LMFDB - Embedded newform 945.1.bg.b.874.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [945,1,Mod(559,945)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(945, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("945.559");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 945 = 3^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	945.bg (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:	$0.471616436938$
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:	$\Z[a_1, \ldots, a_{4}]$
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:	$C_6\times S_3$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			874.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	945.874
Dual form			945.1.bg.b.559.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{11} +(0.500000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{16} +1.00000 q^{17} +(0.500000 + 0.866025i) q^{20} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{28} +(1.00000 + 1.73205i) q^{29} -1.00000 q^{35} +1.00000 q^{44} +(-0.500000 - 0.866025i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(0.500000 + 0.866025i) q^{52} -1.00000 q^{55} +1.00000 q^{64} +(-0.500000 - 0.866025i) q^{65} +(-0.500000 + 0.866025i) q^{68} +1.00000 q^{71} -1.00000 q^{73} +(-0.500000 + 0.866025i) q^{77} +(0.500000 + 0.866025i) q^{79} -1.00000 q^{80} +(-0.500000 - 0.866025i) q^{83} +(0.500000 - 0.866025i) q^{85} -1.00000 q^{91} +(0.500000 + 0.866025i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{4} + q^{5} - q^{7} - q^{11} + q^{13} - q^{16} + 2 q^{17} + q^{20} - q^{25} + 2 q^{28} + 2 q^{29} - 2 q^{35} + 2 q^{44} - q^{47} - q^{49} + q^{52} - 2 q^{55} + 2 q^{64} - q^{65} - q^{68}+ \cdots + q^{97}+O(q^{100}) $ 2 * q - q^4 + q^5 - q^7 - q^11 + q^13 - q^16 + 2 * q^17 + q^20 - q^25 + 2 * q^28 + 2 * q^29 - 2 * q^35 + 2 * q^44 - q^47 - q^49 + q^52 - 2 * q^55 + 2 * q^64 - q^65 - q^68 + 2 * q^71 - 2 * q^73 - q^77 + q^79 - 2 * q^80 - q^83 + q^85 - 2 * q^91 + q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/945\mathbb{Z}\right)^\times$.

$n$	$136$	$596$	$757$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$	$-1$

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 $

$2$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$5$	0.500000	−	0.866025i	0.500000	−	0.866025i
$5$	0.500000	−	0.866025i	0.500000	−	0.866025i
$6$		0			0
$7$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$7$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$11$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$12$		0			0
$13$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$13$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$14$		0			0
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$17$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$18$		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		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$	1.00000			1.00000
$29$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$29$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$30$		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			0
$34$		0			0
$35$	−1.00000			−1.00000
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$	1.00000			1.00000
$45$		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
$52$	0.500000	+	0.866025i	0.500000	+	0.866025i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		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	$0.666667\pi$
$60$		0			0
$61$		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	$0.666667\pi$
$68$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$69$		0			0
$70$		0			0
$71$	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
$73$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$73$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$78$		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	$0.666667\pi$
$80$	−1.00000			−1.00000
$81$		0			0
$82$		0			0
$83$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$83$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$84$		0			0
$85$	0.500000	−	0.866025i	0.500000	−	0.866025i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−1.00000			−1.00000
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$97$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$98$		0			0
$99$		0			0
$100$	1.00000			1.00000
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$103$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$109$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$110$		0			0
$111$		0			0
$112$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$113$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$113$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$114$		0			0
$115$		0			0
$116$	−2.00000			−2.00000
$117$		0			0
$118$		0			0
$119$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$120$		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		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$139$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$140$	0.500000	−	0.866025i	0.500000	−	0.866025i
$141$		0			0
$142$		0			0
$143$	−1.00000			−1.00000
$144$		0			0
$145$	2.00000			2.00000
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$149$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$150$		0			0
$151$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$151$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		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			$0$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$167$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$168$		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	$0.333333\pi$
$173$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$176$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$177$		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
$181$		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	$0.333333\pi$
$191$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	1.00000	−	1.73205i	1.00000	−	1.73205i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.00000			−1.00000
$209$		0			0
$210$		0			0
$211$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$211$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	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	1.00000			$0$
$223$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$227$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$228$		0			0
$229$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$229$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$	−1.00000			−1.00000
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$239$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$240$		0			0
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.500000	+	0.866025i	0.500000	+	0.866025i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		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			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$257$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$	1.00000			1.00000
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$273$		0			0
$274$		0			0
$275$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$281$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$282$		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
$291$		0			0
$292$	0.500000	−	0.866025i	0.500000	−	0.866025i
$293$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$293$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$294$		0			0
$295$		0			0
$296$		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
$305$		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	$0.666667\pi$
$308$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$309$		0			0
$310$		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
$320$	0.500000	−	0.866025i	0.500000	−	0.866025i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−1.00000			−1.00000
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$330$		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	$0.666667\pi$
$332$	1.00000			1.00000
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$337$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$338$		0			0
$339$		0			0
$340$	0.500000	+	0.866025i	0.500000	+	0.866025i
$341$		0			0
$342$		0			0
$343$	1.00000			1.00000
$344$		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	$0.666667\pi$
$348$		0			0
$349$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$349$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$353$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$	0.500000	−	0.866025i	0.500000	−	0.866025i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2.00000			−2.00000			−1.00000			$\pi$
$359$	−2.00000			−2.00000			−1.00000			$\pi$
$360$		0			0
$361$	1.00000			1.00000
$362$		0			0
$363$		0			0
$364$	0.500000	−	0.866025i	0.500000	−	0.866025i
$365$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$366$		0			0
$367$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$367$	0.500000	+	0.866025i	0.500000	+	0.866025i	−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		0.500000	−	0.866025i	$-0.333333\pi$
$373$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$374$		0			0
$375$		0			0
$376$		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
$383$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$383$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$384$		0			0
$385$	0.500000	+	0.866025i	0.500000	+	0.866025i
$386$		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
$394$		0			0
$395$	1.00000			1.00000
$396$		0			0
$397$	2.00000			2.00000			1.00000			$0$
$397$	2.00000			2.00000			1.00000			$0$
$398$		0			0
$399$		0			0
$400$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$401$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$401$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		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
$415$	−1.00000			−1.00000
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$421$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$431$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$432$		0			0
$433$	2.00000			2.00000			1.00000			$0$
$433$	2.00000			2.00000			1.00000			$0$
$434$		0			0
$435$		0			0
$436$	0.500000	−	0.866025i	0.500000	−	0.866025i
$437$		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			0
$448$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$449$	−2.00000			−2.00000			−1.00000			$\pi$
$449$	−2.00000			−2.00000			−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$456$		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			0
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$463$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$464$	1.00000	−	1.73205i	1.00000	−	1.73205i
$465$		0			0
$466$		0			0
$467$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$467$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$	1.00000			1.00000
$477$		0			0
$478$		0			0
$479$		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
$484$		0			0
$485$	1.00000			1.00000
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$491$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$492$		0			0
$493$	1.00000	+	1.73205i	1.00000	+	1.73205i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$498$		0			0
$499$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$499$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$500$	0.500000	−	0.866025i	0.500000	−	0.866025i
$501$		0			0
$502$		0			0
$503$	−2.00000			−2.00000			−1.00000			$\pi$
$503$	−2.00000			−2.00000			−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	0.500000	+	0.866025i	0.500000	+	0.866025i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.00000	+	1.73205i	1.00000	+	1.73205i
$516$		0			0
$517$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$523$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	1.00000			1.00000
$540$		0			0
$541$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$541$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$542$		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	$-0.666667\pi$
$547$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.500000	−	0.866025i	0.500000	−	0.866025i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$	0.500000	+	0.866025i	0.500000	+	0.866025i
$561$		0			0
$562$		0			0
$563$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$563$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$569$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$570$		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			$0$
$572$	0.500000	−	0.866025i	0.500000	−	0.866025i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$577$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$578$		0			0
$579$		0			0
$580$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$581$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$587$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.00000			−2.00000			−1.00000			$\pi$
$593$	−2.00000			−2.00000			−1.00000			$\pi$
$594$		0			0
$595$	−1.00000			−1.00000
$596$	−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	$0.333333\pi$
$599$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$600$		0			0
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$		0			0
$604$	−1.00000			−1.00000
$605$		0			0
$606$		0			0
$607$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$607$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.00000			−1.00000
$612$		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			0		0.500000	−	0.866025i	$-0.333333\pi$
$617$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		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	$-0.666667\pi$
$631$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.500000	+	0.866025i	0.500000	+	0.866025i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$641$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$643$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−2.00000			−2.00000			−1.00000			$\pi$
$647$	−2.00000			−2.00000			−1.00000			$\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$659$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$660$		0			0
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$673$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$677$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$678$		0			0
$679$	0.500000	−	0.866025i	0.500000	−	0.866025i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		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
$700$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$701$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$701$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$702$		0			0
$703$		0			0
$704$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$705$		0			0
$706$		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
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$716$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$	2.00000			2.00000
$722$		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	1.00000			$0$
$727$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$733$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	2.00000			2.00000			1.00000			$0$
$739$	2.00000			2.00000			1.00000			$0$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$	0.500000	+	0.866025i	0.500000	+	0.866025i
$746$		0			0
$747$		0			0
$748$	1.00000			1.00000
$749$		0			0
$750$		0			0
$751$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$751$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$752$	−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		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	0.500000	+	0.866025i	0.500000	+	0.866025i
$764$	−2.00000			−2.00000
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$773$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$782$		0			0
$783$		0			0
$784$	1.00000			1.00000
$785$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$786$		0			0
$787$	0.500000	−	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	0.500000	+	0.866025i	$0.333333\pi$
$797$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$798$		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
$809$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$809$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	1.00000	+	1.73205i	1.00000	+	1.73205i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$821$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	0.500000	−	0.866025i	0.500000	−	0.866025i
$833$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$834$		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	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$842$		0			0
$843$		0			0
$844$	0.500000	+	0.866025i	0.500000	+	0.866025i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−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
$857$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$857$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$858$		0			0
$859$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$859$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$	2.00000			2.00000
$866$		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			0
$874$		0			0
$875$	0.500000	+	0.866025i	0.500000	+	0.866025i
$876$		0			0
$877$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$877$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$		0			0
$880$	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
$887$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$887$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−1.00000			−1.00000
$893$		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			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$907$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$908$	1.00000			1.00000
$909$		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
$913$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	2.00000			2.00000			1.00000			$0$
$919$	2.00000			2.00000			1.00000			$0$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.500000	−	0.866025i	0.500000	−	0.866025i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.00000			−1.00000
$936$		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
$939$		0			0
$940$	0.500000	−	0.866025i	0.500000	−	0.866025i
$941$		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			0
$949$	−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
$955$	2.00000			2.00000
$956$	1.00000	+	1.73205i	1.00000	+	1.73205i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−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	$-0.333333\pi$
$967$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$	−1.00000			−1.00000
$981$		0			0
$982$		0			0
$983$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$983$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$991$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$997$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	945.1.bg.b.874.1		2
3.2	odd	2		315.1.bg.b.139.1	yes	2
5.4	even	2		945.1.bg.a.874.1		2
7.6	odd	2		945.1.bg.a.874.1		2
9.2	odd	6		315.1.bg.b.34.1	yes	2
9.4	even	3		2835.1.e.b.244.1		1
9.5	odd	6		2835.1.e.d.244.1		1
9.7	even	3	inner	945.1.bg.b.559.1		2
15.2	even	4		1575.1.y.a.76.1		4
15.8	even	4		1575.1.y.a.76.2		4
15.14	odd	2		315.1.bg.a.139.1	yes	2
21.2	odd	6		2205.1.bn.a.1354.1		2
21.5	even	6		2205.1.bn.b.1354.1		2
21.11	odd	6		2205.1.q.a.1489.1		2
21.17	even	6		2205.1.q.b.1489.1		2
21.20	even	2		315.1.bg.a.139.1	yes	2
35.34	odd	2	CM	945.1.bg.b.874.1		2
45.2	even	12		1575.1.y.a.601.1		4
45.4	even	6		2835.1.e.c.244.1		1
45.14	odd	6		2835.1.e.a.244.1		1
45.29	odd	6		315.1.bg.a.34.1	✓	2
45.34	even	6		945.1.bg.a.559.1		2
45.38	even	12		1575.1.y.a.601.2		4
63.2	odd	6		2205.1.q.a.619.1		2
63.11	odd	6		2205.1.bn.a.754.1		2
63.13	odd	6		2835.1.e.c.244.1		1
63.20	even	6		315.1.bg.a.34.1	✓	2
63.34	odd	6		945.1.bg.a.559.1		2
63.38	even	6		2205.1.bn.b.754.1		2
63.41	even	6		2835.1.e.a.244.1		1
63.47	even	6		2205.1.q.b.619.1		2
105.44	odd	6		2205.1.bn.b.1354.1		2
105.59	even	6		2205.1.q.a.1489.1		2
105.62	odd	4		1575.1.y.a.76.2		4
105.74	odd	6		2205.1.q.b.1489.1		2
105.83	odd	4		1575.1.y.a.76.1		4
105.89	even	6		2205.1.bn.a.1354.1		2
105.104	even	2		315.1.bg.b.139.1	yes	2
315.34	odd	6	inner	945.1.bg.b.559.1		2
315.74	odd	6		2205.1.bn.b.754.1		2
315.83	odd	12		1575.1.y.a.601.1		4
315.104	even	6		2835.1.e.d.244.1		1
315.139	odd	6		2835.1.e.b.244.1		1
315.164	even	6		2205.1.bn.a.754.1		2
315.209	even	6		315.1.bg.b.34.1	yes	2
315.254	odd	6		2205.1.q.b.619.1		2
315.272	odd	12		1575.1.y.a.601.2		4
315.299	even	6		2205.1.q.a.619.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.1.bg.a.34.1	✓	2	45.29	odd	6
315.1.bg.a.34.1	✓	2	63.20	even	6
315.1.bg.a.139.1	yes	2	15.14	odd	2
315.1.bg.a.139.1	yes	2	21.20	even	2
315.1.bg.b.34.1	yes	2	9.2	odd	6
315.1.bg.b.34.1	yes	2	315.209	even	6
315.1.bg.b.139.1	yes	2	3.2	odd	2
315.1.bg.b.139.1	yes	2	105.104	even	2
945.1.bg.a.559.1		2	45.34	even	6
945.1.bg.a.559.1		2	63.34	odd	6
945.1.bg.a.874.1		2	5.4	even	2
945.1.bg.a.874.1		2	7.6	odd	2
945.1.bg.b.559.1		2	9.7	even	3	inner
945.1.bg.b.559.1		2	315.34	odd	6	inner
945.1.bg.b.874.1		2	1.1	even	1	trivial
945.1.bg.b.874.1		2	35.34	odd	2	CM
1575.1.y.a.76.1		4	15.2	even	4
1575.1.y.a.76.1		4	105.83	odd	4
1575.1.y.a.76.2		4	15.8	even	4
1575.1.y.a.76.2		4	105.62	odd	4
1575.1.y.a.601.1		4	45.2	even	12
1575.1.y.a.601.1		4	315.83	odd	12
1575.1.y.a.601.2		4	45.38	even	12
1575.1.y.a.601.2		4	315.272	odd	12
2205.1.q.a.619.1		2	63.2	odd	6
2205.1.q.a.619.1		2	315.299	even	6
2205.1.q.a.1489.1		2	21.11	odd	6
2205.1.q.a.1489.1		2	105.59	even	6
2205.1.q.b.619.1		2	63.47	even	6
2205.1.q.b.619.1		2	315.254	odd	6
2205.1.q.b.1489.1		2	21.17	even	6
2205.1.q.b.1489.1		2	105.74	odd	6
2205.1.bn.a.754.1		2	63.11	odd	6
2205.1.bn.a.754.1		2	315.164	even	6
2205.1.bn.a.1354.1		2	21.2	odd	6
2205.1.bn.a.1354.1		2	105.89	even	6
2205.1.bn.b.754.1		2	63.38	even	6
2205.1.bn.b.754.1		2	315.74	odd	6
2205.1.bn.b.1354.1		2	21.5	even	6
2205.1.bn.b.1354.1		2	105.44	odd	6
2835.1.e.a.244.1		1	45.14	odd	6
2835.1.e.a.244.1		1	63.41	even	6
2835.1.e.b.244.1		1	9.4	even	3
2835.1.e.b.244.1		1	315.139	odd	6
2835.1.e.c.244.1		1	45.4	even	6
2835.1.e.c.244.1		1	63.13	odd	6
2835.1.e.d.244.1		1	9.5	odd	6
2835.1.e.d.244.1		1	315.104	even	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:	\( N \)	\(=\)	\( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	945.bg (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{4} +(0.500000 - 0.866025i) q^{5} +(-0.500000 - 0.866025i) q^{7} +(-0.500000 - 0.866025i) q^{11} +(0.500000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{16} +1.00000 q^{17} +(0.500000 + 0.866025i) q^{20} +(-0.500000 - 0.866025i) q^{25} +1.00000 q^{28} +(1.00000 + 1.73205i) q^{29} -1.00000 q^{35} +1.00000 q^{44} +(-0.500000 - 0.866025i) q^{47} +(-0.500000 + 0.866025i) q^{49} +(0.500000 + 0.866025i) q^{52} -1.00000 q^{55} +1.00000 q^{64} +(-0.500000 - 0.866025i) q^{65} +(-0.500000 + 0.866025i) q^{68} +1.00000 q^{71} -1.00000 q^{73} +(-0.500000 + 0.866025i) q^{77} +(0.500000 + 0.866025i) q^{79} -1.00000 q^{80} +(-0.500000 - 0.866025i) q^{83} +(0.500000 - 0.866025i) q^{85} -1.00000 q^{91} +(0.500000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{4} + q^{5} - q^{7} - q^{11} + q^{13} - q^{16} + 2 q^{17} + q^{20} - q^{25} + 2 q^{28} + 2 q^{29} - 2 q^{35} + 2 q^{44} - q^{47} - q^{49} + q^{52} - 2 q^{55} + 2 q^{64} - q^{65} - q^{68}+ \cdots + q^{97}+O(q^{100}) \) 2 * q - q^4 + q^5 - q^7 - q^11 + q^13 - q^16 + 2 * q^17 + q^20 - q^25 + 2 * q^28 + 2 * q^29 - 2 * q^35 + 2 * q^44 - q^47 - q^49 + q^52 - 2 * q^55 + 2 * q^64 - q^65 - q^68 + 2 * q^71 - 2 * q^73 - q^77 + q^79 - 2 * q^80 - q^83 + q^85 - 2 * q^91 + q^97

Introduction

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