LMFDB - Embedded newform 520.1.cd.a.459.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [520,1,Mod(179,520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(520, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("520.179");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 520 = 2^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	520.cd (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:	$0.259513806569$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{6}$
Projective field:	Galois closure of 6.2.2970344000.1

Embedding invariants

Embedding label			459.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	520.459
Dual form			520.1.cd.a.179.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.00000i q^{5} +(0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +(-0.500000 - 0.866025i) q^{9} +(-0.500000 + 0.866025i) q^{10} +(-1.50000 - 0.866025i) q^{11} +(-0.866025 + 0.500000i) q^{13} +1.00000 q^{14} +(-0.500000 + 0.866025i) q^{16} -1.00000i q^{18} +(1.50000 - 0.866025i) q^{19} +(-0.866025 + 0.500000i) q^{20} +(-0.866025 - 1.50000i) q^{22} -1.00000 q^{25} -1.00000 q^{26} +(0.866025 + 0.500000i) q^{28} +(-0.866025 + 0.500000i) q^{32} +(0.500000 + 0.866025i) q^{35} +(0.500000 - 0.866025i) q^{36} +(0.866025 + 0.500000i) q^{37} +1.73205 q^{38} -1.00000 q^{40} -1.73205i q^{44} +(0.866025 - 0.500000i) q^{45} -1.00000i q^{47} +(-0.866025 - 0.500000i) q^{50} +(-0.866025 - 0.500000i) q^{52} -1.73205 q^{53} +(0.866025 - 1.50000i) q^{55} +(0.500000 + 0.866025i) q^{56} +(-0.866025 - 0.500000i) q^{63} -1.00000 q^{64} +(-0.500000 - 0.866025i) q^{65} +1.00000i q^{70} +(0.866025 - 0.500000i) q^{72} +(0.500000 + 0.866025i) q^{74} +(1.50000 + 0.866025i) q^{76} -1.73205 q^{77} +(-0.866025 - 0.500000i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(0.866025 - 1.50000i) q^{88} +(1.50000 + 0.866025i) q^{89} +1.00000 q^{90} +(-0.500000 + 0.866025i) q^{91} +(0.500000 - 0.866025i) q^{94} +(0.866025 + 1.50000i) q^{95} +1.73205i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{9} - 2 q^{10} - 6 q^{11} + 4 q^{14} - 2 q^{16} + 6 q^{19} - 4 q^{25} - 4 q^{26} + 2 q^{35} + 2 q^{36} - 4 q^{40} + 2 q^{56} - 4 q^{64} - 2 q^{65} + 2 q^{74} + 6 q^{76} - 2 q^{81} + 6 q^{89}+ \cdots + 2 q^{94}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^9 - 2 * q^10 - 6 * q^11 + 4 * q^14 - 2 * q^16 + 6 * q^19 - 4 * q^25 - 4 * q^26 + 2 * q^35 + 2 * q^36 - 4 * q^40 + 2 * q^56 - 4 * q^64 - 2 * q^65 + 2 * q^74 + 6 * q^76 - 2 * q^81 + 6 * q^89 + 4 * q^90 - 2 * q^91 + 2 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/520\mathbb{Z}\right)^\times$.

$n$	$41$	$261$	$391$	$417$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$	$-1$	$-1$

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 $

$2$	0.866025	+	0.500000i	0.866025	+	0.500000i
$2$	0.866025	+	0.500000i	0.866025	+	0.500000i
$3$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$3$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$4$	0.500000	+	0.866025i	0.500000	+	0.866025i
$5$			1.00000i			1.00000i
$5$			1.00000i			1.00000i
$6$		0			0
$7$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$7$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$8$			1.00000i			1.00000i
$9$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$10$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$11$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$11$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$12$		0			0
$13$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$13$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$14$	1.00000			1.00000
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		−	1.00000i		−	1.00000i
$19$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$19$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$20$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$21$		0			0
$22$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$23$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$23$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−1.00000			−1.00000
$26$	−1.00000			−1.00000
$27$		0			0
$28$	0.866025	+	0.500000i	0.866025	+	0.500000i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$33$		0			0
$34$		0			0
$35$	0.500000	+	0.866025i	0.500000	+	0.866025i
$36$	0.500000	−	0.866025i	0.500000	−	0.866025i
$37$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$37$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$38$	1.73205			1.73205
$39$		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			0
$43$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$43$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$44$		−	1.73205i		−	1.73205i
$45$	0.866025	−	0.500000i	0.866025	−	0.500000i
$46$		0			0
$47$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$47$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$48$		0			0
$49$		0			0
$50$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$51$		0			0
$52$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$53$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$53$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$54$		0			0
$55$	0.866025	−	1.50000i	0.866025	−	1.50000i
$56$	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	$0.333333\pi$
$60$		0			0
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$64$	−1.00000			−1.00000
$65$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$66$		0			0
$67$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$67$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$68$		0			0
$69$		0			0
$70$			1.00000i			1.00000i
$71$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$71$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$72$	0.866025	−	0.500000i	0.866025	−	0.500000i
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$	0.500000	+	0.866025i	0.500000	+	0.866025i
$75$		0			0
$76$	1.50000	+	0.866025i	1.50000	+	0.866025i
$77$	−1.73205			−1.73205
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$81$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$	0.866025	−	1.50000i	0.866025	−	1.50000i
$89$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$89$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$90$	1.00000			1.00000
$91$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$92$		0			0
$93$		0			0
$94$	0.500000	−	0.866025i	0.500000	−	0.866025i
$95$	0.866025	+	1.50000i	0.866025	+	1.50000i
$96$		0			0
$97$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$97$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$98$		0			0
$99$			1.73205i			1.73205i
$100$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$103$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$104$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$105$		0			0
$106$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	1.50000	−	0.866025i	1.50000	−	0.866025i
$111$		0			0
$112$			1.00000i			1.00000i
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.866025	+	0.500000i	0.866025	+	0.500000i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.00000	+	1.73205i	1.00000	+	1.73205i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	1.00000i		−	1.00000i
$126$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$127$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$127$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$128$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$129$		0			0
$130$		−	1.00000i		−	1.00000i
$131$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$131$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	0.866025	−	1.50000i	0.866025	−	1.50000i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$137$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$139$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$140$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$141$		0			0
$142$		0			0
$143$	1.73205			1.73205
$144$	1.00000			1.00000
$145$		0			0
$146$		0			0
$147$		0			0
$148$			1.00000i			1.00000i
$149$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		0			0
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	0.866025	+	1.50000i	0.866025	+	1.50000i
$153$		0			0
$154$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$155$		0			0
$156$		0			0
$157$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$157$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$158$		0			0
$159$		0			0
$160$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$161$		0			0
$162$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$163$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$163$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$167$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$168$		0			0
$169$	0.500000	−	0.866025i	0.500000	−	0.866025i
$170$		0			0
$171$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$172$		0			0
$173$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$173$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$174$		0			0
$175$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$176$	1.50000	−	0.866025i	1.50000	−	0.866025i
$177$		0			0
$178$	0.866025	+	1.50000i	0.866025	+	1.50000i
$179$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$179$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$180$	0.866025	+	0.500000i	0.866025	+	0.500000i
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$183$		0			0
$184$		0			0
$185$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$186$		0			0
$187$		0			0
$188$	0.866025	−	0.500000i	0.866025	−	0.500000i
$189$		0			0
$190$			1.73205i			1.73205i
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$193$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$197$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$198$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$199$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$199$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$200$		−	1.00000i		−	1.00000i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$207$		0			0
$208$		−	1.00000i		−	1.00000i
$209$	−3.00000			−3.00000
$210$		0			0
$211$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$211$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$212$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	1.73205			1.73205
$221$		0			0
$222$		0			0
$223$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$223$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$224$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$225$	0.500000	+	0.866025i	0.500000	+	0.866025i
$226$		0			0
$227$		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			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$	0.500000	+	0.866025i	0.500000	+	0.866025i
$235$	1.00000			1.00000
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$241$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$242$			2.00000i			2.00000i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$248$		0			0
$249$		0			0
$250$	0.500000	−	0.866025i	0.500000	−	0.866025i
$251$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$251$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$252$		−	1.00000i		−	1.00000i
$253$		0			0
$254$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	1.00000			1.00000
$260$	0.500000	−	0.866025i	0.500000	−	0.866025i
$261$		0			0
$262$	0.866025	+	0.500000i	0.866025	+	0.500000i
$263$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$263$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$		−	1.73205i		−	1.73205i
$266$	1.50000	−	0.866025i	1.50000	−	0.866025i
$267$		0			0
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.50000	+	0.866025i	1.50000	+	0.866025i
$276$		0			0
$277$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$277$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$278$		−	1.00000i		−	1.00000i
$279$		0			0
$280$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$283$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$284$		0			0
$285$		0			0
$286$	1.50000	+	0.866025i	1.50000	+	0.866025i
$287$		0			0
$288$	0.866025	+	0.500000i	0.866025	+	0.500000i
$289$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$293$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$294$		0			0
$295$		0			0
$296$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$			1.73205i			1.73205i
$305$		0			0
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$	1.50000	+	0.866025i	1.50000	+	0.866025i
$315$	0.500000	−	0.866025i	0.500000	−	0.866025i
$316$		0			0
$317$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$317$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$318$		0			0
$319$		0			0
$320$		−	1.00000i		−	1.00000i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.00000			−1.00000
$325$	0.866025	−	0.500000i	0.866025	−	0.500000i
$326$		0			0
$327$		0			0
$328$		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
$333$		−	1.00000i		−	1.00000i
$334$	0.500000	+	0.866025i	0.500000	+	0.866025i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	0.866025	−	0.500000i	0.866025	−	0.500000i
$339$		0			0
$340$		0			0
$341$		0			0
$342$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$343$			1.00000i			1.00000i
$344$		0			0
$345$		0			0
$346$			1.73205i			1.73205i
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$349$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$350$	−1.00000			−1.00000
$351$		0			0
$352$	1.73205			1.73205
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$			1.73205i			1.73205i
$357$		0			0
$358$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	0.500000	+	0.866025i	0.500000	+	0.866025i
$361$	1.00000	−	1.73205i	1.00000	−	1.73205i
$362$		0			0
$363$		0			0
$364$	−1.00000			−1.00000
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$		0			0
$370$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$371$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$372$		0			0
$373$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$373$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$374$		0			0
$375$		0			0
$376$	1.00000			1.00000
$377$		0			0
$378$		0			0
$379$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$379$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$380$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$381$		0			0
$382$		0			0
$383$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$383$	1.73205	−	1.00000i	1.73205	−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$384$		0			0
$385$		−	1.73205i		−	1.73205i
$386$		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			0
$394$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$395$		0			0
$396$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$397$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$397$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$398$		0			0
$399$		0			0
$400$	0.500000	−	0.866025i	0.500000	−	0.866025i
$401$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$401$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$406$		0			0
$407$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$408$		0			0
$409$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$409$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.500000	−	0.866025i	0.500000	−	0.866025i
$417$		0			0
$418$	−2.59808	−	1.50000i	−2.59808	−	1.50000i
$419$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$419$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$	0.866025	−	0.500000i	0.866025	−	0.500000i
$423$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$424$		−	1.73205i		−	1.73205i
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$		0			0
$433$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$433$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$439$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$440$	1.50000	+	0.866025i	1.50000	+	0.866025i
$441$		0			0
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		0			0
$445$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$446$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$447$		0			0
$448$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$449$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$449$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$450$			1.00000i			1.00000i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$456$		0			0
$457$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$457$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$461$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$462$		0			0
$463$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$463$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$			1.00000i			1.00000i
$469$		0			0
$470$	0.866025	+	0.500000i	0.866025	+	0.500000i
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$476$		0			0
$477$	0.866025	+	1.50000i	0.866025	+	1.50000i
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	−1.00000			−1.00000
$482$	1.73205			1.73205
$483$		0			0
$484$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$485$		0			0
$486$		0			0
$487$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$487$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$488$		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
$494$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$495$	−1.73205			−1.73205
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		1.00000			$0$
$499$		0			0		−1.00000			$\pi$
$500$	0.866025	−	0.500000i	0.866025	−	0.500000i
$501$		0			0
$502$		−	1.00000i		−	1.00000i
$503$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$503$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$504$	0.500000	−	0.866025i	0.500000	−	0.866025i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−1.73205			−1.73205
$509$		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
$512$		−	1.00000i		−	1.00000i
$513$		0			0
$514$		0			0
$515$		−	1.73205i		−	1.73205i
$516$		0			0
$517$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$518$	0.866025	+	0.500000i	0.866025	+	0.500000i
$519$		0			0
$520$	0.866025	−	0.500000i	0.866025	−	0.500000i
$521$	−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
$523$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$523$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$524$	0.500000	+	0.866025i	0.500000	+	0.866025i
$525$		0			0
$526$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$527$		0			0
$528$		0			0
$529$	0.500000	+	0.866025i	0.500000	+	0.866025i
$530$	0.866025	−	1.50000i	0.866025	−	1.50000i
$531$		0			0
$532$	1.73205			1.73205
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$	0.866025	+	1.50000i	0.866025	+	1.50000i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		−	1.73205i		−	1.73205i
$555$		0			0
$556$	0.500000	−	0.866025i	0.500000	−	0.866025i
$557$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$557$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$558$		0			0
$559$		0			0
$560$	−1.00000			−1.00000
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$			1.00000i			1.00000i
$568$		0			0
$569$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$569$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$570$		0			0
$571$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$571$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$572$	0.866025	+	1.50000i	0.866025	+	1.50000i
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.500000	+	0.866025i	0.500000	+	0.866025i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	2.59808	+	1.50000i	2.59808	+	1.50000i
$584$		0			0
$585$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$586$	1.00000			1.00000
$587$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$587$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		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	$0.333333\pi$
$601$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$606$		0			0
$607$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$607$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$608$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$609$		0			0
$610$		0			0
$611$	0.500000	+	0.866025i	0.500000	+	0.866025i
$612$		0			0
$613$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$613$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		−	1.73205i		−	1.73205i
$617$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$617$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$618$		0			0
$619$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$619$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.73205			1.73205
$624$		0			0
$625$	1.00000			1.00000
$626$		0			0
$627$		0			0
$628$	0.866025	+	1.50000i	0.866025	+	1.50000i
$629$		0			0
$630$	0.866025	−	0.500000i	0.866025	−	0.500000i
$631$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$631$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$632$		0			0
$633$		0			0
$634$	0.500000	−	0.866025i	0.500000	−	0.866025i
$635$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.500000	−	0.866025i	0.500000	−	0.866025i
$641$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$641$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$642$		0			0
$643$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$643$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$647$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$648$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$649$		0			0
$650$	1.00000			1.00000
$651$		0			0
$652$		0			0
$653$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$653$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$654$		0			0
$655$			1.00000i			1.00000i
$656$		0			0
$657$		0			0
$658$		−	1.00000i		−	1.00000i
$659$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$659$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$661$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.50000	+	0.866025i	1.50000	+	0.866025i
$666$	0.500000	−	0.866025i	0.500000	−	0.866025i
$667$		0			0
$668$			1.00000i			1.00000i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$673$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$674$		0			0
$675$		0			0
$676$	1.00000			1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$683$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$684$		−	1.73205i		−	1.73205i
$685$		0			0
$686$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$687$		0			0
$688$		0			0
$689$	1.50000	−	0.866025i	1.50000	−	0.866025i
$690$		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	$0.333333\pi$
$692$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$693$	0.866025	+	1.50000i	0.866025	+	1.50000i
$694$		0			0
$695$	0.866025	−	0.500000i	0.866025	−	0.500000i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	1.73205			1.73205
$704$	1.50000	+	0.866025i	1.50000	+	0.866025i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$709$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$710$		0			0
$711$		0			0
$712$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$713$		0			0
$714$		0			0
$715$			1.73205i			1.73205i
$716$	−2.00000			−2.00000
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$			1.00000i			1.00000i
$721$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$722$	1.73205	−	1.00000i	1.73205	−	1.00000i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$727$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$728$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$729$	1.00000			1.00000
$730$		0			0
$731$		0			0
$732$		0			0
$733$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$733$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$739$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$740$	−1.00000			−1.00000
$741$		0			0
$742$	−1.73205			−1.73205
$743$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$743$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		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	$0.666667\pi$
$752$	0.866025	+	0.500000i	0.866025	+	0.500000i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$757$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$758$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$759$		0			0
$760$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$761$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$761$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	2.00000			2.00000
$767$		0			0
$768$		0			0
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$	0.866025	−	1.50000i	0.866025	−	1.50000i
$771$		0			0
$772$		0			0
$773$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$773$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$			1.73205i			1.73205i
$786$		0			0
$787$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$787$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$788$		−	1.00000i		−	1.00000i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−1.73205			−1.73205
$793$		0			0
$794$	1.00000			1.00000
$795$		0			0
$796$		0			0
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$	0.866025	−	0.500000i	0.866025	−	0.500000i
$801$		−	1.73205i		−	1.73205i
$802$	0.866025	+	1.50000i	0.866025	+	1.50000i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$809$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$810$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$811$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$811$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$812$		0			0
$813$		0			0
$814$		−	1.73205i		−	1.73205i
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.73205			−1.73205
$819$	1.00000			1.00000
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$		0			0
$823$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$823$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$824$		−	1.73205i		−	1.73205i
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		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
$832$	0.866025	−	0.500000i	0.866025	−	0.500000i
$833$		0			0
$834$		0			0
$835$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$836$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$837$		0			0
$838$	1.73205	−	1.00000i	1.73205	−	1.00000i
$839$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$839$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$840$		0			0
$841$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	1.00000			1.00000
$845$	0.866025	+	0.500000i	0.866025	+	0.500000i
$846$	−1.00000			−1.00000
$847$	1.73205	+	1.00000i	1.73205	+	1.00000i
$848$	0.866025	−	1.50000i	0.866025	−	1.50000i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$853$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$854$		0			0
$855$	0.866025	−	1.50000i	0.866025	−	1.50000i
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$859$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$863$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$864$		0			0
$865$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$876$		0			0
$877$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$877$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$878$		0			0
$879$		0			0
$880$	0.866025	+	1.50000i	0.866025	+	1.50000i
$881$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$881$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$887$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$888$		0			0
$889$			1.73205i			1.73205i
$890$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$891$	1.50000	−	0.866025i	1.50000	−	0.866025i
$892$		−	1.00000i		−	1.00000i
$893$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$894$		0			0
$895$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$896$	−1.00000			−1.00000
$897$		0			0
$898$	1.73205			1.73205
$899$		0			0
$900$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$		0			0
$909$		0			0
$910$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$911$		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
$917$	0.866025	−	0.500000i	0.866025	−	0.500000i
$918$		0			0
$919$		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
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$926$	1.00000	−	1.73205i	1.00000	−	1.73205i
$927$	0.866025	+	1.50000i	0.866025	+	1.50000i
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$	0.500000	+	0.866025i	0.500000	+	0.866025i
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$947$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$		0			0
$950$	−1.73205			−1.73205
$951$		0			0
$952$		0			0
$953$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$953$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$954$			1.73205i			1.73205i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.00000			−1.00000
$962$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$963$		0			0
$964$	1.50000	+	0.866025i	1.50000	+	0.866025i
$965$		0			0
$966$		0			0
$967$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$967$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$968$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$969$		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
$973$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$974$	1.00000			1.00000
$975$		0			0
$976$		0			0
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$	−1.50000	−	2.59808i	−1.50000	−	2.59808i
$980$		0			0
$981$		0			0
$982$	0.866025	−	0.500000i	0.866025	−	0.500000i
$983$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$983$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$	0.500000	−	0.866025i	0.500000	−	0.866025i
$986$		0			0
$987$		0			0
$988$	−1.73205			−1.73205
$989$		0			0
$990$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$991$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$991$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$997$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	520.1.cd.a.459.2	yes	4
4.3	odd	2		2080.1.dj.a.719.2		4
5.2	odd	4		2600.1.br.a.251.1		2
5.3	odd	4		2600.1.br.b.251.1		2
5.4	even	2	inner	520.1.cd.a.459.1	yes	4
8.3	odd	2	inner	520.1.cd.a.459.1	yes	4
8.5	even	2		2080.1.dj.a.719.1		4
13.10	even	6	inner	520.1.cd.a.179.2	yes	4
20.19	odd	2		2080.1.dj.a.719.1		4
40.3	even	4		2600.1.br.a.251.1		2
40.19	odd	2	CM	520.1.cd.a.459.2	yes	4
40.27	even	4		2600.1.br.b.251.1		2
40.29	even	2		2080.1.dj.a.719.2		4
52.23	odd	6		2080.1.dj.a.1999.1		4
65.23	odd	12		2600.1.br.a.2051.1		2
65.49	even	6	inner	520.1.cd.a.179.1	✓	4
65.62	odd	12		2600.1.br.b.2051.1		2
104.75	odd	6	inner	520.1.cd.a.179.1	✓	4
104.101	even	6		2080.1.dj.a.1999.2		4
260.179	odd	6		2080.1.dj.a.1999.2		4
520.179	odd	6	inner	520.1.cd.a.179.2	yes	4
520.283	even	12		2600.1.br.b.2051.1		2
520.309	even	6		2080.1.dj.a.1999.1		4
520.387	even	12		2600.1.br.a.2051.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.1.cd.a.179.1	✓	4	65.49	even	6	inner
520.1.cd.a.179.1	✓	4	104.75	odd	6	inner
520.1.cd.a.179.2	yes	4	13.10	even	6	inner
520.1.cd.a.179.2	yes	4	520.179	odd	6	inner
520.1.cd.a.459.1	yes	4	5.4	even	2	inner
520.1.cd.a.459.1	yes	4	8.3	odd	2	inner
520.1.cd.a.459.2	yes	4	1.1	even	1	trivial
520.1.cd.a.459.2	yes	4	40.19	odd	2	CM
2080.1.dj.a.719.1		4	8.5	even	2
2080.1.dj.a.719.1		4	20.19	odd	2
2080.1.dj.a.719.2		4	4.3	odd	2
2080.1.dj.a.719.2		4	40.29	even	2
2080.1.dj.a.1999.1		4	52.23	odd	6
2080.1.dj.a.1999.1		4	520.309	even	6
2080.1.dj.a.1999.2		4	104.101	even	6
2080.1.dj.a.1999.2		4	260.179	odd	6
2600.1.br.a.251.1		2	5.2	odd	4
2600.1.br.a.251.1		2	40.3	even	4
2600.1.br.a.2051.1		2	65.23	odd	12
2600.1.br.a.2051.1		2	520.387	even	12
2600.1.br.b.251.1		2	5.3	odd	4
2600.1.br.b.251.1		2	40.27	even	4
2600.1.br.b.2051.1		2	65.62	odd	12
2600.1.br.b.2051.1		2	520.283	even	12

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:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	520.cd (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.866025 + 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +1.00000i q^{5} +(0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +(-0.500000 - 0.866025i) q^{9} +(-0.500000 + 0.866025i) q^{10} +(-1.50000 - 0.866025i) q^{11} +(-0.866025 + 0.500000i) q^{13} +1.00000 q^{14} +(-0.500000 + 0.866025i) q^{16} -1.00000i q^{18} +(1.50000 - 0.866025i) q^{19} +(-0.866025 + 0.500000i) q^{20} +(-0.866025 - 1.50000i) q^{22} -1.00000 q^{25} -1.00000 q^{26} +(0.866025 + 0.500000i) q^{28} +(-0.866025 + 0.500000i) q^{32} +(0.500000 + 0.866025i) q^{35} +(0.500000 - 0.866025i) q^{36} +(0.866025 + 0.500000i) q^{37} +1.73205 q^{38} -1.00000 q^{40} -1.73205i q^{44} +(0.866025 - 0.500000i) q^{45} -1.00000i q^{47} +(-0.866025 - 0.500000i) q^{50} +(-0.866025 - 0.500000i) q^{52} -1.73205 q^{53} +(0.866025 - 1.50000i) q^{55} +(0.500000 + 0.866025i) q^{56} +(-0.866025 - 0.500000i) q^{63} -1.00000 q^{64} +(-0.500000 - 0.866025i) q^{65} +1.00000i q^{70} +(0.866025 - 0.500000i) q^{72} +(0.500000 + 0.866025i) q^{74} +(1.50000 + 0.866025i) q^{76} -1.73205 q^{77} +(-0.866025 - 0.500000i) q^{80} +(-0.500000 + 0.866025i) q^{81} +(0.866025 - 1.50000i) q^{88} +(1.50000 + 0.866025i) q^{89} +1.00000 q^{90} +(-0.500000 + 0.866025i) q^{91} +(0.500000 - 0.866025i) q^{94} +(0.866025 + 1.50000i) q^{95} +1.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{9} - 2 q^{10} - 6 q^{11} + 4 q^{14} - 2 q^{16} + 6 q^{19} - 4 q^{25} - 4 q^{26} + 2 q^{35} + 2 q^{36} - 4 q^{40} + 2 q^{56} - 4 q^{64} - 2 q^{65} + 2 q^{74} + 6 q^{76} - 2 q^{81} + 6 q^{89}+ \cdots + 2 q^{94}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^9 - 2 * q^10 - 6 * q^11 + 4 * q^14 - 2 * q^16 + 6 * q^19 - 4 * q^25 - 4 * q^26 + 2 * q^35 + 2 * q^36 - 4 * q^40 + 2 * q^56 - 4 * q^64 - 2 * q^65 + 2 * q^74 + 6 * q^76 - 2 * q^81 + 6 * q^89 + 4 * q^90 - 2 * q^91 + 2 * q^94

Introduction

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