LMFDB - Embedded newform 2280.1.t.j.1139.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2280,1,Mod(1139,2280)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2280, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1, 1, 1])) N = Newforms(chi, 1, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2280.1139"); S:= CuspForms(chi, 1); N := Newforms(S);

Level:	$N$	$=$	$2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2280.t (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,4,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$1.13786822880$
Analytic rank:	$0$
Dimension:	$4$
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:	$D_{4}$
Projective field:	Galois closure of 4.0.1039680.3

Embedding invariants

Embedding label			1139.2
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2280.1139
Dual form			2280.1.t.j.1139.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{2} +(0.707107 - 0.707107i) q^{3} -1.00000i q^{4} +1.00000 q^{5} +1.00000i q^{6} +(0.707107 + 0.707107i) q^{8} -1.00000i q^{9} +(-0.707107 + 0.707107i) q^{10} +2.00000i q^{11} +(-0.707107 - 0.707107i) q^{12} -1.41421i q^{13} +(0.707107 - 0.707107i) q^{15} -1.00000 q^{16} +(0.707107 + 0.707107i) q^{18} +1.00000 q^{19} -1.00000i q^{20} +(-1.41421 - 1.41421i) q^{22} +1.00000 q^{24} +1.00000 q^{25} +(1.00000 + 1.00000i) q^{26} +(-0.707107 - 0.707107i) q^{27} +1.00000i q^{30} +(0.707107 - 0.707107i) q^{32} +(1.41421 + 1.41421i) q^{33} -1.00000 q^{36} +1.41421i q^{37} +(-0.707107 + 0.707107i) q^{38} +(-1.00000 - 1.00000i) q^{39} +(0.707107 + 0.707107i) q^{40} +2.00000 q^{44} -1.00000i q^{45} +(-0.707107 + 0.707107i) q^{48} -1.00000 q^{49} +(-0.707107 + 0.707107i) q^{50} -1.41421 q^{52} -1.41421 q^{53} +1.00000 q^{54} +2.00000i q^{55} +(0.707107 - 0.707107i) q^{57} +(-0.707107 - 0.707107i) q^{60} -2.00000i q^{61} +1.00000i q^{64} -1.41421i q^{65} -2.00000 q^{66} +1.41421 q^{67} +(0.707107 - 0.707107i) q^{72} +(-1.00000 - 1.00000i) q^{74} +(0.707107 - 0.707107i) q^{75} -1.00000i q^{76} +1.41421 q^{78} -1.00000 q^{80} -1.00000 q^{81} +(-1.41421 + 1.41421i) q^{88} +(0.707107 + 0.707107i) q^{90} +1.00000 q^{95} -1.00000i q^{96} -1.41421 q^{97} +(0.707107 - 0.707107i) q^{98} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q + 4 q^{5} - 4 q^{16} + 4 q^{19} + 4 q^{24} + 4 q^{25} + 4 q^{26} - 4 q^{36} - 4 q^{39} + 8 q^{44} - 4 q^{49} + 4 q^{54} - 8 q^{66} - 4 q^{74} - 4 q^{80} - 4 q^{81} + 4 q^{95} + 8 q^{99}+O(q^{100})$ 4 * q + 4 * q^5 - 4 * q^16 + 4 * q^19 + 4 * q^24 + 4 * q^25 + 4 * q^26 - 4 * q^36 - 4 * q^39 + 8 * q^44 - 4 * q^49 + 4 * q^54 - 8 * q^66 - 4 * q^74 - 4 * q^80 - 4 * q^81 + 4 * q^95 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2280\mathbb{Z}\right)^\times$ .

$n$	$457$	$761$	$1141$	$1711$	$1921$
$\chi(n)$	$-1$	$-1$	$-1$	$-1$	$-1$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$2$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$3$	0.707107	−	0.707107i	0.707107	−	0.707107i
$3$	0.707107	−	0.707107i	0.707107	−	0.707107i
$4$		−	1.00000i		−	1.00000i
$5$	1.00000			1.00000
$5$	1.00000			1.00000
$6$			1.00000i			1.00000i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	0.707107	+	0.707107i	0.707107	+	0.707107i
$9$		−	1.00000i		−	1.00000i
$10$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$11$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$11$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$12$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$13$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$13$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$14$		0			0
$15$	0.707107	−	0.707107i	0.707107	−	0.707107i
$16$	−1.00000			−1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	0.707107	+	0.707107i	0.707107	+	0.707107i
$19$	1.00000			1.00000
$19$	1.00000			1.00000
$20$		−	1.00000i		−	1.00000i
$21$		0			0
$22$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$	1.00000			1.00000
$25$	1.00000			1.00000
$26$	1.00000	+	1.00000i	1.00000	+	1.00000i
$27$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$28$		0			0
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$			1.00000i			1.00000i
$31$		0			0			−	1.00000i	$-0.5\pi$
$31$		0			0				1.00000i	$0.5\pi$
$32$	0.707107	−	0.707107i	0.707107	−	0.707107i
$33$	1.41421	+	1.41421i	1.41421	+	1.41421i
$34$		0			0
$35$		0			0
$36$	−1.00000			−1.00000
$37$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$37$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$38$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$39$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$40$	0.707107	+	0.707107i	0.707107	+	0.707107i
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	2.00000			2.00000
$45$		−	1.00000i		−	1.00000i
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$49$	−1.00000			−1.00000
$50$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$51$		0			0
$52$	−1.41421			−1.41421
$53$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$53$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$54$	1.00000			1.00000
$55$			2.00000i			2.00000i
$56$		0			0
$57$	0.707107	−	0.707107i	0.707107	−	0.707107i
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$61$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$61$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$62$		0			0
$63$		0			0
$64$			1.00000i			1.00000i
$65$		−	1.41421i		−	1.41421i
$66$	−2.00000			−2.00000
$67$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$67$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$	0.707107	−	0.707107i	0.707107	−	0.707107i
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$75$	0.707107	−	0.707107i	0.707107	−	0.707107i
$76$		−	1.00000i		−	1.00000i
$77$		0			0
$78$	1.41421			1.41421
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	−1.00000			−1.00000
$81$	−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			0
$86$		0			0
$87$		0			0
$88$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$	0.707107	+	0.707107i	0.707107	+	0.707107i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.00000			1.00000
$96$		−	1.00000i		−	1.00000i
$97$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$97$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$98$	0.707107	−	0.707107i	0.707107	−	0.707107i
$99$	2.00000			2.00000
$100$		−	1.00000i		−	1.00000i
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$103$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$104$	1.00000	−	1.00000i	1.00000	−	1.00000i
$105$		0			0
$106$	1.00000	−	1.00000i	1.00000	−	1.00000i
$107$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$107$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$108$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$111$	1.00000	+	1.00000i	1.00000	+	1.00000i
$112$		0			0
$113$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$113$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$114$			1.00000i			1.00000i
$115$		0			0
$116$		0			0
$117$	−1.41421			−1.41421
$118$		0			0
$119$		0			0
$120$	1.00000			1.00000
$121$	−3.00000			−3.00000
$122$	1.41421	+	1.41421i	1.41421	+	1.41421i
$123$		0			0
$124$		0			0
$125$	1.00000			1.00000
$126$		0			0
$127$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$127$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$128$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$129$		0			0
$130$	1.00000	+	1.00000i	1.00000	+	1.00000i
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$	1.41421	−	1.41421i	1.41421	−	1.41421i
$133$		0			0
$134$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$135$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$136$		0			0
$137$		0			0			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.82843			2.82843
$144$			1.00000i			1.00000i
$145$		0			0
$146$		0			0
$147$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$148$	1.41421			1.41421
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$			1.00000i			1.00000i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	0.707107	+	0.707107i	0.707107	+	0.707107i
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$160$	0.707107	−	0.707107i	0.707107	−	0.707107i
$161$		0			0
$162$	0.707107	−	0.707107i	0.707107	−	0.707107i
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	1.41421	+	1.41421i	1.41421	+	1.41421i
$166$		0			0
$167$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$167$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$		0			0
$171$		−	1.00000i		−	1.00000i
$172$		0			0
$173$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$173$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$		−	2.00000i		−	2.00000i
$177$		0			0
$178$		0			0
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	−1.00000			−1.00000
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$184$		0			0
$185$			1.41421i			1.41421i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$191$	−2.00000			−2.00000			−1.00000			$\pi$
$191$	−2.00000			−2.00000			−1.00000			$\pi$
$192$	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
$195$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$196$			1.00000i			1.00000i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$	0.707107	+	0.707107i	0.707107	+	0.707107i
$201$	1.00000	−	1.00000i	1.00000	−	1.00000i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$207$		0			0
$208$			1.41421i			1.41421i
$209$			2.00000i			2.00000i
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$			1.41421i			1.41421i
$213$		0			0
$214$	1.00000	+	1.00000i	1.00000	+	1.00000i
$215$		0			0
$216$		−	1.00000i		−	1.00000i
$217$		0			0
$218$		0			0
$219$		0			0
$220$	2.00000			2.00000
$221$		0			0
$222$	−1.41421			−1.41421
$223$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$223$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$224$		0			0
$225$		−	1.00000i		−	1.00000i
$226$	1.00000	+	1.00000i	1.00000	+	1.00000i
$227$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$227$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$228$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$229$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$229$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$	1.00000	−	1.00000i	1.00000	−	1.00000i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−2.00000			−2.00000			−1.00000			$\pi$
$239$	−2.00000			−2.00000			−1.00000			$\pi$
$240$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	2.12132	−	2.12132i	2.12132	−	2.12132i
$243$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$244$	−2.00000			−2.00000
$245$	−1.00000			−1.00000
$246$		0			0
$247$		−	1.41421i		−	1.41421i
$248$		0			0
$249$		0			0
$250$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$255$		0			0
$256$	1.00000			1.00000
$257$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$257$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$258$		0			0
$259$		0			0
$260$	−1.41421			−1.41421
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$			2.00000i			2.00000i
$265$	−1.41421			−1.41421
$266$		0			0
$267$		0			0
$268$		−	1.41421i		−	1.41421i
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$	1.00000			1.00000
$271$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$271$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			2.00000i			2.00000i
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$	0.707107	−	0.707107i	0.707107	−	0.707107i
$286$	−2.00000	+	2.00000i	−2.00000	+	2.00000i
$287$		0			0
$288$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$289$	−1.00000			−1.00000
$290$		0			0
$291$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$292$		0			0
$293$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$293$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$294$		−	1.00000i		−	1.00000i
$295$		0			0
$296$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$297$	1.41421	−	1.41421i	1.41421	−	1.41421i
$298$		0			0
$299$		0			0
$300$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	−1.00000			−1.00000
$305$		−	2.00000i		−	2.00000i
$306$		0			0
$307$	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
$309$	1.00000	+	1.00000i	1.00000	+	1.00000i
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		−	1.41421i		−	1.41421i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$317$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$318$		−	1.41421i		−	1.41421i
$319$		0			0
$320$			1.00000i			1.00000i
$321$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$322$		0			0
$323$		0			0
$324$			1.00000i			1.00000i
$325$		−	1.41421i		−	1.41421i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$	−2.00000			−2.00000
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$		0			0
$333$	1.41421			1.41421
$334$	1.00000	−	1.00000i	1.00000	−	1.00000i
$335$	1.41421			1.41421
$336$		0			0
$337$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$337$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$338$	0.707107	−	0.707107i	0.707107	−	0.707107i
$339$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$340$		0			0
$341$		0			0
$342$	0.707107	+	0.707107i	0.707107	+	0.707107i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$352$	1.41421	+	1.41421i	1.41421	+	1.41421i
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		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
$363$	−2.12132	+	2.12132i	−2.12132	+	2.12132i
$364$		0			0
$365$		0			0
$366$	2.00000			2.00000
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$371$		0			0
$372$		0			0
$373$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$373$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$374$		0			0
$375$	0.707107	−	0.707107i	0.707107	−	0.707107i
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		−	1.00000i		−	1.00000i
$381$	1.00000	+	1.00000i	1.00000	+	1.00000i
$382$	1.41421	−	1.41421i	1.41421	−	1.41421i
$383$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$383$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$384$	−1.00000			−1.00000
$385$		0			0
$386$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$387$		0			0
$388$			1.41421i			1.41421i
$389$	2.00000			2.00000			1.00000			$0$
$389$	2.00000			2.00000			1.00000			$0$
$390$	1.41421			1.41421
$391$		0			0
$392$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		−	2.00000i		−	2.00000i
$397$		0			0			−	1.00000i	$-0.5\pi$
$397$		0			0				1.00000i	$0.5\pi$
$398$		0			0
$399$		0			0
$400$	−1.00000			−1.00000
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$			1.41421i			1.41421i
$403$		0			0
$404$		0			0
$405$	−1.00000			−1.00000
$406$		0			0
$407$	−2.82843			−2.82843
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$	1.41421			1.41421
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$417$		0			0
$418$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$		0			0
$423$		0			0
$424$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.41421			−1.41421
$429$	2.00000	−	2.00000i	2.00000	−	2.00000i
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$	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
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$441$			1.00000i			1.00000i
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$	1.00000	−	1.00000i	1.00000	−	1.00000i
$445$		0			0
$446$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$447$		0			0
$448$		0			0
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$	0.707107	+	0.707107i	0.707107	+	0.707107i
$451$		0			0
$452$	−1.41421			−1.41421
$453$		0			0
$454$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$455$		0			0
$456$	1.00000			1.00000
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$459$		0			0
$460$		0			0
$461$	−2.00000			−2.00000			−1.00000			$\pi$
$461$	−2.00000			−2.00000			−1.00000			$\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$			1.41421i			1.41421i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.00000			1.00000
$476$		0			0
$477$			1.41421i			1.41421i
$478$	1.41421	−	1.41421i	1.41421	−	1.41421i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		−	1.00000i		−	1.00000i
$481$	2.00000			2.00000
$482$		0			0
$483$		0			0
$484$			3.00000i			3.00000i
$485$	−1.41421			−1.41421
$486$		−	1.00000i		−	1.00000i
$487$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$487$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$488$	1.41421	−	1.41421i	1.41421	−	1.41421i
$489$		0			0
$490$	0.707107	−	0.707107i	0.707107	−	0.707107i
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$	1.00000	+	1.00000i	1.00000	+	1.00000i
$495$	2.00000			2.00000
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0			−	1.00000i	$-0.5\pi$
$499$		0			0				1.00000i	$0.5\pi$
$500$		−	1.00000i		−	1.00000i
$501$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$508$	1.41421			1.41421
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$513$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$514$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$515$			1.41421i			1.41421i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	1.00000	−	1.00000i	1.00000	−	1.00000i
$520$	1.00000	−	1.00000i	1.00000	−	1.00000i
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$523$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$529$	1.00000			1.00000
$530$	1.00000	−	1.00000i	1.00000	−	1.00000i
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		−	1.41421i		−	1.41421i
$536$	1.00000	+	1.00000i	1.00000	+	1.00000i
$537$		0			0
$538$		0			0
$539$		−	2.00000i		−	2.00000i
$540$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$541$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$541$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$542$	1.41421	+	1.41421i	1.41421	+	1.41421i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$547$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$548$		0			0
$549$	−2.00000			−2.00000
$550$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	1.00000	+	1.00000i	1.00000	+	1.00000i
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$563$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$564$		0			0
$565$		−	1.41421i		−	1.41421i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$			1.00000i			1.00000i
$571$		0			0			−	1.00000i	$-0.5\pi$
$571$		0			0				1.00000i	$0.5\pi$
$572$		−	2.82843i		−	2.82843i
$573$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$574$		0			0
$575$		0			0
$576$	1.00000			1.00000
$577$		0			0		1.00000			$0$
$577$		0			0		−1.00000			$\pi$
$578$	0.707107	−	0.707107i	0.707107	−	0.707107i
$579$	1.00000	−	1.00000i	1.00000	−	1.00000i
$580$		0			0
$581$		0			0
$582$		−	1.41421i		−	1.41421i
$583$		−	2.82843i		−	2.82843i
$584$		0			0
$585$	−1.41421			−1.41421
$586$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$587$		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
$590$		0			0
$591$		0			0
$592$		−	1.41421i		−	1.41421i
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$			2.00000i			2.00000i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$	1.00000			1.00000
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		−	1.41421i		−	1.41421i
$604$		0			0
$605$	−3.00000			−3.00000
$606$		0			0
$607$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$607$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$608$	0.707107	−	0.707107i	0.707107	−	0.707107i
$609$		0			0
$610$	1.41421	+	1.41421i	1.41421	+	1.41421i
$611$		0			0
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$	−1.41421			−1.41421
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.00000	+	1.00000i	1.00000	+	1.00000i
$625$	1.00000			1.00000
$626$		0			0
$627$	1.41421	+	1.41421i	1.41421	+	1.41421i
$628$		0			0
$629$		0			0
$630$		0			0
$631$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$631$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$	1.00000	−	1.00000i	1.00000	−	1.00000i
$635$			1.41421i			1.41421i
$636$	1.00000	+	1.00000i	1.00000	+	1.00000i
$637$			1.41421i			1.41421i
$638$		0			0
$639$		0			0
$640$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$	1.41421			1.41421
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$649$		0			0
$650$	1.00000	+	1.00000i	1.00000	+	1.00000i
$651$		0			0
$652$		0			0
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$	1.41421	−	1.41421i	1.41421	−	1.41421i
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$667$		0			0
$668$			1.41421i			1.41421i
$669$	1.00000	+	1.00000i	1.00000	+	1.00000i
$670$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$671$	4.00000			4.00000
$672$		0			0
$673$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$673$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$674$	1.00000	−	1.00000i	1.00000	−	1.00000i
$675$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$676$			1.00000i			1.00000i
$677$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$677$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$678$	1.41421			1.41421
$679$		0			0
$680$		0			0
$681$	1.00000	+	1.00000i	1.00000	+	1.00000i
$682$		0			0
$683$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$683$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$684$	−1.00000			−1.00000
$685$		0			0
$686$		0			0
$687$	1.41421	+	1.41421i	1.41421	+	1.41421i
$688$		0			0
$689$			2.00000i			2.00000i
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		−	1.41421i		−	1.41421i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		−	1.41421i		−	1.41421i
$703$			1.41421i			1.41421i
$704$	−2.00000			−2.00000
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	2.82843			2.82843
$716$		0			0
$717$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$			1.00000i			1.00000i
$721$		0			0
$722$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		−	3.00000i		−	3.00000i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$		0			0
$731$		0			0
$732$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$736$		0			0
$737$			2.82843i			2.82843i
$738$		0			0
$739$	−2.00000			−2.00000			−1.00000			$\pi$
$739$	−2.00000			−2.00000			−1.00000			$\pi$
$740$	1.41421			1.41421
$741$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$742$		0			0
$743$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$743$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$	1.00000	+	1.00000i	1.00000	+	1.00000i
$747$		0			0
$748$		0			0
$749$		0			0
$750$			1.00000i			1.00000i
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$	0.707107	+	0.707107i	0.707107	+	0.707107i
$761$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$761$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$762$	−1.41421			−1.41421
$763$		0			0
$764$			2.00000i			2.00000i
$765$		0			0
$766$	1.00000	−	1.00000i	1.00000	−	1.00000i
$767$		0			0
$768$	0.707107	−	0.707107i	0.707107	−	0.707107i
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	1.00000	+	1.00000i	1.00000	+	1.00000i
$772$		−	1.41421i		−	1.41421i
$773$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$773$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$774$		0			0
$775$		0			0
$776$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$777$		0			0
$778$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$779$		0			0
$780$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.00000			1.00000
$785$		0			0
$786$		0			0
$787$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$787$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	1.41421	+	1.41421i	1.41421	+	1.41421i
$793$	−2.82843			−2.82843
$794$		0			0
$795$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$796$		0			0
$797$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$797$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$	0.707107	−	0.707107i	0.707107	−	0.707107i
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$	0.707107	−	0.707107i	0.707107	−	0.707107i
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$814$	2.00000	−	2.00000i	2.00000	−	2.00000i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	2.00000			2.00000			1.00000			$0$
$821$	2.00000			2.00000			1.00000			$0$
$822$		0			0
$823$		0			0			−	1.00000i	$-0.5\pi$
$823$		0			0				1.00000i	$0.5\pi$
$824$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$825$	1.41421	+	1.41421i	1.41421	+	1.41421i
$826$		0			0
$827$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$827$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$828$		0			0
$829$		0			0			−	1.00000i	$-0.5\pi$
$829$		0			0				1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$	1.41421			1.41421
$833$		0			0
$834$		0			0
$835$	−1.41421			−1.41421
$836$	2.00000			2.00000
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	1.00000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.00000			−1.00000
$846$		0			0
$847$		0			0
$848$	1.41421			1.41421
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		0			0
$855$		−	1.00000i		−	1.00000i
$856$	1.00000	−	1.00000i	1.00000	−	1.00000i
$857$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$857$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$858$			2.82843i			2.82843i
$859$	2.00000			2.00000			1.00000			$0$
$859$	2.00000			2.00000			1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$863$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$864$	−1.00000			−1.00000
$865$	1.41421			1.41421
$866$	1.00000	−	1.00000i	1.00000	−	1.00000i
$867$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		−	2.00000i		−	2.00000i
$872$		0			0
$873$			1.41421i			1.41421i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$877$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$878$		0			0
$879$	1.00000	−	1.00000i	1.00000	−	1.00000i
$880$		−	2.00000i		−	2.00000i
$881$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$881$		−	2.00000i		−	2.00000i		−	1.00000i	$-0.5\pi$
$882$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$887$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$888$			1.41421i			1.41421i
$889$		0			0
$890$		0			0
$891$		−	2.00000i		−	2.00000i
$892$	1.41421			1.41421
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		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			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$907$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$908$	1.41421			1.41421
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$913$		0			0
$914$		0			0
$915$	−1.41421	−	1.41421i	−1.41421	−	1.41421i
$916$	2.00000			2.00000
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	1.00000	−	1.00000i	1.00000	−	1.00000i
$922$	1.41421	−	1.41421i	1.41421	−	1.41421i
$923$		0			0
$924$		0			0
$925$			1.41421i			1.41421i
$926$		0			0
$927$	1.41421			1.41421
$928$		0			0
$929$		0			0		1.00000			$0$
$929$		0			0		−1.00000			$\pi$
$930$		0			0
$931$	−1.00000			−1.00000
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0			−	1.00000i	$-0.5\pi$
$947$		0			0				1.00000i	$0.5\pi$
$948$		0			0
$949$		0			0
$950$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$951$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$952$		0			0
$953$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$953$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$954$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$955$	−2.00000			−2.00000
$956$			2.00000i			2.00000i
$957$		0			0
$958$		0			0
$959$		0			0
$960$	0.707107	+	0.707107i	0.707107	+	0.707107i
$961$	−1.00000			−1.00000
$962$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$963$	−1.41421			−1.41421
$964$		0			0
$965$	1.41421			1.41421
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	−2.12132	−	2.12132i	−2.12132	−	2.12132i
$969$		0			0
$970$	1.00000	−	1.00000i	1.00000	−	1.00000i
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	0.707107	+	0.707107i	0.707107	+	0.707107i
$973$		0			0
$974$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$975$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$976$			2.00000i			2.00000i
$977$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$977$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$978$		0			0
$979$		0			0
$980$			1.00000i			1.00000i
$981$		0			0
$982$		0			0
$983$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$983$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.41421			−1.41421
$989$		0			0
$990$	−1.41421	+	1.41421i	−1.41421	+	1.41421i
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		0			0
$999$	1.00000	−	1.00000i	1.00000	−	1.00000i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2280.1.t.j.1139.2	yes	4
3.2	odd	2		2280.1.t.i.1139.3	yes	4
5.4	even	2	inner	2280.1.t.j.1139.3	yes	4
8.3	odd	2		2280.1.t.i.1139.4	yes	4
15.14	odd	2		2280.1.t.i.1139.2	yes	4
19.18	odd	2	inner	2280.1.t.j.1139.3	yes	4
24.11	even	2	inner	2280.1.t.j.1139.1	yes	4
40.19	odd	2		2280.1.t.i.1139.1	✓	4
57.56	even	2		2280.1.t.i.1139.2	yes	4
95.94	odd	2	CM	2280.1.t.j.1139.2	yes	4
120.59	even	2	inner	2280.1.t.j.1139.4	yes	4
152.75	even	2		2280.1.t.i.1139.1	✓	4
285.284	even	2		2280.1.t.i.1139.3	yes	4
456.227	odd	2	inner	2280.1.t.j.1139.4	yes	4
760.379	even	2		2280.1.t.i.1139.4	yes	4
2280.1139	odd	2	inner	2280.1.t.j.1139.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.1.t.i.1139.1	✓	4	40.19	odd	2
2280.1.t.i.1139.1	✓	4	152.75	even	2
2280.1.t.i.1139.2	yes	4	15.14	odd	2
2280.1.t.i.1139.2	yes	4	57.56	even	2
2280.1.t.i.1139.3	yes	4	3.2	odd	2
2280.1.t.i.1139.3	yes	4	285.284	even	2
2280.1.t.i.1139.4	yes	4	8.3	odd	2
2280.1.t.i.1139.4	yes	4	760.379	even	2
2280.1.t.j.1139.1	yes	4	24.11	even	2	inner
2280.1.t.j.1139.1	yes	4	2280.1139	odd	2	inner
2280.1.t.j.1139.2	yes	4	1.1	even	1	trivial
2280.1.t.j.1139.2	yes	4	95.94	odd	2	CM
2280.1.t.j.1139.3	yes	4	5.4	even	2	inner
2280.1.t.j.1139.3	yes	4	19.18	odd	2	inner
2280.1.t.j.1139.4	yes	4	120.59	even	2	inner
2280.1.t.j.1139.4	yes	4	456.227	odd	2	inner

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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2280.1.t.j.1139.2

Level	$2280$
Weight	$1$
Character	2280.1139
Analytic conductor	$1.138$
Analytic rank	$0$
Dimension	$4$
Projective image	$D_{4}$
CM discriminant	-95
Inner twists	$8$