LMFDB - Embedded newform 616.1.z.b.131.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [616,1,Mod(131,616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(616, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("616.131");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 616 = 2^{3} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	616.z (of order $6$, degree $2$, 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:	$0.307424047782$
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:	$D_{6}$
Projective field:	Galois closure of 6.0.130153408.1

Embedding invariants

Embedding label			131.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	616.131
Dual form			616.1.z.b.395.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.500000 - 0.866025i) q^{2} +(-1.50000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +1.73205i q^{6} +(-0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +(1.00000 - 1.73205i) q^{9} +(0.500000 + 0.866025i) q^{11} +(1.50000 + 0.866025i) q^{12} +1.73205i q^{13} +(0.500000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-1.00000 - 1.73205i) q^{18} -1.73205i q^{21} +1.00000 q^{22} +(1.50000 - 0.866025i) q^{24} +(0.500000 + 0.866025i) q^{25} +(1.50000 + 0.866025i) q^{26} +1.73205i q^{27} +1.00000 q^{28} -1.00000 q^{29} +(0.500000 + 0.866025i) q^{32} +(-1.50000 - 0.866025i) q^{33} -2.00000 q^{36} +(-1.50000 - 2.59808i) q^{39} +(-1.50000 - 0.866025i) q^{42} +(0.500000 - 0.866025i) q^{44} -1.73205i q^{48} +(-0.500000 - 0.866025i) q^{49} +1.00000 q^{50} +(1.50000 - 0.866025i) q^{52} +(1.50000 + 0.866025i) q^{54} +(0.500000 - 0.866025i) q^{56} +(-0.500000 + 0.866025i) q^{58} +(1.50000 - 0.866025i) q^{59} +(-1.50000 - 0.866025i) q^{61} +(1.00000 + 1.73205i) q^{63} +1.00000 q^{64} +(-1.50000 + 0.866025i) q^{66} +(-0.500000 - 0.866025i) q^{67} +(-1.00000 + 1.73205i) q^{72} +(-1.50000 - 0.866025i) q^{75} -1.00000 q^{77} -3.00000 q^{78} +(-0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(-1.50000 + 0.866025i) q^{84} +(1.50000 - 0.866025i) q^{87} +(-0.500000 - 0.866025i) q^{88} +(-1.50000 - 0.866025i) q^{91} +(-1.50000 - 0.866025i) q^{96} +1.73205i q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 3 q^{3} - q^{4} - q^{7} - 2 q^{8} + 2 q^{9} + q^{11} + 3 q^{12} + q^{14} - q^{16} - 2 q^{18} + 2 q^{22} + 3 q^{24} + q^{25} + 3 q^{26} + 2 q^{28} - 2 q^{29} + q^{32} - 3 q^{33} - 4 q^{36}+ \cdots + 4 q^{99}+O(q^{100}) $ 2 * q + q^2 - 3 * q^3 - q^4 - q^7 - 2 * q^8 + 2 * q^9 + q^11 + 3 * q^12 + q^14 - q^16 - 2 * q^18 + 2 * q^22 + 3 * q^24 + q^25 + 3 * q^26 + 2 * q^28 - 2 * q^29 + q^32 - 3 * q^33 - 4 * q^36 - 3 * q^39 - 3 * q^42 + q^44 - q^49 + 2 * q^50 + 3 * q^52 + 3 * q^54 + q^56 - q^58 + 3 * q^59 - 3 * q^61 + 2 * q^63 + 2 * q^64 - 3 * q^66 - q^67 - 2 * q^72 - 3 * q^75 - 2 * q^77 - 6 * q^78 - q^79 - q^81 - 3 * q^84 + 3 * q^87 - q^88 - 3 * q^91 - 3 * q^96 - 2 * q^98 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/616\mathbb{Z}\right)^\times$.

$n$	$57$	$309$	$353$	$463$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{5}{6}\right)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.500000	−	0.866025i	0.500000	−	0.866025i
$2$	0.500000	−	0.866025i	0.500000	−	0.866025i
$3$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$3$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$4$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$			1.73205i			1.73205i
$7$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$7$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$8$	−1.00000			−1.00000
$9$	1.00000	−	1.73205i	1.00000	−	1.73205i
$10$		0			0
$11$	0.500000	+	0.866025i	0.500000	+	0.866025i
$11$	0.500000	+	0.866025i	0.500000	+	0.866025i
$12$	1.50000	+	0.866025i	1.50000	+	0.866025i
$13$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$13$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$14$	0.500000	+	0.866025i	0.500000	+	0.866025i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$19$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$19$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$20$		0			0
$21$		−	1.73205i		−	1.73205i
$22$	1.00000			1.00000
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$	1.50000	−	0.866025i	1.50000	−	0.866025i
$25$	0.500000	+	0.866025i	0.500000	+	0.866025i
$26$	1.50000	+	0.866025i	1.50000	+	0.866025i
$27$			1.73205i			1.73205i
$28$	1.00000			1.00000
$29$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$29$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	0.500000	+	0.866025i	0.500000	+	0.866025i
$33$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$34$		0			0
$35$		0			0
$36$	−2.00000			−2.00000
$37$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$37$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$38$		0			0
$39$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	0.500000	−	0.866025i	0.500000	−	0.866025i
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		−	1.73205i		−	1.73205i
$49$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$50$	1.00000			1.00000
$51$		0			0
$52$	1.50000	−	0.866025i	1.50000	−	0.866025i
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$	1.50000	+	0.866025i	1.50000	+	0.866025i
$55$		0			0
$56$	0.500000	−	0.866025i	0.500000	−	0.866025i
$57$		0			0
$58$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$59$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$59$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$60$		0			0
$61$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$61$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$62$		0			0
$63$	1.00000	+	1.73205i	1.00000	+	1.73205i
$64$	1.00000			1.00000
$65$		0			0
$66$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$67$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$67$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$73$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$73$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$74$		0			0
$75$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$76$		0			0
$77$	−1.00000			−1.00000
$78$	−3.00000			−3.00000
$79$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$79$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$85$		0			0
$86$		0			0
$87$	1.50000	−	0.866025i	1.50000	−	0.866025i
$88$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$97$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$97$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$98$	−1.00000			−1.00000
$99$	2.00000			2.00000
$100$	0.500000	−	0.866025i	0.500000	−	0.866025i
$101$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$101$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$		−	1.73205i		−	1.73205i
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	1.50000	−	0.866025i	1.50000	−	0.866025i
$109$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$109$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$113$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$113$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$		0			0
$116$	0.500000	+	0.866025i	0.500000	+	0.866025i
$117$	3.00000	+	1.73205i	3.00000	+	1.73205i
$118$		−	1.73205i		−	1.73205i
$119$		0			0
$120$		0			0
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$123$		0			0
$124$		0			0
$125$		0			0
$126$	2.00000			2.00000
$127$	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			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$			1.73205i			1.73205i
$133$		0			0
$134$	−1.00000			−1.00000
$135$		0			0
$136$		0			0
$137$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$137$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$144$	1.00000	+	1.73205i	1.00000	+	1.73205i
$145$		0			0
$146$		0			0
$147$	1.50000	+	0.866025i	1.50000	+	0.866025i
$148$		0			0
$149$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$149$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$150$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$151$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$151$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$155$		0			0
$156$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$157$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$157$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$158$	0.500000	+	0.866025i	0.500000	+	0.866025i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−1.00000			−1.00000
$163$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$163$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$167$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$168$			1.73205i			1.73205i
$169$	−2.00000			−2.00000
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$173$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$174$		−	1.73205i		−	1.73205i
$175$	−1.00000			−1.00000
$176$	−1.00000			−1.00000
$177$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$178$		0			0
$179$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$179$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$183$	3.00000			3.00000
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$193$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$193$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$194$	1.50000	+	0.866025i	1.50000	+	0.866025i
$195$		0			0
$196$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$197$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$197$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$198$	1.00000	−	1.73205i	1.00000	−	1.73205i
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$201$	1.50000	+	0.866025i	1.50000	+	0.866025i
$202$		−	1.73205i		−	1.73205i
$203$	0.500000	−	0.866025i	0.500000	−	0.866025i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$209$		0			0
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		−	1.73205i		−	1.73205i
$217$		0			0
$218$	2.00000			2.00000
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−1.00000			−1.00000
$225$	2.00000			2.00000
$226$	0.500000	−	0.866025i	0.500000	−	0.866025i
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$	1.50000	−	0.866025i	1.50000	−	0.866025i
$232$	1.00000			1.00000
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$	3.00000	−	1.73205i	3.00000	−	1.73205i
$235$		0			0
$236$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$237$		−	1.73205i		−	1.73205i
$238$		0			0
$239$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$239$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$241$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$242$	0.500000	+	0.866025i	0.500000	+	0.866025i
$243$		0			0
$244$			1.73205i			1.73205i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	1.00000	−	1.73205i	1.00000	−	1.73205i
$253$		0			0
$254$	0.500000	−	0.866025i	0.500000	−	0.866025i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$257$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$262$		0			0
$263$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$263$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$264$	1.50000	+	0.866025i	1.50000	+	0.866025i
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$271$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$272$		0			0
$273$	3.00000			3.00000
$274$	−1.00000			−1.00000
$275$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$276$		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			$\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$283$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$284$		0			0
$285$		0			0
$286$			1.73205i			1.73205i
$287$		0			0
$288$	2.00000			2.00000
$289$	0.500000	−	0.866025i	0.500000	−	0.866025i
$290$		0			0
$291$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	1.50000	−	0.866025i	1.50000	−	0.866025i
$295$		0			0
$296$		0			0
$297$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$298$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$299$		0			0
$300$			1.73205i			1.73205i
$301$		0			0
$302$	1.00000			1.00000
$303$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	0.500000	+	0.866025i	0.500000	+	0.866025i
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$	1.50000	+	2.59808i	1.50000	+	2.59808i
$313$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$313$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$314$		0			0
$315$		0			0
$316$	1.00000			1.00000
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$325$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$326$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$327$	−3.00000	−	1.73205i	−3.00000	−	1.73205i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$331$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$332$		0			0
$333$		0			0
$334$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$335$		0			0
$336$	1.50000	+	0.866025i	1.50000	+	0.866025i
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$339$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.00000			1.00000
$344$		0			0
$345$		0			0
$346$	1.50000	−	0.866025i	1.50000	−	0.866025i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$351$	−3.00000			−3.00000
$352$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$	1.50000	+	2.59808i	1.50000	+	2.59808i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	1.00000			1.00000
$359$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$359$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$360$		0			0
$361$	0.500000	+	0.866025i	0.500000	+	0.866025i
$362$		0			0
$363$		−	1.73205i		−	1.73205i
$364$			1.73205i			1.73205i
$365$		0			0
$366$	1.50000	−	2.59808i	1.50000	−	2.59808i
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$373$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.73205i		−	1.73205i
$378$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$379$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$380$		0			0
$381$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$			1.73205i			1.73205i
$385$		0			0
$386$		0			0
$387$		0			0
$388$	1.50000	−	0.866025i	1.50000	−	0.866025i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	0.500000	+	0.866025i	0.500000	+	0.866025i
$393$		0			0
$394$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$395$		0			0
$396$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$397$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$397$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$	−1.00000			−1.00000
$401$	−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			$\pi$
$402$	1.50000	−	0.866025i	1.50000	−	0.866025i
$403$		0			0
$404$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$405$		0			0
$406$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$407$		0			0
$408$		0			0
$409$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$409$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$410$		0			0
$411$	1.50000	+	0.866025i	1.50000	+	0.866025i
$412$		0			0
$413$			1.73205i			1.73205i
$414$		0			0
$415$		0			0
$416$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.50000	−	0.866025i	1.50000	−	0.866025i
$428$		0			0
$429$	1.50000	−	2.59808i	1.50000	−	2.59808i
$430$		0			0
$431$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$431$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$432$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$	1.00000	−	1.73205i	1.00000	−	1.73205i
$437$		0			0
$438$		0			0
$439$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$439$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	−2.00000			−2.00000
$442$		0			0
$443$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$443$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$			3.46410i			3.46410i
$448$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$449$	−2.00000			−2.00000			−1.00000			$\pi$
$449$	−2.00000			−2.00000			−1.00000			$\pi$
$450$	1.00000	−	1.73205i	1.00000	−	1.73205i
$451$		0			0
$452$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$453$	−1.50000	−	0.866025i	−1.50000	−	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			0
$459$		0			0
$460$		0			0
$461$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$461$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$462$		−	1.73205i		−	1.73205i
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	0.500000	−	0.866025i	0.500000	−	0.866025i
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		−	3.46410i		−	3.46410i
$469$	1.00000			1.00000
$470$		0			0
$471$		0			0
$472$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$473$		0			0
$474$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$475$		0			0
$476$		0			0
$477$		0			0
$478$	0.500000	−	0.866025i	0.500000	−	0.866025i
$479$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$479$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.00000			1.00000
$485$		0			0
$486$		0			0
$487$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$487$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$488$	1.50000	+	0.866025i	1.50000	+	0.866025i
$489$			1.73205i			1.73205i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$499$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$500$		0			0
$501$	1.50000	+	2.59808i	1.50000	+	2.59808i
$502$		0			0
$503$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$503$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$504$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$505$		0			0
$506$		0			0
$507$	3.00000	−	1.73205i	3.00000	−	1.73205i
$508$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−1.00000
$513$		0			0
$514$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−3.00000			−3.00000
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$	1.00000	+	1.73205i	1.00000	+	1.73205i
$523$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$523$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$524$		0			0
$525$	1.50000	−	0.866025i	1.50000	−	0.866025i
$526$	1.00000			1.00000
$527$		0			0
$528$	1.50000	−	0.866025i	1.50000	−	0.866025i
$529$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$530$		0			0
$531$		−	3.46410i		−	3.46410i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.500000	+	0.866025i	0.500000	+	0.866025i
$537$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$538$		0			0
$539$	0.500000	−	0.866025i	0.500000	−	0.866025i
$540$		0			0
$541$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$541$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$542$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$543$		0			0
$544$		0			0
$545$		0			0
$546$	1.50000	−	2.59808i	1.50000	−	2.59808i
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$549$	−3.00000	+	1.73205i	−3.00000	+	1.73205i
$550$	0.500000	+	0.866025i	0.500000	+	0.866025i
$551$		0			0
$552$		0			0
$553$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$554$	−1.00000			−1.00000
$555$		0			0
$556$		0			0
$557$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$557$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.00000			1.00000
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$	1.50000	+	0.866025i	1.50000	+	0.866025i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	1.00000	−	1.73205i	1.00000	−	1.73205i
$577$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$577$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$578$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$579$		0			0
$580$		0			0
$581$		0			0
$582$	−3.00000			−3.00000
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$587$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$588$		−	1.73205i		−	1.73205i
$589$		0			0
$590$		0			0
$591$	1.50000	−	0.866025i	1.50000	−	0.866025i
$592$		0			0
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$			1.73205i			1.73205i
$595$		0			0
$596$	−2.00000			−2.00000
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$	1.50000	+	0.866025i	1.50000	+	0.866025i
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$	−2.00000			−2.00000
$604$	0.500000	−	0.866025i	0.500000	−	0.866025i
$605$		0			0
$606$	1.50000	+	2.59808i	1.50000	+	2.59808i
$607$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$607$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$608$		0			0
$609$			1.73205i			1.73205i
$610$		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
$616$	1.00000			1.00000
$617$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$617$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$618$		0			0
$619$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$619$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	3.00000			3.00000
$625$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$626$	1.50000	−	0.866025i	1.50000	−	0.866025i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	0.500000	−	0.866025i	0.500000	−	0.866025i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.50000	−	0.866025i	1.50000	−	0.866025i
$638$	−1.00000			−1.00000
$639$		0			0
$640$		0			0
$641$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$641$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$643$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$	0.500000	+	0.866025i	0.500000	+	0.866025i
$649$	1.50000	+	0.866025i	1.50000	+	0.866025i
$650$			1.73205i			1.73205i
$651$		0			0
$652$	−1.00000			−1.00000
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$	−3.00000	+	1.73205i	−3.00000	+	1.73205i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$661$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$662$	0.500000	+	0.866025i	0.500000	+	0.866025i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$669$		0			0
$670$		0			0
$671$		−	1.73205i		−	1.73205i
$672$	1.50000	−	0.866025i	1.50000	−	0.866025i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$676$	1.00000	+	1.73205i	1.00000	+	1.73205i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$			1.73205i			1.73205i
$679$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$683$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$	0.500000	−	0.866025i	0.500000	−	0.866025i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$691$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$692$		−	1.73205i		−	1.73205i
$693$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$694$		0			0
$695$		0			0
$696$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.500000	+	0.866025i	0.500000	+	0.866025i
$701$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$701$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$702$	−1.50000	+	2.59808i	−1.50000	+	2.59808i
$703$		0			0
$704$	0.500000	+	0.866025i	0.500000	+	0.866025i
$705$		0			0
$706$		0			0
$707$			1.73205i			1.73205i
$708$	3.00000			3.00000
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$	1.00000	+	1.73205i	1.00000	+	1.73205i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	0.500000	−	0.866025i	0.500000	−	0.866025i
$717$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$718$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$		0			0
$722$	1.00000			1.00000
$723$		0			0
$724$		0			0
$725$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$726$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$	1.50000	+	0.866025i	1.50000	+	0.866025i
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$733$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$733$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	0.500000	−	0.866025i	0.500000	−	0.866025i
$738$		0			0
$739$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$739$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.00000			−2.00000			−1.00000			$\pi$
$743$	−2.00000			−2.00000			−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$751$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$752$		0			0
$753$		0			0
$754$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$755$		0			0
$756$			1.73205i			1.73205i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$			1.73205i			1.73205i
$763$	−2.00000			−2.00000
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.50000	+	2.59808i	1.50000	+	2.59808i
$768$	1.50000	+	0.866025i	1.50000	+	0.866025i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	3.00000			3.00000
$772$		0			0
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$		0			0
$776$		−	1.73205i		−	1.73205i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	1.73205i		−	1.73205i
$784$	1.00000			1.00000
$785$		0			0
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$	0.500000	+	0.866025i	0.500000	+	0.866025i
$789$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$790$		0			0
$791$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$792$	−2.00000			−2.00000
$793$	1.50000	−	2.59808i	1.50000	−	2.59808i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$801$		0			0
$802$	0.500000	+	0.866025i	0.500000	+	0.866025i
$803$		0			0
$804$		−	1.73205i		−	1.73205i
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−1.00000			−1.00000
$813$	3.00000			3.00000
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−3.00000	+	1.73205i	−3.00000	+	1.73205i
$820$		0			0
$821$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$821$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$822$	1.50000	−	0.866025i	1.50000	−	0.866025i
$823$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$823$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$824$		0			0
$825$		−	1.73205i		−	1.73205i
$826$	1.50000	+	0.866025i	1.50000	+	0.866025i
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$829$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$830$		0			0
$831$	1.50000	+	0.866025i	1.50000	+	0.866025i
$832$			1.73205i			1.73205i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$		0			0
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		−	1.73205i		−	1.73205i
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$859$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$859$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$	1.00000			1.00000
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$865$		0			0
$866$		0			0
$867$			1.73205i			1.73205i
$868$		0			0
$869$	−1.00000			−1.00000
$870$		0			0
$871$	1.50000	−	0.866025i	1.50000	−	0.866025i
$872$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$873$	3.00000	+	1.73205i	3.00000	+	1.73205i
$874$		0			0
$875$		0			0
$876$		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			$\pi$
$878$	1.50000	−	0.866025i	1.50000	−	0.866025i
$879$		0			0
$880$		0			0
$881$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$881$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$882$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$883$	−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
$886$	1.00000	+	1.73205i	1.00000	+	1.73205i
$887$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$887$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$888$		0			0
$889$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$890$		0			0
$891$	0.500000	−	0.866025i	0.500000	−	0.866025i
$892$		0			0
$893$		0			0
$894$	3.00000	+	1.73205i	3.00000	+	1.73205i
$895$		0			0
$896$	0.500000	+	0.866025i	0.500000	+	0.866025i
$897$		0			0
$898$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$899$		0			0
$900$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1.00000			−1.00000
$905$		0			0
$906$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$907$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$907$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$908$		0			0
$909$		−	3.46410i		−	3.46410i
$910$		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			0
$917$		0			0
$918$		0			0
$919$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$919$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$920$		0			0
$921$		0			0
$922$	1.50000	+	0.866025i	1.50000	+	0.866025i
$923$		0			0
$924$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$929$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$929$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−3.00000	−	1.73205i	−3.00000	−	1.73205i
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$	0.500000	−	0.866025i	0.500000	−	0.866025i
$939$	−3.00000			−3.00000
$940$		0			0
$941$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$941$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$942$		0			0
$943$		0			0
$944$			1.73205i			1.73205i
$945$		0			0
$946$		0			0
$947$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$947$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$948$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$957$	1.50000	+	0.866025i	1.50000	+	0.866025i
$958$		−	1.73205i		−	1.73205i
$959$	1.00000			1.00000
$960$		0			0
$961$	0.500000	−	0.866025i	0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	2.00000			2.00000			1.00000			$0$
$967$	2.00000			2.00000			1.00000			$0$
$968$	0.500000	−	0.866025i	0.500000	−	0.866025i
$969$		0			0
$970$		0			0
$971$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$971$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	1.50000	−	2.59808i	1.50000	−	2.59808i
$976$	1.50000	−	0.866025i	1.50000	−	0.866025i
$977$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$977$	−1.00000	−	1.73205i	−1.00000	−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$978$	1.50000	+	0.866025i	1.50000	+	0.866025i
$979$		0			0
$980$		0			0
$981$	4.00000			4.00000
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$991$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$992$		0			0
$993$		−	1.73205i		−	1.73205i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$997$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$998$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	616.1.z.b.131.1	yes	2
4.3	odd	2		2464.1.bh.b.2287.1		2
7.3	odd	6	inner	616.1.z.b.395.1	yes	2
8.3	odd	2		616.1.z.a.131.1	✓	2
8.5	even	2		2464.1.bh.a.2287.1		2
11.10	odd	2		616.1.z.a.131.1	✓	2
28.3	even	6		2464.1.bh.b.1935.1		2
44.43	even	2		2464.1.bh.a.2287.1		2
56.3	even	6		616.1.z.a.395.1	yes	2
56.45	odd	6		2464.1.bh.a.1935.1		2
77.10	even	6		616.1.z.a.395.1	yes	2
88.21	odd	2		2464.1.bh.b.2287.1		2
88.43	even	2	RM	616.1.z.b.131.1	yes	2
308.87	odd	6		2464.1.bh.a.1935.1		2
616.395	odd	6	inner	616.1.z.b.395.1	yes	2
616.549	even	6		2464.1.bh.b.1935.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.1.z.a.131.1	✓	2	8.3	odd	2
616.1.z.a.131.1	✓	2	11.10	odd	2
616.1.z.a.395.1	yes	2	56.3	even	6
616.1.z.a.395.1	yes	2	77.10	even	6
616.1.z.b.131.1	yes	2	1.1	even	1	trivial
616.1.z.b.131.1	yes	2	88.43	even	2	RM
616.1.z.b.395.1	yes	2	7.3	odd	6	inner
616.1.z.b.395.1	yes	2	616.395	odd	6	inner
2464.1.bh.a.1935.1		2	56.45	odd	6
2464.1.bh.a.1935.1		2	308.87	odd	6
2464.1.bh.a.2287.1		2	8.5	even	2
2464.1.bh.a.2287.1		2	44.43	even	2
2464.1.bh.b.1935.1		2	28.3	even	6
2464.1.bh.b.1935.1		2	616.549	even	6
2464.1.bh.b.2287.1		2	4.3	odd	2
2464.1.bh.b.2287.1		2	88.21	odd	2

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:	\( N \)	\(=\)	\( 616 = 2^{3} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	616.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(-1.50000 + 0.866025i) q^{3} +(-0.500000 - 0.866025i) q^{4} +1.73205i q^{6} +(-0.500000 + 0.866025i) q^{7} -1.00000 q^{8} +(1.00000 - 1.73205i) q^{9} +(0.500000 + 0.866025i) q^{11} +(1.50000 + 0.866025i) q^{12} +1.73205i q^{13} +(0.500000 + 0.866025i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(-1.00000 - 1.73205i) q^{18} -1.73205i q^{21} +1.00000 q^{22} +(1.50000 - 0.866025i) q^{24} +(0.500000 + 0.866025i) q^{25} +(1.50000 + 0.866025i) q^{26} +1.73205i q^{27} +1.00000 q^{28} -1.00000 q^{29} +(0.500000 + 0.866025i) q^{32} +(-1.50000 - 0.866025i) q^{33} -2.00000 q^{36} +(-1.50000 - 2.59808i) q^{39} +(-1.50000 - 0.866025i) q^{42} +(0.500000 - 0.866025i) q^{44} -1.73205i q^{48} +(-0.500000 - 0.866025i) q^{49} +1.00000 q^{50} +(1.50000 - 0.866025i) q^{52} +(1.50000 + 0.866025i) q^{54} +(0.500000 - 0.866025i) q^{56} +(-0.500000 + 0.866025i) q^{58} +(1.50000 - 0.866025i) q^{59} +(-1.50000 - 0.866025i) q^{61} +(1.00000 + 1.73205i) q^{63} +1.00000 q^{64} +(-1.50000 + 0.866025i) q^{66} +(-0.500000 - 0.866025i) q^{67} +(-1.00000 + 1.73205i) q^{72} +(-1.50000 - 0.866025i) q^{75} -1.00000 q^{77} -3.00000 q^{78} +(-0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} +(-1.50000 + 0.866025i) q^{84} +(1.50000 - 0.866025i) q^{87} +(-0.500000 - 0.866025i) q^{88} +(-1.50000 - 0.866025i) q^{91} +(-1.50000 - 0.866025i) q^{96} +1.73205i q^{97} -1.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 3 q^{3} - q^{4} - q^{7} - 2 q^{8} + 2 q^{9} + q^{11} + 3 q^{12} + q^{14} - q^{16} - 2 q^{18} + 2 q^{22} + 3 q^{24} + q^{25} + 3 q^{26} + 2 q^{28} - 2 q^{29} + q^{32} - 3 q^{33} - 4 q^{36}+ \cdots + 4 q^{99}+O(q^{100}) \) 2 * q + q^2 - 3 * q^3 - q^4 - q^7 - 2 * q^8 + 2 * q^9 + q^11 + 3 * q^12 + q^14 - q^16 - 2 * q^18 + 2 * q^22 + 3 * q^24 + q^25 + 3 * q^26 + 2 * q^28 - 2 * q^29 + q^32 - 3 * q^33 - 4 * q^36 - 3 * q^39 - 3 * q^42 + q^44 - q^49 + 2 * q^50 + 3 * q^52 + 3 * q^54 + q^56 - q^58 + 3 * q^59 - 3 * q^61 + 2 * q^63 + 2 * q^64 - 3 * q^66 - q^67 - 2 * q^72 - 3 * q^75 - 2 * q^77 - 6 * q^78 - q^79 - q^81 - 3 * q^84 + 3 * q^87 - q^88 - 3 * q^91 - 3 * q^96 - 2 * q^98 + 4 * q^99

Introduction

L-functions

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