LMFDB - Embedded newform 420.1.be.a.59.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [420,1,Mod(59,420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(420, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("420.59");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	420.be (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:	$0.209607305306$
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 6.0.726062400.1

Embedding invariants

Embedding label			59.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	420.59
Dual form			420.1.be.a.299.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 + 0.500000i) q^{2} +(0.866025 - 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})$	\(q+(-0.866025 + 0.500000i) q^{2} +(0.866025 - 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +(0.866025 + 0.500000i) q^{10} -1.00000i q^{12} +1.00000 q^{14} +(-0.866025 - 0.500000i) q^{15} +(-0.500000 - 0.866025i) q^{16} +1.00000i q^{18} -1.00000 q^{20} -1.00000 q^{21} +(0.866025 - 0.500000i) q^{23} +(0.500000 + 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} -1.00000i q^{27} +(-0.866025 + 0.500000i) q^{28} +1.73205i q^{29} +1.00000 q^{30} +(0.866025 + 0.500000i) q^{32} +1.00000i q^{35} +(-0.500000 - 0.866025i) q^{36} +(0.866025 - 0.500000i) q^{40} +1.00000 q^{41} +(0.866025 - 0.500000i) q^{42} +1.73205 q^{43} -1.00000 q^{45} +(-0.500000 + 0.866025i) q^{46} +(-0.866025 - 0.500000i) q^{48} +(0.500000 + 0.866025i) q^{49} -1.00000i q^{50} +(0.500000 + 0.866025i) q^{54} +(0.500000 - 0.866025i) q^{56} +(-0.866025 - 1.50000i) q^{58} +(-0.866025 + 0.500000i) q^{60} +(-1.50000 + 0.866025i) q^{61} +(-0.866025 + 0.500000i) q^{63} -1.00000 q^{64} +(-0.866025 + 1.50000i) q^{67} +(0.500000 - 0.866025i) q^{69} +(-0.500000 - 0.866025i) q^{70} +(0.866025 + 0.500000i) q^{72} +1.00000i q^{75} +(-0.500000 + 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.866025 + 0.500000i) q^{82} -1.73205 q^{83} +(-0.500000 + 0.866025i) q^{84} +(-1.50000 + 0.866025i) q^{86} +(0.866025 + 1.50000i) q^{87} +(-0.500000 - 0.866025i) q^{89} +(0.866025 - 0.500000i) q^{90} -1.00000i q^{92} +1.00000 q^{96} +(-0.866025 - 0.500000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^9	$ 4 q + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{9} + 4 q^{14} - 2 q^{16} - 4 q^{20} - 4 q^{21} + 2 q^{24} - 2 q^{25} + 4 q^{30} - 2 q^{36} + 4 q^{41} - 4 q^{45} - 2 q^{46} + 2 q^{49} + 2 q^{54} + 2 q^{56} - 6 q^{61} - 4 q^{64} + 2 q^{69} - 2 q^{70} - 2 q^{80} - 2 q^{81} - 2 q^{84} - 6 q^{86} - 2 q^{89} + 4 q^{96}+O(q^{100}) $ 4 * q + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^9 + 4 * q^14 - 2 * q^16 - 4 * q^20 - 4 * q^21 + 2 * q^24 - 2 * q^25 + 4 * q^30 - 2 * q^36 + 4 * q^41 - 4 * q^45 - 2 * q^46 + 2 * q^49 + 2 * q^54 + 2 * q^56 - 6 * q^61 - 4 * q^64 + 2 * q^69 - 2 * q^70 - 2 * q^80 - 2 * q^81 - 2 * q^84 - 6 * q^86 - 2 * q^89 + 4 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$-1$	$-1$

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
$3$	0.866025	−	0.500000i	0.866025	−	0.500000i
$3$	0.866025	−	0.500000i	0.866025	−	0.500000i
$4$	0.500000	−	0.866025i	0.500000	−	0.866025i
$5$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$5$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$6$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$7$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$7$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$8$			1.00000i			1.00000i
$9$	0.500000	−	0.866025i	0.500000	−	0.866025i
$10$	0.866025	+	0.500000i	0.866025	+	0.500000i
$11$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$11$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$12$		−	1.00000i		−	1.00000i
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$	1.00000			1.00000
$15$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$16$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$			1.00000i			1.00000i
$19$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$19$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$20$	−1.00000			−1.00000
$21$	−1.00000			−1.00000
$22$		0			0
$23$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$23$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$24$	0.500000	+	0.866025i	0.500000	+	0.866025i
$25$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$26$		0			0
$27$		−	1.00000i		−	1.00000i
$28$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$29$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$29$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$30$	1.00000			1.00000
$31$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$31$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$32$	0.866025	+	0.500000i	0.866025	+	0.500000i
$33$		0			0
$34$		0			0
$35$			1.00000i			1.00000i
$36$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$		0			0
$40$	0.866025	−	0.500000i	0.866025	−	0.500000i
$41$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$41$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$42$	0.866025	−	0.500000i	0.866025	−	0.500000i
$43$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$43$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$44$		0			0
$45$	−1.00000			−1.00000
$46$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$49$	0.500000	+	0.866025i	0.500000	+	0.866025i
$50$		−	1.00000i		−	1.00000i
$51$		0			0
$52$		0			0
$53$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$53$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$54$	0.500000	+	0.866025i	0.500000	+	0.866025i
$55$		0			0
$56$	0.500000	−	0.866025i	0.500000	−	0.866025i
$57$		0			0
$58$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$61$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$61$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$62$		0			0
$63$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$64$	−1.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$67$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$68$		0			0
$69$	0.500000	−	0.866025i	0.500000	−	0.866025i
$70$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	0.866025	+	0.500000i	0.866025	+	0.500000i
$73$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$73$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$74$		0			0
$75$			1.00000i			1.00000i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$79$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$80$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$83$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$83$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$84$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$85$		0			0
$86$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$87$	0.866025	+	1.50000i	0.866025	+	1.50000i
$88$		0			0
$89$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$89$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$90$	0.866025	−	0.500000i	0.866025	−	0.500000i
$91$		0			0
$92$		−	1.00000i		−	1.00000i
$93$		0			0
$94$		0			0
$95$		0			0
$96$	1.00000			1.00000
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$99$		0			0
$100$	0.500000	+	0.866025i	0.500000	+	0.866025i
$101$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$101$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$102$		0			0
$103$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$103$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$104$		0			0
$105$	0.500000	+	0.866025i	0.500000	+	0.866025i
$106$		0			0
$107$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$107$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$108$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$109$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$109$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$110$		0			0
$111$		0			0
$112$			1.00000i			1.00000i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$116$	1.50000	+	0.866025i	1.50000	+	0.866025i
$117$		0			0
$118$		0			0
$119$		0			0
$120$	0.500000	−	0.866025i	0.500000	−	0.866025i
$121$	0.500000	+	0.866025i	0.500000	+	0.866025i
$122$	0.866025	−	1.50000i	0.866025	−	1.50000i
$123$	0.866025	−	0.500000i	0.866025	−	0.500000i
$124$		0			0
$125$	1.00000			1.00000
$126$	0.500000	−	0.866025i	0.500000	−	0.866025i
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	0.866025	−	0.500000i	0.866025	−	0.500000i
$129$	1.50000	−	0.866025i	1.50000	−	0.866025i
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$		0			0
$134$		−	1.73205i		−	1.73205i
$135$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$136$		0			0
$137$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$137$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$138$			1.00000i			1.00000i
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$	0.866025	+	0.500000i	0.866025	+	0.500000i
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1.00000			−1.00000
$145$	1.50000	−	0.866025i	1.50000	−	0.866025i
$146$		0			0
$147$	0.866025	+	0.500000i	0.866025	+	0.500000i
$148$		0			0
$149$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$149$	−1.50000	+	0.866025i	−1.50000	+	0.866025i	−1.00000			$\pi$
$150$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$151$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$151$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$157$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$158$		0			0
$159$		0			0
$160$		−	1.00000i		−	1.00000i
$161$	−1.00000			−1.00000
$162$	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	$0.833333\pi$
$164$	0.500000	−	0.866025i	0.500000	−	0.866025i
$165$		0			0
$166$	1.50000	−	0.866025i	1.50000	−	0.866025i
$167$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$167$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$168$		−	1.00000i		−	1.00000i
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$	0.866025	−	1.50000i	0.866025	−	1.50000i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$175$	0.866025	−	0.500000i	0.866025	−	0.500000i
$176$		0			0
$177$		0			0
$178$	0.866025	+	0.500000i	0.866025	+	0.500000i
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$181$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$181$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$182$		0			0
$183$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$184$	0.500000	+	0.866025i	0.500000	+	0.866025i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$		0			0
$196$	1.00000			1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		0			0
$199$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$199$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$200$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$201$			1.73205i			1.73205i
$202$			1.00000i			1.00000i
$203$	0.866025	−	1.50000i	0.866025	−	1.50000i
$204$		0			0
$205$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$206$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$207$		−	1.00000i		−	1.00000i
$208$		0			0
$209$		0			0
$210$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$	0.500000	−	0.866025i	0.500000	−	0.866025i
$215$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$216$	1.00000			1.00000
$217$		0			0
$218$			1.00000i			1.00000i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$223$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$224$	−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	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$	1.00000			1.00000
$231$		0			0
$232$	−1.73205			−1.73205
$233$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$233$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$			1.00000i			1.00000i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$243$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$244$			1.73205i			1.73205i
$245$	0.500000	−	0.866025i	0.500000	−	0.866025i
$246$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$247$		0			0
$248$		0			0
$249$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$250$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$			1.00000i			1.00000i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$259$		0			0
$260$		0			0
$261$	1.50000	+	0.866025i	1.50000	+	0.866025i
$262$		0			0
$263$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$263$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$268$	0.866025	+	1.50000i	0.866025	+	1.50000i
$269$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$269$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$270$	0.500000	−	0.866025i	0.500000	−	0.866025i
$271$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$271$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$	−1.00000			−1.00000
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$283$	−1.73205	−	1.00000i	−1.73205	−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$288$	0.866025	−	0.500000i	0.866025	−	0.500000i
$289$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$290$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	−1.00000			−1.00000
$295$		0			0
$296$		0			0
$297$		0			0
$298$	0.866025	−	1.50000i	0.866025	−	1.50000i
$299$		0			0
$300$	0.866025	+	0.500000i	0.866025	+	0.500000i
$301$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$302$		0			0
$303$		−	1.00000i		−	1.00000i
$304$		0			0
$305$	1.50000	+	0.866025i	1.50000	+	0.866025i
$306$		0			0
$307$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$307$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$308$		0			0
$309$	0.500000	−	0.866025i	0.500000	−	0.866025i
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$313$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$314$		0			0
$315$	0.866025	+	0.500000i	0.866025	+	0.500000i
$316$		0			0
$317$		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
$320$	0.500000	+	0.866025i	0.500000	+	0.866025i
$321$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$322$	0.866025	−	0.500000i	0.866025	−	0.500000i
$323$		0			0
$324$	−1.00000			−1.00000
$325$		0			0
$326$		0			0
$327$		−	1.00000i		−	1.00000i
$328$			1.00000i			1.00000i
$329$		0			0
$330$		0			0
$331$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$331$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$332$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$333$		0			0
$334$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$335$	1.73205			1.73205
$336$	0.500000	+	0.866025i	0.500000	+	0.866025i
$337$		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			0
$343$		−	1.00000i		−	1.00000i
$344$			1.73205i			1.73205i
$345$	−1.00000			−1.00000
$346$		0			0
$347$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$347$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$348$	1.73205			1.73205
$349$			1.73205i			1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$349$			1.73205i			1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$350$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$351$		0			0
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$	−1.00000			−1.00000
$357$		0			0
$358$		0			0
$359$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$359$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$360$		−	1.00000i		−	1.00000i
$361$	0.500000	−	0.866025i	0.500000	−	0.866025i
$362$	0.866025	+	1.50000i	0.866025	+	1.50000i
$363$	0.866025	+	0.500000i	0.866025	+	0.500000i
$364$		0			0
$365$		0			0
$366$		−	1.73205i		−	1.73205i
$367$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$367$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$368$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$369$	0.500000	−	0.866025i	0.500000	−	0.866025i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	0.866025	−	0.500000i	0.866025	−	0.500000i
$376$		0			0
$377$		0			0
$378$		−	1.00000i		−	1.00000i
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$383$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$384$	0.500000	−	0.866025i	0.500000	−	0.866025i
$385$		0			0
$386$		0			0
$387$	0.866025	−	1.50000i	0.866025	−	1.50000i
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$397$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$398$		0			0
$399$		0			0
$400$	1.00000			1.00000
$401$	1.50000	−	0.866025i	1.50000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$401$	1.50000	−	0.866025i	1.50000	−	0.866025i	1.00000			$0$
$402$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$403$		0			0
$404$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$405$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$406$			1.73205i			1.73205i
$407$		0			0
$408$		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			$\pi$
$410$	0.866025	+	0.500000i	0.866025	+	0.500000i
$411$		0			0
$412$		−	1.00000i		−	1.00000i
$413$		0			0
$414$	0.500000	+	0.866025i	0.500000	+	0.866025i
$415$	0.866025	+	1.50000i	0.866025	+	1.50000i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$	1.00000			1.00000
$421$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$421$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.73205			1.73205
$428$			1.00000i			1.00000i
$429$		0			0
$430$	1.50000	+	0.866025i	1.50000	+	0.866025i
$431$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$431$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$432$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$	0.866025	−	1.50000i	0.866025	−	1.50000i
$436$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$437$		0			0
$438$		0			0
$439$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$439$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$440$		0			0
$441$	1.00000			1.00000
$442$		0			0
$443$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$443$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$444$		0			0
$445$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$446$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$447$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$448$	0.866025	+	0.500000i	0.866025	+	0.500000i
$449$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$449$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$450$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$		0			0
$459$		0			0
$460$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$461$	−2.00000			−2.00000			−1.00000			$\pi$
$461$	−2.00000			−2.00000			−1.00000			$\pi$
$462$		0			0
$463$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$463$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$464$	1.50000	−	0.866025i	1.50000	−	0.866025i
$465$		0			0
$466$		0			0
$467$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$467$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$468$		0			0
$469$	1.50000	−	0.866025i	1.50000	−	0.866025i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$481$		0			0
$482$		0			0
$483$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$484$	1.00000			1.00000
$485$		0			0
$486$	1.00000			1.00000
$487$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$487$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$488$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$489$		0			0
$490$			1.00000i			1.00000i
$491$		0			0			−	1.00000i	$-0.5\pi$
$491$		0			0				1.00000i	$0.5\pi$
$492$		−	1.00000i		−	1.00000i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	0.866025	−	1.50000i	0.866025	−	1.50000i
$499$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$499$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$500$	0.500000	−	0.866025i	0.500000	−	0.866025i
$501$	1.50000	−	0.866025i	1.50000	−	0.866025i
$502$		0			0
$503$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$503$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$504$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$505$	−1.00000			−1.00000
$506$		0			0
$507$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$508$		0			0
$509$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$509$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$		−	1.00000i		−	1.00000i
$513$		0			0
$514$		0			0
$515$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$516$		−	1.73205i		−	1.73205i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$521$	−1.00000	+	1.73205i	−1.00000	+	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$522$	−1.73205			−1.73205
$523$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$523$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$524$		0			0
$525$	0.500000	−	0.866025i	0.500000	−	0.866025i
$526$	−1.00000			−1.00000
$527$		0			0
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$	1.00000			1.00000
$535$	0.866025	+	0.500000i	0.866025	+	0.500000i
$536$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$537$		0			0
$538$		−	1.00000i		−	1.00000i
$539$		0			0
$540$			1.00000i			1.00000i
$541$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$541$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$544$		0			0
$545$	−1.00000			−1.00000
$546$		0			0
$547$	1.73205			1.73205			0.866025	−	0.500000i	$-0.166667\pi$
$547$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$548$		0			0
$549$			1.73205i			1.73205i
$550$		0			0
$551$		0			0
$552$	0.866025	+	0.500000i	0.866025	+	0.500000i
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$		0			0
$560$	0.866025	−	0.500000i	0.866025	−	0.500000i
$561$		0			0
$562$		0			0
$563$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$563$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$	2.00000			2.00000
$567$			1.00000i			1.00000i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$571$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$572$		0			0
$573$		0			0
$574$	1.00000			1.00000
$575$			1.00000i			1.00000i
$576$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$577$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$577$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$578$	0.866025	+	0.500000i	0.866025	+	0.500000i
$579$		0			0
$580$		−	1.73205i		−	1.73205i
$581$	1.50000	+	0.866025i	1.50000	+	0.866025i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	0.866025	−	0.500000i	0.866025	−	0.500000i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$			1.73205i			1.73205i
$597$		0			0
$598$		0			0
$599$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$599$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$600$	−1.00000			−1.00000
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$	1.73205			1.73205
$603$	0.866025	+	1.50000i	0.866025	+	1.50000i
$604$		0			0
$605$	0.500000	−	0.866025i	0.500000	−	0.866025i
$606$	0.500000	+	0.866025i	0.500000	+	0.866025i
$607$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$607$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		−	1.73205i		−	1.73205i
$610$	−1.73205			−1.73205
$611$		0			0
$612$		0			0
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$	0.500000	+	0.866025i	0.500000	+	0.866025i
$615$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$			1.00000i			1.00000i
$619$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$619$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$620$		0			0
$621$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$622$		0			0
$623$			1.00000i			1.00000i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$	−1.00000			−1.00000
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$641$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	−1.50000	−	0.866025i	−1.50000	−	0.866025i	−1.00000			$\pi$
$642$		−	1.00000i		−	1.00000i
$643$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$643$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$644$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$645$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$646$		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
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$	0.500000	+	0.866025i	0.500000	+	0.866025i
$655$		0			0
$656$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$661$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		−	1.73205i		−	1.73205i
$665$		0			0
$666$		0			0
$667$	0.866025	+	1.50000i	0.866025	+	1.50000i
$668$	0.866025	−	1.50000i	0.866025	−	1.50000i
$669$	1.00000	+	1.73205i	1.00000	+	1.73205i
$670$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$671$		0			0
$672$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	0.866025	+	0.500000i	0.866025	+	0.500000i
$676$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$683$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$684$		0			0
$685$		0			0
$686$	0.500000	+	0.866025i	0.500000	+	0.866025i
$687$		0			0
$688$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$689$		0			0
$690$	0.866025	−	0.500000i	0.866025	−	0.500000i
$691$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$691$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$692$		0			0
$693$		0			0
$694$	1.00000			1.00000
$695$		0			0
$696$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$697$		0			0
$698$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$699$		0			0
$700$		−	1.00000i		−	1.00000i
$701$		−	1.73205i		−	1.73205i	−0.500000	−	0.866025i	$-0.666667\pi$
$701$		−	1.73205i		−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$708$		0			0
$709$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$709$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$	0.866025	−	0.500000i	0.866025	−	0.500000i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$	0.500000	+	0.866025i	0.500000	+	0.866025i
$721$	−1.00000			−1.00000
$722$			1.00000i			1.00000i
$723$		0			0
$724$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$725$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$726$	−1.00000			−1.00000
$727$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$727$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$728$		0			0
$729$	−1.00000			−1.00000
$730$		0			0
$731$		0			0
$732$	0.866025	+	1.50000i	0.866025	+	1.50000i
$733$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$733$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$734$	1.00000			1.00000
$735$		−	1.00000i		−	1.00000i
$736$	1.00000			1.00000
$737$		0			0
$738$			1.00000i			1.00000i
$739$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$739$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$743$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$744$		0			0
$745$	1.50000	+	0.866025i	1.50000	+	0.866025i
$746$		0			0
$747$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$748$		0			0
$749$	1.00000			1.00000
$750$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$751$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$751$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.500000	+	0.866025i	0.500000	+	0.866025i
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$761$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$764$		0			0
$765$		0			0
$766$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$767$		0			0
$768$			1.00000i			1.00000i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$			1.73205i			1.73205i
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	1.73205			1.73205
$784$	0.500000	−	0.866025i	0.500000	−	0.866025i
$785$		0			0
$786$		0			0
$787$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$787$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$788$		0			0
$789$	1.00000			1.00000
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$801$	−1.00000			−1.00000
$802$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$803$		0			0
$804$	1.50000	+	0.866025i	1.50000	+	0.866025i
$805$	0.500000	+	0.866025i	0.500000	+	0.866025i
$806$		0			0
$807$			1.00000i			1.00000i
$808$	0.866025	+	0.500000i	0.866025	+	0.500000i
$809$	1.50000	+	0.866025i	1.50000	+	0.866025i	1.00000			$0$
$809$	1.50000	+	0.866025i	1.50000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$810$		−	1.00000i		−	1.00000i
$811$		0			0			−	1.00000i	$-0.5\pi$
$811$		0			0				1.00000i	$0.5\pi$
$812$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.73205			1.73205
$819$		0			0
$820$	−1.00000			−1.00000
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	−0.866025	+	1.50000i	−0.866025	+	1.50000i			1.00000i	$0.5\pi$
$823$	−0.866025	+	1.50000i	−0.866025	+	1.50000i	−0.866025	+	0.500000i	$0.833333\pi$
$824$	0.500000	+	0.866025i	0.500000	+	0.866025i
$825$		0			0
$826$		0			0
$827$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$827$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$828$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−0.866025	−	1.50000i	−0.866025	−	1.50000i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$841$	−2.00000			−2.00000
$842$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$843$		0			0
$844$		0			0
$845$	0.500000	+	0.866025i	0.500000	+	0.866025i
$846$		0			0
$847$		−	1.00000i		−	1.00000i
$848$		0			0
$849$	−2.00000			−2.00000
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$855$		0			0
$856$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$859$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$860$	−1.73205			−1.73205
$861$	−1.00000			−1.00000
$862$		0			0
$863$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$863$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$864$	0.500000	−	0.866025i	0.500000	−	0.866025i
$865$		0			0
$866$		0			0
$867$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$868$		0			0
$869$		0			0
$870$			1.73205i			1.73205i
$871$		0			0
$872$	0.866025	+	0.500000i	0.866025	+	0.500000i
$873$		0			0
$874$		0			0
$875$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$881$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$882$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$883$		0			0			−	1.00000i	$-0.5\pi$
$883$		0			0				1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$	0.500000	−	0.866025i	0.500000	−	0.866025i
$887$	−0.866025	−	1.50000i	−0.866025	−	1.50000i	−0.866025	−	0.500000i	$-0.833333\pi$
$887$	−0.866025	−	1.50000i	−0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$888$		0			0
$889$		0			0
$890$		−	1.00000i		−	1.00000i
$891$		0			0
$892$	1.73205	+	1.00000i	1.73205	+	1.00000i
$893$		0			0
$894$		−	1.73205i		−	1.73205i
$895$		0			0
$896$	−1.00000			−1.00000
$897$		0			0
$898$	0.866025	+	1.50000i	0.866025	+	1.50000i
$899$		0			0
$900$	1.00000			1.00000
$901$		0			0
$902$		0			0
$903$	−1.73205			−1.73205
$904$		0			0
$905$	−1.50000	+	0.866025i	−1.50000	+	0.866025i
$906$		0			0
$907$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$907$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$908$		0			0
$909$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	1.73205			1.73205
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$919$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$920$	0.500000	−	0.866025i	0.500000	−	0.866025i
$921$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$922$	1.73205	−	1.00000i	1.73205	−	1.00000i
$923$		0			0
$924$		0			0
$925$		0			0
$926$	1.50000	−	0.866025i	1.50000	−	0.866025i
$927$		−	1.00000i		−	1.00000i
$928$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$929$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$929$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$	−1.50000	−	0.866025i	−1.50000	−	0.866025i
$935$		0			0
$936$		0			0
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$	−0.866025	+	1.50000i	−0.866025	+	1.50000i
$939$		0			0
$940$		0			0
$941$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$941$	1.00000	−	1.73205i	1.00000	−	1.73205i	0.500000	−	0.866025i	$-0.333333\pi$
$942$		0			0
$943$	0.866025	−	0.500000i	0.866025	−	0.500000i
$944$		0			0
$945$	1.00000			1.00000
$946$		0			0
$947$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$947$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	0.866025	+	0.500000i	0.866025	+	0.500000i
$961$	0.500000	+	0.866025i	0.500000	+	0.866025i
$962$		0			0
$963$			1.00000i			1.00000i
$964$		0			0
$965$		0			0
$966$	0.500000	−	0.866025i	0.500000	−	0.866025i
$967$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$967$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$968$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.50000	+	0.866025i	1.50000	+	0.866025i
$977$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$977$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$978$		0			0
$979$		0			0
$980$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$981$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$982$		0			0
$983$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$983$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$984$	0.500000	+	0.866025i	0.500000	+	0.866025i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.50000	−	0.866025i	1.50000	−	0.866025i
$990$		0			0
$991$		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
$996$			1.73205i			1.73205i
$997$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$997$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.1.be.a.59.1	✓	4
3.2	odd	2		420.1.be.b.59.2	yes	4
4.3	odd	2	inner	420.1.be.a.59.2	yes	4
5.2	odd	4		2100.1.ba.b.1151.1		2
5.3	odd	4		2100.1.ba.c.1151.1		2
5.4	even	2	inner	420.1.be.a.59.2	yes	4
7.2	even	3		2940.1.be.a.1979.2		4
7.3	odd	6		2940.1.o.a.2939.2		4
7.4	even	3		2940.1.o.b.2939.1		4
7.5	odd	6		420.1.be.b.299.2	yes	4
7.6	odd	2		2940.1.be.d.2579.1		4
12.11	even	2		420.1.be.b.59.1	yes	4
15.2	even	4		2100.1.ba.d.1151.1		2
15.8	even	4		2100.1.ba.a.1151.1		2
15.14	odd	2		420.1.be.b.59.1	yes	4
20.3	even	4		2100.1.ba.b.1151.1		2
20.7	even	4		2100.1.ba.c.1151.1		2
20.19	odd	2	CM	420.1.be.a.59.1	✓	4
21.2	odd	6		2940.1.be.d.1979.1		4
21.5	even	6	inner	420.1.be.a.299.1	yes	4
21.11	odd	6		2940.1.o.a.2939.4		4
21.17	even	6		2940.1.o.b.2939.3		4
21.20	even	2		2940.1.be.a.2579.2		4
28.3	even	6		2940.1.o.a.2939.3		4
28.11	odd	6		2940.1.o.b.2939.4		4
28.19	even	6		420.1.be.b.299.1	yes	4
28.23	odd	6		2940.1.be.a.1979.1		4
28.27	even	2		2940.1.be.d.2579.2		4
35.4	even	6		2940.1.o.b.2939.4		4
35.9	even	6		2940.1.be.a.1979.1		4
35.12	even	12		2100.1.ba.a.551.1		2
35.19	odd	6		420.1.be.b.299.1	yes	4
35.24	odd	6		2940.1.o.a.2939.3		4
35.33	even	12		2100.1.ba.d.551.1		2
35.34	odd	2		2940.1.be.d.2579.2		4
60.23	odd	4		2100.1.ba.d.1151.1		2
60.47	odd	4		2100.1.ba.a.1151.1		2
60.59	even	2		420.1.be.b.59.2	yes	4
84.11	even	6		2940.1.o.a.2939.1		4
84.23	even	6		2940.1.be.d.1979.2		4
84.47	odd	6	inner	420.1.be.a.299.2	yes	4
84.59	odd	6		2940.1.o.b.2939.2		4
84.83	odd	2		2940.1.be.a.2579.1		4
105.44	odd	6		2940.1.be.d.1979.2		4
105.47	odd	12		2100.1.ba.c.551.1		2
105.59	even	6		2940.1.o.b.2939.2		4
105.68	odd	12		2100.1.ba.b.551.1		2
105.74	odd	6		2940.1.o.a.2939.1		4
105.89	even	6	inner	420.1.be.a.299.2	yes	4
105.104	even	2		2940.1.be.a.2579.1		4
140.19	even	6		420.1.be.b.299.2	yes	4
140.39	odd	6		2940.1.o.b.2939.1		4
140.47	odd	12		2100.1.ba.d.551.1		2
140.59	even	6		2940.1.o.a.2939.2		4
140.79	odd	6		2940.1.be.a.1979.2		4
140.103	odd	12		2100.1.ba.a.551.1		2
140.139	even	2		2940.1.be.d.2579.1		4
420.47	even	12		2100.1.ba.b.551.1		2
420.59	odd	6		2940.1.o.b.2939.3		4
420.179	even	6		2940.1.o.a.2939.4		4
420.299	odd	6	inner	420.1.be.a.299.1	yes	4
420.359	even	6		2940.1.be.d.1979.1		4
420.383	even	12		2100.1.ba.c.551.1		2
420.419	odd	2		2940.1.be.a.2579.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.1.be.a.59.1	✓	4	1.1	even	1	trivial
420.1.be.a.59.1	✓	4	20.19	odd	2	CM
420.1.be.a.59.2	yes	4	4.3	odd	2	inner
420.1.be.a.59.2	yes	4	5.4	even	2	inner
420.1.be.a.299.1	yes	4	21.5	even	6	inner
420.1.be.a.299.1	yes	4	420.299	odd	6	inner
420.1.be.a.299.2	yes	4	84.47	odd	6	inner
420.1.be.a.299.2	yes	4	105.89	even	6	inner
420.1.be.b.59.1	yes	4	12.11	even	2
420.1.be.b.59.1	yes	4	15.14	odd	2
420.1.be.b.59.2	yes	4	3.2	odd	2
420.1.be.b.59.2	yes	4	60.59	even	2
420.1.be.b.299.1	yes	4	28.19	even	6
420.1.be.b.299.1	yes	4	35.19	odd	6
420.1.be.b.299.2	yes	4	7.5	odd	6
420.1.be.b.299.2	yes	4	140.19	even	6
2100.1.ba.a.551.1		2	35.12	even	12
2100.1.ba.a.551.1		2	140.103	odd	12
2100.1.ba.a.1151.1		2	15.8	even	4
2100.1.ba.a.1151.1		2	60.47	odd	4
2100.1.ba.b.551.1		2	105.68	odd	12
2100.1.ba.b.551.1		2	420.47	even	12
2100.1.ba.b.1151.1		2	5.2	odd	4
2100.1.ba.b.1151.1		2	20.3	even	4
2100.1.ba.c.551.1		2	105.47	odd	12
2100.1.ba.c.551.1		2	420.383	even	12
2100.1.ba.c.1151.1		2	5.3	odd	4
2100.1.ba.c.1151.1		2	20.7	even	4
2100.1.ba.d.551.1		2	35.33	even	12
2100.1.ba.d.551.1		2	140.47	odd	12
2100.1.ba.d.1151.1		2	15.2	even	4
2100.1.ba.d.1151.1		2	60.23	odd	4
2940.1.o.a.2939.1		4	84.11	even	6
2940.1.o.a.2939.1		4	105.74	odd	6
2940.1.o.a.2939.2		4	7.3	odd	6
2940.1.o.a.2939.2		4	140.59	even	6
2940.1.o.a.2939.3		4	28.3	even	6
2940.1.o.a.2939.3		4	35.24	odd	6
2940.1.o.a.2939.4		4	21.11	odd	6
2940.1.o.a.2939.4		4	420.179	even	6
2940.1.o.b.2939.1		4	7.4	even	3
2940.1.o.b.2939.1		4	140.39	odd	6
2940.1.o.b.2939.2		4	84.59	odd	6
2940.1.o.b.2939.2		4	105.59	even	6
2940.1.o.b.2939.3		4	21.17	even	6
2940.1.o.b.2939.3		4	420.59	odd	6
2940.1.o.b.2939.4		4	28.11	odd	6
2940.1.o.b.2939.4		4	35.4	even	6
2940.1.be.a.1979.1		4	28.23	odd	6
2940.1.be.a.1979.1		4	35.9	even	6
2940.1.be.a.1979.2		4	7.2	even	3
2940.1.be.a.1979.2		4	140.79	odd	6
2940.1.be.a.2579.1		4	84.83	odd	2
2940.1.be.a.2579.1		4	105.104	even	2
2940.1.be.a.2579.2		4	21.20	even	2
2940.1.be.a.2579.2		4	420.419	odd	2
2940.1.be.d.1979.1		4	21.2	odd	6
2940.1.be.d.1979.1		4	420.359	even	6
2940.1.be.d.1979.2		4	84.23	even	6
2940.1.be.d.1979.2		4	105.44	odd	6
2940.1.be.d.2579.1		4	7.6	odd	2
2940.1.be.d.2579.1		4	140.139	even	2
2940.1.be.d.2579.2		4	28.27	even	2
2940.1.be.d.2579.2		4	35.34	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:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	420.be (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 + 0.500000i) q^{2} +(0.866025 - 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +O(q^{10})\)	\(q+(-0.866025 + 0.500000i) q^{2} +(0.866025 - 0.500000i) q^{3} +(0.500000 - 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +(-0.500000 + 0.866025i) q^{6} +(-0.866025 - 0.500000i) q^{7} +1.00000i q^{8} +(0.500000 - 0.866025i) q^{9} +(0.866025 + 0.500000i) q^{10} -1.00000i q^{12} +1.00000 q^{14} +(-0.866025 - 0.500000i) q^{15} +(-0.500000 - 0.866025i) q^{16} +1.00000i q^{18} -1.00000 q^{20} -1.00000 q^{21} +(0.866025 - 0.500000i) q^{23} +(0.500000 + 0.866025i) q^{24} +(-0.500000 + 0.866025i) q^{25} -1.00000i q^{27} +(-0.866025 + 0.500000i) q^{28} +1.73205i q^{29} +1.00000 q^{30} +(0.866025 + 0.500000i) q^{32} +1.00000i q^{35} +(-0.500000 - 0.866025i) q^{36} +(0.866025 - 0.500000i) q^{40} +1.00000 q^{41} +(0.866025 - 0.500000i) q^{42} +1.73205 q^{43} -1.00000 q^{45} +(-0.500000 + 0.866025i) q^{46} +(-0.866025 - 0.500000i) q^{48} +(0.500000 + 0.866025i) q^{49} -1.00000i q^{50} +(0.500000 + 0.866025i) q^{54} +(0.500000 - 0.866025i) q^{56} +(-0.866025 - 1.50000i) q^{58} +(-0.866025 + 0.500000i) q^{60} +(-1.50000 + 0.866025i) q^{61} +(-0.866025 + 0.500000i) q^{63} -1.00000 q^{64} +(-0.866025 + 1.50000i) q^{67} +(0.500000 - 0.866025i) q^{69} +(-0.500000 - 0.866025i) q^{70} +(0.866025 + 0.500000i) q^{72} +1.00000i q^{75} +(-0.500000 + 0.866025i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.866025 + 0.500000i) q^{82} -1.73205 q^{83} +(-0.500000 + 0.866025i) q^{84} +(-1.50000 + 0.866025i) q^{86} +(0.866025 + 1.50000i) q^{87} +(-0.500000 - 0.866025i) q^{89} +(0.866025 - 0.500000i) q^{90} -1.00000i q^{92} +1.00000 q^{96} +(-0.866025 - 0.500000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^9	\( 4 q + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{9} + 4 q^{14} - 2 q^{16} - 4 q^{20} - 4 q^{21} + 2 q^{24} - 2 q^{25} + 4 q^{30} - 2 q^{36} + 4 q^{41} - 4 q^{45} - 2 q^{46} + 2 q^{49} + 2 q^{54} + 2 q^{56} - 6 q^{61} - 4 q^{64} + 2 q^{69} - 2 q^{70} - 2 q^{80} - 2 q^{81} - 2 q^{84} - 6 q^{86} - 2 q^{89} + 4 q^{96}+O(q^{100}) \) 4 * q + 2 * q^4 - 2 * q^5 - 2 * q^6 + 2 * q^9 + 4 * q^14 - 2 * q^16 - 4 * q^20 - 4 * q^21 + 2 * q^24 - 2 * q^25 + 4 * q^30 - 2 * q^36 + 4 * q^41 - 4 * q^45 - 2 * q^46 + 2 * q^49 + 2 * q^54 + 2 * q^56 - 6 * q^61 - 4 * q^64 + 2 * q^69 - 2 * q^70 - 2 * q^80 - 2 * q^81 - 2 * q^84 - 6 * q^86 - 2 * q^89 + 4 * q^96

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