LMFDB - Embedded newform 2240.1.dl.b.737.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,1,Mod(417,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 6, 3, 8]))

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

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

chi := DirichletCharacter("2240.417");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$2240 = 2^{6} \cdot 5 \cdot 7$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	2240.dl (of order $12$ , degree $4$ , 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:	$1.11790562830$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - x^{4} + 1$ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$S_{4}$
Projective field:	Galois closure of 4.2.49000.1

Embedding invariants

Embedding label			737.1
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2240.737
Dual form			2240.1.dl.b.1313.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.366025 + 1.36603i) q^{3} +(-0.258819 - 0.965926i) q^{5} +(-0.707107 - 0.707107i) q^{7} +(-0.866025 - 0.500000i) q^{9} +(-0.866025 + 0.500000i) q^{11} +(0.707107 + 0.707107i) q^{13} +1.41421 q^{15} +(-0.500000 + 0.866025i) q^{19} +(1.22474 - 0.707107i) q^{21} +(0.258819 + 0.965926i) q^{23} +(-0.866025 + 0.500000i) q^{25} -1.41421 q^{29} +(-0.707107 - 1.22474i) q^{31} +(-0.366025 - 1.36603i) q^{33} +(-0.500000 + 0.866025i) q^{35} +(0.258819 + 0.965926i) q^{37} +(-1.22474 + 0.707107i) q^{39} -1.00000 q^{41} +(-0.258819 + 0.965926i) q^{45} +(-0.965926 + 0.258819i) q^{47} +1.00000i q^{49} +(-0.258819 + 0.965926i) q^{53} +(0.707107 + 0.707107i) q^{55} +(-1.00000 - 1.00000i) q^{57} +(-1.22474 - 0.707107i) q^{61} +(0.258819 + 0.965926i) q^{63} +(0.500000 - 0.866025i) q^{65} +(1.36603 + 0.366025i) q^{67} -1.41421 q^{69} +1.41421 q^{71} +(-0.366025 - 1.36603i) q^{75} +(0.965926 + 0.258819i) q^{77} +(-0.500000 - 0.866025i) q^{81} +(-1.00000 - 1.00000i) q^{83} +(0.517638 - 1.93185i) q^{87} -1.00000i q^{91} +(1.93185 - 0.517638i) q^{93} +(0.965926 + 0.258819i) q^{95} +(1.00000 + 1.00000i) q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$8 q + 4 q^{3} - 4 q^{19} + 4 q^{33} - 4 q^{35} - 8 q^{41} - 8 q^{57} + 4 q^{65} + 4 q^{67} + 4 q^{75} - 4 q^{81} - 8 q^{83} + 8 q^{97} + 8 q^{99}+O(q^{100})$ 8 * q + 4 * q^3 - 4 * q^19 + 4 * q^33 - 4 * q^35 - 8 * q^41 - 8 * q^57 + 4 * q^65 + 4 * q^67 + 4 * q^75 - 4 * q^81 - 8 * q^83 + 8 * q^97 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times$ .

$n$	$897$	$1471$	$1541$	$1921$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$	$e\left(\frac{1}{3}\right)$

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			0
$2$		0			0
$3$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$3$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$4$		0			0
$5$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$5$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$6$		0			0
$7$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$7$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$8$		0			0
$9$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$10$		0			0
$11$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$11$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$12$		0			0
$13$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$13$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$14$		0			0
$15$	1.41421			1.41421
$16$		0			0
$17$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$17$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$18$		0			0
$19$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$19$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$20$		0			0
$21$	1.22474	−	0.707107i	1.22474	−	0.707107i
$22$		0			0
$23$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$23$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$24$		0			0
$25$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$29$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$30$		0			0
$31$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$31$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$32$		0			0
$33$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$34$		0			0
$35$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$36$		0			0
$37$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$37$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$38$		0			0
$39$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$40$		0			0
$41$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$41$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$43$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$44$		0			0
$45$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$46$		0			0
$47$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$47$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$48$		0			0
$49$			1.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$53$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$54$		0			0
$55$	0.707107	+	0.707107i	0.707107	+	0.707107i
$56$		0			0
$57$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$61$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$62$		0			0
$63$	0.258819	+	0.965926i	0.258819	+	0.965926i
$64$		0			0
$65$	0.500000	−	0.866025i	0.500000	−	0.866025i
$66$		0			0
$67$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$67$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$	−1.41421			−1.41421
$70$		0			0
$71$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$71$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$72$		0			0
$73$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$73$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$74$		0			0
$75$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$76$		0			0
$77$	0.965926	+	0.258819i	0.965926	+	0.258819i
$78$		0			0
$79$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$79$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$82$		0			0
$83$	−1.00000	−	1.00000i	−1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$83$	−1.00000	−	1.00000i	−1.00000	−	1.00000i	−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	0.517638	−	1.93185i	0.517638	−	1.93185i
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$		−	1.00000i		−	1.00000i
$92$		0			0
$93$	1.93185	−	0.517638i	1.93185	−	0.517638i
$94$		0			0
$95$	0.965926	+	0.258819i	0.965926	+	0.258819i
$96$		0			0
$97$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$97$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$98$		0			0
$99$	1.00000			1.00000
$100$		0			0
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$103$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$104$		0			0
$105$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$106$		0			0
$107$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$107$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$108$		0			0
$109$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$109$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$110$		0			0
$111$	−1.41421			−1.41421
$112$		0			0
$113$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$113$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$114$		0			0
$115$	0.866025	−	0.500000i	0.866025	−	0.500000i
$116$		0			0
$117$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$118$		0			0
$119$		0			0
$120$		0			0
$121$		0			0
$122$		0			0
$123$	0.366025	−	1.36603i	0.366025	−	1.36603i
$124$		0			0
$125$	0.707107	+	0.707107i	0.707107	+	0.707107i
$126$		0			0
$127$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$127$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$131$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$132$		0			0
$133$	0.965926	−	0.258819i	0.965926	−	0.258819i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$137$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		−	1.41421i		−	1.41421i
$142$		0			0
$143$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$144$		0			0
$145$	0.366025	+	1.36603i	0.366025	+	1.36603i
$146$		0			0
$147$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$151$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$156$		0			0
$157$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$157$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$158$		0			0
$159$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$160$		0			0
$161$	0.500000	−	0.866025i	0.500000	−	0.866025i
$162$		0			0
$163$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$163$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$164$		0			0
$165$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$166$		0			0
$167$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$167$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$168$		0			0
$169$		0			0
$170$		0			0
$171$	0.866025	−	0.500000i	0.866025	−	0.500000i
$172$		0			0
$173$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$173$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$174$		0			0
$175$	0.965926	+	0.258819i	0.965926	+	0.258819i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$179$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$	1.41421	−	1.41421i	1.41421	−	1.41421i
$184$		0			0
$185$	0.866025	−	0.500000i	0.866025	−	0.500000i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$193$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$	1.00000	+	1.00000i	1.00000	+	1.00000i
$196$		0			0
$197$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$197$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$198$		0			0
$199$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$199$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$200$		0			0
$201$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$202$		0			0
$203$	1.00000	+	1.00000i	1.00000	+	1.00000i
$204$		0			0
$205$	0.258819	+	0.965926i	0.258819	+	0.965926i
$206$		0			0
$207$	0.258819	−	0.965926i	0.258819	−	0.965926i
$208$		0			0
$209$		−	1.00000i		−	1.00000i
$210$		0			0
$211$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$211$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$212$		0			0
$213$	−0.517638	+	1.93185i	−0.517638	+	1.93185i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$223$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$224$		0			0
$225$	1.00000			1.00000
$226$		0			0
$227$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$227$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$229$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$232$		0			0
$233$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$233$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$	0.500000	+	0.866025i	0.500000	+	0.866025i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			1.41421i			1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$239$			1.41421i			1.41421i	−0.707107	+	0.707107i	$0.750000\pi$
$240$		0			0
$241$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$241$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$	1.36603	−	0.366025i	1.36603	−	0.366025i
$244$		0			0
$245$	0.965926	−	0.258819i	0.965926	−	0.258819i
$246$		0			0
$247$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$248$		0			0
$249$	1.73205	−	1.00000i	1.73205	−	1.00000i
$250$		0			0
$251$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$251$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$252$		0			0
$253$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$257$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	0.500000	−	0.866025i	0.500000	−	0.866025i
$260$		0			0
$261$	1.22474	+	0.707107i	1.22474	+	0.707107i
$262$		0			0
$263$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$263$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$264$		0			0
$265$	1.00000			1.00000
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$269$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$270$		0			0
$271$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$271$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$272$		0			0
$273$	1.36603	+	0.366025i	1.36603	+	0.366025i
$274$		0			0
$275$	0.500000	−	0.866025i	0.500000	−	0.866025i
$276$		0			0
$277$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$277$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$278$		0			0
$279$			1.41421i			1.41421i
$280$		0			0
$281$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$281$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$282$		0			0
$283$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$283$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$284$		0			0
$285$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$286$		0			0
$287$	0.707107	+	0.707107i	0.707107	+	0.707107i
$288$		0			0
$289$	0.866025	+	0.500000i	0.866025	+	0.500000i
$290$		0			0
$291$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$292$		0			0
$293$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$293$	−0.707107	−	0.707107i	−0.707107	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$306$		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
$309$		0			0
$310$		0			0
$311$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$311$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$312$		0			0
$313$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$313$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$314$		0			0
$315$	0.866025	−	0.500000i	0.866025	−	0.500000i
$316$		0			0
$317$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$317$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$318$		0			0
$319$	1.22474	−	0.707107i	1.22474	−	0.707107i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$326$		0			0
$327$	1.93185	−	0.517638i	1.93185	−	0.517638i
$328$		0			0
$329$	0.866025	+	0.500000i	0.866025	+	0.500000i
$330$		0			0
$331$	0.866025	+	0.500000i	0.866025	+	0.500000i	0.866025	−	0.500000i	$-0.166667\pi$
$331$	0.866025	+	0.500000i	0.866025	+	0.500000i			1.00000i	$0.5\pi$
$332$		0			0
$333$	0.258819	−	0.965926i	0.258819	−	0.965926i
$334$		0			0
$335$		−	1.41421i		−	1.41421i
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$		0			0
$339$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$340$		0			0
$341$	1.22474	+	0.707107i	1.22474	+	0.707107i
$342$		0			0
$343$	0.707107	−	0.707107i	0.707107	−	0.707107i
$344$		0			0
$345$	0.366025	+	1.36603i	0.366025	+	1.36603i
$346$		0			0
$347$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$347$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$348$		0			0
$349$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$349$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$353$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$354$		0			0
$355$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$359$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$360$		0			0
$361$		0			0
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$367$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$368$		0			0
$369$	0.866025	+	0.500000i	0.866025	+	0.500000i
$370$		0			0
$371$	0.866025	−	0.500000i	0.866025	−	0.500000i
$372$		0			0
$373$	−1.93185	+	0.517638i	−1.93185	+	0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$373$	−1.93185	+	0.517638i	−1.93185	+	0.517638i	−0.965926	+	0.258819i	$0.916667\pi$
$374$		0			0
$375$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$376$		0			0
$377$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$378$		0			0
$379$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$379$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$380$		0			0
$381$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$382$		0			0
$383$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	−0.965926	−	0.258819i	$-0.916667\pi$
$383$	−0.258819	−	0.965926i	−0.258819	−	0.965926i	0.707107	−	0.707107i	$-0.250000\pi$
$384$		0			0
$385$		−	1.00000i		−	1.00000i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$389$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$397$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$398$		0			0
$399$			1.41421i			1.41421i
$400$		0			0
$401$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$401$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$402$		0			0
$403$	0.366025	−	1.36603i	0.366025	−	1.36603i
$404$		0			0
$405$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$406$		0			0
$407$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$408$		0			0
$409$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$409$	−1.73205	+	1.00000i	−1.73205	+	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$419$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$421$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$422$		0			0
$423$	0.965926	+	0.258819i	0.965926	+	0.258819i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	0.366025	+	1.36603i	0.366025	+	1.36603i
$428$		0			0
$429$	0.707107	−	1.22474i	0.707107	−	1.22474i
$430$		0			0
$431$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$431$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$432$		0			0
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$	−2.00000			−2.00000
$436$		0			0
$437$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$438$		0			0
$439$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$439$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$440$		0			0
$441$	0.500000	−	0.866025i	0.500000	−	0.866025i
$442$		0			0
$443$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$443$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$449$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$450$		0			0
$451$	0.866025	−	0.500000i	0.866025	−	0.500000i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$456$		0			0
$457$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$457$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			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
$463$	0.707107	−	0.707107i	0.707107	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$463$	0.707107	−	0.707107i	0.707107	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$464$		0			0
$465$	−1.00000	−	1.73205i	−1.00000	−	1.73205i
$466$		0			0
$467$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$467$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$468$		0			0
$469$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$470$		0			0
$471$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	1.00000i		−	1.00000i
$476$		0			0
$477$	0.707107	−	0.707107i	0.707107	−	0.707107i
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$482$		0			0
$483$	1.00000	+	1.00000i	1.00000	+	1.00000i
$484$		0			0
$485$	0.707107	−	1.22474i	0.707107	−	1.22474i
$486$		0			0
$487$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$487$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$496$		0			0
$497$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$498$		0			0
$499$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$499$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$500$		0			0
$501$	1.22474	−	0.707107i	1.22474	−	0.707107i
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	0.707107	−	0.707107i	0.707107	−	0.707107i
$518$		0			0
$519$		−	1.41421i		−	1.41421i
$520$		0			0
$521$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$521$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$522$		0			0
$523$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$523$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$524$		0			0
$525$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$526$		0			0
$527$		0			0
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	1.36603	−	0.366025i	1.36603	−	0.366025i
$538$		0			0
$539$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$540$		0			0
$541$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$541$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$546$		0			0
$547$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$547$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$548$		0			0
$549$	0.707107	+	1.22474i	0.707107	+	1.22474i
$550$		0			0
$551$	0.707107	−	1.22474i	0.707107	−	1.22474i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.366025	+	1.36603i	0.366025	+	1.36603i
$556$		0			0
$557$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.707107	−	0.707107i	$-0.250000\pi$
$557$	0.965926	+	0.258819i	0.965926	+	0.258819i	0.258819	+	0.965926i	$0.416667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$563$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$564$		0			0
$565$	1.22474	+	0.707107i	1.22474	+	0.707107i
$566$		0			0
$567$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$568$		0			0
$569$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$569$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$570$		0			0
$571$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$571$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$576$		0			0
$577$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$577$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$578$		0			0
$579$	−1.00000	+	1.73205i	−1.00000	+	1.73205i
$580$		0			0
$581$			1.41421i			1.41421i
$582$		0			0
$583$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$584$		0			0
$585$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$586$		0			0
$587$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$587$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$588$		0			0
$589$	1.41421			1.41421
$590$		0			0
$591$	0.707107	+	1.22474i	0.707107	+	1.22474i
$592$		0			0
$593$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$593$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.517638	−	1.93185i	−0.517638	−	1.93185i
$598$		0			0
$599$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$599$	−1.22474	+	0.707107i	−1.22474	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$600$		0			0
$601$		0			0			−	1.00000i	$-0.5\pi$
$601$		0			0				1.00000i	$0.5\pi$
$602$		0			0
$603$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$607$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$608$		0			0
$609$	−1.73205	+	1.00000i	−1.73205	+	1.00000i
$610$		0			0
$611$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$612$		0			0
$613$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$613$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$614$		0			0
$615$	−1.41421			−1.41421
$616$		0			0
$617$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$617$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$618$		0			0
$619$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$619$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.500000	−	0.866025i	0.500000	−	0.866025i
$626$		0			0
$627$	1.36603	+	0.366025i	1.36603	+	0.366025i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$631$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$632$		0			0
$633$	1.36603	+	0.366025i	1.36603	+	0.366025i
$634$		0			0
$635$	0.500000	−	0.866025i	0.500000	−	0.866025i
$636$		0			0
$637$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$638$		0			0
$639$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$640$		0			0
$641$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	0.500000	−	0.866025i	$-0.333333\pi$
$641$	−0.500000	−	0.866025i	−0.500000	−	0.866025i	−1.00000			$\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$647$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$652$		0			0
$653$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.707107	−	0.707107i	$-0.750000\pi$
$653$	−0.965926	+	0.258819i	−0.965926	+	0.258819i	−0.258819	+	0.965926i	$0.583333\pi$
$654$		0			0
$655$	0.258819	−	0.965926i	0.258819	−	0.965926i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$666$		0			0
$667$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.41421			1.41421
$672$		0			0
$673$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$673$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$677$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$678$		0			0
$679$		−	1.41421i		−	1.41421i
$680$		0			0
$681$	1.00000	−	1.73205i	1.00000	−	1.73205i
$682$		0			0
$683$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$683$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$690$		0			0
$691$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$691$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$692$		0			0
$693$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$	−2.00000			−2.00000
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$704$		0			0
$705$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$709$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.00000	−	1.00000i	1.00000	−	1.00000i
$714$		0			0
$715$			1.00000i			1.00000i
$716$		0			0
$717$	−1.93185	−	0.517638i	−1.93185	−	0.517638i
$718$		0			0
$719$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$719$	1.22474	+	0.707107i	1.22474	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$724$		0			0
$725$	1.22474	−	0.707107i	1.22474	−	0.707107i
$726$		0			0
$727$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$727$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$733$	0.965926	−	0.258819i	0.965926	−	0.258819i	0.707107	+	0.707107i	$0.250000\pi$
$734$		0			0
$735$			1.41421i			1.41421i
$736$		0			0
$737$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$738$		0			0
$739$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$739$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$740$		0			0
$741$		−	1.41421i		−	1.41421i
$742$		0			0
$743$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$743$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.366025	+	1.36603i	0.366025	+	1.36603i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	0.258819	+	0.965926i	$0.416667\pi$
$751$	−0.707107	+	1.22474i	−0.707107	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$752$		0			0
$753$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$757$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$	1.22474	−	0.707107i	1.22474	−	0.707107i
$760$		0			0
$761$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$761$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$762$		0			0
$763$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		−	1.00000i		−	1.00000i	−0.866025	−	0.500000i	$-0.833333\pi$
$769$		−	1.00000i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$770$		0			0
$771$			2.00000i			2.00000i
$772$		0			0
$773$	0.258819	−	0.965926i	0.258819	−	0.965926i	−0.707107	−	0.707107i	$-0.750000\pi$
$773$	0.258819	−	0.965926i	0.258819	−	0.965926i	0.965926	−	0.258819i	$-0.0833333\pi$
$774$		0			0
$775$	1.22474	+	0.707107i	1.22474	+	0.707107i
$776$		0			0
$777$	1.00000	+	1.00000i	1.00000	+	1.00000i
$778$		0			0
$779$	0.500000	−	0.866025i	0.500000	−	0.866025i
$780$		0			0
$781$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	1.00000i		−	1.00000i
$786$		0			0
$787$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$787$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.41421			1.41421
$792$		0			0
$793$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$794$		0			0
$795$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$796$		0			0
$797$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$797$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$806$		0			0
$807$	1.93185	−	0.517638i	1.93185	−	0.517638i
$808$		0			0
$809$	0.866025	−	0.500000i	0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$809$	0.866025	−	0.500000i	0.866025	−	0.500000i	0.866025	+	0.500000i	$0.166667\pi$
$810$		0			0
$811$			1.00000i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$811$			1.00000i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$823$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$824$		0			0
$825$	1.00000	+	1.00000i	1.00000	+	1.00000i
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$		0			0
$829$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$829$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$836$		0			0
$837$		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
$843$	−0.366025	+	1.36603i	−0.366025	+	1.36603i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$852$		0			0
$853$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$853$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$854$		0			0
$855$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$856$		0			0
$857$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$857$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$859$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$860$		0			0
$861$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$862$		0			0
$863$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$863$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$864$		0			0
$865$	0.500000	+	0.866025i	0.500000	+	0.866025i
$866$		0			0
$867$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$868$		0			0
$869$		0			0
$870$		0			0
$871$	0.707107	+	1.22474i	0.707107	+	1.22474i
$872$		0			0
$873$	−0.366025	−	1.36603i	−0.366025	−	1.36603i
$874$		0			0
$875$		−	1.00000i		−	1.00000i
$876$		0			0
$877$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$877$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$	1.22474	−	0.707107i	1.22474	−	0.707107i
$880$		0			0
$881$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$881$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$882$		0			0
$883$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$883$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$887$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$888$		0			0
$889$		−	1.00000i		−	1.00000i
$890$		0			0
$891$	0.866025	+	0.500000i	0.866025	+	0.500000i
$892$		0			0
$893$	0.258819	−	0.965926i	0.258819	−	0.965926i
$894$		0			0
$895$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$896$		0			0
$897$	−1.00000	−	1.00000i	−1.00000	−	1.00000i
$898$		0			0
$899$	1.00000	+	1.73205i	1.00000	+	1.73205i
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.500000	−	0.866025i	$-0.666667\pi$
$907$	−1.36603	−	0.366025i	−1.36603	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	1.36603	+	0.366025i	1.36603	+	0.366025i
$914$		0			0
$915$	−1.73205	−	1.00000i	−1.73205	−	1.00000i
$916$		0			0
$917$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$918$		0			0
$919$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$919$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.00000	+	1.00000i	1.00000	+	1.00000i
$924$		0			0
$925$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$929$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$932$		0			0
$933$	1.93185	−	0.517638i	1.93185	−	0.517638i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$937$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$938$		0			0
$939$	2.00000			2.00000
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$947$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$953$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	0.517638	+	1.93185i	0.517638	+	1.93185i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1.41421i		−	1.41421i
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.866025	−	0.500000i	−0.866025	−	0.500000i		−	1.00000i	$-0.5\pi$
$971$	−0.866025	−	0.500000i	−0.866025	−	0.500000i	−0.866025	+	0.500000i	$0.833333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	0.707107	−	1.22474i	0.707107	−	1.22474i
$976$		0			0
$977$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$977$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$			1.41421i			1.41421i
$982$		0			0
$983$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$983$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$986$		0			0
$987$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$991$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$992$		0			0
$993$	−1.00000	+	1.00000i	−1.00000	+	1.00000i
$994$		0			0
$995$	1.00000	+	1.00000i	1.00000	+	1.00000i
$996$		0			0
$997$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$997$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$998$		0			0
$999$		0			0

Display

a_p

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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	2240.1.dl.b.737.1	yes	8
4.3	odd	2		2240.1.dl.a.737.1	yes	8
5.3	odd	4		2240.1.dl.a.1633.1	yes	8
7.4	even	3	inner	2240.1.dl.b.417.2	yes	8
8.3	odd	2	inner	2240.1.dl.b.737.2	yes	8
8.5	even	2		2240.1.dl.a.737.2	yes	8
20.3	even	4	inner	2240.1.dl.b.1633.1	yes	8
28.11	odd	6		2240.1.dl.a.417.2	yes	8
35.18	odd	12		2240.1.dl.a.1313.2	yes	8
40.3	even	4		2240.1.dl.a.1633.2	yes	8
40.13	odd	4	inner	2240.1.dl.b.1633.2	yes	8
56.11	odd	6	inner	2240.1.dl.b.417.1	yes	8
56.53	even	6		2240.1.dl.a.417.1	✓	8
140.123	even	12	inner	2240.1.dl.b.1313.2	yes	8
280.53	odd	12	inner	2240.1.dl.b.1313.1	yes	8
280.123	even	12		2240.1.dl.a.1313.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2240.1.dl.a.417.1	✓	8	56.53	even	6
2240.1.dl.a.417.2	yes	8	28.11	odd	6
2240.1.dl.a.737.1	yes	8	4.3	odd	2
2240.1.dl.a.737.2	yes	8	8.5	even	2
2240.1.dl.a.1313.1	yes	8	280.123	even	12
2240.1.dl.a.1313.2	yes	8	35.18	odd	12
2240.1.dl.a.1633.1	yes	8	5.3	odd	4
2240.1.dl.a.1633.2	yes	8	40.3	even	4
2240.1.dl.b.417.1	yes	8	56.11	odd	6	inner
2240.1.dl.b.417.2	yes	8	7.4	even	3	inner
2240.1.dl.b.737.1	yes	8	1.1	even	1	trivial
2240.1.dl.b.737.2	yes	8	8.3	odd	2	inner
2240.1.dl.b.1313.1	yes	8	280.53	odd	12	inner
2240.1.dl.b.1313.2	yes	8	140.123	even	12	inner
2240.1.dl.b.1633.1	yes	8	20.3	even	4	inner
2240.1.dl.b.1633.2	yes	8	40.13	odd	4	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	2240.1.dl.b.737.1

Level	$2240$
Weight	$1$
Character	2240.737
Analytic conductor	$1.118$
Analytic rank	$0$
Dimension	$8$
Projective image	$S_{4}$
CM/RM	no
Inner twists	$8$