517$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$518$		0			0
$519$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$523$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.500000	−	0.866025i	0.500000	−	0.866025i
$529$	1.00000			1.00000
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.500000	−	0.866025i	0.500000	−	0.866025i
$538$		0			0
$539$		0			0
$540$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$541$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$541$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.500000	+	0.866025i	0.500000	+	0.866025i
$546$		0			0
$547$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$547$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$562$		0			0
$563$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$563$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$564$	0.500000	+	0.866025i	0.500000	+	0.866025i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	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
$571$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$571$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$572$	1.00000			1.00000
$573$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$574$		0			0
$575$		0			0
$576$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$577$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$577$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$578$		0			0
$579$		0			0
$580$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−1.00000			−1.00000
$586$		0			0
$587$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$587$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$593$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$	0.500000	−	0.866025i	0.500000	−	0.866025i
$597$		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			$0$
$600$		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
$604$	0.500000	−	0.866025i	0.500000	−	0.866025i
$605$		0			0
$606$		0			0
$607$	2.00000			2.00000			1.00000			$0$
$607$	2.00000			2.00000			1.00000			$0$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$612$	0.500000	−	0.866025i	0.500000	−	0.866025i
$613$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$613$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.00000			−1.00000
$625$	1.00000			1.00000
$626$		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	$-0.666667\pi$
$631$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$	0.500000	−	0.866025i	0.500000	−	0.866025i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	0.500000	−	0.866025i	0.500000	−	0.866025i
$640$		0			0
$641$	2.00000			2.00000			1.00000			$0$
$641$	2.00000			2.00000			1.00000			$0$
$642$		0			0
$643$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$643$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.00000			−1.00000
$658$		0			0
$659$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$659$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$660$	−1.00000			−1.00000
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$	1.00000			1.00000
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.00000			−1.00000
$669$	−1.00000			−1.00000
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$		0			0
$675$	1.00000			1.00000
$676$		0			0
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$677$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	0.500000	+	0.866025i	0.500000	+	0.866025i
$682$		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
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	2.00000			2.00000
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$701$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$702$		0			0
$703$		0			0
$704$	−1.00000			−1.00000
$705$	0.500000	−	0.866025i	0.500000	−	0.866025i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$709$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$710$		0			0
$711$	−1.00000			−1.00000
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$716$	−1.00000			−1.00000
$717$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$	1.00000			1.00000
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$726$		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
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$	2.00000			2.00000			1.00000			$0$
$733$	2.00000			2.00000			1.00000			$0$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−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
$745$	−1.00000			−1.00000
$746$		0			0
$747$	0.500000	+	0.866025i	0.500000	+	0.866025i
$748$	1.00000			1.00000
$749$		0			0
$750$		0			0
$751$	2.00000			2.00000			1.00000			$0$
$751$	2.00000			2.00000			1.00000			$0$
$752$	0.500000	−	0.866025i	0.500000	−	0.866025i
$753$		0			0
$754$		0			0
$755$	−1.00000			−1.00000
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$	2.00000			2.00000
$765$	−1.00000			−1.00000
$766$		0			0
$767$		0			0
$768$	1.00000			1.00000
$769$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$769$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$	0.500000	+	0.866025i	0.500000	+	0.866025i
$772$		0			0
$773$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$773$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	0.500000	+	0.866025i	0.500000	+	0.866025i
$781$	1.00000			1.00000
$782$		0			0
$783$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$784$		0			0
$785$	0.500000	−	0.866025i	0.500000	−	0.866025i
$786$		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
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$797$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$798$		0			0
$799$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$809$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−1.00000			−1.00000
$817$		0			0
$818$		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	$0.666667\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$	0.500000	+	0.866025i	0.500000	+	0.866025i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$829$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$830$		0			0
$831$		0			0
$832$	0.500000	+	0.866025i	0.500000	+	0.866025i
$833$		0			0
$834$		0			0
$835$	0.500000	+	0.866025i	0.500000	+	0.866025i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$842$		0			0
$843$	−1.00000			−1.00000
$844$	−1.00000			−1.00000
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.00000			−1.00000
$850$		0			0
$851$		0			0
$852$	−1.00000			−1.00000
$853$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$853$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$857$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$		0			0		1.00000			$0$
$859$		0			0		−1.00000			$\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.500000	+	0.866025i	0.500000	+	0.866025i
$874$		0			0
$875$		0			0
$876$	0.500000	+	0.866025i	0.500000	+	0.866025i
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$880$	0.500000	+	0.866025i	0.500000	+	0.866025i
$881$		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			$\pi$
$884$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$885$		0			0
$886$		0			0
$887$	2.00000			2.00000			1.00000			$0$
$887$	2.00000			2.00000			1.00000			$0$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.500000	+	0.866025i	0.500000	+	0.866025i
$892$	0.500000	+	0.866025i	0.500000	+	0.866025i
$893$		0			0
$894$		0			0
$895$	0.500000	+	0.866025i	0.500000	+	0.866025i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$	0.500000	−	0.866025i	0.500000	−	0.866025i
$909$		0			0
$910$		0			0
$911$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$911$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$912$		0			0
$913$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$919$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$	0.500000	+	0.866025i	0.500000	+	0.866025i
$922$		0			0
$923$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$936$		0			0
$937$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$937$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$938$		0			0
$939$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$940$	−1.00000			−1.00000
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		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	$0.333333\pi$
$948$	0.500000	+	0.866025i	0.500000	+	0.866025i
$949$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$956$	2.00000			2.00000
$957$	−2.00000			−2.00000
$958$		0			0
$959$		0			0
$960$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		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
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$973$		0			0
$974$		0			0
$975$	0.500000	−	0.866025i	0.500000	−	0.866025i
$976$		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
$981$	−1.00000			−1.00000
$982$		0			0
$983$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$983$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$991$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$	0.500000	−	0.866025i	0.500000	−	0.866025i
$994$		0			0
$995$		0			0
$996$	0.500000	−	0.866025i	0.500000	−	0.866025i
$997$	2.00000			2.00000			1.00000			$0$
$997$	2.00000			2.00000			1.00000			$0$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2205.1.bn.a.754.1		2
5.4	even	2		2205.1.bn.b.754.1		2
7.2	even	3		315.1.bg.b.34.1	yes	2
7.3	odd	6		2205.1.q.b.619.1		2
7.4	even	3		2205.1.q.a.619.1		2
7.5	odd	6		315.1.bg.a.34.1	✓	2
7.6	odd	2		2205.1.bn.b.754.1		2
9.4	even	3		2205.1.q.a.1489.1		2
21.2	odd	6		945.1.bg.b.559.1		2
21.5	even	6		945.1.bg.a.559.1		2
35.2	odd	12		1575.1.y.a.601.1		4
35.4	even	6		2205.1.q.b.619.1		2
35.9	even	6		315.1.bg.a.34.1	✓	2
35.12	even	12		1575.1.y.a.601.2		4
35.19	odd	6		315.1.bg.b.34.1	yes	2
35.23	odd	12		1575.1.y.a.601.2		4
35.24	odd	6		2205.1.q.a.619.1		2
35.33	even	12		1575.1.y.a.601.1		4
35.34	odd	2	CM	2205.1.bn.a.754.1		2
45.4	even	6		2205.1.q.b.1489.1		2
63.2	odd	6		2835.1.e.b.244.1		1
63.4	even	3	inner	2205.1.bn.a.1354.1		2
63.5	even	6		945.1.bg.a.874.1		2
63.13	odd	6		2205.1.q.b.1489.1		2
63.16	even	3		2835.1.e.d.244.1		1
63.23	odd	6		945.1.bg.b.874.1		2
63.31	odd	6		2205.1.bn.b.1354.1		2
63.40	odd	6		315.1.bg.a.139.1	yes	2
63.47	even	6		2835.1.e.c.244.1		1
63.58	even	3		315.1.bg.b.139.1	yes	2
63.61	odd	6		2835.1.e.a.244.1		1
105.44	odd	6		945.1.bg.a.559.1		2
105.89	even	6		945.1.bg.b.559.1		2
315.4	even	6		2205.1.bn.b.1354.1		2
315.58	odd	12		1575.1.y.a.76.2		4
315.79	even	6		2835.1.e.a.244.1		1
315.94	odd	6	inner	2205.1.bn.a.1354.1		2
315.103	even	12		1575.1.y.a.76.1		4
315.124	odd	6		2835.1.e.d.244.1		1
315.139	odd	6		2205.1.q.a.1489.1		2
315.149	odd	6		945.1.bg.a.874.1		2
315.184	even	6		315.1.bg.a.139.1	yes	2
315.194	even	6		945.1.bg.b.874.1		2
315.229	odd	6		315.1.bg.b.139.1	yes	2
315.247	odd	12		1575.1.y.a.76.1		4
315.254	odd	6		2835.1.e.c.244.1		1
315.292	even	12		1575.1.y.a.76.2		4
315.299	even	6		2835.1.e.b.244.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.1.bg.a.34.1	✓	2	7.5	odd	6
315.1.bg.a.34.1	✓	2	35.9	even	6
315.1.bg.a.139.1	yes	2	63.40	odd	6
315.1.bg.a.139.1	yes	2	315.184	even	6
315.1.bg.b.34.1	yes	2	7.2	even	3
315.1.bg.b.34.1	yes	2	35.19	odd	6
315.1.bg.b.139.1	yes	2	63.58	even	3
315.1.bg.b.139.1	yes	2	315.229	odd	6
945.1.bg.a.559.1		2	21.5	even	6
945.1.bg.a.559.1		2	105.44	odd	6
945.1.bg.a.874.1		2	63.5	even	6
945.1.bg.a.874.1		2	315.149	odd	6
945.1.bg.b.559.1		2	21.2	odd	6
945.1.bg.b.559.1		2	105.89	even	6
945.1.bg.b.874.1		2	63.23	odd	6
945.1.bg.b.874.1		2	315.194	even	6
1575.1.y.a.76.1		4	315.103	even	12
1575.1.y.a.76.1		4	315.247	odd	12
1575.1.y.a.76.2		4	315.58	odd	12
1575.1.y.a.76.2		4	315.292	even	12
1575.1.y.a.601.1		4	35.2	odd	12
1575.1.y.a.601.1		4	35.33	even	12
1575.1.y.a.601.2		4	35.12	even	12
1575.1.y.a.601.2		4	35.23	odd	12
2205.1.q.a.619.1		2	7.4	even	3
2205.1.q.a.619.1		2	35.24	odd	6
2205.1.q.a.1489.1		2	9.4	even	3
2205.1.q.a.1489.1		2	315.139	odd	6
2205.1.q.b.619.1		2	7.3	odd	6
2205.1.q.b.619.1		2	35.4	even	6
2205.1.q.b.1489.1		2	45.4	even	6
2205.1.q.b.1489.1		2	63.13	odd	6
2205.1.bn.a.754.1		2	1.1	even	1	trivial
2205.1.bn.a.754.1		2	35.34	odd	2	CM
2205.1.bn.a.1354.1		2	63.4	even	3	inner
2205.1.bn.a.1354.1		2	315.94	odd	6	inner
2205.1.bn.b.754.1		2	5.4	even	2
2205.1.bn.b.754.1		2	7.6	odd	2
2205.1.bn.b.1354.1		2	63.31	odd	6
2205.1.bn.b.1354.1		2	315.4	even	6
2835.1.e.a.244.1		1	63.61	odd	6
2835.1.e.a.244.1		1	315.79	even	6
2835.1.e.b.244.1		1	63.2	odd	6
2835.1.e.b.244.1		1	315.299	even	6
2835.1.e.c.244.1		1	63.47	even	6
2835.1.e.c.244.1		1	315.254	odd	6
2835.1.e.d.244.1		1	63.16	even	3
2835.1.e.d.244.1		1	315.124	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2205.bn (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 - 0.866025i) q^{3} +(-0.500000 + 0.866025i) q^{4} +1.00000 q^{5} +(-0.500000 + 0.866025i) q^{9} -1.00000 q^{11} +1.00000 q^{12} +(0.500000 + 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{15} +(-0.500000 - 0.866025i) q^{16} +(0.500000 + 0.866025i) q^{17} +(-0.500000 + 0.866025i) q^{20} +1.00000 q^{25} +1.00000 q^{27} +(-1.00000 + 1.73205i) q^{29} +(0.500000 + 0.866025i) q^{33} +(-0.500000 - 0.866025i) q^{36} +(0.500000 - 0.866025i) q^{39} +(0.500000 - 0.866025i) q^{44} +(-0.500000 + 0.866025i) q^{45} +(0.500000 + 0.866025i) q^{47} +(-0.500000 + 0.866025i) q^{48} +(0.500000 - 0.866025i) q^{51} -1.00000 q^{52} -1.00000 q^{55} +1.00000 q^{60} +1.00000 q^{64} +(0.500000 + 0.866025i) q^{65} -1.00000 q^{68} -1.00000 q^{71} +(0.500000 + 0.866025i) q^{73} +(-0.500000 - 0.866025i) q^{75} +(0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(0.500000 - 0.866025i) q^{83} +(0.500000 + 0.866025i) q^{85} +2.00000 q^{87} +(0.500000 - 0.866025i) q^{97} +(0.500000 - 0.866025i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{4} + 2 q^{5} - q^{9} - 2 q^{11} + 2 q^{12} + q^{13} - q^{15} - q^{16} + q^{17} - q^{20} + 2 q^{25} + 2 q^{27} - 2 q^{29} + q^{33} - q^{36} + q^{39} + q^{44} - q^{45} + q^{47}+ \cdots + q^{99}+O(q^{100}) \) 2 * q - q^3 - q^4 + 2 * q^5 - q^9 - 2 * q^11 + 2 * q^12 + q^13 - q^15 - q^16 + q^17 - q^20 + 2 * q^25 + 2 * q^27 - 2 * q^29 + q^33 - q^36 + q^39 + q^44 - q^45 + q^47 - q^48 + q^51 - 2 * q^52 - 2 * q^55 + 2 * q^60 + 2 * q^64 + q^65 - 2 * q^68 - 2 * q^71 + q^73 - q^75 + q^79 - q^80 - q^81 + q^83 + q^85 + 4 * q^87 + q^97 + q^99

Introduction

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