LMFDB - Embedded newform 1472.1.s.b.597.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,1,Mod(45,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, base_ring=CyclotomicField(16))

chi = DirichletCharacter(H, H._module([0, 7, 8]))

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

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

chi := DirichletCharacter("1472.45");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1472 = 2^{6} \cdot 23$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1472.s (of order $16$ , degree $8$ , minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$0.734623698596$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$2$ over $\Q(\zeta_{16})$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - x^{8} + 1$ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{48}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{48} - \cdots)$

Embedding invariants

Embedding label			597.2
Root			$0.608761 - 0.793353i$ of defining polynomial
Character	$\chi$	$=$	1472.597
Dual form			1472.1.s.b.1149.2

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.608761 - 0.793353i) q^{2} +(-0.125419 - 0.630526i) q^{3} +(-0.258819 - 0.965926i) q^{4} +(-0.576581 - 0.284338i) q^{6} +(-0.923880 - 0.382683i) q^{8} +(0.542046 - 0.224523i) q^{9} +(-0.576581 + 0.284338i) q^{12} +(-1.09645 - 0.732626i) q^{13} +(-0.866025 + 0.500000i) q^{16} +(0.151851 - 0.566715i) q^{18} +(-0.382683 - 0.923880i) q^{23} +(-0.125419 + 0.630526i) q^{24} +(0.382683 - 0.923880i) q^{25} +(-1.24871 + 0.423880i) q^{26} +(-0.566715 - 0.848149i) q^{27} +(0.867580 - 0.172572i) q^{29} +1.21752i q^{31} +(-0.130526 + 0.991445i) q^{32} +(-0.357164 - 0.465466i) q^{36} +(-0.324423 + 0.783227i) q^{39} +(0.382683 + 0.923880i) q^{41} +(-0.965926 - 0.258819i) q^{46} +(-0.366025 + 0.366025i) q^{47} +(0.423880 + 0.483342i) q^{48} +(0.707107 + 0.707107i) q^{49} +(-0.500000 - 0.866025i) q^{50} +(-0.423880 + 1.24871i) q^{52} +(-1.01788 - 0.0667151i) q^{54} +(0.391239 - 0.793353i) q^{58} +(-0.324423 + 0.216773i) q^{59} +(0.965926 + 0.741181i) q^{62} +(0.707107 + 0.707107i) q^{64} +(-0.534534 + 0.357164i) q^{69} +(-0.241181 - 0.0999004i) q^{71} -0.586707 q^{72} +(1.78480 - 0.739288i) q^{73} +(-0.630526 - 0.125419i) q^{75} +(0.423880 + 0.734181i) q^{78} +(-0.0488389 + 0.0488389i) q^{81} +(0.965926 + 0.258819i) q^{82} +(-0.217623 - 0.525388i) q^{87} +(-0.793353 + 0.608761i) q^{92} +(0.767680 - 0.152701i) q^{93} +(0.0675653 + 0.513210i) q^{94} +(0.641502 - 0.0420463i) q^{96} +(0.991445 - 0.130526i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 8 q^{9} + 8 q^{47} - 8 q^{48} - 8 q^{50} + 8 q^{52} + 16 q^{58} - 8 q^{71} + 16 q^{72} - 8 q^{75} - 8 q^{78} - 8 q^{81} - 8 q^{87}+O(q^{100})$ 16 * q + 8 * q^9 + 8 * q^47 - 8 * q^48 - 8 * q^50 + 8 * q^52 + 16 * q^58 - 8 * q^71 + 16 * q^72 - 8 * q^75 - 8 * q^78 - 8 * q^81 - 8 * q^87

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times$ .

$n$	$645$	$833$	$1151$
$\chi(n)$	$e\left(\frac{13}{16}\right)$	$-1$	$1$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0.608761	−	0.793353i	0.608761	−	0.793353i
$2$	0.608761	−	0.793353i	0.608761	−	0.793353i
$3$	−0.125419	−	0.630526i	−0.125419	−	0.630526i	−0.991445	−	0.130526i	$-0.958333\pi$
$3$	−0.125419	−	0.630526i	−0.125419	−	0.630526i	0.866025	−	0.500000i	$-0.166667\pi$
$4$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$5$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$5$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$6$	−0.576581	−	0.284338i	−0.576581	−	0.284338i
$7$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$7$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$8$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$9$	0.542046	−	0.224523i	0.542046	−	0.224523i
$10$		0			0
$11$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$11$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$12$	−0.576581	+	0.284338i	−0.576581	+	0.284338i
$13$	−1.09645	−	0.732626i	−1.09645	−	0.732626i	−0.130526	−	0.991445i	$-0.541667\pi$
$13$	−1.09645	−	0.732626i	−1.09645	−	0.732626i	−0.965926	+	0.258819i	$0.916667\pi$
$14$		0			0
$15$		0			0
$16$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$17$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$	0.151851	−	0.566715i	0.151851	−	0.566715i
$19$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$19$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$23$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$24$	−0.125419	+	0.630526i	−0.125419	+	0.630526i
$25$	0.382683	−	0.923880i	0.382683	−	0.923880i
$26$	−1.24871	+	0.423880i	−1.24871	+	0.423880i
$27$	−0.566715	−	0.848149i	−0.566715	−	0.848149i
$28$		0			0
$29$	0.867580	−	0.172572i	0.867580	−	0.172572i	0.258819	−	0.965926i	$-0.416667\pi$
$29$	0.867580	−	0.172572i	0.867580	−	0.172572i	0.608761	+	0.793353i	$0.291667\pi$
$30$		0			0
$31$			1.21752i			1.21752i	0.793353	+	0.608761i	$0.208333\pi$
$31$			1.21752i			1.21752i	−0.793353	+	0.608761i	$0.791667\pi$
$32$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.357164	−	0.465466i	−0.357164	−	0.465466i
$37$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$37$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$38$		0			0
$39$	−0.324423	+	0.783227i	−0.324423	+	0.783227i
$40$		0			0
$41$	0.382683	+	0.923880i	0.382683	+	0.923880i	0.991445	+	0.130526i	$0.0416667\pi$
$41$	0.382683	+	0.923880i	0.382683	+	0.923880i	−0.608761	+	0.793353i	$0.708333\pi$
$42$		0			0
$43$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$43$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$44$		0			0
$45$		0			0
$46$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$47$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$47$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$48$	0.423880	+	0.483342i	0.423880	+	0.483342i
$49$	0.707107	+	0.707107i	0.707107	+	0.707107i
$50$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$51$		0			0
$52$	−0.423880	+	1.24871i	−0.423880	+	1.24871i
$53$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$53$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$54$	−1.01788	−	0.0667151i	−1.01788	−	0.0667151i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.391239	−	0.793353i	0.391239	−	0.793353i
$59$	−0.324423	+	0.216773i	−0.324423	+	0.216773i	−0.707107	−	0.707107i	$-0.750000\pi$
$59$	−0.324423	+	0.216773i	−0.324423	+	0.216773i	0.382683	+	0.923880i	$0.375000\pi$
$60$		0			0
$61$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$61$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$62$	0.965926	+	0.741181i	0.965926	+	0.741181i
$63$		0			0
$64$	0.707107	+	0.707107i	0.707107	+	0.707107i
$65$		0			0
$66$		0			0
$67$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$67$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$68$		0			0
$69$	−0.534534	+	0.357164i	−0.534534	+	0.357164i
$70$		0			0
$71$	−0.241181	−	0.0999004i	−0.241181	−	0.0999004i	0.258819	−	0.965926i	$-0.416667\pi$
$71$	−0.241181	−	0.0999004i	−0.241181	−	0.0999004i	−0.500000	+	0.866025i	$0.666667\pi$
$72$	−0.586707			−0.586707
$73$	1.78480	−	0.739288i	1.78480	−	0.739288i	0.793353	−	0.608761i	$-0.208333\pi$
$73$	1.78480	−	0.739288i	1.78480	−	0.739288i	0.991445	−	0.130526i	$-0.0416667\pi$
$74$		0			0
$75$	−0.630526	−	0.125419i	−0.630526	−	0.125419i
$76$		0			0
$77$		0			0
$78$	0.423880	+	0.734181i	0.423880	+	0.734181i
$79$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$79$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$80$		0			0
$81$	−0.0488389	+	0.0488389i	−0.0488389	+	0.0488389i
$82$	0.965926	+	0.258819i	0.965926	+	0.258819i
$83$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$83$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.217623	−	0.525388i	−0.217623	−	0.525388i
$88$		0			0
$89$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$89$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$90$		0			0
$91$		0			0
$92$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$93$	0.767680	−	0.152701i	0.767680	−	0.152701i
$94$	0.0675653	+	0.513210i	0.0675653	+	0.513210i
$95$		0			0
$96$	0.641502	−	0.0420463i	0.641502	−	0.0420463i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	0.991445	−	0.130526i	0.991445	−	0.130526i
$99$		0			0
$100$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$101$	0.216773	+	0.324423i	0.216773	+	0.324423i	0.923880	−	0.382683i	$-0.125000\pi$
$101$	0.216773	+	0.324423i	0.216773	+	0.324423i	−0.707107	+	0.707107i	$0.750000\pi$
$102$		0			0
$103$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$103$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$104$	0.732626	+	1.09645i	0.732626	+	1.09645i
$105$		0			0
$106$		0			0
$107$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$107$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$108$	−0.672572	+	0.766922i	−0.672572	+	0.766922i
$109$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$109$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$113$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$114$		0			0
$115$		0			0
$116$	−0.391239	−	0.793353i	−0.391239	−	0.793353i
$117$	−0.758819	−	0.150938i	−0.758819	−	0.150938i
$118$	−0.0255190	+	0.389345i	−0.0255190	+	0.389345i
$119$		0			0
$120$		0			0
$121$	0.923880	+	0.382683i	0.923880	+	0.382683i
$122$		0			0
$123$	0.534534	−	0.357164i	0.534534	−	0.357164i
$124$	1.17604	−	0.315118i	1.17604	−	0.315118i
$125$		0			0
$126$		0			0
$127$	1.93185			1.93185			0.965926	−	0.258819i	$-0.0833333\pi$
$127$	1.93185			1.93185			0.965926	+	0.258819i	$0.0833333\pi$
$128$	0.991445	−	0.130526i	0.991445	−	0.130526i
$129$		0			0
$130$		0			0
$131$	0.389345	+	1.95737i	0.389345	+	1.95737i	0.258819	+	0.965926i	$0.416667\pi$
$131$	0.389345	+	1.95737i	0.389345	+	1.95737i	0.130526	+	0.991445i	$0.458333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$137$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$138$	−0.0420463	+	0.641502i	−0.0420463	+	0.641502i
$139$	−0.128293	−	0.0255190i	−0.128293	−	0.0255190i	0.130526	−	0.991445i	$-0.458333\pi$
$139$	−0.128293	−	0.0255190i	−0.128293	−	0.0255190i	−0.258819	+	0.965926i	$0.583333\pi$
$140$		0			0
$141$	0.276695	+	0.184882i	0.276695	+	0.184882i
$142$	−0.226078	+	0.130526i	−0.226078	+	0.130526i
$143$		0			0
$144$	−0.357164	+	0.465466i	−0.357164	+	0.465466i
$145$		0			0
$146$	0.500000	−	1.86603i	0.500000	−	1.86603i
$147$	0.357164	−	0.534534i	0.357164	−	0.534534i
$148$		0			0
$149$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$149$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$150$	−0.483342	+	0.423880i	−0.483342	+	0.423880i
$151$	−0.662827	−	1.60021i	−0.662827	−	1.60021i	−0.793353	−	0.608761i	$-0.791667\pi$
$151$	−0.662827	−	1.60021i	−0.662827	−	1.60021i	0.130526	−	0.991445i	$-0.458333\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.840506	+	0.110655i	0.840506	+	0.110655i
$157$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$157$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.00901526	+	0.0684777i	0.00901526	+	0.0684777i
$163$	1.47479	−	0.293353i	1.47479	−	0.293353i	0.608761	−	0.793353i	$-0.291667\pi$
$163$	1.47479	−	0.293353i	1.47479	−	0.293353i	0.866025	+	0.500000i	$0.166667\pi$
$164$	0.793353	−	0.608761i	0.793353	−	0.608761i
$165$		0			0
$166$		0			0
$167$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$167$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	−1.00000			$\pi$
$168$		0			0
$169$	0.282783	+	0.682699i	0.282783	+	0.682699i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.923880	−	1.38268i	0.923880	−	1.38268i		−	1.00000i	$-0.5\pi$
$173$	0.923880	−	1.38268i	0.923880	−	1.38268i	0.923880	−	0.382683i	$-0.125000\pi$
$174$	−0.549299	−	0.147184i	−0.549299	−	0.147184i
$175$		0			0
$176$		0			0
$177$	0.177370	+	0.177370i	0.177370	+	0.177370i
$178$		0			0
$179$	−1.25026	−	0.835400i	−1.25026	−	0.835400i	−0.258819	−	0.965926i	$-0.583333\pi$
$179$	−1.25026	−	0.835400i	−1.25026	−	0.835400i	−0.991445	+	0.130526i	$0.958333\pi$
$180$		0			0
$181$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$181$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$182$		0			0
$183$		0			0
$184$			1.00000i			1.00000i
$185$		0			0
$186$	0.346188	−	0.702000i	0.346188	−	0.702000i
$187$		0			0
$188$	0.448288	+	0.258819i	0.448288	+	0.258819i
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	0.357164	−	0.534534i	0.357164	−	0.534534i
$193$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$193$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$194$		0			0
$195$		0			0
$196$	0.500000	−	0.866025i	0.500000	−	0.866025i
$197$	−1.65938	+	1.10876i	−1.65938	+	1.10876i	−0.793353	+	0.608761i	$0.791667\pi$
$197$	−1.65938	+	1.10876i	−1.65938	+	1.10876i	−0.866025	+	0.500000i	$0.833333\pi$
$198$		0			0
$199$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$199$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$200$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$201$		0			0
$202$	0.389345	+	0.0255190i	0.389345	+	0.0255190i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.414864	−	0.414864i	−0.414864	−	0.414864i
$208$	1.31587	+	0.0862466i	1.31587	+	0.0862466i
$209$		0			0
$210$		0			0
$211$	1.08979	−	1.63099i	1.08979	−	1.63099i	0.382683	−	0.923880i	$-0.375000\pi$
$211$	1.08979	−	1.63099i	1.08979	−	1.63099i	0.707107	−	0.707107i	$-0.250000\pi$
$212$		0			0
$213$	−0.0327410	+	0.164600i	−0.0327410	+	0.164600i
$214$		0			0
$215$		0			0
$216$	0.199004	+	1.00046i	0.199004	+	1.00046i
$217$		0			0
$218$		0			0
$219$	−0.689989	−	1.03264i	−0.689989	−	1.03264i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			0.765367i			0.765367i	0.923880	+	0.382683i	$0.125000\pi$
$223$			0.765367i			0.765367i	−0.923880	+	0.382683i	$0.875000\pi$
$224$		0			0
$225$		−	0.586707i		−	0.586707i
$226$		0			0
$227$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$227$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$228$		0			0
$229$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$229$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$230$		0			0
$231$		0			0
$232$	−0.867580	−	0.172572i	−0.867580	−	0.172572i
$233$	0.758819	+	1.83195i	0.758819	+	1.83195i	0.500000	+	0.866025i	$0.333333\pi$
$233$	0.758819	+	1.83195i	0.758819	+	1.83195i	0.258819	+	0.965926i	$0.416667\pi$
$234$	−0.581687	+	0.510126i	−0.581687	+	0.510126i
$235$		0			0
$236$	0.293353	+	0.257264i	0.293353	+	0.257264i
$237$		0			0
$238$		0			0
$239$	−1.12197	+	1.12197i	−1.12197	+	1.12197i	−0.130526	+	0.991445i	$0.541667\pi$
$239$	−1.12197	+	1.12197i	−1.12197	+	1.12197i	−0.991445	+	0.130526i	$0.958333\pi$
$240$		0			0
$241$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$241$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$242$	0.866025	−	0.500000i	0.866025	−	0.500000i
$243$	−0.811230	−	0.542046i	−0.811230	−	0.542046i
$244$		0			0
$245$		0			0
$246$	0.0420463	−	0.641502i	0.0420463	−	0.641502i
$247$		0			0
$248$	0.465926	−	1.12484i	0.465926	−	1.12484i
$249$		0			0
$250$		0			0
$251$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$251$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$252$		0			0
$253$		0			0
$254$	1.17604	−	1.53264i	1.17604	−	1.53264i
$255$		0			0
$256$	0.500000	−	0.866025i	0.500000	−	0.866025i
$257$	0.261052			0.261052			0.130526	−	0.991445i	$-0.458333\pi$
$257$	0.261052			0.261052			0.130526	+	0.991445i	$0.458333\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.431522	−	0.288334i	0.431522	−	0.288334i
$262$	1.78990	+	0.882683i	1.78990	+	0.882683i
$263$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$263$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.108761	−	0.0726721i	−0.108761	−	0.0726721i	0.500000	−	0.866025i	$-0.333333\pi$
$269$	−0.108761	−	0.0726721i	−0.108761	−	0.0726721i	−0.608761	+	0.793353i	$0.708333\pi$
$270$		0			0
$271$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$271$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	0.483342	+	0.423880i	0.483342	+	0.423880i
$277$	−0.369474	+	1.85747i	−0.369474	+	1.85747i	0.130526	+	0.991445i	$0.458333\pi$
$277$	−0.369474	+	1.85747i	−0.369474	+	1.85747i	−0.500000	+	0.866025i	$0.666667\pi$
$278$	−0.0983454	+	0.0862466i	−0.0983454	+	0.0862466i
$279$	0.273362	+	0.659954i	0.273362	+	0.659954i
$280$		0			0
$281$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$281$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$282$	0.315118	−	0.106968i	0.315118	−	0.106968i
$283$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$283$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$284$	−0.0340742	+	0.258819i	−0.0340742	+	0.258819i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.151851	+	0.566715i	0.151851	+	0.566715i
$289$		−	1.00000i		−	1.00000i
$290$		0			0
$291$		0			0
$292$	−1.17604	−	1.53264i	−1.17604	−	1.53264i
$293$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$293$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$294$	−0.206647	−	0.608761i	−0.206647	−	0.608761i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.257264	+	1.29335i	−0.257264	+	1.29335i
$300$	0.0420463	+	0.641502i	0.0420463	+	0.641502i
$301$		0			0
$302$	−1.67303	−	0.448288i	−1.67303	−	0.448288i
$303$	0.177370	−	0.177370i	0.177370	−	0.177370i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.923880	−	0.617317i	−0.923880	−	0.617317i		−	1.00000i	$-0.5\pi$
$307$	−0.923880	−	0.617317i	−0.923880	−	0.617317i	−0.923880	+	0.382683i	$0.875000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1.60021	+	0.662827i	−1.60021	+	0.662827i	−0.991445	−	0.130526i	$-0.958333\pi$
$311$	−1.60021	+	0.662827i	−1.60021	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$312$	0.599456	−	0.599456i	0.599456	−	0.599456i
$313$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$313$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.216773	+	1.08979i	0.216773	+	1.08979i	0.923880	+	0.382683i	$0.125000\pi$
$317$	0.216773	+	1.08979i	0.216773	+	1.08979i	−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.0598152	+	0.0345343i	0.0598152	+	0.0345343i
$325$	−1.09645	+	0.732626i	−1.09645	+	0.732626i
$326$	0.665060	−	1.34861i	0.665060	−	1.34861i
$327$		0			0
$328$		−	1.00000i		−	1.00000i
$329$		0			0
$330$		0			0
$331$	−1.29335	−	0.257264i	−1.29335	−	0.257264i	−0.500000	−	0.866025i	$-0.666667\pi$
$331$	−1.29335	−	0.257264i	−1.29335	−	0.257264i	−0.793353	+	0.608761i	$0.791667\pi$
$332$		0			0
$333$		0			0
$334$	0.382683	+	0.662827i	0.382683	+	0.662827i
$335$		0			0
$336$		0			0
$337$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$337$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$338$	0.713769	+	0.191254i	0.713769	+	0.191254i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.534534	−	1.57469i	−0.534534	−	1.57469i
$347$	0.617317	+	0.923880i	0.617317	+	0.923880i	1.00000			$0$
$347$	0.617317	+	0.923880i	0.617317	+	0.923880i	−0.382683	+	0.923880i	$0.625000\pi$
$348$	−0.451161	+	0.346188i	−0.451161	+	0.346188i
$349$	−0.630526	+	0.125419i	−0.630526	+	0.125419i	−0.500000	−	0.866025i	$-0.666667\pi$
$349$	−0.630526	+	0.125419i	−0.630526	+	0.125419i	−0.130526	+	0.991445i	$0.541667\pi$
$350$		0			0
$351$			1.34514i			1.34514i
$352$		0			0
$353$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$353$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$354$	0.248693	−	0.0327410i	0.248693	−	0.0327410i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−1.42388	+	0.483342i	−1.42388	+	0.483342i
$359$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$359$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$360$		0			0
$361$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$362$		0			0
$363$	0.125419	−	0.630526i	0.125419	−	0.630526i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$367$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$368$	0.793353	+	0.608761i	0.793353	+	0.608761i
$369$	0.414864	+	0.414864i	0.414864	+	0.414864i
$370$		0			0
$371$		0			0
$372$	−0.346188	−	0.702000i	−0.346188	−	0.702000i
$373$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$373$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$374$		0			0
$375$		0			0
$376$	0.478235	−	0.198092i	0.478235	−	0.198092i
$377$	−1.07769	−	0.446394i	−1.07769	−	0.446394i
$378$		0			0
$379$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$379$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$380$		0			0
$381$	−0.242292	−	1.21808i	−0.242292	−	1.21808i
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	−0.206647	−	0.608761i	−0.206647	−	0.608761i
$385$		0			0
$386$	−1.05441	+	1.37413i	−1.05441	+	1.37413i
$387$		0			0
$388$		0			0
$389$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$389$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$390$		0			0
$391$		0			0
$392$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$393$	1.18534	−	0.490985i	1.18534	−	0.490985i
$394$	−0.130526	+	1.99144i	−0.130526	+	1.99144i
$395$		0			0
$396$		0			0
$397$	−1.49144	−	0.996552i	−1.49144	−	0.996552i	−0.991445	−	0.130526i	$-0.958333\pi$
$397$	−1.49144	−	0.996552i	−1.49144	−	0.996552i	−0.500000	−	0.866025i	$-0.666667\pi$
$398$		0			0
$399$		0			0
$400$	0.130526	+	0.991445i	0.130526	+	0.991445i
$401$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$401$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$402$		0			0
$403$	0.891989	−	1.33496i	0.891989	−	1.33496i
$404$	0.257264	−	0.293353i	0.257264	−	0.293353i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.758819	+	1.83195i	−0.758819	+	1.83195i	−0.258819	+	0.965926i	$0.583333\pi$
$409$	−0.758819	+	1.83195i	−0.758819	+	1.83195i	−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.581687	+	0.0765806i	−0.581687	+	0.0765806i
$415$		0			0
$416$	0.869474	−	0.991445i	0.869474	−	0.991445i
$417$			0.0840926i			0.0840926i
$418$		0			0
$419$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$419$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$420$		0			0
$421$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$421$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$422$	−0.630526	−	1.85747i	−0.630526	−	1.85747i
$423$	−0.116222	+	0.280584i	−0.116222	+	0.280584i
$424$		0			0
$425$		0			0
$426$	0.110655	+	0.126178i	0.110655	+	0.126178i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$431$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$432$	0.914864	+	0.451161i	0.914864	+	0.451161i
$433$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$433$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−1.23929	−	0.0812272i	−1.23929	−	0.0812272i
$439$	1.83195	−	0.758819i	1.83195	−	0.758819i	0.866025	−	0.500000i	$-0.166667\pi$
$439$	1.83195	−	0.758819i	1.83195	−	0.758819i	0.965926	−	0.258819i	$-0.0833333\pi$
$440$		0			0
$441$	0.542046	+	0.224523i	0.542046	+	0.224523i
$442$		0			0
$443$	−0.108761	+	0.0726721i	−0.108761	+	0.0726721i	−0.608761	−	0.793353i	$-0.708333\pi$
$443$	−0.108761	+	0.0726721i	−0.108761	+	0.0726721i	0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$	0.607206	+	0.465926i	0.607206	+	0.465926i
$447$		0			0
$448$		0			0
$449$	−1.84776			−1.84776			−0.923880	−	0.382683i	$-0.875000\pi$
$449$	−1.84776			−1.84776			−0.923880	+	0.382683i	$0.875000\pi$
$450$	−0.465466	−	0.357164i	−0.465466	−	0.357164i
$451$		0			0
$452$		0			0
$453$	−0.925841	+	0.618627i	−0.925841	+	0.618627i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$457$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.57469	−	1.05217i	−1.57469	−	1.05217i	−0.965926	−	0.258819i	$-0.916667\pi$
$461$	−1.57469	−	1.05217i	−1.57469	−	1.05217i	−0.608761	−	0.793353i	$-0.708333\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	−0.665060	+	0.583242i	−0.665060	+	0.583242i
$465$		0			0
$466$	1.91532	+	0.513210i	1.91532	+	0.513210i
$467$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$467$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$468$	0.0506014	+	0.772029i	0.0506014	+	0.772029i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	0.207107	+	1.57313i	0.207107	+	1.57313i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.130526	−	0.991445i	0.130526	−	0.991445i
$485$		0			0
$486$	−0.923880	+	0.313615i	−0.923880	+	0.313615i
$487$	−0.0999004	+	0.241181i	−0.0999004	+	0.241181i	−0.965926	−	0.258819i	$-0.916667\pi$
$487$	−0.0999004	+	0.241181i	−0.0999004	+	0.241181i	0.866025	+	0.500000i	$0.166667\pi$
$488$		0			0
$489$	−0.369934	−	0.893100i	−0.369934	−	0.893100i
$490$		0			0
$491$	−0.172572	+	0.867580i	−0.172572	+	0.867580i	0.793353	+	0.608761i	$0.208333\pi$
$491$	−0.172572	+	0.867580i	−0.172572	+	0.867580i	−0.965926	+	0.258819i	$0.916667\pi$
$492$	−0.483342	−	0.423880i	−0.483342	−	0.423880i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.608761	−	1.05441i	−0.608761	−	1.05441i
$497$		0			0
$498$		0			0
$499$	1.49144	+	0.996552i	1.49144	+	0.996552i	0.991445	+	0.130526i	$0.0416667\pi$
$499$	1.49144	+	0.996552i	1.49144	+	0.996552i	0.500000	+	0.866025i	$0.333333\pi$
$500$		0			0
$501$	0.482584	+	0.0959919i	0.482584	+	0.0959919i
$502$		0			0
$503$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$503$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.394993	−	0.263926i	0.394993	−	0.263926i
$508$	−0.500000	−	1.86603i	−0.500000	−	1.86603i
$509$	0.293353	+	1.47479i	0.293353	+	1.47479i	0.793353	+	0.608761i	$0.208333\pi$
$509$	0.293353	+	1.47479i	0.293353	+	1.47479i	−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$		0			0
$512$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$513$		0			0
$514$	0.158919	−	0.207107i	0.158919	−	0.207107i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−0.987691	−	0.409115i	−0.987691	−	0.409115i
$520$		0			0
$521$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$521$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$522$	0.0339434	−	0.517876i	0.0339434	−	0.517876i
$523$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$523$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$524$	1.78990	−	0.882683i	1.78990	−	0.882683i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$530$		0			0
$531$	−0.127182	+	0.190341i	−0.127182	+	0.190341i
$532$		0			0
$533$	0.257264	−	1.29335i	0.257264	−	1.29335i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−0.369934	+	0.893100i	−0.369934	+	0.893100i
$538$	−0.123864	+	0.0420463i	−0.123864	+	0.0420463i
$539$		0			0
$540$		0			0
$541$	1.75928	−	0.349942i	1.75928	−	0.349942i	0.793353	−	0.608761i	$-0.208333\pi$
$541$	1.75928	−	0.349942i	1.75928	−	0.349942i	0.965926	+	0.258819i	$0.0833333\pi$
$542$	1.40211	−	0.184592i	1.40211	−	0.184592i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.867580	−	0.172572i	0.867580	−	0.172572i	0.258819	−	0.965926i	$-0.416667\pi$
$547$	0.867580	−	0.172572i	0.867580	−	0.172572i	0.608761	+	0.793353i	$0.291667\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	0.630526	−	0.125419i	0.630526	−	0.125419i
$553$		0			0
$554$	1.24871	+	1.42388i	1.24871	+	1.42388i
$555$		0			0
$556$	0.00855514	+	0.130526i	0.00855514	+	0.130526i
$557$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$557$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$558$	0.689989	+	0.184882i	0.689989	+	0.184882i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$563$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$564$	0.106968	−	0.315118i	0.106968	−	0.315118i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	0.184592	+	0.184592i	0.184592	+	0.184592i
$569$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$569$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$570$		0			0
$571$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$571$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.00000			−1.00000
$576$	0.542046	+	0.224523i	0.542046	+	0.224523i
$577$	−1.98289			−1.98289			−0.991445	−	0.130526i	$-0.958333\pi$
$577$	−1.98289			−1.98289			−0.991445	+	0.130526i	$0.958333\pi$
$578$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$579$	0.217233	+	1.09210i	0.217233	+	1.09210i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−1.93185			−1.93185
$585$		0			0
$586$		0			0
$587$	−1.75928	−	0.349942i	−1.75928	−	0.349942i	−0.793353	−	0.608761i	$-0.791667\pi$
$587$	−1.75928	−	0.349942i	−1.75928	−	0.349942i	−0.965926	+	0.258819i	$0.916667\pi$
$588$	−0.608761	−	0.206647i	−0.608761	−	0.206647i
$589$		0			0
$590$		0			0
$591$	0.907222	+	0.907222i	0.907222	+	0.907222i
$592$		0			0
$593$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$593$	−1.30656	+	1.30656i	−1.30656	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	0.869474	+	0.991445i	0.869474	+	0.991445i
$599$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$599$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$600$	0.534534	+	0.357164i	0.534534	+	0.357164i
$601$	0.382683	−	0.923880i	0.382683	−	0.923880i	−0.608761	−	0.793353i	$-0.708333\pi$
$601$	0.382683	−	0.923880i	0.382683	−	0.923880i	0.991445	−	0.130526i	$-0.0416667\pi$
$602$		0			0
$603$		0			0
$604$	−1.37413	+	1.05441i	−1.37413	+	1.05441i
$605$		0			0
$606$	−0.0327410	−	0.248693i	−0.0327410	−	0.248693i
$607$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$607$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.669489	−	0.133170i	0.669489	−	0.133170i
$612$		0			0
$613$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$613$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$614$	−1.05217	+	0.357164i	−1.05217	+	0.357164i
$615$		0			0
$616$		0			0
$617$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$617$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$618$		0			0
$619$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$619$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$620$		0			0
$621$	−0.566715	+	0.848149i	−0.566715	+	0.848149i
$622$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$623$		0			0
$624$	−0.110655	−	0.840506i	−0.110655	−	0.840506i
$625$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$631$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$632$		0			0
$633$	−1.16506	−	0.482584i	−1.16506	−	0.482584i
$634$	0.996552	+	0.491445i	0.996552	+	0.491445i
$635$		0			0
$636$		0			0
$637$	−0.257264	−	1.29335i	−0.257264	−	1.29335i
$638$		0			0
$639$	−0.153161			−0.153161
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$643$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.923880	+	0.382683i	0.923880	+	0.382683i	0.793353	−	0.608761i	$-0.208333\pi$
$647$	0.923880	+	0.382683i	0.923880	+	0.382683i	0.130526	+	0.991445i	$0.458333\pi$
$648$	0.0638111	−	0.0264314i	0.0638111	−	0.0264314i
$649$		0			0
$650$	−0.0862466	+	1.31587i	−0.0862466	+	1.31587i
$651$		0			0
$652$	−0.665060	−	1.34861i	−0.665060	−	1.34861i
$653$	−1.25026	−	0.835400i	−1.25026	−	0.835400i	−0.258819	−	0.965926i	$-0.583333\pi$
$653$	−1.25026	−	0.835400i	−1.25026	−	0.835400i	−0.991445	+	0.130526i	$0.958333\pi$
$654$		0			0
$655$		0			0
$656$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$657$	0.801456	−	0.801456i	0.801456	−	0.801456i
$658$		0			0
$659$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$659$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$660$		0			0
$661$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$661$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$662$	−0.991445	+	0.869474i	−0.991445	+	0.869474i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−0.491445	−	0.735499i	−0.491445	−	0.735499i
$668$	0.758819	+	0.0999004i	0.758819	+	0.0999004i
$669$	0.482584	−	0.0959919i	0.482584	−	0.0959919i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	1.21752i		−	1.21752i	−0.793353	−	0.608761i	$-0.791667\pi$
$673$		−	1.21752i		−	1.21752i	0.793353	−	0.608761i	$-0.208333\pi$
$674$		0			0
$675$	−1.00046	+	0.199004i	−1.00046	+	0.199004i
$676$	0.586247	−	0.449843i	0.586247	−	0.449843i
$677$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$677$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.0255190	−	0.128293i	0.0255190	−	0.128293i	−0.965926	−	0.258819i	$-0.916667\pi$
$683$	0.0255190	−	0.128293i	0.0255190	−	0.128293i	0.991445	+	0.130526i	$0.0416667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.923880	+	0.617317i	0.923880	+	0.617317i	0.923880	−	0.382683i	$-0.125000\pi$
$691$	0.923880	+	0.617317i	0.923880	+	0.617317i			1.00000i	$0.5\pi$
$692$	−1.57469	−	0.534534i	−1.57469	−	0.534534i
$693$		0			0
$694$	1.10876	+	0.0726721i	1.10876	+	0.0726721i
$695$		0			0
$696$			0.568676i			0.568676i
$697$		0			0
$698$	−0.284338	+	0.576581i	−0.284338	+	0.576581i
$699$	1.05992	−	0.708218i	1.05992	−	0.708218i
$700$		0			0
$701$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$701$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$702$	1.06718	+	0.818872i	1.06718	+	0.818872i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.410670	−	0.315118i	−0.410670	−	0.315118i
$707$		0			0
$708$	0.125419	−	0.217233i	0.125419	−	0.217233i
$709$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$709$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.12484	−	0.465926i	1.12484	−	0.465926i
$714$		0			0
$715$		0			0
$716$	−0.483342	+	1.42388i	−0.483342	+	1.42388i
$717$	0.848149	+	0.566715i	0.848149	+	0.566715i
$718$		0			0
$719$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$719$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$720$		0			0
$721$		0			0
$722$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$723$		0			0
$724$		0			0
$725$	0.172572	−	0.867580i	0.172572	−	0.867580i
$726$	−0.423880	−	0.483342i	−0.423880	−	0.483342i
$727$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$727$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$728$		0			0
$729$	−0.266462	+	0.643296i	−0.266462	+	0.643296i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$733$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$734$		0			0
$735$		0			0
$736$	0.965926	−	0.258819i	0.965926	−	0.258819i
$737$		0			0
$738$	0.581687	−	0.0765806i	0.581687	−	0.0765806i
$739$	1.95737	−	0.389345i	1.95737	−	0.389345i	0.965926	−	0.258819i	$-0.0833333\pi$
$739$	1.95737	−	0.389345i	1.95737	−	0.389345i	0.991445	−	0.130526i	$-0.0416667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$743$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$744$	−0.767680	−	0.152701i	−0.767680	−	0.152701i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$751$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$752$	0.133975	−	0.500000i	0.133975	−	0.500000i
$753$		0			0
$754$	−1.01021	+	0.583242i	−1.01021	+	0.583242i
$755$		0			0
$756$		0			0
$757$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$757$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.12484	−	0.465926i	−1.12484	−	0.465926i	−0.258819	−	0.965926i	$-0.583333\pi$
$761$	−1.12484	−	0.465926i	−1.12484	−	0.465926i	−0.866025	+	0.500000i	$0.833333\pi$
$762$	−1.11387	−	0.549299i	−1.11387	−	0.549299i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.514528			0.514528
$768$	−0.608761	−	0.206647i	−0.608761	−	0.206647i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−0.0327410	−	0.164600i	−0.0327410	−	0.164600i
$772$	0.448288	+	1.67303i	0.448288	+	1.67303i
$773$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$773$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$774$		0			0
$775$	1.12484	+	0.465926i	1.12484	+	0.465926i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−0.638038	−	0.638038i	−0.638038	−	0.638038i
$784$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$785$		0			0
$786$	0.332066	−	1.23929i	0.332066	−	1.23929i
$787$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$787$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$788$	1.50046	+	1.31587i	1.50046	+	1.31587i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	−1.69855	+	0.576581i	−1.69855	+	0.576581i
$795$		0			0
$796$		0			0
$797$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$797$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$798$		0			0
$799$		0			0
$800$	0.866025	+	0.500000i	0.866025	+	0.500000i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−0.516083	−	1.52033i	−0.516083	−	1.52033i
$807$	−0.0321808	+	0.0776914i	−0.0321808	+	0.0776914i
$808$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i
$809$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	0.707107	−	0.707107i	$-0.250000\pi$
$809$	−0.292893	−	0.707107i	−0.292893	−	0.707107i	−1.00000			$\pi$
$810$		0			0
$811$	−0.349942	+	1.75928i	−0.349942	+	1.75928i	0.258819	+	0.965926i	$0.416667\pi$
$811$	−0.349942	+	1.75928i	−0.349942	+	1.75928i	−0.608761	+	0.793353i	$0.708333\pi$
$812$		0			0
$813$	0.505107	−	0.755946i	0.505107	−	0.755946i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.991445	+	1.71723i	0.991445	+	1.71723i
$819$		0			0
$820$		0			0
$821$	1.92388	+	0.382683i	1.92388	+	0.382683i	1.00000			$0$
$821$	1.92388	+	0.382683i	1.92388	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$822$		0			0
$823$	0.478235	−	0.198092i	0.478235	−	0.198092i	−0.130526	−	0.991445i	$-0.541667\pi$
$823$	0.478235	−	0.198092i	0.478235	−	0.198092i	0.608761	+	0.793353i	$0.291667\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$827$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$828$	−0.293353	+	0.508103i	−0.293353	+	0.508103i
$829$	0.382683	+	1.92388i	0.382683	+	1.92388i	0.382683	+	0.923880i	$0.375000\pi$
$829$	0.382683	+	1.92388i	0.382683	+	1.92388i			1.00000i	$0.5\pi$
$830$		0			0
$831$	1.21752			1.21752
$832$	−0.257264	−	1.29335i	−0.257264	−	1.29335i
$833$		0			0
$834$	0.0667151	+	0.0511923i	0.0667151	+	0.0511923i
$835$		0			0
$836$		0			0
$837$	1.03264	−	0.689989i	1.03264	−	0.689989i
$838$		0			0
$839$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$839$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$840$		0			0
$841$	−0.200965	+	0.0832424i	−0.200965	+	0.0832424i
$842$		0			0
$843$		0			0
$844$	−1.85747	−	0.630526i	−1.85747	−	0.630526i
$845$		0			0
$846$	0.151851	+	0.263013i	0.151851	+	0.263013i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	0.167466	−	0.0109763i	0.167466	−	0.0109763i
$853$	0.324423	−	1.63099i	0.324423	−	1.63099i	−0.382683	−	0.923880i	$-0.625000\pi$
$853$	0.324423	−	1.63099i	0.324423	−	1.63099i	0.707107	−	0.707107i	$-0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.0999004	−	0.241181i	0.0999004	−	0.241181i	−0.866025	−	0.500000i	$-0.833333\pi$
$857$	0.0999004	−	0.241181i	0.0999004	−	0.241181i	0.965926	+	0.258819i	$0.0833333\pi$
$858$		0			0
$859$	0.732626	+	1.09645i	0.732626	+	1.09645i	0.991445	+	0.130526i	$0.0416667\pi$
$859$	0.732626	+	1.09645i	0.732626	+	1.09645i	−0.258819	+	0.965926i	$0.583333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1.98289i		−	1.98289i	−0.130526	−	0.991445i	$-0.541667\pi$
$863$		−	1.98289i		−	1.98289i	0.130526	−	0.991445i	$-0.458333\pi$
$864$	0.914864	−	0.451161i	0.914864	−	0.451161i
$865$		0			0
$866$		0			0
$867$	−0.630526	+	0.125419i	−0.630526	+	0.125419i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−0.818872	+	0.933745i	−0.818872	+	0.933745i
$877$	−0.216773	+	0.324423i	−0.216773	+	0.324423i	−0.923880	−	0.382683i	$-0.875000\pi$
$877$	−0.216773	+	0.324423i	−0.216773	+	0.324423i	0.707107	+	0.707107i	$0.250000\pi$
$878$	0.513210	−	1.91532i	0.513210	−	1.91532i
$879$		0			0
$880$		0			0
$881$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$881$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$882$	0.508103	−	0.293353i	0.508103	−	0.293353i
$883$	−1.38268	−	0.923880i	−1.38268	−	0.923880i	−0.382683	−	0.923880i	$-0.625000\pi$
$883$	−1.38268	−	0.923880i	−1.38268	−	0.923880i	−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−0.00855514	+	0.130526i	−0.00855514	+	0.130526i
$887$	1.46593	−	0.607206i	1.46593	−	0.607206i	0.500000	−	0.866025i	$-0.333333\pi$
$887$	1.46593	−	0.607206i	1.46593	−	0.607206i	0.965926	+	0.258819i	$0.0833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	0.739288	−	0.198092i	0.739288	−	0.198092i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	0.847759			0.847759
$898$	−1.12484	+	1.46593i	−1.12484	+	1.46593i
$899$	0.210111	+	1.05630i	0.210111	+	1.05630i
$900$	−0.566715	+	0.151851i	−0.566715	+	0.151851i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−0.0728263	+	1.11111i	−0.0728263	+	1.11111i
$907$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$907$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$908$		0			0
$909$	0.190341	+	0.127182i	0.190341	+	0.127182i
$910$		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			0
$919$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$919$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$920$		0			0
$921$	−0.273362	+	0.659954i	−0.273362	+	0.659954i
$922$	−1.79335	+	0.608761i	−1.79335	+	0.608761i
$923$	0.191254	+	0.286231i	0.191254	+	0.286231i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	0.0578541	+	0.882683i	0.0578541	+	0.882683i
$929$		−	1.98289i		−	1.98289i	−0.130526	−	0.991445i	$-0.541667\pi$
$929$		−	1.98289i		−	1.98289i	0.130526	−	0.991445i	$-0.458333\pi$
$930$		0			0
$931$		0			0
$932$	1.57313	−	1.20711i	1.57313	−	1.20711i
$933$	0.618627	+	0.925841i	0.618627	+	0.925841i
$934$		0			0
$935$		0			0
$936$	0.643296	+	0.429836i	0.643296	+	0.429836i
$937$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$937$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$941$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$942$		0			0
$943$	0.707107	−	0.707107i	0.707107	−	0.707107i
$944$	0.172572	−	0.349942i	0.172572	−	0.349942i
$945$		0			0
$946$		0			0
$947$	0.735499	+	0.491445i	0.735499	+	0.491445i	0.866025	−	0.500000i	$-0.166667\pi$
$947$	0.735499	+	0.491445i	0.735499	+	0.491445i	−0.130526	+	0.991445i	$0.541667\pi$
$948$		0			0
$949$	−2.49857	−	0.496996i	−2.49857	−	0.496996i
$950$		0			0
$951$	0.659954	−	0.273362i	0.659954	−	0.273362i
$952$		0			0
$953$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$953$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$954$		0			0
$955$		0			0
$956$	1.37413	+	0.793353i	1.37413	+	0.793353i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.482362			−0.482362
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.923880	−	0.382683i	−0.923880	−	0.382683i	−0.130526	−	0.991445i	$-0.541667\pi$
$967$	−0.923880	−	0.382683i	−0.923880	−	0.382683i	−0.793353	+	0.608761i	$0.791667\pi$
$968$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$969$		0			0
$970$		0			0
$971$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$971$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$972$	−0.313615	+	0.923880i	−0.313615	+	0.923880i
$973$		0			0
$974$	0.130526	+	0.226078i	0.130526	+	0.226078i
$975$	0.599456	+	0.599456i	0.599456	+	0.599456i
$976$		0			0
$977$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$977$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$978$	−0.933745	−	0.250196i	−0.933745	−	0.250196i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	0.583242	+	0.665060i	0.583242	+	0.665060i
$983$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$983$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$984$	−0.630526	+	0.125419i	−0.630526	+	0.125419i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$991$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$992$	−1.20711	−	0.158919i	−1.20711	−	0.158919i
$993$			0.847759i			0.847759i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.617317	+	0.923880i	0.617317	+	0.923880i	1.00000			$0$
$997$	0.617317	+	0.923880i	0.617317	+	0.923880i	−0.382683	+	0.923880i	$0.625000\pi$
$998$	1.69855	−	0.576581i	1.69855	−	0.576581i
$999$		0			0

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	1472.1.s.b.597.2	✓	16
23.22	odd	2	CM	1472.1.s.b.597.2	✓	16
64.61	even	16	inner	1472.1.s.b.1149.2	yes	16
1472.1149	odd	16	inner	1472.1.s.b.1149.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1472.1.s.b.597.2	✓	16	1.1	even	1	trivial
1472.1.s.b.597.2	✓	16	23.22	odd	2	CM
1472.1.s.b.1149.2	yes	16	64.61	even	16	inner
1472.1.s.b.1149.2	yes	16	1472.1149	odd	16	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

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	1472.1.s.b.597.2

Level	$1472$
Weight	$1$
Character	1472.597
Analytic conductor	$0.735$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{48}$
CM discriminant	-23
Inner twists	$4$