LMFDB - Embedded newform 2520.1.fw.a.2029.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2520,1,Mod(349,2520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2520, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4, 3, 3]))

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

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

chi := DirichletCharacter("2520.349");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2520.fw (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:	$1.25764383184$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{12}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{12} - \cdots)$

Embedding invariants

Embedding label			2029.1
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2520.2029
Dual form			2520.1.fw.a.349.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 0.500000i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(0.500000 - 0.866025i) q^{4} +(0.707107 - 0.707107i) q^{5} +(0.965926 + 0.258819i) q^{6} +(0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +1.00000i q^{9} +O(q^{10})$	\(q+(-0.866025 + 0.500000i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(0.500000 - 0.866025i) q^{4} +(0.707107 - 0.707107i) q^{5} +(0.965926 + 0.258819i) q^{6} +(0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +1.00000i q^{9} +(-0.258819 + 0.965926i) q^{10} +(-0.965926 + 0.258819i) q^{12} +(-1.22474 - 0.707107i) q^{13} +(-0.500000 + 0.866025i) q^{14} -1.00000 q^{15} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 - 0.866025i) q^{18} +1.93185 q^{19} +(-0.258819 - 0.965926i) q^{20} +(-0.965926 - 0.258819i) q^{21} +(1.50000 + 0.866025i) q^{23} +(0.707107 - 0.707107i) q^{24} -1.00000i q^{25} +1.41421 q^{26} +(0.707107 - 0.707107i) q^{27} -1.00000i q^{28} +(0.866025 - 0.500000i) q^{30} +(0.866025 + 0.500000i) q^{32} +(0.258819 - 0.965926i) q^{35} +(0.866025 + 0.500000i) q^{36} +(-1.67303 + 0.965926i) q^{38} +(0.366025 + 1.36603i) q^{39} +(0.707107 + 0.707107i) q^{40} +(0.965926 - 0.258819i) q^{42} +(0.707107 + 0.707107i) q^{45} -1.73205 q^{46} +(-0.258819 + 0.965926i) q^{48} +(0.500000 - 0.866025i) q^{49} +(0.500000 + 0.866025i) q^{50} +(-1.22474 + 0.707107i) q^{52} +(-0.258819 + 0.965926i) q^{54} +(0.500000 + 0.866025i) q^{56} +(-1.36603 - 1.36603i) q^{57} +(-0.707107 + 1.22474i) q^{59} +(-0.500000 + 0.866025i) q^{60} +(-0.965926 - 1.67303i) q^{61} +(0.500000 + 0.866025i) q^{63} -1.00000 q^{64} +(-1.36603 + 0.366025i) q^{65} +(-0.448288 - 1.67303i) q^{69} +(0.258819 + 0.965926i) q^{70} -1.73205 q^{71} -1.00000 q^{72} +(-0.707107 + 0.707107i) q^{75} +(0.965926 - 1.67303i) q^{76} +(-1.00000 - 1.00000i) q^{78} +(0.866025 + 1.50000i) q^{79} +(-0.965926 - 0.258819i) q^{80} -1.00000 q^{81} +(1.22474 - 0.707107i) q^{83} +(-0.707107 + 0.707107i) q^{84} +(-0.965926 - 0.258819i) q^{90} -1.41421 q^{91} +(1.50000 - 0.866025i) q^{92} +(1.36603 - 1.36603i) q^{95} +(-0.258819 - 0.965926i) q^{96} +1.00000i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4}+O(q^{10}) $ 8 * q + 4 * q^4	$ 8 q + 4 q^{4} - 4 q^{14} - 8 q^{15} - 4 q^{16} - 4 q^{18} + 12 q^{23} - 4 q^{39} + 4 q^{49} + 4 q^{50} + 4 q^{56} - 4 q^{57} - 4 q^{60} + 4 q^{63} - 8 q^{64} - 4 q^{65} - 8 q^{72} - 8 q^{78} - 8 q^{81} + 12 q^{92} + 4 q^{95}+O(q^{100}) $ 8 * q + 4 * q^4 - 4 * q^14 - 8 * q^15 - 4 * q^16 - 4 * q^18 + 12 * q^23 - 4 * q^39 + 4 * q^49 + 4 * q^50 + 4 * q^56 - 4 * q^57 - 4 * q^60 + 4 * q^63 - 8 * q^64 - 4 * q^65 - 8 * q^72 - 8 * q^78 - 8 * q^81 + 12 * q^92 + 4 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times$.

$n$	$281$	$631$	$1081$	$1261$	$2017$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$1$	$-1$	$-1$	$-1$

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
$3$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$3$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$4$	0.500000	−	0.866025i	0.500000	−	0.866025i
$5$	0.707107	−	0.707107i	0.707107	−	0.707107i
$5$	0.707107	−	0.707107i	0.707107	−	0.707107i
$6$	0.965926	+	0.258819i	0.965926	+	0.258819i
$7$	0.866025	−	0.500000i	0.866025	−	0.500000i
$7$	0.866025	−	0.500000i	0.866025	−	0.500000i
$8$			1.00000i			1.00000i
$9$			1.00000i			1.00000i
$10$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$13$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$13$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$14$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$15$	−1.00000			−1.00000
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$19$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$19$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$20$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$21$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$22$		0			0
$23$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$23$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$24$	0.707107	−	0.707107i	0.707107	−	0.707107i
$25$		−	1.00000i		−	1.00000i
$26$	1.41421			1.41421
$27$	0.707107	−	0.707107i	0.707107	−	0.707107i
$28$		−	1.00000i		−	1.00000i
$29$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$29$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$30$	0.866025	−	0.500000i	0.866025	−	0.500000i
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$	0.866025	+	0.500000i	0.866025	+	0.500000i
$33$		0			0
$34$		0			0
$35$	0.258819	−	0.965926i	0.258819	−	0.965926i
$36$	0.866025	+	0.500000i	0.866025	+	0.500000i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$39$	0.366025	+	1.36603i	0.366025	+	1.36603i
$40$	0.707107	+	0.707107i	0.707107	+	0.707107i
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$	0.965926	−	0.258819i	0.965926	−	0.258819i
$43$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$43$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$44$		0			0
$45$	0.707107	+	0.707107i	0.707107	+	0.707107i
$46$	−1.73205			−1.73205
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$49$	0.500000	−	0.866025i	0.500000	−	0.866025i
$50$	0.500000	+	0.866025i	0.500000	+	0.866025i
$51$		0			0
$52$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$55$		0			0
$56$	0.500000	+	0.866025i	0.500000	+	0.866025i
$57$	−1.36603	−	1.36603i	−1.36603	−	1.36603i
$58$		0			0
$59$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$59$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$60$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$61$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.707107	−	0.707107i	$-0.750000\pi$
$61$	−0.965926	−	1.67303i	−0.965926	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$62$		0			0
$63$	0.500000	+	0.866025i	0.500000	+	0.866025i
$64$	−1.00000			−1.00000
$65$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$66$		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
$69$	−0.448288	−	1.67303i	−0.448288	−	1.67303i
$70$	0.258819	+	0.965926i	0.258819	+	0.965926i
$71$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$71$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$72$	−1.00000			−1.00000
$73$		0			0			−	1.00000i	$-0.5\pi$
$73$		0			0				1.00000i	$0.5\pi$
$74$		0			0
$75$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$76$	0.965926	−	1.67303i	0.965926	−	1.67303i
$77$		0			0
$78$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$79$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$79$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$80$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$81$	−1.00000			−1.00000
$82$		0			0
$83$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$83$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$84$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$91$	−1.41421			−1.41421
$92$	1.50000	−	0.866025i	1.50000	−	0.866025i
$93$		0			0
$94$		0			0
$95$	1.36603	−	1.36603i	1.36603	−	1.36603i
$96$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$97$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$97$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$98$			1.00000i			1.00000i
$99$		0			0
$100$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$101$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$101$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$	0.707107	−	1.22474i	0.707107	−	1.22474i
$105$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$113$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$113$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$114$	1.86603	+	0.500000i	1.86603	+	0.500000i
$115$	1.67303	−	0.448288i	1.67303	−	0.448288i
$116$		0			0
$117$	0.707107	−	1.22474i	0.707107	−	1.22474i
$118$		−	1.41421i		−	1.41421i
$119$		0			0
$120$		−	1.00000i		−	1.00000i
$121$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$122$	1.67303	+	0.965926i	1.67303	+	0.965926i
$123$		0			0
$124$		0			0
$125$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$126$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$127$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$127$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$128$	0.866025	−	0.500000i	0.866025	−	0.500000i
$129$		0			0
$130$	1.00000	−	1.00000i	1.00000	−	1.00000i
$131$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$131$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$	1.67303	−	0.965926i	1.67303	−	0.965926i
$134$		0			0
$135$		−	1.00000i		−	1.00000i
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$	1.22474	+	1.22474i	1.22474	+	1.22474i
$139$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$139$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$140$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$141$		0			0
$142$	1.50000	−	0.866025i	1.50000	−	0.866025i
$143$		0			0
$144$	0.866025	−	0.500000i	0.866025	−	0.500000i
$145$		0			0
$146$		0			0
$147$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$148$		0			0
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$	0.258819	−	0.965926i	0.258819	−	0.965926i
$151$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$151$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$152$			1.93185i			1.93185i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.36603	+	0.366025i	1.36603	+	0.366025i
$157$	−0.448288	−	0.258819i	−0.448288	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$157$	−0.448288	−	0.258819i	−0.448288	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$158$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$159$		0			0
$160$	0.965926	−	0.258819i	0.965926	−	0.258819i
$161$	1.73205			1.73205
$162$	0.866025	−	0.500000i	0.866025	−	0.500000i
$163$		0			0			−	1.00000i	$-0.5\pi$
$163$		0			0				1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$	0.258819	−	0.965926i	0.258819	−	0.965926i
$169$	0.500000	+	0.866025i	0.500000	+	0.866025i
$170$		0			0
$171$			1.93185i			1.93185i
$172$		0			0
$173$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$173$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$174$		0			0
$175$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$176$		0			0
$177$	1.36603	−	0.366025i	1.36603	−	0.366025i
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$	0.965926	−	0.258819i	0.965926	−	0.258819i
$181$	0.517638			0.517638			0.258819	−	0.965926i	$-0.416667\pi$
$181$	0.517638			0.517638			0.258819	+	0.965926i	$0.416667\pi$
$182$	1.22474	−	0.707107i	1.22474	−	0.707107i
$183$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$184$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.258819	−	0.965926i	0.258819	−	0.965926i
$190$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$191$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$191$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$192$	0.707107	+	0.707107i	0.707107	+	0.707107i
$193$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$193$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$194$		0			0
$195$	1.22474	+	0.707107i	1.22474	+	0.707107i
$196$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	1.00000			1.00000
$201$		0			0
$202$	−0.448288	−	0.258819i	−0.448288	−	0.258819i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$208$			1.41421i			1.41421i
$209$		0			0
$210$	0.500000	−	0.866025i	0.500000	−	0.866025i
$211$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$211$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$212$		0			0
$213$	1.22474	+	1.22474i	1.22474	+	1.22474i
$214$		0			0
$215$		0			0
$216$	0.707107	+	0.707107i	0.707107	+	0.707107i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$223$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$224$	1.00000			1.00000
$225$	1.00000			1.00000
$226$	1.00000			1.00000
$227$	−0.448288	+	0.258819i	−0.448288	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$227$	−0.448288	+	0.258819i	−0.448288	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$228$	−1.86603	+	0.500000i	−1.86603	+	0.500000i
$229$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$229$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$230$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$231$		0			0
$232$		0			0
$233$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$233$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$234$			1.41421i			1.41421i
$235$		0			0
$236$	0.707107	+	1.22474i	0.707107	+	1.22474i
$237$	0.448288	−	1.67303i	0.448288	−	1.67303i
$238$		0			0
$239$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$239$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$240$	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	$0.333333\pi$
$242$		−	1.00000i		−	1.00000i
$243$	0.707107	+	0.707107i	0.707107	+	0.707107i
$244$	−1.93185			−1.93185
$245$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$246$		0			0
$247$	−2.36603	−	1.36603i	−2.36603	−	1.36603i
$248$		0			0
$249$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$250$	0.965926	+	0.258819i	0.965926	+	0.258819i
$251$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$251$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$252$	1.00000			1.00000
$253$		0			0
$254$	0.866025	+	1.50000i	0.866025	+	1.50000i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$261$		0			0
$262$		−	0.517638i		−	0.517638i
$263$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$263$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$267$		0			0
$268$		0			0
$269$	−0.517638			−0.517638			−0.258819	−	0.965926i	$-0.583333\pi$
$269$	−0.517638			−0.517638			−0.258819	+	0.965926i	$0.583333\pi$
$270$	0.500000	+	0.866025i	0.500000	+	0.866025i
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		0			0
$273$	1.00000	+	1.00000i	1.00000	+	1.00000i
$274$		0			0
$275$		0			0
$276$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$277$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$277$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$278$		−	0.517638i		−	0.517638i
$279$		0			0
$280$	0.965926	+	0.258819i	0.965926	+	0.258819i
$281$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$281$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$282$		0			0
$283$	−0.448288	−	0.258819i	−0.448288	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$283$	−0.448288	−	0.258819i	−0.448288	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$284$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$285$	−1.93185			−1.93185
$286$		0			0
$287$		0			0
$288$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$289$	−1.00000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.67303	−	0.965926i	−1.67303	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$293$	−1.67303	−	0.965926i	−1.67303	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$294$	0.707107	−	0.707107i	0.707107	−	0.707107i
$295$	0.366025	+	1.36603i	0.366025	+	1.36603i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−1.22474	−	2.12132i	−1.22474	−	2.12132i
$300$	0.258819	+	0.965926i	0.258819	+	0.965926i
$301$		0			0
$302$	0.866025	+	0.500000i	0.866025	+	0.500000i
$303$	0.133975	−	0.500000i	0.133975	−	0.500000i
$304$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$305$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$306$		0			0
$307$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$307$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$313$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$313$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$314$	0.517638			0.517638
$315$	0.965926	+	0.258819i	0.965926	+	0.258819i
$316$	1.73205			1.73205
$317$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$317$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$318$		0			0
$319$		0			0
$320$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$321$		0			0
$322$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$323$		0			0
$324$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$325$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$		−	1.41421i		−	1.41421i
$333$		0			0
$334$		0			0
$335$		0			0
$336$	0.258819	+	0.965926i	0.258819	+	0.965926i
$337$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$337$	1.73205	+	1.00000i	1.73205	+	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$338$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$339$	0.258819	+	0.965926i	0.258819	+	0.965926i
$340$		0			0
$341$		0			0
$342$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$343$		−	1.00000i		−	1.00000i
$344$		0			0
$345$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$346$	0.707107	−	1.22474i	0.707107	−	1.22474i
$347$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$347$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$348$		0			0
$349$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$349$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$350$	0.866025	+	0.500000i	0.866025	+	0.500000i
$351$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$355$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$359$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$360$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$361$	2.73205			2.73205
$362$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$363$	0.965926	−	0.258819i	0.965926	−	0.258819i
$364$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$365$		0			0
$366$	−0.500000	−	1.86603i	−0.500000	−	1.86603i
$367$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$367$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$368$		−	1.73205i		−	1.73205i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$373$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$374$		0			0
$375$			1.00000i			1.00000i
$376$		0			0
$377$		0			0
$378$	0.258819	+	0.965926i	0.258819	+	0.965926i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	−0.500000	−	1.86603i	−0.500000	−	1.86603i
$381$	−1.22474	+	1.22474i	−1.22474	+	1.22474i
$382$	1.50000	+	0.866025i	1.50000	+	0.866025i
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$385$		0			0
$386$	−1.00000			−1.00000
$387$		0			0
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$	−1.41421			−1.41421
$391$		0			0
$392$	0.866025	+	0.500000i	0.866025	+	0.500000i
$393$	0.500000	−	0.133975i	0.500000	−	0.133975i
$394$		0			0
$395$	1.67303	+	0.448288i	1.67303	+	0.448288i
$396$		0			0
$397$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$397$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$	−1.86603	−	0.500000i	−1.86603	−	0.500000i
$400$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$401$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$401$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$402$		0			0
$403$		0			0
$404$	0.517638			0.517638
$405$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			1.41421i			1.41421i
$414$		−	1.73205i		−	1.73205i
$415$	0.366025	−	1.36603i	0.366025	−	1.36603i
$416$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$417$	0.500000	−	0.133975i	0.500000	−	0.133975i
$418$		0			0
$419$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.258819	+	0.965926i	$0.583333\pi$
$419$	−0.965926	+	1.67303i	−0.965926	+	1.67303i	−0.707107	+	0.707107i	$0.750000\pi$
$420$			1.00000i			1.00000i
$421$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$421$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$427$	−1.67303	−	0.965926i	−1.67303	−	0.965926i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.00000			2.00000			1.00000			$0$
$431$	2.00000			2.00000			1.00000			$0$
$432$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.89778	+	1.67303i	2.89778	+	1.67303i
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$	0.866025	+	0.500000i	0.866025	+	0.500000i
$442$		0			0
$443$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$443$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$449$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$449$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$450$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$451$		0			0
$452$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$453$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$454$	0.258819	−	0.448288i	0.258819	−	0.448288i
$455$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$456$	1.36603	−	1.36603i	1.36603	−	1.36603i
$457$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$457$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$458$		−	0.517638i		−	0.517638i
$459$		0			0
$460$	0.448288	−	1.67303i	0.448288	−	1.67303i
$461$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$461$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$462$		0			0
$463$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$463$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$467$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$467$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$468$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$469$		0			0
$470$		0			0
$471$	0.133975	+	0.500000i	0.133975	+	0.500000i
$472$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$473$		0			0
$474$	0.448288	+	1.67303i	0.448288	+	1.67303i
$475$		−	1.93185i		−	1.93185i
$476$		0			0
$477$		0			0
$478$			1.00000i			1.00000i
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$481$		0			0
$482$		0			0
$483$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$484$	0.500000	+	0.866025i	0.500000	+	0.866025i
$485$		0			0
$486$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$487$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$487$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$488$	1.67303	−	0.965926i	1.67303	−	0.965926i
$489$		0			0
$490$	0.707107	+	0.707107i	0.707107	+	0.707107i
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$	2.73205			2.73205
$495$		0			0
$496$		0			0
$497$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$498$	1.36603	−	0.366025i	1.36603	−	0.366025i
$499$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$499$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$500$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$501$		0			0
$502$	−1.67303	+	0.965926i	−1.67303	+	0.965926i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$505$	0.500000	+	0.133975i	0.500000	+	0.133975i
$506$		0			0
$507$	0.258819	−	0.965926i	0.258819	−	0.965926i
$508$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$509$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$509$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$510$		0			0
$511$		0			0
$512$		−	1.00000i		−	1.00000i
$513$	1.36603	−	1.36603i	1.36603	−	1.36603i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	1.36603	+	0.366025i	1.36603	+	0.366025i
$520$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$			1.93185i			1.93185i	0.258819	+	0.965926i	$0.416667\pi$
$523$			1.93185i			1.93185i	−0.258819	+	0.965926i	$0.583333\pi$
$524$	0.258819	+	0.448288i	0.258819	+	0.448288i
$525$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$526$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$527$		0			0
$528$		0			0
$529$	1.00000	+	1.73205i	1.00000	+	1.73205i
$530$		0			0
$531$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$532$		−	1.93185i		−	1.93185i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	0.448288	−	0.258819i	0.448288	−	0.258819i
$539$		0			0
$540$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$	−0.366025	−	0.366025i	−0.366025	−	0.366025i
$544$		0			0
$545$		0			0
$546$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$547$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$547$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$548$		0			0
$549$	1.67303	−	0.965926i	1.67303	−	0.965926i
$550$		0			0
$551$		0			0
$552$	1.67303	−	0.448288i	1.67303	−	0.448288i
$553$	1.50000	+	0.866025i	1.50000	+	0.866025i
$554$		0			0
$555$		0			0
$556$	0.258819	+	0.448288i	0.258819	+	0.448288i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$561$		0			0
$562$	0.866025	+	0.500000i	0.866025	+	0.500000i
$563$	0.448288	+	0.258819i	0.448288	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$563$	0.448288	+	0.258819i	0.448288	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$564$		0			0
$565$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$566$	0.517638			0.517638
$567$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$568$		−	1.73205i		−	1.73205i
$569$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$569$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$570$	1.67303	−	0.965926i	1.67303	−	0.965926i
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$574$		0			0
$575$	0.866025	−	1.50000i	0.866025	−	1.50000i
$576$		−	1.00000i		−	1.00000i
$577$		0			0			−	1.00000i	$-0.5\pi$
$577$		0			0				1.00000i	$0.5\pi$
$578$	0.866025	−	0.500000i	0.866025	−	0.500000i
$579$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$580$		0			0
$581$	0.707107	−	1.22474i	0.707107	−	1.22474i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$586$	1.93185			1.93185
$587$	−1.67303	+	0.965926i	−1.67303	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$587$	−1.67303	+	0.965926i	−1.67303	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$588$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$589$		0			0
$590$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$591$		0			0
$592$		0			0
$593$		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
$598$	2.12132	+	1.22474i	2.12132	+	1.22474i
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$601$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$601$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$602$		0			0
$603$		0			0
$604$	−1.00000			−1.00000
$605$	0.258819	+	0.965926i	0.258819	+	0.965926i
$606$	0.133975	+	0.500000i	0.133975	+	0.500000i
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$	1.67303	+	0.965926i	1.67303	+	0.965926i
$609$		0			0
$610$	1.86603	−	0.500000i	1.86603	−	0.500000i
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	−0.965926	−	1.67303i	−0.965926	−	1.67303i
$615$		0			0
$616$		0			0
$617$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$617$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$618$		0			0
$619$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$619$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$620$		0			0
$621$	1.67303	−	0.448288i	1.67303	−	0.448288i
$622$		0			0
$623$		0			0
$624$	1.00000	−	1.00000i	1.00000	−	1.00000i
$625$	−1.00000			−1.00000
$626$		0			0
$627$		0			0
$628$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$629$		0			0
$630$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$631$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$631$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$632$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$633$		0			0
$634$		0			0
$635$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$636$		0			0
$637$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$638$		0			0
$639$		−	1.73205i		−	1.73205i
$640$	0.258819	−	0.965926i	0.258819	−	0.965926i
$641$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$641$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$642$		0			0
$643$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$643$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$644$	0.866025	−	1.50000i	0.866025	−	1.50000i
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		−	1.00000i		−	1.00000i
$649$		0			0
$650$		−	1.41421i		−	1.41421i
$651$		0			0
$652$		0			0
$653$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$653$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$654$		0			0
$655$	0.133975	+	0.500000i	0.133975	+	0.500000i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$661$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$662$		0			0
$663$		0			0
$664$	0.707107	+	1.22474i	0.707107	+	1.22474i
$665$	0.500000	−	1.86603i	0.500000	−	1.86603i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$673$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$673$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$674$	−2.00000			−2.00000
$675$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$676$	1.00000			1.00000
$677$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$677$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$678$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$679$		0			0
$680$		0			0
$681$	0.500000	+	0.133975i	0.500000	+	0.133975i
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$	1.67303	+	0.965926i	1.67303	+	0.965926i
$685$		0			0
$686$	0.500000	+	0.866025i	0.500000	+	0.866025i
$687$	0.500000	−	0.133975i	0.500000	−	0.133975i
$688$		0			0
$689$		0			0
$690$	1.73205			1.73205
$691$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	0.707107	−	0.707107i	$-0.250000\pi$
$691$	−0.258819	−	0.448288i	−0.258819	−	0.448288i	−0.965926	+	0.258819i	$0.916667\pi$
$692$			1.41421i			1.41421i
$693$		0			0
$694$		0			0
$695$	0.133975	+	0.500000i	0.133975	+	0.500000i
$696$		0			0
$697$		0			0
$698$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$699$	1.22474	−	1.22474i	1.22474	−	1.22474i
$700$	−1.00000			−1.00000
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$	1.00000	−	1.00000i	1.00000	−	1.00000i
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.448288	+	0.258819i	0.448288	+	0.258819i
$708$	0.366025	−	1.36603i	0.366025	−	1.36603i
$709$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$709$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$710$	0.448288	−	1.67303i	0.448288	−	1.67303i
$711$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$718$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$	0.258819	−	0.965926i	0.258819	−	0.965926i
$721$		0			0
$722$	−2.36603	+	1.36603i	−2.36603	+	1.36603i
$723$		0			0
$724$	0.258819	−	0.448288i	0.258819	−	0.448288i
$725$		0			0
$726$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$727$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$727$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$728$		−	1.41421i		−	1.41421i
$729$		−	1.00000i		−	1.00000i
$730$		0			0
$731$		0			0
$732$	1.36603	+	1.36603i	1.36603	+	1.36603i
$733$	0.448288	+	0.258819i	0.448288	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$733$	0.448288	+	0.258819i	0.448288	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$734$		0			0
$735$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$736$	0.866025	+	1.50000i	0.866025	+	1.50000i
$737$		0			0
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$	0.707107	+	2.63896i	0.707107	+	2.63896i
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.707107	+	1.22474i	0.707107	+	1.22474i
$748$		0			0
$749$		0			0
$750$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$751$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$751$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$752$		0			0
$753$	−1.36603	−	1.36603i	−1.36603	−	1.36603i
$754$		0			0
$755$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$756$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$	1.36603	+	1.36603i	1.36603	+	1.36603i
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$	0.448288	−	1.67303i	0.448288	−	1.67303i
$763$		0			0
$764$	−1.73205			−1.73205
$765$		0			0
$766$		0			0
$767$	1.73205	−	1.00000i	1.73205	−	1.00000i
$768$	0.965926	−	0.258819i	0.965926	−	0.258819i
$769$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$769$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$770$		0			0
$771$		0			0
$772$	0.866025	−	0.500000i	0.866025	−	0.500000i
$773$		−	1.93185i		−	1.93185i	−0.258819	−	0.965926i	$-0.583333\pi$
$773$		−	1.93185i		−	1.93185i	0.258819	−	0.965926i	$-0.416667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	1.22474	−	0.707107i	1.22474	−	0.707107i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.00000			−1.00000
$785$	−0.500000	+	0.133975i	−0.500000	+	0.133975i
$786$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$787$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$787$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$788$		0			0
$789$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$790$	−1.67303	+	0.448288i	−1.67303	+	0.448288i
$791$	−1.00000			−1.00000
$792$		0			0
$793$			2.73205i			2.73205i
$794$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$795$		0			0
$796$		0			0
$797$	−1.67303	−	0.965926i	−1.67303	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$797$	−1.67303	−	0.965926i	−1.67303	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$798$	1.86603	−	0.500000i	1.86603	−	0.500000i
$799$		0			0
$800$	0.500000	−	0.866025i	0.500000	−	0.866025i
$801$		0			0
$802$			1.73205i			1.73205i
$803$		0			0
$804$		0			0
$805$	1.22474	−	1.22474i	1.22474	−	1.22474i
$806$		0			0
$807$	0.366025	+	0.366025i	0.366025	+	0.366025i
$808$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$	0.258819	−	0.965926i	0.258819	−	0.965926i
$811$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$811$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		−	1.41421i		−	1.41421i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$823$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$824$		0			0
$825$		0			0
$826$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	0.866025	+	1.50000i	0.866025	+	1.50000i
$829$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$829$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$830$	0.366025	+	1.36603i	0.366025	+	1.36603i
$831$		0			0
$832$	1.22474	+	0.707107i	1.22474	+	0.707107i
$833$		0			0
$834$	−0.366025	+	0.366025i	−0.366025	+	0.366025i
$835$		0			0
$836$		0			0
$837$		0			0
$838$		−	1.93185i		−	1.93185i
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$841$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$842$		0			0
$843$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$844$		0			0
$845$	0.965926	+	0.258819i	0.965926	+	0.258819i
$846$		0			0
$847$			1.00000i			1.00000i
$848$		0			0
$849$	0.133975	+	0.500000i	0.133975	+	0.500000i
$850$		0			0
$851$		0			0
$852$	1.67303	−	0.448288i	1.67303	−	0.448288i
$853$	1.67303	−	0.965926i	1.67303	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$853$	1.67303	−	0.965926i	1.67303	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$854$	1.93185			1.93185
$855$	1.36603	+	1.36603i	1.36603	+	1.36603i
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	0.707107	−	1.22474i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$859$	0.707107	−	1.22474i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$860$		0			0
$861$		0			0
$862$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$863$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$863$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$864$	0.965926	−	0.258819i	0.965926	−	0.258819i
$865$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$866$		0			0
$867$	0.707107	+	0.707107i	0.707107	+	0.707107i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$	−3.34607			−3.34607
$875$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$876$		0			0
$877$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$877$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$878$		0			0
$879$	0.500000	+	1.86603i	0.500000	+	1.86603i
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	−1.00000			−1.00000
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$	0.707107	−	1.22474i	0.707107	−	1.22474i
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.500000	−	0.866025i	0.500000	−	0.866025i
$897$	−0.633975	+	2.36603i	−0.633975	+	2.36603i
$898$	0.866025	−	0.500000i	0.866025	−	0.500000i
$899$		0			0
$900$	0.500000	−	0.866025i	0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.500000	−	0.866025i	0.500000	−	0.866025i
$905$	0.366025	−	0.366025i	0.366025	−	0.366025i
$906$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$907$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$907$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$908$			0.517638i			0.517638i
$909$	−0.448288	+	0.258819i	−0.448288	+	0.258819i
$910$	0.366025	−	1.36603i	0.366025	−	1.36603i
$911$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$911$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$912$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$913$		0			0
$914$	0.500000	−	0.866025i	0.500000	−	0.866025i
$915$	0.965926	+	1.67303i	0.965926	+	1.67303i
$916$	0.258819	+	0.448288i	0.258819	+	0.448288i
$917$			0.517638i			0.517638i
$918$		0			0
$919$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$919$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$920$	0.448288	+	1.67303i	0.448288	+	1.67303i
$921$	1.36603	−	1.36603i	1.36603	−	1.36603i
$922$	0.448288	+	0.258819i	0.448288	+	0.258819i
$923$	2.12132	+	1.22474i	2.12132	+	1.22474i
$924$		0			0
$925$		0			0
$926$	−1.00000			−1.00000
$927$		0			0
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	0.965926	−	1.67303i	0.965926	−	1.67303i
$932$	1.50000	+	0.866025i	1.50000	+	0.866025i
$933$		0			0
$934$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$935$		0			0
$936$	1.22474	+	0.707107i	1.22474	+	0.707107i
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.258819	−	0.965926i	$-0.416667\pi$
$941$	0.965926	−	1.67303i	0.965926	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$942$	−0.366025	−	0.366025i	−0.366025	−	0.366025i
$943$		0			0
$944$	1.41421			1.41421
$945$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$946$		0			0
$947$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$947$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$948$	−1.22474	−	1.22474i	−1.22474	−	1.22474i
$949$		0			0
$950$	0.965926	+	1.67303i	0.965926	+	1.67303i
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$956$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$957$		0			0
$958$		0			0
$959$		0			0
$960$	1.00000			1.00000
$961$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.965926	−	0.258819i	0.965926	−	0.258819i
$966$	1.67303	+	0.448288i	1.67303	+	0.448288i
$967$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$967$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$968$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$969$		0			0
$970$		0			0
$971$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$971$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$972$	0.965926	−	0.258819i	0.965926	−	0.258819i
$973$			0.517638i			0.517638i
$974$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$975$	1.36603	−	0.366025i	1.36603	−	0.366025i
$976$	−0.965926	+	1.67303i	−0.965926	+	1.67303i
$977$		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
$980$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$981$		0			0
$982$		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
$988$	−2.36603	+	1.36603i	−2.36603	+	1.36603i
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$	0.866025	−	1.50000i	0.866025	−	1.50000i
$995$		0			0
$996$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$997$	1.67303	−	0.965926i	1.67303	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$997$	1.67303	−	0.965926i	1.67303	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2520.1.fw.a.2029.1	yes	8
5.4	even	2		2520.1.fw.b.2029.4	yes	8
7.6	odd	2	inner	2520.1.fw.a.2029.2	yes	8
8.5	even	2	inner	2520.1.fw.a.2029.2	yes	8
9.7	even	3		2520.1.fw.b.349.4	yes	8
35.34	odd	2		2520.1.fw.b.2029.3	yes	8
40.29	even	2		2520.1.fw.b.2029.3	yes	8
45.34	even	6	inner	2520.1.fw.a.349.1	✓	8
56.13	odd	2	CM	2520.1.fw.a.2029.1	yes	8
63.34	odd	6		2520.1.fw.b.349.3	yes	8
72.61	even	6		2520.1.fw.b.349.3	yes	8
280.69	odd	2		2520.1.fw.b.2029.4	yes	8
315.34	odd	6	inner	2520.1.fw.a.349.2	yes	8
360.349	even	6	inner	2520.1.fw.a.349.2	yes	8
504.349	odd	6		2520.1.fw.b.349.4	yes	8
2520.349	odd	6	inner	2520.1.fw.a.349.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2520.1.fw.a.349.1	✓	8	45.34	even	6	inner
2520.1.fw.a.349.1	✓	8	2520.349	odd	6	inner
2520.1.fw.a.349.2	yes	8	315.34	odd	6	inner
2520.1.fw.a.349.2	yes	8	360.349	even	6	inner
2520.1.fw.a.2029.1	yes	8	1.1	even	1	trivial
2520.1.fw.a.2029.1	yes	8	56.13	odd	2	CM
2520.1.fw.a.2029.2	yes	8	7.6	odd	2	inner
2520.1.fw.a.2029.2	yes	8	8.5	even	2	inner
2520.1.fw.b.349.3	yes	8	63.34	odd	6
2520.1.fw.b.349.3	yes	8	72.61	even	6
2520.1.fw.b.349.4	yes	8	9.7	even	3
2520.1.fw.b.349.4	yes	8	504.349	odd	6
2520.1.fw.b.2029.3	yes	8	35.34	odd	2
2520.1.fw.b.2029.3	yes	8	40.29	even	2
2520.1.fw.b.2029.4	yes	8	5.4	even	2
2520.1.fw.b.2029.4	yes	8	280.69	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:	\( N \)	\(=\)	\( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2520.fw (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 0.500000i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(0.500000 - 0.866025i) q^{4} +(0.707107 - 0.707107i) q^{5} +(0.965926 + 0.258819i) q^{6} +(0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +1.00000i q^{9} +O(q^{10})\)	\(q+(-0.866025 + 0.500000i) q^{2} +(-0.707107 - 0.707107i) q^{3} +(0.500000 - 0.866025i) q^{4} +(0.707107 - 0.707107i) q^{5} +(0.965926 + 0.258819i) q^{6} +(0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +1.00000i q^{9} +(-0.258819 + 0.965926i) q^{10} +(-0.965926 + 0.258819i) q^{12} +(-1.22474 - 0.707107i) q^{13} +(-0.500000 + 0.866025i) q^{14} -1.00000 q^{15} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 - 0.866025i) q^{18} +1.93185 q^{19} +(-0.258819 - 0.965926i) q^{20} +(-0.965926 - 0.258819i) q^{21} +(1.50000 + 0.866025i) q^{23} +(0.707107 - 0.707107i) q^{24} -1.00000i q^{25} +1.41421 q^{26} +(0.707107 - 0.707107i) q^{27} -1.00000i q^{28} +(0.866025 - 0.500000i) q^{30} +(0.866025 + 0.500000i) q^{32} +(0.258819 - 0.965926i) q^{35} +(0.866025 + 0.500000i) q^{36} +(-1.67303 + 0.965926i) q^{38} +(0.366025 + 1.36603i) q^{39} +(0.707107 + 0.707107i) q^{40} +(0.965926 - 0.258819i) q^{42} +(0.707107 + 0.707107i) q^{45} -1.73205 q^{46} +(-0.258819 + 0.965926i) q^{48} +(0.500000 - 0.866025i) q^{49} +(0.500000 + 0.866025i) q^{50} +(-1.22474 + 0.707107i) q^{52} +(-0.258819 + 0.965926i) q^{54} +(0.500000 + 0.866025i) q^{56} +(-1.36603 - 1.36603i) q^{57} +(-0.707107 + 1.22474i) q^{59} +(-0.500000 + 0.866025i) q^{60} +(-0.965926 - 1.67303i) q^{61} +(0.500000 + 0.866025i) q^{63} -1.00000 q^{64} +(-1.36603 + 0.366025i) q^{65} +(-0.448288 - 1.67303i) q^{69} +(0.258819 + 0.965926i) q^{70} -1.73205 q^{71} -1.00000 q^{72} +(-0.707107 + 0.707107i) q^{75} +(0.965926 - 1.67303i) q^{76} +(-1.00000 - 1.00000i) q^{78} +(0.866025 + 1.50000i) q^{79} +(-0.965926 - 0.258819i) q^{80} -1.00000 q^{81} +(1.22474 - 0.707107i) q^{83} +(-0.707107 + 0.707107i) q^{84} +(-0.965926 - 0.258819i) q^{90} -1.41421 q^{91} +(1.50000 - 0.866025i) q^{92} +(1.36603 - 1.36603i) q^{95} +(-0.258819 - 0.965926i) q^{96} +1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4}+O(q^{10}) \) 8 * q + 4 * q^4	\( 8 q + 4 q^{4} - 4 q^{14} - 8 q^{15} - 4 q^{16} - 4 q^{18} + 12 q^{23} - 4 q^{39} + 4 q^{49} + 4 q^{50} + 4 q^{56} - 4 q^{57} - 4 q^{60} + 4 q^{63} - 8 q^{64} - 4 q^{65} - 8 q^{72} - 8 q^{78} - 8 q^{81} + 12 q^{92} + 4 q^{95}+O(q^{100}) \) 8 * q + 4 * q^4 - 4 * q^14 - 8 * q^15 - 4 * q^16 - 4 * q^18 + 12 * q^23 - 4 * q^39 + 4 * q^49 + 4 * q^50 + 4 * q^56 - 4 * q^57 - 4 * q^60 + 4 * q^63 - 8 * q^64 - 4 * q^65 - 8 * q^72 - 8 * q^78 - 8 * q^81 + 12 * q^92 + 4 * q^95

Introduction

L-functions

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