LMFDB - Embedded newform 1520.1.bh.b.949.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,1,Mod(189,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2, 2]))

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

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

chi := DirichletCharacter("1520.189");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1520.bh (of order $4$, 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:	$0.758578819202$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.972800.1

Embedding invariants

Embedding label			949.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1520.949
Dual form			1520.1.bh.b.189.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 + 0.707107i) q^{2} +(-0.707107 + 0.707107i) q^{3} +1.00000i q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +(-0.707107 + 0.707107i) q^{8} +O(q^{10})$	\(q+(0.707107 + 0.707107i) q^{2} +(-0.707107 + 0.707107i) q^{3} +1.00000i q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +(-0.707107 + 0.707107i) q^{8} +(0.707107 + 0.707107i) q^{10} +(-0.707107 - 0.707107i) q^{12} +(0.707107 - 0.707107i) q^{13} +(0.707107 + 0.707107i) q^{14} +(-0.707107 + 0.707107i) q^{15} -1.00000 q^{16} -1.00000i q^{17} +(0.707107 + 0.707107i) q^{19} +1.00000i q^{20} +(-0.707107 + 0.707107i) q^{21} -1.00000 q^{23} -1.00000i q^{24} +1.00000 q^{25} +1.00000 q^{26} +(-0.707107 - 0.707107i) q^{27} +1.00000i q^{28} +(-0.707107 - 0.707107i) q^{29} -1.00000 q^{30} +(-0.707107 - 0.707107i) q^{32} +(0.707107 - 0.707107i) q^{34} +1.00000 q^{35} +1.00000i q^{38} +1.00000i q^{39} +(-0.707107 + 0.707107i) q^{40} -1.00000 q^{42} +(-1.00000 + 1.00000i) q^{43} +(-0.707107 - 0.707107i) q^{46} -2.00000i q^{47} +(0.707107 - 0.707107i) q^{48} +(0.707107 + 0.707107i) q^{50} +(0.707107 + 0.707107i) q^{51} +(0.707107 + 0.707107i) q^{52} +(-0.707107 - 0.707107i) q^{53} -1.00000i q^{54} +(-0.707107 + 0.707107i) q^{56} -1.00000 q^{57} -1.00000i q^{58} +(-0.707107 + 0.707107i) q^{59} +(-0.707107 - 0.707107i) q^{60} +(-1.00000 + 1.00000i) q^{61} -1.00000i q^{64} +(0.707107 - 0.707107i) q^{65} +(-0.707107 + 0.707107i) q^{67} +1.00000 q^{68} +(0.707107 - 0.707107i) q^{69} +(0.707107 + 0.707107i) q^{70} -1.00000 q^{73} +(-0.707107 + 0.707107i) q^{75} +(-0.707107 + 0.707107i) q^{76} +(-0.707107 + 0.707107i) q^{78} -1.41421i q^{79} -1.00000 q^{80} +1.00000 q^{81} +(-0.707107 - 0.707107i) q^{84} -1.00000i q^{85} -1.41421 q^{86} +1.00000 q^{87} +1.41421 q^{89} +(0.707107 - 0.707107i) q^{91} -1.00000i q^{92} +(1.41421 - 1.41421i) q^{94} +(0.707107 + 0.707107i) q^{95} +1.00000 q^{96} +1.41421 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{6} + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 - 4 * q^6 + 4 * q^7	$ 4 q + 4 q^{5} - 4 q^{6} + 4 q^{7} - 4 q^{16} - 4 q^{23} + 4 q^{25} + 4 q^{26} - 4 q^{30} + 4 q^{35} - 4 q^{42} - 4 q^{43} - 4 q^{57} - 4 q^{61} + 4 q^{68} - 4 q^{73} - 4 q^{80} + 4 q^{81} + 4 q^{87} + 4 q^{96}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^6 + 4 * q^7 - 4 * q^16 - 4 * q^23 + 4 * q^25 + 4 * q^26 - 4 * q^30 + 4 * q^35 - 4 * q^42 - 4 * q^43 - 4 * q^57 - 4 * q^61 + 4 * q^68 - 4 * q^73 - 4 * q^80 + 4 * q^81 + 4 * q^87 + 4 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{4}\right)$	$-1$

Coefficient data

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

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

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

$2$	0.707107	+	0.707107i	0.707107	+	0.707107i
$2$	0.707107	+	0.707107i	0.707107	+	0.707107i
$3$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$3$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$4$			1.00000i			1.00000i
$5$	1.00000			1.00000
$5$	1.00000			1.00000
$6$	−1.00000			−1.00000
$7$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$7$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$8$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$9$		0			0
$10$	0.707107	+	0.707107i	0.707107	+	0.707107i
$11$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$11$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$12$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$13$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$13$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$14$	0.707107	+	0.707107i	0.707107	+	0.707107i
$15$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$16$	−1.00000			−1.00000
$17$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$17$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$18$		0			0
$19$	0.707107	+	0.707107i	0.707107	+	0.707107i
$19$	0.707107	+	0.707107i	0.707107	+	0.707107i
$20$			1.00000i			1.00000i
$21$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$22$		0			0
$23$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$23$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$24$		−	1.00000i		−	1.00000i
$25$	1.00000			1.00000
$26$	1.00000			1.00000
$27$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$28$			1.00000i			1.00000i
$29$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$29$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$30$	−1.00000			−1.00000
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$33$		0			0
$34$	0.707107	−	0.707107i	0.707107	−	0.707107i
$35$	1.00000			1.00000
$36$		0			0
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$			1.00000i			1.00000i
$39$			1.00000i			1.00000i
$40$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$	−1.00000			−1.00000
$43$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$43$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$47$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$47$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$48$	0.707107	−	0.707107i	0.707107	−	0.707107i
$49$		0			0
$50$	0.707107	+	0.707107i	0.707107	+	0.707107i
$51$	0.707107	+	0.707107i	0.707107	+	0.707107i
$52$	0.707107	+	0.707107i	0.707107	+	0.707107i
$53$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$53$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$54$		−	1.00000i		−	1.00000i
$55$		0			0
$56$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$57$	−1.00000			−1.00000
$58$		−	1.00000i		−	1.00000i
$59$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$59$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$60$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$61$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$61$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$		−	1.00000i		−	1.00000i
$65$	0.707107	−	0.707107i	0.707107	−	0.707107i
$66$		0			0
$67$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$67$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$68$	1.00000			1.00000
$69$	0.707107	−	0.707107i	0.707107	−	0.707107i
$70$	0.707107	+	0.707107i	0.707107	+	0.707107i
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$73$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$74$		0			0
$75$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$76$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$77$		0			0
$78$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$79$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$79$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$80$	−1.00000			−1.00000
$81$	1.00000			1.00000
$82$		0			0
$83$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$83$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$84$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$85$		−	1.00000i		−	1.00000i
$86$	−1.41421			−1.41421
$87$	1.00000			1.00000
$88$		0			0
$89$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$89$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$90$		0			0
$91$	0.707107	−	0.707107i	0.707107	−	0.707107i
$92$		−	1.00000i		−	1.00000i
$93$		0			0
$94$	1.41421	−	1.41421i	1.41421	−	1.41421i
$95$	0.707107	+	0.707107i	0.707107	+	0.707107i
$96$	1.00000			1.00000
$97$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$97$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$98$		0			0
$99$		0			0
$100$			1.00000i			1.00000i
$101$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$101$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$102$			1.00000i			1.00000i
$103$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$103$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$104$			1.00000i			1.00000i
$105$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$106$		−	1.00000i		−	1.00000i
$107$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$107$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$108$	0.707107	−	0.707107i	0.707107	−	0.707107i
$109$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$109$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$110$		0			0
$111$		0			0
$112$	−1.00000			−1.00000
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$115$	−1.00000			−1.00000
$116$	0.707107	−	0.707107i	0.707107	−	0.707107i
$117$		0			0
$118$	−1.00000			−1.00000
$119$		−	1.00000i		−	1.00000i
$120$		−	1.00000i		−	1.00000i
$121$		−	1.00000i		−	1.00000i
$122$	−1.41421			−1.41421
$123$		0			0
$124$		0			0
$125$	1.00000			1.00000
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$	0.707107	−	0.707107i	0.707107	−	0.707107i
$129$		−	1.41421i		−	1.41421i
$130$	1.00000			1.00000
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$	0.707107	+	0.707107i	0.707107	+	0.707107i
$134$	−1.00000			−1.00000
$135$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$136$	0.707107	+	0.707107i	0.707107	+	0.707107i
$137$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$137$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$138$	1.00000			1.00000
$139$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$139$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$140$			1.00000i			1.00000i
$141$	1.41421	+	1.41421i	1.41421	+	1.41421i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$146$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$147$		0			0
$148$		0			0
$149$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$149$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$150$	−1.00000			−1.00000
$151$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$151$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$152$	−1.00000			−1.00000
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.00000			−1.00000
$157$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$157$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$158$	1.00000	−	1.00000i	1.00000	−	1.00000i
$159$	1.00000			1.00000
$160$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$161$	−1.00000			−1.00000
$162$	0.707107	+	0.707107i	0.707107	+	0.707107i
$163$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$163$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		−	1.00000i		−	1.00000i
$169$		0			0
$170$	0.707107	−	0.707107i	0.707107	−	0.707107i
$171$		0			0
$172$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$	0.707107	+	0.707107i	0.707107	+	0.707107i
$175$	1.00000			1.00000
$176$		0			0
$177$		−	1.00000i		−	1.00000i
$178$	1.00000	+	1.00000i	1.00000	+	1.00000i
$179$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$179$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$180$		0			0
$181$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$181$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$182$	1.00000			1.00000
$183$		−	1.41421i		−	1.41421i
$184$	0.707107	−	0.707107i	0.707107	−	0.707107i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	2.00000			2.00000
$189$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$190$			1.00000i			1.00000i
$191$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$191$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$192$	0.707107	+	0.707107i	0.707107	+	0.707107i
$193$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$193$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$194$	1.00000	+	1.00000i	1.00000	+	1.00000i
$195$			1.00000i			1.00000i
$196$		0			0
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$199$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$200$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$201$		−	1.00000i		−	1.00000i
$202$		0			0
$203$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$204$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$205$		0			0
$206$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$207$		0			0
$208$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$209$		0			0
$210$	−1.00000			−1.00000
$211$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$211$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$212$	0.707107	−	0.707107i	0.707107	−	0.707107i
$213$		0			0
$214$			1.00000i			1.00000i
$215$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$216$	1.00000			1.00000
$217$		0			0
$218$		−	1.00000i		−	1.00000i
$219$	0.707107	−	0.707107i	0.707107	−	0.707107i
$220$		0			0
$221$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$225$		0			0
$226$		0			0
$227$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$227$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$228$		−	1.00000i		−	1.00000i
$229$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$229$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$230$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$231$		0			0
$232$	1.00000			1.00000
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		−	2.00000i		−	2.00000i
$236$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$237$	1.00000	+	1.00000i	1.00000	+	1.00000i
$238$	0.707107	−	0.707107i	0.707107	−	0.707107i
$239$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$239$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$240$	0.707107	−	0.707107i	0.707107	−	0.707107i
$241$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$241$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$242$	0.707107	−	0.707107i	0.707107	−	0.707107i
$243$		0			0
$244$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$245$		0			0
$246$		0			0
$247$	1.00000			1.00000
$248$		0			0
$249$		0			0
$250$	0.707107	+	0.707107i	0.707107	+	0.707107i
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	0.707107	+	0.707107i	0.707107	+	0.707107i
$256$	1.00000			1.00000
$257$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$257$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$258$	1.00000	−	1.00000i	1.00000	−	1.00000i
$259$		0			0
$260$	0.707107	+	0.707107i	0.707107	+	0.707107i
$261$		0			0
$262$		0			0
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$266$			1.00000i			1.00000i
$267$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$268$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$269$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$269$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$270$		−	1.00000i		−	1.00000i
$271$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$271$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$272$			1.00000i			1.00000i
$273$			1.00000i			1.00000i
$274$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$275$		0			0
$276$	0.707107	+	0.707107i	0.707107	+	0.707107i
$277$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$277$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$278$		−	1.41421i		−	1.41421i
$279$		0			0
$280$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$281$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$281$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$282$			2.00000i			2.00000i
$283$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$283$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$284$		0			0
$285$	−1.00000			−1.00000
$286$		0			0
$287$		0			0
$288$		0			0
$289$		0			0
$290$		−	1.00000i		−	1.00000i
$291$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$292$		−	1.00000i		−	1.00000i
$293$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$293$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$294$		0			0
$295$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$296$		0			0
$297$		0			0
$298$			1.41421i			1.41421i
$299$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$300$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$301$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$302$	1.00000	+	1.00000i	1.00000	+	1.00000i
$303$		0			0
$304$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$305$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$306$		0			0
$307$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$307$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$310$		0			0
$311$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$311$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$312$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$313$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$313$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$314$			1.41421i			1.41421i
$315$		0			0
$316$	1.41421			1.41421
$317$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$317$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$318$	0.707107	+	0.707107i	0.707107	+	0.707107i
$319$		0			0
$320$		−	1.00000i		−	1.00000i
$321$	−1.00000			−1.00000
$322$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$323$	0.707107	−	0.707107i	0.707107	−	0.707107i
$324$			1.00000i			1.00000i
$325$	0.707107	−	0.707107i	0.707107	−	0.707107i
$326$		0			0
$327$	1.00000			1.00000
$328$		0			0
$329$		−	2.00000i		−	2.00000i
$330$		0			0
$331$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$331$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$336$	0.707107	−	0.707107i	0.707107	−	0.707107i
$337$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$337$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$338$		0			0
$339$		0			0
$340$	1.00000			1.00000
$341$		0			0
$342$		0			0
$343$	−1.00000			−1.00000
$344$		−	1.41421i		−	1.41421i
$345$	0.707107	−	0.707107i	0.707107	−	0.707107i
$346$		0			0
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$			1.00000i			1.00000i
$349$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$349$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$350$	0.707107	+	0.707107i	0.707107	+	0.707107i
$351$	−1.00000			−1.00000
$352$		0			0
$353$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$353$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$354$	0.707107	−	0.707107i	0.707107	−	0.707107i
$355$		0			0
$356$			1.41421i			1.41421i
$357$	0.707107	+	0.707107i	0.707107	+	0.707107i
$358$		0			0
$359$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$359$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$360$		0			0
$361$			1.00000i			1.00000i
$362$		0			0
$363$	0.707107	+	0.707107i	0.707107	+	0.707107i
$364$	0.707107	+	0.707107i	0.707107	+	0.707107i
$365$	−1.00000			−1.00000
$366$	1.00000	−	1.00000i	1.00000	−	1.00000i
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$	1.00000			1.00000
$369$		0			0
$370$		0			0
$371$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$372$		0			0
$373$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$373$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$374$		0			0
$375$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$376$	1.41421	+	1.41421i	1.41421	+	1.41421i
$377$	−1.00000			−1.00000
$378$		−	1.00000i		−	1.00000i
$379$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$379$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$380$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$381$		0			0
$382$	0.707107	+	0.707107i	0.707107	+	0.707107i
$383$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$383$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$384$			1.00000i			1.00000i
$385$		0			0
$386$	1.00000	+	1.00000i	1.00000	+	1.00000i
$387$		0			0
$388$			1.41421i			1.41421i
$389$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$389$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$390$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$391$			1.00000i			1.00000i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	1.41421i		−	1.41421i
$396$		0			0
$397$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$397$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$398$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$399$	−1.00000			−1.00000
$400$	−1.00000			−1.00000
$401$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$401$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$402$	0.707107	−	0.707107i	0.707107	−	0.707107i
$403$		0			0
$404$		0			0
$405$	1.00000			1.00000
$406$		−	1.00000i		−	1.00000i
$407$		0			0
$408$	−1.00000			−1.00000
$409$		0			0			−	1.00000i	$-0.5\pi$
$409$		0			0				1.00000i	$0.5\pi$
$410$		0			0
$411$	0.707107	−	0.707107i	0.707107	−	0.707107i
$412$	−1.41421			−1.41421
$413$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$414$		0			0
$415$		0			0
$416$	−1.00000			−1.00000
$417$	1.41421			1.41421
$418$		0			0
$419$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$419$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$420$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$421$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$421$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$422$		−	1.00000i		−	1.00000i
$423$		0			0
$424$	1.00000			1.00000
$425$		−	1.00000i		−	1.00000i
$426$		0			0
$427$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$428$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$429$		0			0
$430$	−1.41421			−1.41421
$431$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$431$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$432$	0.707107	+	0.707107i	0.707107	+	0.707107i
$433$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$433$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$	1.00000			1.00000
$436$	0.707107	−	0.707107i	0.707107	−	0.707107i
$437$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$438$	1.00000			1.00000
$439$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$439$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$440$		0			0
$441$		0			0
$442$		−	1.00000i		−	1.00000i
$443$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$443$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$444$		0			0
$445$	1.41421			1.41421
$446$		0			0
$447$	−1.41421			−1.41421
$448$		−	1.00000i		−	1.00000i
$449$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$449$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$454$	1.00000			1.00000
$455$	0.707107	−	0.707107i	0.707107	−	0.707107i
$456$	0.707107	−	0.707107i	0.707107	−	0.707107i
$457$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$457$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$458$		0			0
$459$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$460$		−	1.00000i		−	1.00000i
$461$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$461$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	0.707107	+	0.707107i	0.707107	+	0.707107i
$465$		0			0
$466$		0			0
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$470$	1.41421	−	1.41421i	1.41421	−	1.41421i
$471$	−1.41421			−1.41421
$472$		−	1.00000i		−	1.00000i
$473$		0			0
$474$			1.41421i			1.41421i
$475$	0.707107	+	0.707107i	0.707107	+	0.707107i
$476$	1.00000			1.00000
$477$		0			0
$478$	0.707107	+	0.707107i	0.707107	+	0.707107i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$	1.00000			1.00000
$481$		0			0
$482$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$483$	0.707107	−	0.707107i	0.707107	−	0.707107i
$484$	1.00000			1.00000
$485$	1.41421			1.41421
$486$		0			0
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		−	1.41421i		−	1.41421i
$489$		0			0
$490$		0			0
$491$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$491$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$492$		0			0
$493$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$494$	0.707107	+	0.707107i	0.707107	+	0.707107i
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$499$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$500$			1.00000i			1.00000i
$501$		0			0
$502$		0			0
$503$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$503$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$509$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$510$			1.00000i			1.00000i
$511$	−1.00000			−1.00000
$512$	0.707107	+	0.707107i	0.707107	+	0.707107i
$513$		−	1.00000i		−	1.00000i
$514$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$515$			1.41421i			1.41421i
$516$	1.41421			1.41421
$517$		0			0
$518$		0			0
$519$		0			0
$520$			1.00000i			1.00000i
$521$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$521$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$522$		0			0
$523$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$523$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$524$		0			0
$525$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$526$		0			0
$527$		0			0
$528$		0			0
$529$		0			0
$530$		−	1.00000i		−	1.00000i
$531$		0			0
$532$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$533$		0			0
$534$	−1.41421			−1.41421
$535$	0.707107	+	0.707107i	0.707107	+	0.707107i
$536$		−	1.00000i		−	1.00000i
$537$		0			0
$538$		0			0
$539$		0			0
$540$	0.707107	−	0.707107i	0.707107	−	0.707107i
$541$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$541$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$542$	0.707107	+	0.707107i	0.707107	+	0.707107i
$543$		0			0
$544$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$545$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$546$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		−	1.00000i		−	1.00000i
$549$		0			0
$550$		0			0
$551$		−	1.00000i		−	1.00000i
$552$			1.00000i			1.00000i
$553$		−	1.41421i		−	1.41421i
$554$	−1.41421			−1.41421
$555$		0			0
$556$	1.00000	−	1.00000i	1.00000	−	1.00000i
$557$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$557$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$558$		0			0
$559$			1.41421i			1.41421i
$560$	−1.00000			−1.00000
$561$		0			0
$562$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$563$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$563$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$564$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$565$		0			0
$566$		0			0
$567$	1.00000			1.00000
$568$		0			0
$569$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$569$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$570$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$571$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$571$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$572$		0			0
$573$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$574$		0			0
$575$	−1.00000			−1.00000
$576$		0			0
$577$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$577$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$578$		0			0
$579$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$580$	0.707107	−	0.707107i	0.707107	−	0.707107i
$581$		0			0
$582$	−1.41421			−1.41421
$583$		0			0
$584$	0.707107	−	0.707107i	0.707107	−	0.707107i
$585$		0			0
$586$			1.00000i			1.00000i
$587$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$587$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$588$		0			0
$589$		0			0
$590$	−1.00000			−1.00000
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		−	1.00000i		−	1.00000i
$596$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$597$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$598$	−1.00000			−1.00000
$599$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$599$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$600$		−	1.00000i		−	1.00000i
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$	−1.41421			−1.41421
$603$		0			0
$604$			1.41421i			1.41421i
$605$		−	1.00000i		−	1.00000i
$606$		0			0
$607$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$607$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$608$		−	1.00000i		−	1.00000i
$609$	1.00000			1.00000
$610$	−1.41421			−1.41421
$611$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$612$		0			0
$613$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$613$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		−	1.41421i		−	1.41421i
$619$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$619$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$620$		0			0
$621$	0.707107	+	0.707107i	0.707107	+	0.707107i
$622$	0.707107	−	0.707107i	0.707107	−	0.707107i
$623$	1.41421			1.41421
$624$		−	1.00000i		−	1.00000i
$625$	1.00000			1.00000
$626$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$627$		0			0
$628$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	1.00000	+	1.00000i	1.00000	+	1.00000i
$633$	1.00000			1.00000
$634$	1.00000			1.00000
$635$		0			0
$636$			1.00000i			1.00000i
$637$		0			0
$638$		0			0
$639$		0			0
$640$	0.707107	−	0.707107i	0.707107	−	0.707107i
$641$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$641$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$642$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$643$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$643$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$644$		−	1.00000i		−	1.00000i
$645$		−	1.41421i		−	1.41421i
$646$	1.00000			1.00000
$647$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$647$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$648$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$649$		0			0
$650$	1.00000			1.00000
$651$		0			0
$652$		0			0
$653$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$653$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$654$	0.707107	+	0.707107i	0.707107	+	0.707107i
$655$		0			0
$656$		0			0
$657$		0			0
$658$	1.41421	−	1.41421i	1.41421	−	1.41421i
$659$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$659$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$660$		0			0
$661$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$661$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$662$	1.00000			1.00000
$663$	1.00000			1.00000
$664$		0			0
$665$	0.707107	+	0.707107i	0.707107	+	0.707107i
$666$		0			0
$667$	0.707107	+	0.707107i	0.707107	+	0.707107i
$668$		0			0
$669$		0			0
$670$	−1.00000			−1.00000
$671$		0			0
$672$	1.00000			1.00000
$673$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$673$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$674$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$675$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$676$		0			0
$677$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$677$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$678$		0			0
$679$	1.41421			1.41421
$680$	0.707107	+	0.707107i	0.707107	+	0.707107i
$681$			1.00000i			1.00000i
$682$		0			0
$683$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$683$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$684$		0			0
$685$	−1.00000			−1.00000
$686$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$687$		0			0
$688$	1.00000	−	1.00000i	1.00000	−	1.00000i
$689$	−1.00000			−1.00000
$690$	1.00000			1.00000
$691$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$691$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$696$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$697$		0			0
$698$	−1.41421			−1.41421
$699$		0			0
$700$			1.00000i			1.00000i
$701$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$701$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$702$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$703$		0			0
$704$		0			0
$705$	1.41421	+	1.41421i	1.41421	+	1.41421i
$706$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$707$		0			0
$708$	1.00000			1.00000
$709$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$709$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$710$		0			0
$711$		0			0
$712$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$713$		0			0
$714$			1.00000i			1.00000i
$715$		0			0
$716$		0			0
$717$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$718$	0.707107	−	0.707107i	0.707107	−	0.707107i
$719$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$719$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$			1.41421i			1.41421i
$722$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$723$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$724$		0			0
$725$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$726$			1.00000i			1.00000i
$727$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$727$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$728$			1.00000i			1.00000i
$729$			1.00000i			1.00000i
$730$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$731$	1.00000	+	1.00000i	1.00000	+	1.00000i
$732$	1.41421			1.41421
$733$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$733$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$		0			0
$736$	0.707107	+	0.707107i	0.707107	+	0.707107i
$737$		0			0
$738$		0			0
$739$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$739$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$740$		0			0
$741$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$742$		−	1.00000i		−	1.00000i
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	1.00000	+	1.00000i	1.00000	+	1.00000i
$746$		−	1.00000i		−	1.00000i
$747$		0			0
$748$		0			0
$749$	0.707107	+	0.707107i	0.707107	+	0.707107i
$750$	−1.00000			−1.00000
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			2.00000i			2.00000i
$753$		0			0
$754$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$755$	1.41421			1.41421
$756$	0.707107	−	0.707107i	0.707107	−	0.707107i
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$	−1.00000			−1.00000
$759$		0			0
$760$	−1.00000			−1.00000
$761$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$761$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$762$		0			0
$763$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$764$			1.00000i			1.00000i
$765$		0			0
$766$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$767$			1.00000i			1.00000i
$768$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$769$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$769$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$770$		0			0
$771$	1.00000	−	1.00000i	1.00000	−	1.00000i
$772$			1.41421i			1.41421i
$773$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$773$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$774$		0			0
$775$		0			0
$776$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$777$		0			0
$778$		0			0
$779$		0			0
$780$	−1.00000			−1.00000
$781$		0			0
$782$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$783$			1.00000i			1.00000i
$784$		0			0
$785$	1.00000	+	1.00000i	1.00000	+	1.00000i
$786$		0			0
$787$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$787$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$788$		0			0
$789$		0			0
$790$	1.00000	−	1.00000i	1.00000	−	1.00000i
$791$		0			0
$792$		0			0
$793$			1.41421i			1.41421i
$794$			1.41421i			1.41421i
$795$	1.00000			1.00000
$796$	−1.00000			−1.00000
$797$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$797$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$798$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$799$	−2.00000			−2.00000
$800$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$801$		0			0
$802$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$803$		0			0
$804$	1.00000			1.00000
$805$	−1.00000			−1.00000
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$809$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$810$	0.707107	+	0.707107i	0.707107	+	0.707107i
$811$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$811$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$812$	0.707107	−	0.707107i	0.707107	−	0.707107i
$813$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$814$		0			0
$815$		0			0
$816$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$817$	−1.41421			−1.41421
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$821$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$822$	1.00000			1.00000
$823$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$823$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$824$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$825$		0			0
$826$	−1.00000			−1.00000
$827$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$827$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$828$		0			0
$829$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$829$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$830$		0			0
$831$		−	1.41421i		−	1.41421i
$832$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$833$		0			0
$834$	1.00000	+	1.00000i	1.00000	+	1.00000i
$835$		0			0
$836$		0			0
$837$		0			0
$838$	−1.41421			−1.41421
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		−	1.00000i		−	1.00000i
$841$		0			0
$842$	−1.00000			−1.00000
$843$	1.00000	−	1.00000i	1.00000	−	1.00000i
$844$	0.707107	−	0.707107i	0.707107	−	0.707107i
$845$		0			0
$846$		0			0
$847$		−	1.00000i		−	1.00000i
$848$	0.707107	+	0.707107i	0.707107	+	0.707107i
$849$		0			0
$850$	0.707107	−	0.707107i	0.707107	−	0.707107i
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$	−1.41421			−1.41421
$855$		0			0
$856$	−1.00000			−1.00000
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$859$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$860$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$861$		0			0
$862$	1.00000	−	1.00000i	1.00000	−	1.00000i
$863$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$863$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$864$			1.00000i			1.00000i
$865$		0			0
$866$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$867$		0			0
$868$		0			0
$869$		0			0
$870$	0.707107	+	0.707107i	0.707107	+	0.707107i
$871$			1.00000i			1.00000i
$872$	1.00000			1.00000
$873$		0			0
$874$		−	1.00000i		−	1.00000i
$875$	1.00000			1.00000
$876$	0.707107	+	0.707107i	0.707107	+	0.707107i
$877$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$877$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$878$	1.00000	+	1.00000i	1.00000	+	1.00000i
$879$	−1.00000			−1.00000
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$	0.707107	−	0.707107i	0.707107	−	0.707107i
$885$		−	1.00000i		−	1.00000i
$886$		0			0
$887$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$887$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$	1.00000	+	1.00000i	1.00000	+	1.00000i
$891$		0			0
$892$		0			0
$893$	1.41421	−	1.41421i	1.41421	−	1.41421i
$894$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$895$		0			0
$896$	0.707107	−	0.707107i	0.707107	−	0.707107i
$897$		−	1.00000i		−	1.00000i
$898$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$899$		0			0
$900$		0			0
$901$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$902$		0			0
$903$		−	1.41421i		−	1.41421i
$904$		0			0
$905$		0			0
$906$	−1.41421			−1.41421
$907$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$907$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$908$	0.707107	+	0.707107i	0.707107	+	0.707107i
$909$		0			0
$910$	1.00000			1.00000
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	1.00000			1.00000
$913$		0			0
$914$	0.707107	+	0.707107i	0.707107	+	0.707107i
$915$		−	1.41421i		−	1.41421i
$916$		0			0
$917$		0			0
$918$	−1.00000			−1.00000
$919$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$919$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$920$	0.707107	−	0.707107i	0.707107	−	0.707107i
$921$		0			0
$922$	1.41421			1.41421
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$			1.00000i			1.00000i
$929$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$929$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	0.707107	+	0.707107i	0.707107	+	0.707107i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$937$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$938$	−1.00000			−1.00000
$939$	0.707107	−	0.707107i	0.707107	−	0.707107i
$940$	2.00000			2.00000
$941$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$941$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$942$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$943$		0			0
$944$	0.707107	−	0.707107i	0.707107	−	0.707107i
$945$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$949$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$950$			1.00000i			1.00000i
$951$			1.00000i			1.00000i
$952$	0.707107	+	0.707107i	0.707107	+	0.707107i
$953$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$953$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$954$		0			0
$955$	1.00000			1.00000
$956$			1.00000i			1.00000i
$957$		0			0
$958$		0			0
$959$	−1.00000			−1.00000
$960$	0.707107	+	0.707107i	0.707107	+	0.707107i
$961$	1.00000			1.00000
$962$		0			0
$963$		0			0
$964$	−1.41421			−1.41421
$965$	1.41421			1.41421
$966$	1.00000			1.00000
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	0.707107	+	0.707107i	0.707107	+	0.707107i
$969$			1.00000i			1.00000i
$970$	1.00000	+	1.00000i	1.00000	+	1.00000i
$971$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$971$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$972$		0			0
$973$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$974$		0			0
$975$			1.00000i			1.00000i
$976$	1.00000	−	1.00000i	1.00000	−	1.00000i
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		−	1.41421i		−	1.41421i
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$	−1.00000			−1.00000
$987$	1.41421	+	1.41421i	1.41421	+	1.41421i
$988$			1.00000i			1.00000i
$989$	1.00000	−	1.00000i	1.00000	−	1.00000i
$990$		0			0
$991$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$991$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$992$		0			0
$993$			1.00000i			1.00000i
$994$		0			0
$995$			1.00000i			1.00000i
$996$		0			0
$997$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$997$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$998$	1.41421			1.41421
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.1.bh.b.949.2	yes	4
5.4	even	2		1520.1.bh.a.949.1	yes	4
16.13	even	4		1520.1.bh.a.189.2	yes	4
19.18	odd	2	inner	1520.1.bh.b.949.1	yes	4
80.29	even	4	inner	1520.1.bh.b.189.1	yes	4
95.94	odd	2		1520.1.bh.a.949.2	yes	4
304.189	odd	4		1520.1.bh.a.189.1	✓	4
1520.189	odd	4	inner	1520.1.bh.b.189.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1520.1.bh.a.189.1	✓	4	304.189	odd	4
1520.1.bh.a.189.2	yes	4	16.13	even	4
1520.1.bh.a.949.1	yes	4	5.4	even	2
1520.1.bh.a.949.2	yes	4	95.94	odd	2
1520.1.bh.b.189.1	yes	4	80.29	even	4	inner
1520.1.bh.b.189.2	yes	4	1520.189	odd	4	inner
1520.1.bh.b.949.1	yes	4	19.18	odd	2	inner
1520.1.bh.b.949.2	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1520.bh (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{2} +(-0.707107 + 0.707107i) q^{3} +1.00000i q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +(-0.707107 + 0.707107i) q^{8} +O(q^{10})\)	\(q+(0.707107 + 0.707107i) q^{2} +(-0.707107 + 0.707107i) q^{3} +1.00000i q^{4} +1.00000 q^{5} -1.00000 q^{6} +1.00000 q^{7} +(-0.707107 + 0.707107i) q^{8} +(0.707107 + 0.707107i) q^{10} +(-0.707107 - 0.707107i) q^{12} +(0.707107 - 0.707107i) q^{13} +(0.707107 + 0.707107i) q^{14} +(-0.707107 + 0.707107i) q^{15} -1.00000 q^{16} -1.00000i q^{17} +(0.707107 + 0.707107i) q^{19} +1.00000i q^{20} +(-0.707107 + 0.707107i) q^{21} -1.00000 q^{23} -1.00000i q^{24} +1.00000 q^{25} +1.00000 q^{26} +(-0.707107 - 0.707107i) q^{27} +1.00000i q^{28} +(-0.707107 - 0.707107i) q^{29} -1.00000 q^{30} +(-0.707107 - 0.707107i) q^{32} +(0.707107 - 0.707107i) q^{34} +1.00000 q^{35} +1.00000i q^{38} +1.00000i q^{39} +(-0.707107 + 0.707107i) q^{40} -1.00000 q^{42} +(-1.00000 + 1.00000i) q^{43} +(-0.707107 - 0.707107i) q^{46} -2.00000i q^{47} +(0.707107 - 0.707107i) q^{48} +(0.707107 + 0.707107i) q^{50} +(0.707107 + 0.707107i) q^{51} +(0.707107 + 0.707107i) q^{52} +(-0.707107 - 0.707107i) q^{53} -1.00000i q^{54} +(-0.707107 + 0.707107i) q^{56} -1.00000 q^{57} -1.00000i q^{58} +(-0.707107 + 0.707107i) q^{59} +(-0.707107 - 0.707107i) q^{60} +(-1.00000 + 1.00000i) q^{61} -1.00000i q^{64} +(0.707107 - 0.707107i) q^{65} +(-0.707107 + 0.707107i) q^{67} +1.00000 q^{68} +(0.707107 - 0.707107i) q^{69} +(0.707107 + 0.707107i) q^{70} -1.00000 q^{73} +(-0.707107 + 0.707107i) q^{75} +(-0.707107 + 0.707107i) q^{76} +(-0.707107 + 0.707107i) q^{78} -1.41421i q^{79} -1.00000 q^{80} +1.00000 q^{81} +(-0.707107 - 0.707107i) q^{84} -1.00000i q^{85} -1.41421 q^{86} +1.00000 q^{87} +1.41421 q^{89} +(0.707107 - 0.707107i) q^{91} -1.00000i q^{92} +(1.41421 - 1.41421i) q^{94} +(0.707107 + 0.707107i) q^{95} +1.00000 q^{96} +1.41421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{6} + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 - 4 * q^6 + 4 * q^7	\( 4 q + 4 q^{5} - 4 q^{6} + 4 q^{7} - 4 q^{16} - 4 q^{23} + 4 q^{25} + 4 q^{26} - 4 q^{30} + 4 q^{35} - 4 q^{42} - 4 q^{43} - 4 q^{57} - 4 q^{61} + 4 q^{68} - 4 q^{73} - 4 q^{80} + 4 q^{81} + 4 q^{87} + 4 q^{96}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^6 + 4 * q^7 - 4 * q^16 - 4 * q^23 + 4 * q^25 + 4 * q^26 - 4 * q^30 + 4 * q^35 - 4 * q^42 - 4 * q^43 - 4 * q^57 - 4 * q^61 + 4 * q^68 - 4 * q^73 - 4 * q^80 + 4 * q^81 + 4 * q^87 + 4 * q^96

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