LMFDB - Embedded newform 1352.1.p.a.147.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,1,Mod(147,1352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1352, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 1]))

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

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

chi := DirichletCharacter("1352.147");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1352 = 2^{3} \cdot 13^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1352.p (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:	$0.674735897080$
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:	no (minimal twist has level 104)
Projective image:	$D_{3}$
Projective field:	Galois closure of 3.1.104.1
Artin image:	$C_3\times S_3$
Artin field:	Galois closure of 6.0.190102016.3

Embedding invariants

Embedding label			147.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1352.147
Dual form			1352.1.p.a.699.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{2} +(0.500000 - 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} -1.00000 q^{5} +(0.500000 + 0.866025i) q^{6} +(0.500000 + 0.866025i) q^{7} +1.00000 q^{8} +(0.500000 - 0.866025i) q^{10} -1.00000 q^{12} -1.00000 q^{14} +(-0.500000 + 0.866025i) q^{15} +(-0.500000 + 0.866025i) q^{16} +(0.500000 + 0.866025i) q^{17} +(0.500000 + 0.866025i) q^{20} +1.00000 q^{21} +(0.500000 - 0.866025i) q^{24} +1.00000 q^{27} +(0.500000 - 0.866025i) q^{28} +(-0.500000 - 0.866025i) q^{30} +2.00000 q^{31} +(-0.500000 - 0.866025i) q^{32} -1.00000 q^{34} +(-0.500000 - 0.866025i) q^{35} +(0.500000 - 0.866025i) q^{37} -1.00000 q^{40} +(-0.500000 + 0.866025i) q^{42} +(0.500000 + 0.866025i) q^{43} -1.00000 q^{47} +(0.500000 + 0.866025i) q^{48} +1.00000 q^{51} +(-0.500000 + 0.866025i) q^{54} +(0.500000 + 0.866025i) q^{56} +1.00000 q^{60} +(-1.00000 + 1.73205i) q^{62} +1.00000 q^{64} +(0.500000 - 0.866025i) q^{68} +1.00000 q^{70} +(0.500000 + 0.866025i) q^{71} +(0.500000 + 0.866025i) q^{74} +(0.500000 - 0.866025i) q^{80} +(0.500000 - 0.866025i) q^{81} +(-0.500000 - 0.866025i) q^{84} +(-0.500000 - 0.866025i) q^{85} -1.00000 q^{86} +(1.00000 - 1.73205i) q^{93} +(0.500000 - 0.866025i) q^{94} -1.00000 q^{96} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - q^{2} + q^{3} - q^{4} - 2 q^{5} + q^{6} + q^{7} + 2 q^{8} + q^{10} - 2 q^{12} - 2 q^{14} - q^{15} - q^{16} + q^{17} + q^{20} + 2 q^{21} + q^{24} + 2 q^{27} + q^{28} - q^{30} + 4 q^{31}+ \cdots - 2 q^{96}+O(q^{100})$ 2 * q - q^2 + q^3 - q^4 - 2 * q^5 + q^6 + q^7 + 2 * q^8 + q^10 - 2 * q^12 - 2 * q^14 - q^15 - q^16 + q^17 + q^20 + 2 * q^21 + q^24 + 2 * q^27 + q^28 - q^30 + 4 * q^31 - q^32 - 2 * q^34 - q^35 + q^37 - 2 * q^40 - q^42 + q^43 - 2 * q^47 + q^48 + 2 * q^51 - q^54 + q^56 + 2 * q^60 - 2 * q^62 + 2 * q^64 + q^68 + 2 * q^70 + q^71 + q^74 + q^80 + q^81 - q^84 - q^85 - 2 * q^86 + 2 * q^93 + q^94 - 2 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1352\mathbb{Z}\right)^\times$ .

$n$	$677$	$1015$	$1185$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{1}{6}\right)$

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
$3$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$4$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$5$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$6$	0.500000	+	0.866025i	0.500000	+	0.866025i
$7$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$7$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$8$	1.00000			1.00000
$9$		0			0
$10$	0.500000	−	0.866025i	0.500000	−	0.866025i
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	−1.00000			−1.00000
$13$		0			0
$13$		0			0
$14$	−1.00000			−1.00000
$15$	−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	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$	0.500000	+	0.866025i	0.500000	+	0.866025i
$21$	1.00000			1.00000
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	0.500000	−	0.866025i	0.500000	−	0.866025i
$25$		0			0
$26$		0			0
$27$	1.00000			1.00000
$28$	0.500000	−	0.866025i	0.500000	−	0.866025i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$31$	2.00000			2.00000			1.00000			$0$
$31$	2.00000			2.00000			1.00000			$0$
$32$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$33$		0			0
$34$	−1.00000			−1.00000
$35$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$36$		0			0
$37$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$37$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$38$		0			0
$39$		0			0
$40$	−1.00000			−1.00000
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$43$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$43$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$47$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$48$	0.500000	+	0.866025i	0.500000	+	0.866025i
$49$		0			0
$50$		0			0
$51$	1.00000			1.00000
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$55$		0			0
$56$	0.500000	+	0.866025i	0.500000	+	0.866025i
$57$		0			0
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$	1.00000			1.00000
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$63$		0			0
$64$	1.00000			1.00000
$65$		0			0
$66$		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
$70$	1.00000			1.00000
$71$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$71$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	0.500000	+	0.866025i	0.500000	+	0.866025i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	0.500000	−	0.866025i	0.500000	−	0.866025i
$81$	0.500000	−	0.866025i	0.500000	−	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$85$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$86$	−1.00000			−1.00000
$87$		0			0
$88$		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
$93$	1.00000	−	1.73205i	1.00000	−	1.73205i
$94$	0.500000	−	0.866025i	0.500000	−	0.866025i
$95$		0			0
$96$	−1.00000			−1.00000
$97$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$97$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$	−1.00000			−1.00000
$106$		0			0
$107$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$107$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$108$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$109$	−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
$111$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$112$	−1.00000			−1.00000
$113$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$113$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$120$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$121$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$125$	1.00000			1.00000
$126$		0			0
$127$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$127$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$128$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$129$	1.00000			1.00000
$130$		0			0
$131$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$131$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−1.00000			−1.00000
$136$	0.500000	+	0.866025i	0.500000	+	0.866025i
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	0.500000	+	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
$141$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$142$	−1.00000			−1.00000
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−1.00000			−1.00000
$149$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$149$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−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
$153$		0			0
$154$		0			0
$155$	−2.00000			−2.00000
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$	0.500000	+	0.866025i	0.500000	+	0.866025i
$161$		0			0
$162$	0.500000	+	0.866025i	0.500000	+	0.866025i
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$		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	$0.666667\pi$
$168$	1.00000			1.00000
$169$		0			0
$170$	1.00000			1.00000
$171$		0			0
$172$	0.500000	−	0.866025i	0.500000	−	0.866025i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$179$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$186$	1.00000	+	1.73205i	1.00000	+	1.73205i
$187$		0			0
$188$	0.500000	+	0.866025i	0.500000	+	0.866025i
$189$	0.500000	+	0.866025i	0.500000	+	0.866025i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	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			0
$196$		0			0
$197$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$197$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$198$		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			0
$209$		0			0
$210$	0.500000	−	0.866025i	0.500000	−	0.866025i
$211$	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
$213$	1.00000			1.00000
$214$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$215$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$216$	1.00000			1.00000
$217$	1.00000	+	1.73205i	1.00000	+	1.73205i
$218$	0.500000	−	0.866025i	0.500000	−	0.866025i
$219$		0			0
$220$		0			0
$221$		0			0
$222$	1.00000			1.00000
$223$	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			$0$
$224$	0.500000	−	0.866025i	0.500000	−	0.866025i
$225$		0			0
$226$	2.00000			2.00000
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$229$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$233$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$	1.00000			1.00000
$236$		0			0
$237$		0			0
$238$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$239$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$239$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$240$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$241$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$241$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$242$	1.00000			1.00000
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	2.00000			2.00000
$249$		0			0
$250$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$251$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$251$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−1.00000			−1.00000
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$257$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$258$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$259$	1.00000			1.00000
$260$		0			0
$261$		0			0
$262$	0.500000	−	0.866025i	0.500000	−	0.866025i
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			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	$0.333333\pi$
$270$	0.500000	−	0.866025i	0.500000	−	0.866025i
$271$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$271$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$272$	−1.00000			−1.00000
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$277$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$278$	−1.00000			−1.00000
$279$		0			0
$280$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$283$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$283$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$284$	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			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			0
$296$	0.500000	−	0.866025i	0.500000	−	0.866025i
$297$		0			0
$298$	2.00000			2.00000
$299$		0			0
$300$		0			0
$301$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$302$	0.500000	−	0.866025i	0.500000	−	0.866025i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		1.00000			$0$
$307$		0			0		−1.00000			$\pi$
$308$		0			0
$309$		0			0
$310$	1.00000	−	1.73205i	1.00000	−	1.73205i
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$313$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.00000			2.00000			1.00000			$0$
$317$	2.00000			2.00000			1.00000			$0$
$318$		0			0
$319$		0			0
$320$	−1.00000			−1.00000
$321$	1.00000	+	1.73205i	1.00000	+	1.73205i
$322$		0			0
$323$		0			0
$324$	−1.00000			−1.00000
$325$		0			0
$326$		0			0
$327$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$328$		0			0
$329$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		0			0
$333$		0			0
$334$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$335$		0			0
$336$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$337$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$337$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$338$		0			0
$339$	−2.00000			−2.00000
$340$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$341$		0			0
$342$		0			0
$343$	1.00000			1.00000
$344$	0.500000	+	0.866025i	0.500000	+	0.866025i
$345$		0			0
$346$		0			0
$347$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$347$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$349$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$356$		0			0
$357$	0.500000	+	0.866025i	0.500000	+	0.866025i
$358$	0.500000	+	0.866025i	0.500000	+	0.866025i
$359$	2.00000			2.00000			1.00000			$0$
$359$	2.00000			2.00000			1.00000			$0$
$360$		0			0
$361$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$362$		0			0
$363$	−1.00000			−1.00000
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$367$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$368$		0			0
$369$		0			0
$370$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$371$		0			0
$372$	−2.00000			−2.00000
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	0.500000	−	0.866025i	0.500000	−	0.866025i
$376$	−1.00000			−1.00000
$377$		0			0
$378$	−1.00000			−1.00000
$379$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$379$		0			0		−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	1.00000			$0$
$383$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$384$	0.500000	+	0.866025i	0.500000	+	0.866025i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$394$	0.500000	+	0.866025i	0.500000	+	0.866025i
$395$		0			0
$396$		0			0
$397$	−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			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$406$		0			0
$407$		0			0
$408$	1.00000			1.00000
$409$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$409$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.00000			1.00000
$418$		0			0
$419$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$420$	0.500000	+	0.866025i	0.500000	+	0.866025i
$421$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$421$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$422$	0.500000	+	0.866025i	0.500000	+	0.866025i
$423$		0			0
$424$		0			0
$425$		0			0
$426$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$427$		0			0
$428$	2.00000			2.00000
$429$		0			0
$430$	1.00000			1.00000
$431$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$431$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$432$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$433$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$433$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$434$	−2.00000			−2.00000
$435$		0			0
$436$	0.500000	+	0.866025i	0.500000	+	0.866025i
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$443$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$444$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$445$		0			0
$446$	0.500000	+	0.866025i	0.500000	+	0.866025i
$447$	−2.00000			−2.00000
$448$	0.500000	+	0.866025i	0.500000	+	0.866025i
$449$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$449$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$453$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$457$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$458$	0.500000	−	0.866025i	0.500000	−	0.866025i
$459$	0.500000	+	0.866025i	0.500000	+	0.866025i
$460$		0			0
$461$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$461$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$	2.00000			2.00000			1.00000			$0$
$463$	2.00000			2.00000			1.00000			$0$
$464$		0			0
$465$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$466$	0.500000	−	0.866025i	0.500000	−	0.866025i
$467$	2.00000			2.00000			1.00000			$0$
$467$	2.00000			2.00000			1.00000			$0$
$468$		0			0
$469$		0			0
$470$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$	1.00000			1.00000
$477$		0			0
$478$	0.500000	−	0.866025i	0.500000	−	0.866025i
$479$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$479$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$480$	1.00000			1.00000
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$487$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$491$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$497$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$501$	1.00000	+	1.73205i	1.00000	+	1.73205i
$502$	2.00000			2.00000
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$509$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$510$	0.500000	−	0.866025i	0.500000	−	0.866025i
$511$		0			0
$512$	1.00000			1.00000
$513$		0			0
$514$	0.500000	+	0.866025i	0.500000	+	0.866025i
$515$		0			0
$516$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$517$		0			0
$518$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$519$		0			0
$520$		0			0
$521$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$521$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$523$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$524$	0.500000	+	0.866025i	0.500000	+	0.866025i
$525$		0			0
$526$		0			0
$527$	1.00000	+	1.73205i	1.00000	+	1.73205i
$528$		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
$535$	1.00000	−	1.73205i	1.00000	−	1.73205i
$536$		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
$541$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$541$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$542$	0.500000	+	0.866025i	0.500000	+	0.866025i
$543$		0			0
$544$	0.500000	−	0.866025i	0.500000	−	0.866025i
$545$	1.00000			1.00000
$546$		0			0
$547$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$547$	−1.00000			−1.00000			−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
$555$	0.500000	+	0.866025i	0.500000	+	0.866025i
$556$	0.500000	−	0.866025i	0.500000	−	0.866025i
$557$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$557$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$558$		0			0
$559$		0			0
$560$	1.00000			1.00000
$561$		0			0
$562$		0			0
$563$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$563$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$564$	1.00000			1.00000
$565$	1.00000	+	1.73205i	1.00000	+	1.73205i
$566$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$567$	1.00000			1.00000
$568$	0.500000	+	0.866025i	0.500000	+	0.866025i
$569$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$569$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$570$		0			0
$571$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$571$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		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			0
$591$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$592$	0.500000	+	0.866025i	0.500000	+	0.866025i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$	0.500000	−	0.866025i	0.500000	−	0.866025i
$596$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		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			$0$
$602$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$603$		0			0
$604$	0.500000	+	0.866025i	0.500000	+	0.866025i
$605$	0.500000	+	0.866025i	0.500000	+	0.866025i
$606$		0			0
$607$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$607$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$613$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$614$		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			$\pi$
$620$	1.00000	+	1.73205i	1.00000	+	1.73205i
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.00000			−1.00000
$626$	0.500000	−	0.866025i	0.500000	−	0.866025i
$627$		0			0
$628$		0			0
$629$	1.00000			1.00000
$630$		0			0
$631$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$631$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$632$		0			0
$633$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$634$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.500000	−	0.866025i	0.500000	−	0.866025i
$641$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$642$	−2.00000			−2.00000
$643$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$643$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$644$		0			0
$645$	−1.00000			−1.00000
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$	0.500000	−	0.866025i	0.500000	−	0.866025i
$649$		0			0
$650$		0			0
$651$	2.00000			2.00000
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$655$	1.00000			1.00000
$656$		0			0
$657$		0			0
$658$	1.00000			1.00000
$659$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$659$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$660$		0			0
$661$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$661$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−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
$668$	2.00000			2.00000
$669$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$670$		0			0
$671$		0			0
$672$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$673$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$673$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$674$	0.500000	−	0.866025i	0.500000	−	0.866025i
$675$		0			0
$676$		0			0
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$	1.00000	−	1.73205i	1.00000	−	1.73205i
$679$		0			0
$680$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$		0			0
$685$		0			0
$686$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$687$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$688$	−1.00000			−1.00000
$689$		0			0
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$		0			0
$694$	−1.00000			−1.00000
$695$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$696$		0			0
$697$		0			0
$698$	0.500000	+	0.866025i	0.500000	+	0.866025i
$699$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	0.500000	−	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	$-0.666667\pi$
$710$	1.00000			1.00000
$711$		0			0
$712$		0			0
$713$		0			0
$714$	−1.00000			−1.00000
$715$		0			0
$716$	−1.00000			−1.00000
$717$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$718$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$		0			0
$722$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	0.500000	−	0.866025i	0.500000	−	0.866025i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	1.00000			1.00000
$730$		0			0
$731$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$732$		0			0
$733$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$733$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$739$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$740$	1.00000			1.00000
$741$		0			0
$742$		0			0
$743$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$743$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$744$	1.00000	−	1.73205i	1.00000	−	1.73205i
$745$	1.00000	+	1.73205i	1.00000	+	1.73205i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−2.00000			−2.00000
$750$	0.500000	+	0.866025i	0.500000	+	0.866025i
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$	0.500000	−	0.866025i	0.500000	−	0.866025i
$753$	−2.00000			−2.00000
$754$		0			0
$755$	1.00000			1.00000
$756$	0.500000	−	0.866025i	0.500000	−	0.866025i
$757$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$757$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$764$		0			0
$765$		0			0
$766$	−1.00000			−1.00000
$767$		0			0
$768$	−1.00000			−1.00000
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$	−0.500000	−	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
$777$	0.500000	−	0.866025i	0.500000	−	0.866025i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$787$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$787$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$788$	−1.00000			−1.00000
$789$		0			0
$790$		0			0
$791$	1.00000	−	1.73205i	1.00000	−	1.73205i
$792$		0			0
$793$		0			0
$794$	2.00000			2.00000
$795$		0			0
$796$		0			0
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$809$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$810$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$814$		0			0
$815$		0			0
$816$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$821$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$822$		0			0
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		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		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$835$	1.00000	−	1.73205i	1.00000	−	1.73205i
$836$		0			0
$837$	2.00000			2.00000
$838$	0.500000	+	0.866025i	0.500000	+	0.866025i
$839$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$839$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$840$	−1.00000			−1.00000
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$	0.500000	−	0.866025i	0.500000	−	0.866025i
$843$		0			0
$844$	−1.00000			−1.00000
$845$		0			0
$846$		0			0
$847$	0.500000	−	0.866025i	0.500000	−	0.866025i
$848$		0			0
$849$	1.00000	+	1.73205i	1.00000	+	1.73205i
$850$		0			0
$851$		0			0
$852$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$853$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$853$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$857$	2.00000			2.00000			1.00000			$0$
$857$	2.00000			2.00000			1.00000			$0$
$858$		0			0
$859$	2.00000			2.00000			1.00000			$0$
$859$	2.00000			2.00000			1.00000			$0$
$860$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$861$		0			0
$862$	0.500000	+	0.866025i	0.500000	+	0.866025i
$863$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$863$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$864$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$865$		0			0
$866$	−1.00000			−1.00000
$867$		0			0
$868$	1.00000	−	1.73205i	1.00000	−	1.73205i
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−1.00000			−1.00000
$873$		0			0
$874$		0			0
$875$	0.500000	+	0.866025i	0.500000	+	0.866025i
$876$		0			0
$877$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$877$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$878$		0			0
$879$	1.00000			1.00000
$880$		0			0
$881$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$881$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$882$		0			0
$883$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$883$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$884$		0			0
$885$		0			0
$886$	0.500000	−	0.866025i	0.500000	−	0.866025i
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$889$		0			0
$890$		0			0
$891$		0			0
$892$	−1.00000			−1.00000
$893$		0			0
$894$	1.00000	−	1.73205i	1.00000	−	1.73205i
$895$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$896$	−1.00000			−1.00000
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$	0.500000	+	0.866025i	0.500000	+	0.866025i
$904$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$905$		0			0
$906$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$907$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$907$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.500000	+	0.866025i	0.500000	+	0.866025i
$917$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$918$	−1.00000			−1.00000
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$		0			0
$921$		0			0
$922$	−1.00000			−1.00000
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$927$		0			0
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$931$		0			0
$932$	0.500000	+	0.866025i	0.500000	+	0.866025i
$933$		0			0
$934$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$935$		0			0
$936$		0			0
$937$	2.00000			2.00000			1.00000			$0$
$937$	2.00000			2.00000			1.00000			$0$
$938$		0			0
$939$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$940$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$941$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$941$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$946$		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			0
$951$	1.00000	−	1.73205i	1.00000	−	1.73205i
$952$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$953$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$953$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$954$		0			0
$955$		0			0
$956$	0.500000	+	0.866025i	0.500000	+	0.866025i
$957$		0			0
$958$	0.500000	+	0.866025i	0.500000	+	0.866025i
$959$		0			0
$960$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$961$	3.00000			3.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$967$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$968$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$969$		0			0
$970$		0			0
$971$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$971$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$974$	2.00000			2.00000
$975$		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
$980$		0			0
$981$		0			0
$982$	0.500000	+	0.866025i	0.500000	+	0.866025i
$983$	−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
$985$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$986$		0			0
$987$	−1.00000			−1.00000
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$993$		0			0
$994$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$		0			0
$999$	0.500000	−	0.866025i	0.500000	−	0.866025i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.1.p.a.147.1		2
8.3	odd	2		1352.1.p.b.147.1		2
13.2	odd	12		1352.1.n.a.315.1		4
13.3	even	3	inner	1352.1.p.a.699.1		2
13.4	even	6		104.1.h.a.51.1	✓	1
13.5	odd	4		1352.1.n.a.867.2		4
13.6	odd	12		1352.1.g.a.339.1		2
13.7	odd	12		1352.1.g.a.339.2		2
13.8	odd	4		1352.1.n.a.867.1		4
13.9	even	3		104.1.h.b.51.1	yes	1
13.10	even	6		1352.1.p.b.699.1		2
13.11	odd	12		1352.1.n.a.315.2		4
13.12	even	2		1352.1.p.b.147.1		2
39.17	odd	6		936.1.o.b.883.1		1
39.35	odd	6		936.1.o.a.883.1		1
52.35	odd	6		416.1.h.a.207.1		1
52.43	odd	6		416.1.h.b.207.1		1
65.4	even	6		2600.1.o.d.51.1		1
65.9	even	6		2600.1.o.b.51.1		1
65.17	odd	12		2600.1.b.b.1299.1		2
65.22	odd	12		2600.1.b.a.1299.2		2
65.43	odd	12		2600.1.b.b.1299.2		2
65.48	odd	12		2600.1.b.a.1299.1		2
104.3	odd	6		1352.1.p.b.699.1		2
104.11	even	12		1352.1.n.a.315.1		4
104.19	even	12		1352.1.g.a.339.2		2
104.35	odd	6		104.1.h.a.51.1	✓	1
104.43	odd	6		104.1.h.b.51.1	yes	1
104.51	odd	2	CM	1352.1.p.a.147.1		2
104.59	even	12		1352.1.g.a.339.1		2
104.61	even	6		416.1.h.b.207.1		1
104.67	even	12		1352.1.n.a.315.2		4
104.69	even	6		416.1.h.a.207.1		1
104.75	odd	6	inner	1352.1.p.a.699.1		2
104.83	even	4		1352.1.n.a.867.1		4
104.99	even	4		1352.1.n.a.867.2		4
156.35	even	6		3744.1.o.b.2287.1		1
156.95	even	6		3744.1.o.a.2287.1		1
208.35	odd	12		3328.1.c.a.3327.2		2
208.43	odd	12		3328.1.c.e.3327.1		2
208.61	even	12		3328.1.c.e.3327.1		2
208.69	even	12		3328.1.c.a.3327.2		2
208.139	odd	12		3328.1.c.a.3327.1		2
208.147	odd	12		3328.1.c.e.3327.2		2
208.165	even	12		3328.1.c.e.3327.2		2
208.173	even	12		3328.1.c.a.3327.1		2
312.35	even	6		936.1.o.b.883.1		1
312.173	odd	6		3744.1.o.b.2287.1		1
312.251	even	6		936.1.o.a.883.1		1
312.269	odd	6		3744.1.o.a.2287.1		1
520.43	even	12		2600.1.b.a.1299.1		2
520.139	odd	6		2600.1.o.d.51.1		1
520.147	even	12		2600.1.b.a.1299.2		2
520.243	even	12		2600.1.b.b.1299.2		2
520.347	even	12		2600.1.b.b.1299.1		2
520.459	odd	6		2600.1.o.b.51.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.1.h.a.51.1	✓	1	13.4	even	6
104.1.h.a.51.1	✓	1	104.35	odd	6
104.1.h.b.51.1	yes	1	13.9	even	3
104.1.h.b.51.1	yes	1	104.43	odd	6
416.1.h.a.207.1		1	52.35	odd	6
416.1.h.a.207.1		1	104.69	even	6
416.1.h.b.207.1		1	52.43	odd	6
416.1.h.b.207.1		1	104.61	even	6
936.1.o.a.883.1		1	39.35	odd	6
936.1.o.a.883.1		1	312.251	even	6
936.1.o.b.883.1		1	39.17	odd	6
936.1.o.b.883.1		1	312.35	even	6
1352.1.g.a.339.1		2	13.6	odd	12
1352.1.g.a.339.1		2	104.59	even	12
1352.1.g.a.339.2		2	13.7	odd	12
1352.1.g.a.339.2		2	104.19	even	12
1352.1.n.a.315.1		4	13.2	odd	12
1352.1.n.a.315.1		4	104.11	even	12
1352.1.n.a.315.2		4	13.11	odd	12
1352.1.n.a.315.2		4	104.67	even	12
1352.1.n.a.867.1		4	13.8	odd	4
1352.1.n.a.867.1		4	104.83	even	4
1352.1.n.a.867.2		4	13.5	odd	4
1352.1.n.a.867.2		4	104.99	even	4
1352.1.p.a.147.1		2	1.1	even	1	trivial
1352.1.p.a.147.1		2	104.51	odd	2	CM
1352.1.p.a.699.1		2	13.3	even	3	inner
1352.1.p.a.699.1		2	104.75	odd	6	inner
1352.1.p.b.147.1		2	8.3	odd	2
1352.1.p.b.147.1		2	13.12	even	2
1352.1.p.b.699.1		2	13.10	even	6
1352.1.p.b.699.1		2	104.3	odd	6
2600.1.b.a.1299.1		2	65.48	odd	12
2600.1.b.a.1299.1		2	520.43	even	12
2600.1.b.a.1299.2		2	65.22	odd	12
2600.1.b.a.1299.2		2	520.147	even	12
2600.1.b.b.1299.1		2	65.17	odd	12
2600.1.b.b.1299.1		2	520.347	even	12
2600.1.b.b.1299.2		2	65.43	odd	12
2600.1.b.b.1299.2		2	520.243	even	12
2600.1.o.b.51.1		1	65.9	even	6
2600.1.o.b.51.1		1	520.459	odd	6
2600.1.o.d.51.1		1	65.4	even	6
2600.1.o.d.51.1		1	520.139	odd	6
3328.1.c.a.3327.1		2	208.139	odd	12
3328.1.c.a.3327.1		2	208.173	even	12
3328.1.c.a.3327.2		2	208.35	odd	12
3328.1.c.a.3327.2		2	208.69	even	12
3328.1.c.e.3327.1		2	208.43	odd	12
3328.1.c.e.3327.1		2	208.61	even	12
3328.1.c.e.3327.2		2	208.147	odd	12
3328.1.c.e.3327.2		2	208.165	even	12
3744.1.o.a.2287.1		1	156.95	even	6
3744.1.o.a.2287.1		1	312.269	odd	6
3744.1.o.b.2287.1		1	156.35	even	6
3744.1.o.b.2287.1		1	312.173	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 $\F_{q}$

Introduction

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Modular forms

Varieties

Fields

Representations

Groups

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Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

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Label	1352.1.p.a.147.1

Level	$1352$
Weight	$1$
Character	1352.147
Analytic conductor	$0.675$
Analytic rank	$0$
Dimension	$2$
Projective image	$D_{3}$
CM discriminant	-104
Inner twists	$4$