LMFDB - Embedded newform 3240.1.z.b.379.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,1,Mod(379,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("3240.379");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3240.z (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:	$1.61697064093$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.3240.1
Artin image:	$C_6\times D_6$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{24} + \cdots)$

Embedding invariants

Embedding label			379.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3240.379
Dual form			3240.1.z.b.2539.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +1.00000 q^{10} +(1.00000 - 1.73205i) q^{11} +(-0.500000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -1.00000 q^{19} +(-0.500000 + 0.866025i) q^{20} +(1.00000 + 1.73205i) q^{22} +(0.500000 + 0.866025i) q^{23} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{26} +1.00000 q^{28} +(-0.500000 - 0.866025i) q^{32} +1.00000 q^{35} -2.00000 q^{37} +(0.500000 - 0.866025i) q^{38} +(-0.500000 - 0.866025i) q^{40} +(-0.500000 - 0.866025i) q^{41} -2.00000 q^{44} -1.00000 q^{46} +(0.500000 - 0.866025i) q^{47} +(-0.500000 - 0.866025i) q^{50} +(-0.500000 + 0.866025i) q^{52} -1.00000 q^{53} -2.00000 q^{55} +(-0.500000 + 0.866025i) q^{56} +(-0.500000 - 0.866025i) q^{59} +1.00000 q^{64} +(-0.500000 + 0.866025i) q^{65} +(-0.500000 + 0.866025i) q^{70} +(1.00000 - 1.73205i) q^{74} +(0.500000 + 0.866025i) q^{76} +(1.00000 + 1.73205i) q^{77} +1.00000 q^{80} +1.00000 q^{82} +(1.00000 - 1.73205i) q^{88} -2.00000 q^{89} +1.00000 q^{91} +(0.500000 - 0.866025i) q^{92} +(0.500000 + 0.866025i) q^{94} +(0.500000 + 0.866025i) q^{95} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - q^{5} - q^{7} + 2 q^{8} + 2 q^{10} + 2 q^{11} - q^{13} - q^{14} - q^{16} - 2 q^{19} - q^{20} + 2 q^{22} + q^{23} - q^{25} + 2 q^{26} + 2 q^{28} - q^{32} + 2 q^{35} - 4 q^{37}+ \cdots + q^{95}+O(q^{100}) $ 2 * q - q^2 - q^4 - q^5 - q^7 + 2 * q^8 + 2 * q^10 + 2 * q^11 - q^13 - q^14 - q^16 - 2 * q^19 - q^20 + 2 * q^22 + q^23 - q^25 + 2 * q^26 + 2 * q^28 - q^32 + 2 * q^35 - 4 * q^37 + q^38 - q^40 - q^41 - 4 * q^44 - 2 * q^46 + q^47 - q^50 - q^52 - 2 * q^53 - 4 * q^55 - q^56 - q^59 + 2 * q^64 - q^65 - q^70 + 2 * q^74 + q^76 + 2 * q^77 + 2 * q^80 + 2 * q^82 + 2 * q^88 - 4 * q^89 + 2 * q^91 + q^92 + q^94 + q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$-1$	$-1$	$-1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$2$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$3$		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	0.500000	+	0.866025i	$0.333333\pi$
$7$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$8$	1.00000			1.00000
$9$		0			0
$10$	1.00000			1.00000
$11$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$11$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$12$		0			0
$13$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$13$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$14$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$15$		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
$19$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$19$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$20$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$21$		0			0
$22$	1.00000	+	1.73205i	1.00000	+	1.73205i
$23$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$23$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$	1.00000			1.00000
$27$		0			0
$28$	1.00000			1.00000
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$31$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$32$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$33$		0			0
$34$		0			0
$35$	1.00000			1.00000
$36$		0			0
$37$	−2.00000			−2.00000			−1.00000			$\pi$
$37$	−2.00000			−2.00000			−1.00000			$\pi$
$38$	0.500000	−	0.866025i	0.500000	−	0.866025i
$39$		0			0
$40$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$41$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$41$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$42$		0			0
$43$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$43$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$44$	−2.00000			−2.00000
$45$		0			0
$46$	−1.00000			−1.00000
$47$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$47$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$48$		0			0
$49$		0			0
$50$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$51$		0			0
$52$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$53$	−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
$55$	−2.00000			−2.00000
$56$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$57$		0			0
$58$		0			0
$59$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$59$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		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	$-0.666667\pi$
$67$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	1.00000	−	1.73205i	1.00000	−	1.73205i
$75$		0			0
$76$	0.500000	+	0.866025i	0.500000	+	0.866025i
$77$	1.00000	+	1.73205i	1.00000	+	1.73205i
$78$		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
$82$	1.00000			1.00000
$83$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$83$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	1.00000	−	1.73205i	1.00000	−	1.73205i
$89$	−2.00000			−2.00000			−1.00000			$\pi$
$89$	−2.00000			−2.00000			−1.00000			$\pi$
$90$		0			0
$91$	1.00000			1.00000
$92$	0.500000	−	0.866025i	0.500000	−	0.866025i
$93$		0			0
$94$	0.500000	+	0.866025i	0.500000	+	0.866025i
$95$	0.500000	+	0.866025i	0.500000	+	0.866025i
$96$		0			0
$97$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$97$		0			0		−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	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$103$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$104$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$105$		0			0
$106$	0.500000	−	0.866025i	0.500000	−	0.866025i
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$	1.00000	−	1.73205i	1.00000	−	1.73205i
$111$		0			0
$112$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$	0.500000	−	0.866025i	0.500000	−	0.866025i
$116$		0			0
$117$		0			0
$118$	1.00000			1.00000
$119$		0			0
$120$		0			0
$121$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			1.00000
$126$		0			0
$127$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$127$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$128$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$129$		0			0
$130$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$131$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$131$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$132$		0			0
$133$	0.500000	−	0.866025i	0.500000	−	0.866025i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$139$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$140$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$141$		0			0
$142$		0			0
$143$	−2.00000			−2.00000
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	1.00000	+	1.73205i	1.00000	+	1.73205i
$149$		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	$0.666667\pi$
$152$	−1.00000			−1.00000
$153$		0			0
$154$	−2.00000			−2.00000
$155$		0			0
$156$		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
$160$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$161$	−1.00000			−1.00000
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$165$		0			0
$166$		0			0
$167$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$167$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$173$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$174$		0			0
$175$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$176$	1.00000	+	1.73205i	1.00000	+	1.73205i
$177$		0			0
$178$	1.00000	−	1.73205i	1.00000	−	1.73205i
$179$	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			$\pi$
$182$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$183$		0			0
$184$	0.500000	+	0.866025i	0.500000	+	0.866025i
$185$	1.00000	+	1.73205i	1.00000	+	1.73205i
$186$		0			0
$187$		0			0
$188$	−1.00000			−1.00000
$189$		0			0
$190$	−1.00000			−1.00000
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$197$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$206$	1.00000			1.00000
$207$		0			0
$208$	1.00000			1.00000
$209$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$210$		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	$0.666667\pi$
$212$	0.500000	+	0.866025i	0.500000	+	0.866025i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	1.00000	+	1.73205i	1.00000	+	1.73205i
$221$		0			0
$222$		0			0
$223$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$223$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$224$	1.00000			1.00000
$225$		0			0
$226$		0			0
$227$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$	0.500000	+	0.866025i	0.500000	+	0.866025i
$231$		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
$236$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$241$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$242$	3.00000			3.00000
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.500000	+	0.866025i	0.500000	+	0.866025i
$248$		0			0
$249$		0			0
$250$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$251$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$251$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$252$		0			0
$253$	2.00000			2.00000
$254$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	1.00000	−	1.73205i	1.00000	−	1.73205i
$260$	1.00000			1.00000
$261$		0			0
$262$	1.00000			1.00000
$263$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$263$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$264$		0			0
$265$	0.500000	+	0.866025i	0.500000	+	0.866025i
$266$	0.500000	+	0.866025i	0.500000	+	0.866025i
$267$		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			0
$273$		0			0
$274$		0			0
$275$	1.00000	+	1.73205i	1.00000	+	1.73205i
$276$		0			0
$277$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$277$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$278$	−1.00000			−1.00000
$279$		0			0
$280$	1.00000			1.00000
$281$	−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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$283$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$284$		0			0
$285$		0			0
$286$	1.00000	−	1.73205i	1.00000	−	1.73205i
$287$	1.00000			1.00000
$288$		0			0
$289$	1.00000			1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$293$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$294$		0			0
$295$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$296$	−2.00000			−2.00000
$297$		0			0
$298$		0			0
$299$	0.500000	−	0.866025i	0.500000	−	0.866025i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.500000	−	0.866025i	0.500000	−	0.866025i
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$	1.00000	−	1.73205i	1.00000	−	1.73205i
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$313$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$314$	1.00000			1.00000
$315$		0			0
$316$		0			0
$317$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$317$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$318$		0			0
$319$		0			0
$320$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$321$		0			0
$322$	0.500000	−	0.866025i	0.500000	−	0.866025i
$323$		0			0
$324$		0			0
$325$	1.00000			1.00000
$326$		0			0
$327$		0			0
$328$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$329$	0.500000	+	0.866025i	0.500000	+	0.866025i
$330$		0			0
$331$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$331$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$	2.00000			2.00000
$335$		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
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.00000			−1.00000
$344$		0			0
$345$		0			0
$346$	0.500000	+	0.866025i	0.500000	+	0.866025i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$	1.00000			1.00000
$351$		0			0
$352$	−2.00000			−2.00000
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$	1.00000	+	1.73205i	1.00000	+	1.73205i
$357$		0			0
$358$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$365$		0			0
$366$		0			0
$367$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$367$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$368$	−1.00000			−1.00000
$369$		0			0
$370$	−2.00000			−2.00000
$371$	0.500000	−	0.866025i	0.500000	−	0.866025i
$372$		0			0
$373$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$373$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$	0.500000	−	0.866025i	0.500000	−	0.866025i
$377$		0			0
$378$		0			0
$379$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$380$	0.500000	−	0.866025i	0.500000	−	0.866025i
$381$		0			0
$382$		0			0
$383$	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
$385$	1.00000	−	1.73205i	1.00000	−	1.73205i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	0.500000	−	0.866025i	0.500000	−	0.866025i
$395$		0			0
$396$		0			0
$397$	−2.00000			−2.00000			−1.00000			$\pi$
$397$	−2.00000			−2.00000			−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$401$	−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
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$408$		0			0
$409$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$409$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$410$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$411$		0			0
$412$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$413$	1.00000			1.00000
$414$		0			0
$415$		0			0
$416$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$417$		0			0
$418$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$419$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$419$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$	−1.00000			−1.00000
$423$		0			0
$424$	−1.00000			−1.00000
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$	−2.00000			−2.00000
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$	1.00000	+	1.73205i	1.00000	+	1.73205i
$446$	1.00000	+	1.73205i	1.00000	+	1.73205i
$447$		0			0
$448$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$449$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$449$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$	−2.00000			−2.00000
$452$		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	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$		0			0
$459$		0			0
$460$	−1.00000			−1.00000
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$463$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$	0.500000	−	0.866025i	0.500000	−	0.866025i
$471$		0			0
$472$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$473$		0			0
$474$		0			0
$475$	0.500000	−	0.866025i	0.500000	−	0.866025i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	1.00000	+	1.73205i	1.00000	+	1.73205i
$482$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$483$		0			0
$484$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$485$		0			0
$486$		0			0
$487$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$487$	1.00000			1.00000			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	0.500000	−	0.866025i	$-0.333333\pi$
$491$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$	−1.00000			−1.00000
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$499$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$500$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$501$		0			0
$502$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$503$	2.00000			2.00000			1.00000			$0$
$503$	2.00000			2.00000			1.00000			$0$
$504$		0			0
$505$		0			0
$506$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$507$		0			0
$508$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$	1.00000			1.00000
$513$		0			0
$514$		0			0
$515$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$516$		0			0
$517$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$518$	1.00000	+	1.73205i	1.00000	+	1.73205i
$519$		0			0
$520$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$521$	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			$\pi$
$524$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$525$		0			0
$526$	0.500000	+	0.866025i	0.500000	+	0.866025i
$527$		0			0
$528$		0			0
$529$		0			0
$530$	−1.00000			−1.00000
$531$		0			0
$532$	−1.00000			−1.00000
$533$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		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
$550$	−2.00000			−2.00000
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$555$		0			0
$556$	0.500000	−	0.866025i	0.500000	−	0.866025i
$557$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$557$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		0			0
$560$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$561$		0			0
$562$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			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
$571$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$572$	1.00000	+	1.73205i	1.00000	+	1.73205i
$573$		0			0
$574$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$575$	−1.00000			−1.00000
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$584$		0			0
$585$		0			0
$586$	−1.00000			−1.00000
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$		0			0
$589$		0			0
$590$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$591$		0			0
$592$	1.00000	−	1.73205i	1.00000	−	1.73205i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	0.500000	+	0.866025i	0.500000	+	0.866025i
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$601$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$606$		0			0
$607$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$607$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$608$	0.500000	+	0.866025i	0.500000	+	0.866025i
$609$		0			0
$610$		0			0
$611$	−1.00000			−1.00000
$612$		0			0
$613$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$613$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$614$		0			0
$615$		0			0
$616$	1.00000	+	1.73205i	1.00000	+	1.73205i
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$		0			0
$619$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$619$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.00000	−	1.73205i	1.00000	−	1.73205i
$624$		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
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$	0.500000	+	0.866025i	0.500000	+	0.866025i
$635$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	1.00000			1.00000
$641$	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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$	0.500000	+	0.866025i	0.500000	+	0.866025i
$645$		0			0
$646$		0			0
$647$	2.00000			2.00000			1.00000			$0$
$647$	2.00000			2.00000			1.00000			$0$
$648$		0			0
$649$	−2.00000			−2.00000
$650$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$651$		0			0
$652$		0			0
$653$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$653$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$654$		0			0
$655$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$656$	1.00000			1.00000
$657$		0			0
$658$	−1.00000			−1.00000
$659$	−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	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$663$		0			0
$664$		0			0
$665$	−1.00000			−1.00000
$666$		0			0
$667$		0			0
$668$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$669$		0			0
$670$		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
$675$		0			0
$676$		0			0
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$677$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$678$		0			0
$679$		0			0
$680$		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
$686$	0.500000	−	0.866025i	0.500000	−	0.866025i
$687$		0			0
$688$		0			0
$689$	0.500000	+	0.866025i	0.500000	+	0.866025i
$690$		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			$0$
$692$	−1.00000			−1.00000
$693$		0			0
$694$		0			0
$695$	0.500000	−	0.866025i	0.500000	−	0.866025i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	2.00000			2.00000
$704$	1.00000	−	1.73205i	1.00000	−	1.73205i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	−2.00000			−2.00000
$713$		0			0
$714$		0			0
$715$	1.00000	+	1.73205i	1.00000	+	1.73205i
$716$	−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
$721$	1.00000			1.00000
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$727$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$728$	1.00000			1.00000
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$733$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$734$	1.00000	+	1.73205i	1.00000	+	1.73205i
$735$		0			0
$736$	0.500000	−	0.866025i	0.500000	−	0.866025i
$737$		0			0
$738$		0			0
$739$	2.00000			2.00000			1.00000			$0$
$739$	2.00000			2.00000			1.00000			$0$
$740$	1.00000	−	1.73205i	1.00000	−	1.73205i
$741$		0			0
$742$	0.500000	+	0.866025i	0.500000	+	0.866025i
$743$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$744$		0			0
$745$		0			0
$746$	−2.00000			−2.00000
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$	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	$-0.333333\pi$
$757$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$758$	0.500000	−	0.866025i	0.500000	−	0.866025i
$759$		0			0
$760$	0.500000	+	0.866025i	0.500000	+	0.866025i
$761$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$761$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−1.00000			−1.00000
$767$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$768$		0			0
$769$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$769$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$770$	1.00000	+	1.73205i	1.00000	+	1.73205i
$771$		0			0
$772$		0			0
$773$	2.00000			2.00000			1.00000			$0$
$773$	2.00000			2.00000			1.00000			$0$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.500000	+	0.866025i	0.500000	+	0.866025i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−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	$0.333333\pi$
$788$	0.500000	+	0.866025i	0.500000	+	0.866025i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	1.00000	−	1.73205i	1.00000	−	1.73205i
$795$		0			0
$796$		0			0
$797$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$797$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$798$		0			0
$799$		0			0
$800$	1.00000			1.00000
$801$		0			0
$802$	1.00000			1.00000
$803$		0			0
$804$		0			0
$805$	0.500000	+	0.866025i	0.500000	+	0.866025i
$806$		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
$811$	2.00000			2.00000			1.00000			$0$
$811$	2.00000			2.00000			1.00000			$0$
$812$		0			0
$813$		0			0
$814$	−2.00000	−	3.46410i	−2.00000	−	3.46410i
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.00000			−1.00000
$819$		0			0
$820$	1.00000			1.00000
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$823$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$824$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$825$		0			0
$826$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$827$		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			0
$834$		0			0
$835$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$836$	2.00000			2.00000
$837$		0			0
$838$	−2.00000			−2.00000
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	0.500000	−	0.866025i	0.500000	−	0.866025i
$845$		0			0
$846$		0			0
$847$	3.00000			3.00000
$848$	0.500000	−	0.866025i	0.500000	−	0.866025i
$849$		0			0
$850$		0			0
$851$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$852$		0			0
$853$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$853$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$859$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$863$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$	−1.00000			−1.00000
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	1.00000			1.00000
$875$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$876$		0			0
$877$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$877$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$	1.00000	−	1.73205i	1.00000	−	1.73205i
$881$	−2.00000			−2.00000			−1.00000			$\pi$
$881$	−2.00000			−2.00000			−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$887$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$890$	−2.00000			−2.00000
$891$		0			0
$892$	−2.00000			−2.00000
$893$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$894$		0			0
$895$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$896$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$897$		0			0
$898$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$899$		0			0
$900$		0			0
$901$		0			0
$902$	1.00000	−	1.73205i	1.00000	−	1.73205i
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$		0			0
$910$	1.00000			1.00000
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.00000			1.00000
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$	0.500000	−	0.866025i	0.500000	−	0.866025i
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.00000	−	1.73205i	1.00000	−	1.73205i
$926$	1.00000			1.00000
$927$		0			0
$928$		0			0
$929$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$929$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$	0.500000	+	0.866025i	0.500000	+	0.866025i
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$	0.500000	−	0.866025i	0.500000	−	0.866025i
$944$	1.00000			1.00000
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		0			0
$950$	0.500000	+	0.866025i	0.500000	+	0.866025i
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$	−2.00000			−2.00000
$963$		0			0
$964$	2.00000			2.00000
$965$		0			0
$966$		0			0
$967$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$967$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$968$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$969$		0			0
$970$		0			0
$971$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$971$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$	−1.00000			−1.00000
$974$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$975$		0			0
$976$		0			0
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$	−2.00000	+	3.46410i	−2.00000	+	3.46410i
$980$		0			0
$981$		0			0
$982$	1.00000			1.00000
$983$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$983$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$	0.500000	+	0.866025i	0.500000	+	0.866025i
$986$		0			0
$987$		0			0
$988$	0.500000	−	0.866025i	0.500000	−	0.866025i
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$997$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$998$	−1.00000			−1.00000
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.1.z.b.379.1		2
3.2	odd	2		3240.1.z.h.379.1		2
5.4	even	2		3240.1.z.i.379.1		2
8.3	odd	2		3240.1.z.i.379.1		2
9.2	odd	6		3240.1.p.b.1459.1	yes	1
9.4	even	3	inner	3240.1.z.b.2539.1		2
9.5	odd	6		3240.1.z.h.2539.1		2
9.7	even	3		3240.1.p.d.1459.1	yes	1
15.14	odd	2		3240.1.z.c.379.1		2
24.11	even	2		3240.1.z.c.379.1		2
40.19	odd	2	CM	3240.1.z.b.379.1		2
45.4	even	6		3240.1.z.i.2539.1		2
45.14	odd	6		3240.1.z.c.2539.1		2
45.29	odd	6		3240.1.p.c.1459.1	yes	1
45.34	even	6		3240.1.p.a.1459.1	✓	1
72.11	even	6		3240.1.p.c.1459.1	yes	1
72.43	odd	6		3240.1.p.a.1459.1	✓	1
72.59	even	6		3240.1.z.c.2539.1		2
72.67	odd	6		3240.1.z.i.2539.1		2
120.59	even	2		3240.1.z.h.379.1		2
360.59	even	6		3240.1.z.h.2539.1		2
360.139	odd	6	inner	3240.1.z.b.2539.1		2
360.259	odd	6		3240.1.p.d.1459.1	yes	1
360.299	even	6		3240.1.p.b.1459.1	yes	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.1.p.a.1459.1	✓	1	45.34	even	6
3240.1.p.a.1459.1	✓	1	72.43	odd	6
3240.1.p.b.1459.1	yes	1	9.2	odd	6
3240.1.p.b.1459.1	yes	1	360.299	even	6
3240.1.p.c.1459.1	yes	1	45.29	odd	6
3240.1.p.c.1459.1	yes	1	72.11	even	6
3240.1.p.d.1459.1	yes	1	9.7	even	3
3240.1.p.d.1459.1	yes	1	360.259	odd	6
3240.1.z.b.379.1		2	1.1	even	1	trivial
3240.1.z.b.379.1		2	40.19	odd	2	CM
3240.1.z.b.2539.1		2	9.4	even	3	inner
3240.1.z.b.2539.1		2	360.139	odd	6	inner
3240.1.z.c.379.1		2	15.14	odd	2
3240.1.z.c.379.1		2	24.11	even	2
3240.1.z.c.2539.1		2	45.14	odd	6
3240.1.z.c.2539.1		2	72.59	even	6
3240.1.z.h.379.1		2	3.2	odd	2
3240.1.z.h.379.1		2	120.59	even	2
3240.1.z.h.2539.1		2	9.5	odd	6
3240.1.z.h.2539.1		2	360.59	even	6
3240.1.z.i.379.1		2	5.4	even	2
3240.1.z.i.379.1		2	8.3	odd	2
3240.1.z.i.2539.1		2	45.4	even	6
3240.1.z.i.2539.1		2	72.67	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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3240.z (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +1.00000 q^{10} +(1.00000 - 1.73205i) q^{11} +(-0.500000 - 0.866025i) q^{13} +(-0.500000 - 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} -1.00000 q^{19} +(-0.500000 + 0.866025i) q^{20} +(1.00000 + 1.73205i) q^{22} +(0.500000 + 0.866025i) q^{23} +(-0.500000 + 0.866025i) q^{25} +1.00000 q^{26} +1.00000 q^{28} +(-0.500000 - 0.866025i) q^{32} +1.00000 q^{35} -2.00000 q^{37} +(0.500000 - 0.866025i) q^{38} +(-0.500000 - 0.866025i) q^{40} +(-0.500000 - 0.866025i) q^{41} -2.00000 q^{44} -1.00000 q^{46} +(0.500000 - 0.866025i) q^{47} +(-0.500000 - 0.866025i) q^{50} +(-0.500000 + 0.866025i) q^{52} -1.00000 q^{53} -2.00000 q^{55} +(-0.500000 + 0.866025i) q^{56} +(-0.500000 - 0.866025i) q^{59} +1.00000 q^{64} +(-0.500000 + 0.866025i) q^{65} +(-0.500000 + 0.866025i) q^{70} +(1.00000 - 1.73205i) q^{74} +(0.500000 + 0.866025i) q^{76} +(1.00000 + 1.73205i) q^{77} +1.00000 q^{80} +1.00000 q^{82} +(1.00000 - 1.73205i) q^{88} -2.00000 q^{89} +1.00000 q^{91} +(0.500000 - 0.866025i) q^{92} +(0.500000 + 0.866025i) q^{94} +(0.500000 + 0.866025i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - q^{5} - q^{7} + 2 q^{8} + 2 q^{10} + 2 q^{11} - q^{13} - q^{14} - q^{16} - 2 q^{19} - q^{20} + 2 q^{22} + q^{23} - q^{25} + 2 q^{26} + 2 q^{28} - q^{32} + 2 q^{35} - 4 q^{37}+ \cdots + q^{95}+O(q^{100}) \) 2 * q - q^2 - q^4 - q^5 - q^7 + 2 * q^8 + 2 * q^10 + 2 * q^11 - q^13 - q^14 - q^16 - 2 * q^19 - q^20 + 2 * q^22 + q^23 - q^25 + 2 * q^26 + 2 * q^28 - q^32 + 2 * q^35 - 4 * q^37 + q^38 - q^40 - q^41 - 4 * q^44 - 2 * q^46 + q^47 - q^50 - q^52 - 2 * q^53 - 4 * q^55 - q^56 - q^59 + 2 * q^64 - q^65 - q^70 + 2 * q^74 + q^76 + 2 * q^77 + 2 * q^80 + 2 * q^82 + 2 * q^88 - 4 * q^89 + 2 * q^91 + q^92 + q^94 + q^95

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