LMFDB - Embedded newform 2040.1.cj.a.1109.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2040,1,Mod(149,2040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2040, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 2, 2, 1]))

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

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

chi := DirichletCharacter("2040.149");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2040.cj (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:	$1.01809262577$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.2.4716480.5

Embedding invariants

Embedding label			1109.2
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2040.1109
Dual form			2040.1.cj.a.149.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} +(0.707107 - 0.707107i) q^{3} -1.00000 q^{4} +(0.707107 - 0.707107i) q^{5} +(-0.707107 - 0.707107i) q^{6} +1.00000i q^{8} -1.00000i q^{9} +O(q^{10})$	\(q-1.00000i q^{2} +(0.707107 - 0.707107i) q^{3} -1.00000 q^{4} +(0.707107 - 0.707107i) q^{5} +(-0.707107 - 0.707107i) q^{6} +1.00000i q^{8} -1.00000i q^{9} +(-0.707107 - 0.707107i) q^{10} +(-0.707107 + 0.707107i) q^{12} +1.41421 q^{13} -1.00000i q^{15} +1.00000 q^{16} +1.00000i q^{17} -1.00000 q^{18} +(-0.707107 + 0.707107i) q^{20} +(-1.00000 - 1.00000i) q^{23} +(0.707107 + 0.707107i) q^{24} -1.00000i q^{25} -1.41421i q^{26} +(-0.707107 - 0.707107i) q^{27} -1.00000 q^{30} +(1.00000 - 1.00000i) q^{31} -1.00000i q^{32} +1.00000 q^{34} +1.00000i q^{36} +(-1.41421 + 1.41421i) q^{37} +(1.00000 - 1.00000i) q^{39} +(0.707107 + 0.707107i) q^{40} +1.41421i q^{43} +(-0.707107 - 0.707107i) q^{45} +(-1.00000 + 1.00000i) q^{46} -2.00000 q^{47} +(0.707107 - 0.707107i) q^{48} +1.00000i q^{49} -1.00000 q^{50} +(0.707107 + 0.707107i) q^{51} -1.41421 q^{52} +(-0.707107 + 0.707107i) q^{54} +1.41421i q^{59} +1.00000i q^{60} +(-1.00000 - 1.00000i) q^{62} -1.00000 q^{64} +(1.00000 - 1.00000i) q^{65} +1.41421 q^{67} -1.00000i q^{68} -1.41421 q^{69} +1.00000 q^{72} +(1.41421 + 1.41421i) q^{74} +(-0.707107 - 0.707107i) q^{75} +(-1.00000 - 1.00000i) q^{78} +(-1.00000 - 1.00000i) q^{79} +(0.707107 - 0.707107i) q^{80} -1.00000 q^{81} +(0.707107 + 0.707107i) q^{85} +1.41421 q^{86} +(-0.707107 + 0.707107i) q^{90} +(1.00000 + 1.00000i) q^{92} -1.41421i q^{93} +2.00000i q^{94} +(-0.707107 - 0.707107i) q^{96} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4}+O(q^{10}) $ 4 * q - 4 * q^4	$ 4 q - 4 q^{4} + 4 q^{16} - 4 q^{18} - 4 q^{23} - 4 q^{30} + 4 q^{31} + 4 q^{34} + 4 q^{39} - 4 q^{46} - 8 q^{47} - 4 q^{50} - 4 q^{62} - 4 q^{64} + 4 q^{65} + 4 q^{72} - 4 q^{78} - 4 q^{79} - 4 q^{81} + 4 q^{92} + 4 q^{98}+O(q^{100}) $ 4 * q - 4 * q^4 + 4 * q^16 - 4 * q^18 - 4 * q^23 - 4 * q^30 + 4 * q^31 + 4 * q^34 + 4 * q^39 - 4 * q^46 - 8 * q^47 - 4 * q^50 - 4 * q^62 - 4 * q^64 + 4 * q^65 + 4 * q^72 - 4 * q^78 - 4 * q^79 - 4 * q^81 + 4 * q^92 + 4 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2040\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1021$	$1361$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$	$-1$	$-1$	$-1$

Coefficient data

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

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

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

$2$		−	1.00000i		−	1.00000i
$2$		−	1.00000i		−	1.00000i
$3$	0.707107	−	0.707107i	0.707107	−	0.707107i
$3$	0.707107	−	0.707107i	0.707107	−	0.707107i
$4$	−1.00000			−1.00000
$5$	0.707107	−	0.707107i	0.707107	−	0.707107i
$5$	0.707107	−	0.707107i	0.707107	−	0.707107i
$6$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$			1.00000i			1.00000i
$9$		−	1.00000i		−	1.00000i
$10$	−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
$13$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$13$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$		−	1.00000i		−	1.00000i
$16$	1.00000			1.00000
$17$			1.00000i			1.00000i
$17$			1.00000i			1.00000i
$18$	−1.00000			−1.00000
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$21$		0			0
$22$		0			0
$23$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$23$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$24$	0.707107	+	0.707107i	0.707107	+	0.707107i
$25$		−	1.00000i		−	1.00000i
$26$		−	1.41421i		−	1.41421i
$27$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$28$		0			0
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$	−1.00000			−1.00000
$31$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$31$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$32$		−	1.00000i		−	1.00000i
$33$		0			0
$34$	1.00000			1.00000
$35$		0			0
$36$			1.00000i			1.00000i
$37$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$37$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$	1.00000	−	1.00000i	1.00000	−	1.00000i
$40$	0.707107	+	0.707107i	0.707107	+	0.707107i
$41$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$41$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$42$		0			0
$43$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$43$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$46$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$47$	−2.00000			−2.00000			−1.00000			$\pi$
$47$	−2.00000			−2.00000			−1.00000			$\pi$
$48$	0.707107	−	0.707107i	0.707107	−	0.707107i
$49$			1.00000i			1.00000i
$50$	−1.00000			−1.00000
$51$	0.707107	+	0.707107i	0.707107	+	0.707107i
$52$	−1.41421			−1.41421
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$59$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$60$			1.00000i			1.00000i
$61$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$61$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$62$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$63$		0			0
$64$	−1.00000			−1.00000
$65$	1.00000	−	1.00000i	1.00000	−	1.00000i
$66$		0			0
$67$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$67$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$68$		−	1.00000i		−	1.00000i
$69$	−1.41421			−1.41421
$70$		0			0
$71$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$71$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$72$	1.00000			1.00000
$73$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$73$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$74$	1.41421	+	1.41421i	1.41421	+	1.41421i
$75$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$76$		0			0
$77$		0			0
$78$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$79$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$79$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$80$	0.707107	−	0.707107i	0.707107	−	0.707107i
$81$	−1.00000			−1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	0.707107	+	0.707107i	0.707107	+	0.707107i
$86$	1.41421			1.41421
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$91$		0			0
$92$	1.00000	+	1.00000i	1.00000	+	1.00000i
$93$		−	1.41421i		−	1.41421i
$94$			2.00000i			2.00000i
$95$		0			0
$96$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$97$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$97$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$98$	1.00000			1.00000
$99$		0			0
$100$			1.00000i			1.00000i
$101$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$101$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$102$	0.707107	−	0.707107i	0.707107	−	0.707107i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$			1.41421i			1.41421i
$105$		0			0
$106$		0			0
$107$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$107$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$108$	0.707107	+	0.707107i	0.707107	+	0.707107i
$109$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$109$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$110$		0			0
$111$			2.00000i			2.00000i
$112$		0			0
$113$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$113$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$114$		0			0
$115$	−1.41421			−1.41421
$116$		0			0
$117$		−	1.41421i		−	1.41421i
$118$	1.41421			1.41421
$119$		0			0
$120$	1.00000			1.00000
$121$		−	1.00000i		−	1.00000i
$122$		0			0
$123$		0			0
$124$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$125$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$			1.00000i			1.00000i
$129$	1.00000	+	1.00000i	1.00000	+	1.00000i
$130$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$131$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$131$	−1.41421	+	1.41421i	−1.41421	+	1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$132$		0			0
$133$		0			0
$134$		−	1.41421i		−	1.41421i
$135$	−1.00000			−1.00000
$136$	−1.00000			−1.00000
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$			1.41421i			1.41421i
$139$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$139$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$140$		0			0
$141$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$142$		0			0
$143$		0			0
$144$		−	1.00000i		−	1.00000i
$145$		0			0
$146$		0			0
$147$	0.707107	+	0.707107i	0.707107	+	0.707107i
$148$	1.41421	−	1.41421i	1.41421	−	1.41421i
$149$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$149$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$150$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	1.00000			1.00000
$154$		0			0
$155$		−	1.41421i		−	1.41421i
$156$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$157$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$157$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$158$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$159$		0			0
$160$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$161$		0			0
$162$			1.00000i			1.00000i
$163$		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
$167$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$167$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$168$		0			0
$169$	1.00000			1.00000
$170$	0.707107	−	0.707107i	0.707107	−	0.707107i
$171$		0			0
$172$		−	1.41421i		−	1.41421i
$173$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$173$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	1.00000	+	1.00000i	1.00000	+	1.00000i
$178$		0			0
$179$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$179$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$180$	0.707107	+	0.707107i	0.707107	+	0.707107i
$181$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$181$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$182$		0			0
$183$		0			0
$184$	1.00000	−	1.00000i	1.00000	−	1.00000i
$185$			2.00000i			2.00000i
$186$	−1.41421			−1.41421
$187$		0			0
$188$	2.00000			2.00000
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$193$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$193$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$194$		0			0
$195$		−	1.41421i		−	1.41421i
$196$		−	1.00000i		−	1.00000i
$197$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$197$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$198$		0			0
$199$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$199$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$200$	1.00000			1.00000
$201$	1.00000	−	1.00000i	1.00000	−	1.00000i
$202$		−	1.41421i		−	1.41421i
$203$		0			0
$204$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$205$		0			0
$206$		0			0
$207$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$208$	1.41421			1.41421
$209$		0			0
$210$		0			0
$211$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$211$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.00000	+	1.00000i	1.00000	+	1.00000i
$216$	0.707107	−	0.707107i	0.707107	−	0.707107i
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$			1.41421i			1.41421i
$222$	2.00000			2.00000
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$	−1.00000			−1.00000
$226$	1.00000	−	1.00000i	1.00000	−	1.00000i
$227$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$227$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$			1.41421i			1.41421i
$231$		0			0
$232$		0			0
$233$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$233$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$234$	−1.41421			−1.41421
$235$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$236$		−	1.41421i		−	1.41421i
$237$	−1.41421			−1.41421
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		−	1.00000i		−	1.00000i
$241$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$241$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$242$	−1.00000			−1.00000
$243$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$244$		0			0
$245$	0.707107	+	0.707107i	0.707107	+	0.707107i
$246$		0			0
$247$		0			0
$248$	1.00000	+	1.00000i	1.00000	+	1.00000i
$249$		0			0
$250$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$251$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$251$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	1.00000			1.00000
$256$	1.00000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$	1.00000	−	1.00000i	1.00000	−	1.00000i
$259$		0			0
$260$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$261$		0			0
$262$	1.41421	+	1.41421i	1.41421	+	1.41421i
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.41421			−1.41421
$269$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$269$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$270$			1.00000i			1.00000i
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$			1.00000i			1.00000i
$273$		0			0
$274$		0			0
$275$		0			0
$276$	1.41421			1.41421
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		0			0
$279$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	1.41421	+	1.41421i	1.41421	+	1.41421i
$283$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$283$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.00000			−1.00000
$289$	−1.00000			−1.00000
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$	0.707107	−	0.707107i	0.707107	−	0.707107i
$295$	1.00000	+	1.00000i	1.00000	+	1.00000i
$296$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$297$		0			0
$298$		−	1.41421i		−	1.41421i
$299$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$300$	0.707107	+	0.707107i	0.707107	+	0.707107i
$301$		0			0
$302$		0			0
$303$	1.00000	−	1.00000i	1.00000	−	1.00000i
$304$		0			0
$305$		0			0
$306$		−	1.00000i		−	1.00000i
$307$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$307$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$308$		0			0
$309$		0			0
$310$	−1.41421			−1.41421
$311$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$311$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$312$	1.00000	+	1.00000i	1.00000	+	1.00000i
$313$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$313$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$314$			1.41421i			1.41421i
$315$		0			0
$316$	1.00000	+	1.00000i	1.00000	+	1.00000i
$317$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$317$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$318$		0			0
$319$		0			0
$320$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	1.00000			1.00000
$325$		−	1.41421i		−	1.41421i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0			−	1.00000i	$-0.5\pi$
$331$		0			0				1.00000i	$0.5\pi$
$332$		0			0
$333$	1.41421	+	1.41421i	1.41421	+	1.41421i
$334$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$335$	1.00000	−	1.00000i	1.00000	−	1.00000i
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$		−	1.00000i		−	1.00000i
$339$	1.41421			1.41421
$340$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−1.41421			−1.41421
$345$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$346$		0			0
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$352$		0			0
$353$	−2.00000			−2.00000			−1.00000			$\pi$
$353$	−2.00000			−2.00000			−1.00000			$\pi$
$354$	1.00000	−	1.00000i	1.00000	−	1.00000i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.41421			−1.41421
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$	0.707107	−	0.707107i	0.707107	−	0.707107i
$361$	−1.00000			−1.00000
$362$		0			0
$363$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$369$		0			0
$370$	2.00000			2.00000
$371$		0			0
$372$			1.41421i			1.41421i
$373$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$373$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$374$		0			0
$375$	−1.00000			−1.00000
$376$		−	2.00000i		−	2.00000i
$377$		0			0
$378$		0			0
$379$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$379$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	0.707107	+	0.707107i	0.707107	+	0.707107i
$385$		0			0
$386$		0			0
$387$	1.41421			1.41421
$388$		0			0
$389$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$389$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$390$	−1.41421			−1.41421
$391$	1.00000	−	1.00000i	1.00000	−	1.00000i
$392$	−1.00000			−1.00000
$393$			2.00000i			2.00000i
$394$		0			0
$395$	−1.41421			−1.41421
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$	1.00000	+	1.00000i	1.00000	+	1.00000i
$399$		0			0
$400$		−	1.00000i		−	1.00000i
$401$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$401$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$402$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$403$	1.41421	−	1.41421i	1.41421	−	1.41421i
$404$	−1.41421			−1.41421
$405$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$406$		0			0
$407$		0			0
$408$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$409$	2.00000			2.00000			1.00000			$0$
$409$	2.00000			2.00000			1.00000			$0$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	1.00000	+	1.00000i	1.00000	+	1.00000i
$415$		0			0
$416$		−	1.41421i		−	1.41421i
$417$		0			0
$418$		0			0
$419$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$419$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$			2.00000i			2.00000i
$424$		0			0
$425$	1.00000			1.00000
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$	1.00000	−	1.00000i	1.00000	−	1.00000i
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$439$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$440$		0			0
$441$	1.00000			1.00000
$442$	1.41421			1.41421
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$		−	2.00000i		−	2.00000i
$445$		0			0
$446$		0			0
$447$	1.00000	−	1.00000i	1.00000	−	1.00000i
$448$		0			0
$449$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$449$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$450$			1.00000i			1.00000i
$451$		0			0
$452$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0			−	1.00000i	$-0.5\pi$
$457$		0			0				1.00000i	$0.5\pi$
$458$		0			0
$459$	0.707107	−	0.707107i	0.707107	−	0.707107i
$460$	1.41421			1.41421
$461$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$461$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$466$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$			1.41421i			1.41421i
$469$		0			0
$470$	1.41421	+	1.41421i	1.41421	+	1.41421i
$471$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$472$	−1.41421			−1.41421
$473$		0			0
$474$			1.41421i			1.41421i
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$	−1.00000			−1.00000
$481$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$482$	1.00000	+	1.00000i	1.00000	+	1.00000i
$483$		0			0
$484$			1.00000i			1.00000i
$485$		0			0
$486$	0.707107	+	0.707107i	0.707107	+	0.707107i
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$		0			0
$489$		0			0
$490$	0.707107	−	0.707107i	0.707107	−	0.707107i
$491$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$491$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	1.00000	−	1.00000i	1.00000	−	1.00000i
$497$		0			0
$498$		0			0
$499$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$499$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$500$	0.707107	+	0.707107i	0.707107	+	0.707107i
$501$		−	1.41421i		−	1.41421i
$502$		−	1.41421i		−	1.41421i
$503$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$503$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$504$		0			0
$505$	1.00000	−	1.00000i	1.00000	−	1.00000i
$506$		0			0
$507$	0.707107	−	0.707107i	0.707107	−	0.707107i
$508$		0			0
$509$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$509$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$510$		−	1.00000i		−	1.00000i
$511$		0			0
$512$		−	1.00000i		−	1.00000i
$513$		0			0
$514$		0			0
$515$		0			0
$516$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$517$		0			0
$518$		0			0
$519$		0			0
$520$	1.00000	+	1.00000i	1.00000	+	1.00000i
$521$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$521$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$522$		0			0
$523$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$523$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$524$	1.41421	−	1.41421i	1.41421	−	1.41421i
$525$		0			0
$526$		0			0
$527$	1.00000	+	1.00000i	1.00000	+	1.00000i
$528$		0			0
$529$			1.00000i			1.00000i
$530$		0			0
$531$	1.41421			1.41421
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$			1.41421i			1.41421i
$537$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$538$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$539$		0			0
$540$	1.00000			1.00000
$541$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$541$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$542$		0			0
$543$		0			0
$544$	1.00000			1.00000
$545$		0			0
$546$		0			0
$547$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$547$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		−	1.41421i		−	1.41421i
$553$		0			0
$554$		0			0
$555$	1.41421	+	1.41421i	1.41421	+	1.41421i
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$559$			2.00000i			2.00000i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	1.41421	−	1.41421i	1.41421	−	1.41421i
$565$	1.41421			1.41421
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		0			0
$571$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$571$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$576$			1.00000i			1.00000i
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$			1.00000i			1.00000i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$589$		0			0
$590$	1.00000	−	1.00000i	1.00000	−	1.00000i
$591$		0			0
$592$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$593$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$593$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$594$		0			0
$595$		0			0
$596$	−1.41421			−1.41421
$597$			1.41421i			1.41421i
$598$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	0.707107	−	0.707107i	0.707107	−	0.707107i
$601$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$601$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$602$		0			0
$603$		−	1.41421i		−	1.41421i
$604$		0			0
$605$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$606$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.82843			−2.82843
$612$	−1.00000			−1.00000
$613$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$613$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$614$		−	1.41421i		−	1.41421i
$615$		0			0
$616$		0			0
$617$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$617$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$618$		0			0
$619$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$619$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$620$			1.41421i			1.41421i
$621$			1.41421i			1.41421i
$622$		0			0
$623$		0			0
$624$	1.00000	−	1.00000i	1.00000	−	1.00000i
$625$	−1.00000			−1.00000
$626$		0			0
$627$		0			0
$628$	1.41421			1.41421
$629$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	1.00000	−	1.00000i	1.00000	−	1.00000i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$			1.41421i			1.41421i
$638$		0			0
$639$		0			0
$640$	0.707107	+	0.707107i	0.707107	+	0.707107i
$641$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$641$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$	1.41421			1.41421
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		−	1.00000i		−	1.00000i
$649$		0			0
$650$	−1.41421			−1.41421
$651$		0			0
$652$		0			0
$653$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$653$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$654$		0			0
$655$			2.00000i			2.00000i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$659$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$	1.00000	+	1.00000i	1.00000	+	1.00000i
$664$		0			0
$665$		0			0
$666$	1.41421	−	1.41421i	1.41421	−	1.41421i
$667$		0			0
$668$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$669$		0			0
$670$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$671$		0			0
$672$		0			0
$673$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$673$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$674$		0			0
$675$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$676$	−1.00000			−1.00000
$677$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$677$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$678$		−	1.41421i		−	1.41421i
$679$		0			0
$680$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$681$		0			0
$682$		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
$685$		0			0
$686$		0			0
$687$		0			0
$688$			1.41421i			1.41421i
$689$		0			0
$690$	1.00000	+	1.00000i	1.00000	+	1.00000i
$691$		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
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		−	1.41421i		−	1.41421i
$700$		0			0
$701$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$701$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$702$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$703$		0			0
$704$		0			0
$705$			2.00000i			2.00000i
$706$			2.00000i			2.00000i
$707$		0			0
$708$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$709$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$709$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$710$		0			0
$711$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$712$		0			0
$713$	−2.00000			−2.00000
$714$		0			0
$715$		0			0
$716$			1.41421i			1.41421i
$717$		0			0
$718$		0			0
$719$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$719$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$720$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$721$		0			0
$722$			1.00000i			1.00000i
$723$			1.41421i			1.41421i
$724$		0			0
$725$		0			0
$726$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$		0			0
$731$	−1.41421			−1.41421
$732$		0			0
$733$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$733$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$734$		0			0
$735$	1.00000			1.00000
$736$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$737$		0			0
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		−	2.00000i		−	2.00000i
$741$		0			0
$742$		0			0
$743$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$743$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$744$	1.41421			1.41421
$745$	1.00000	−	1.00000i	1.00000	−	1.00000i
$746$		−	1.41421i		−	1.41421i
$747$		0			0
$748$		0			0
$749$		0			0
$750$			1.00000i			1.00000i
$751$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$751$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$752$	−2.00000			−2.00000
$753$	1.00000	−	1.00000i	1.00000	−	1.00000i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$757$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		1.00000			$0$
$761$		0			0		−1.00000			$\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.707107	−	0.707107i	0.707107	−	0.707107i
$766$		0			0
$767$			2.00000i			2.00000i
$768$	0.707107	−	0.707107i	0.707107	−	0.707107i
$769$	−2.00000			−2.00000			−1.00000			$\pi$
$769$	−2.00000			−2.00000			−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		−	1.41421i		−	1.41421i
$775$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$776$		0			0
$777$		0			0
$778$	1.41421			1.41421
$779$		0			0
$780$			1.41421i			1.41421i
$781$		0			0
$782$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$783$		0			0
$784$			1.00000i			1.00000i
$785$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$786$	2.00000			2.00000
$787$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$787$	1.41421	−	1.41421i	1.41421	−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$788$		0			0
$789$		0			0
$790$			1.41421i			1.41421i
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	1.00000	−	1.00000i	1.00000	−	1.00000i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		−	2.00000i		−	2.00000i
$800$	−1.00000			−1.00000
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$805$		0			0
$806$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$807$		−	2.00000i		−	2.00000i
$808$			1.41421i			1.41421i
$809$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$809$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$810$	0.707107	+	0.707107i	0.707107	+	0.707107i
$811$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$811$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	0.707107	+	0.707107i	0.707107	+	0.707107i
$817$		0			0
$818$		−	2.00000i		−	2.00000i
$819$		0			0
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$	1.00000	−	1.00000i	1.00000	−	1.00000i
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$	−1.41421			−1.41421
$833$	−1.00000			−1.00000
$834$		0			0
$835$		−	1.41421i		−	1.41421i
$836$		0			0
$837$	−1.41421			−1.41421
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$			1.00000i			1.00000i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.707107	−	0.707107i	0.707107	−	0.707107i
$846$	2.00000			2.00000
$847$		0			0
$848$		0			0
$849$		0			0
$850$		−	1.00000i		−	1.00000i
$851$	2.82843			2.82843
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$857$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$858$		0			0
$859$		0			0			−	1.00000i	$-0.5\pi$
$859$		0			0				1.00000i	$0.5\pi$
$860$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$865$		0			0
$866$		0			0
$867$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	2.00000			2.00000
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$877$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$878$	1.00000	+	1.00000i	1.00000	+	1.00000i
$879$		0			0
$880$		0			0
$881$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$881$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$882$		−	1.00000i		−	1.00000i
$883$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$883$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$884$		−	1.41421i		−	1.41421i
$885$	1.41421			1.41421
$886$		0			0
$887$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$887$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$888$	−2.00000			−2.00000
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$895$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$896$		0			0
$897$	−2.00000			−2.00000
$898$		0			0
$899$		0			0
$900$	1.00000			1.00000
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$905$		0			0
$906$		0			0
$907$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$907$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$908$		0			0
$909$		−	1.41421i		−	1.41421i
$910$		0			0
$911$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$911$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		−	1.41421i		−	1.41421i
$921$	1.00000	−	1.00000i	1.00000	−	1.00000i
$922$	1.41421			1.41421
$923$		0			0
$924$		0			0
$925$	1.41421	+	1.41421i	1.41421	+	1.41421i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$929$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$930$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$931$		0			0
$932$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	1.41421			1.41421
$937$		0			0			−	1.00000i	$-0.5\pi$
$937$		0			0				1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$	1.41421	−	1.41421i	1.41421	−	1.41421i
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$	1.00000	+	1.00000i	1.00000	+	1.00000i
$943$		0			0
$944$			1.41421i			1.41421i
$945$		0			0
$946$		0			0
$947$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$947$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$948$	1.41421			1.41421
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.00000			−2.00000			−1.00000			$\pi$
$953$	−2.00000			−2.00000			−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$			1.00000i			1.00000i
$961$		−	1.00000i		−	1.00000i
$962$	2.00000	+	2.00000i	2.00000	+	2.00000i
$963$		0			0
$964$	1.00000	−	1.00000i	1.00000	−	1.00000i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	1.00000			1.00000
$969$		0			0
$970$		0			0
$971$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$971$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$972$	0.707107	−	0.707107i	0.707107	−	0.707107i
$973$		0			0
$974$		0			0
$975$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$976$		0			0
$977$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$977$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$978$		0			0
$979$		0			0
$980$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$981$		0			0
$982$	1.41421			1.41421
$983$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$983$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.41421	−	1.41421i	1.41421	−	1.41421i
$990$		0			0
$991$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$991$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$992$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$993$		0			0
$994$		0			0
$995$			1.41421i			1.41421i
$996$		0			0
$997$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$997$	−1.41421	−	1.41421i	−1.41421	−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$998$		0			0
$999$	2.00000			2.00000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2040.1.cj.a.1109.2	yes	4
3.2	odd	2		2040.1.cj.b.1109.2	yes	4
5.4	even	2		2040.1.cj.b.1109.1	yes	4
8.5	even	2	inner	2040.1.cj.a.1109.1	yes	4
15.14	odd	2	inner	2040.1.cj.a.1109.1	yes	4
17.13	even	4	inner	2040.1.cj.a.149.2	yes	4
24.5	odd	2		2040.1.cj.b.1109.1	yes	4
40.29	even	2		2040.1.cj.b.1109.2	yes	4
51.47	odd	4		2040.1.cj.b.149.2	yes	4
85.64	even	4		2040.1.cj.b.149.1	yes	4
120.29	odd	2	CM	2040.1.cj.a.1109.2	yes	4
136.13	even	4	inner	2040.1.cj.a.149.1	✓	4
255.149	odd	4	inner	2040.1.cj.a.149.1	✓	4
408.149	odd	4		2040.1.cj.b.149.1	yes	4
680.149	even	4		2040.1.cj.b.149.2	yes	4
2040.149	odd	4	inner	2040.1.cj.a.149.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.1.cj.a.149.1	✓	4	136.13	even	4	inner
2040.1.cj.a.149.1	✓	4	255.149	odd	4	inner
2040.1.cj.a.149.2	yes	4	17.13	even	4	inner
2040.1.cj.a.149.2	yes	4	2040.149	odd	4	inner
2040.1.cj.a.1109.1	yes	4	8.5	even	2	inner
2040.1.cj.a.1109.1	yes	4	15.14	odd	2	inner
2040.1.cj.a.1109.2	yes	4	1.1	even	1	trivial
2040.1.cj.a.1109.2	yes	4	120.29	odd	2	CM
2040.1.cj.b.149.1	yes	4	85.64	even	4
2040.1.cj.b.149.1	yes	4	408.149	odd	4
2040.1.cj.b.149.2	yes	4	51.47	odd	4
2040.1.cj.b.149.2	yes	4	680.149	even	4
2040.1.cj.b.1109.1	yes	4	5.4	even	2
2040.1.cj.b.1109.1	yes	4	24.5	odd	2
2040.1.cj.b.1109.2	yes	4	3.2	odd	2
2040.1.cj.b.1109.2	yes	4	40.29	even	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:	\( N \)	\(=\)	\( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2040.cj (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} +(0.707107 - 0.707107i) q^{3} -1.00000 q^{4} +(0.707107 - 0.707107i) q^{5} +(-0.707107 - 0.707107i) q^{6} +1.00000i q^{8} -1.00000i q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} +(0.707107 - 0.707107i) q^{3} -1.00000 q^{4} +(0.707107 - 0.707107i) q^{5} +(-0.707107 - 0.707107i) q^{6} +1.00000i q^{8} -1.00000i q^{9} +(-0.707107 - 0.707107i) q^{10} +(-0.707107 + 0.707107i) q^{12} +1.41421 q^{13} -1.00000i q^{15} +1.00000 q^{16} +1.00000i q^{17} -1.00000 q^{18} +(-0.707107 + 0.707107i) q^{20} +(-1.00000 - 1.00000i) q^{23} +(0.707107 + 0.707107i) q^{24} -1.00000i q^{25} -1.41421i q^{26} +(-0.707107 - 0.707107i) q^{27} -1.00000 q^{30} +(1.00000 - 1.00000i) q^{31} -1.00000i q^{32} +1.00000 q^{34} +1.00000i q^{36} +(-1.41421 + 1.41421i) q^{37} +(1.00000 - 1.00000i) q^{39} +(0.707107 + 0.707107i) q^{40} +1.41421i q^{43} +(-0.707107 - 0.707107i) q^{45} +(-1.00000 + 1.00000i) q^{46} -2.00000 q^{47} +(0.707107 - 0.707107i) q^{48} +1.00000i q^{49} -1.00000 q^{50} +(0.707107 + 0.707107i) q^{51} -1.41421 q^{52} +(-0.707107 + 0.707107i) q^{54} +1.41421i q^{59} +1.00000i q^{60} +(-1.00000 - 1.00000i) q^{62} -1.00000 q^{64} +(1.00000 - 1.00000i) q^{65} +1.41421 q^{67} -1.00000i q^{68} -1.41421 q^{69} +1.00000 q^{72} +(1.41421 + 1.41421i) q^{74} +(-0.707107 - 0.707107i) q^{75} +(-1.00000 - 1.00000i) q^{78} +(-1.00000 - 1.00000i) q^{79} +(0.707107 - 0.707107i) q^{80} -1.00000 q^{81} +(0.707107 + 0.707107i) q^{85} +1.41421 q^{86} +(-0.707107 + 0.707107i) q^{90} +(1.00000 + 1.00000i) q^{92} -1.41421i q^{93} +2.00000i q^{94} +(-0.707107 - 0.707107i) q^{96} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4}+O(q^{10}) \) 4 * q - 4 * q^4	\( 4 q - 4 q^{4} + 4 q^{16} - 4 q^{18} - 4 q^{23} - 4 q^{30} + 4 q^{31} + 4 q^{34} + 4 q^{39} - 4 q^{46} - 8 q^{47} - 4 q^{50} - 4 q^{62} - 4 q^{64} + 4 q^{65} + 4 q^{72} - 4 q^{78} - 4 q^{79} - 4 q^{81} + 4 q^{92} + 4 q^{98}+O(q^{100}) \) 4 * q - 4 * q^4 + 4 * q^16 - 4 * q^18 - 4 * q^23 - 4 * q^30 + 4 * q^31 + 4 * q^34 + 4 * q^39 - 4 * q^46 - 8 * q^47 - 4 * q^50 - 4 * q^62 - 4 * q^64 + 4 * q^65 + 4 * q^72 - 4 * q^78 - 4 * q^79 - 4 * q^81 + 4 * q^92 + 4 * q^98

Introduction

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