LMFDB - Embedded newform 1472.1.s.b.45.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,1,Mod(45,1472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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			45.2
Root			$0.793353 - 0.608761i$ of defining polynomial
Character	$\chi$	$=$	1472.45
Dual form			1472.1.s.b.229.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.793353 - 0.608761i) q^{2} +(0.996552 + 1.49144i) q^{3} +(0.258819 - 0.965926i) q^{4} +(1.69855 + 0.576581i) q^{6} +(-0.382683 - 0.923880i) q^{8} +(-0.848609 + 2.04872i) q^{9} +(1.69855 - 0.576581i) q^{12} +(-0.0255190 + 0.128293i) q^{13} +(-0.866025 - 0.500000i) q^{16} +(0.573937 + 2.14196i) q^{18} +(0.923880 + 0.382683i) q^{23} +(0.996552 - 1.49144i) q^{24} +(-0.923880 + 0.382683i) q^{25} +(0.0578541 + 0.117317i) q^{26} +(-2.14196 + 0.426063i) q^{27} +(0.534534 - 0.357164i) q^{29} -1.58671i q^{31} +(-0.991445 + 0.130526i) q^{32} +(1.75928 + 1.34994i) q^{36} +(-0.216773 + 0.0897902i) q^{39} +(-0.923880 - 0.382683i) q^{41} +(0.965926 - 0.258819i) q^{46} +(-0.366025 - 0.366025i) q^{47} +(-0.117317 - 1.78990i) q^{48} +(-0.707107 + 0.707107i) q^{49} +(-0.500000 + 0.866025i) q^{50} +(0.117317 + 0.0578541i) q^{52} +(-1.43996 + 1.64196i) q^{54} +(0.206647 - 0.608761i) q^{58} +(-0.216773 - 1.08979i) q^{59} +(-0.965926 - 1.25882i) q^{62} +(-0.707107 + 0.707107i) q^{64} +(0.349942 + 1.75928i) q^{69} +(-0.758819 - 1.83195i) q^{71} +2.21752 q^{72} +(-0.739288 + 1.78480i) q^{73} +(-1.49144 - 0.996552i) q^{75} +(-0.117317 + 0.203198i) q^{78} +(-1.20200 - 1.20200i) q^{81} +(-0.965926 + 0.258819i) q^{82} +(1.06538 + 0.441296i) q^{87} +(0.608761 - 0.793353i) q^{92} +(2.36649 - 1.58124i) q^{93} +(-0.513210 - 0.0675653i) q^{94} +(-1.18270 - 1.34861i) q^{96} +(-0.130526 + 0.991445i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q + 8 q^{9} + 8 q^{47} - 8 q^{48} - 8 q^{50} + 8 q^{52} + 16 q^{58} - 8 q^{71} + 16 q^{72} - 8 q^{75} - 8 q^{78} - 8 q^{81} - 8 q^{87}+O(q^{100})$ 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 $\left(\mathbb{Z}/1472\mathbb{Z}\right)^\times$ .

$n$	$645$	$833$	$1151$
$\chi(n)$	$e\left(\frac{7}{16}\right)$	$-1$	$1$

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

)

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

$2$	0.793353	−	0.608761i	0.793353	−	0.608761i
$2$	0.793353	−	0.608761i	0.793353	−	0.608761i
$3$	0.996552	+	1.49144i	0.996552	+	1.49144i	0.866025	+	0.500000i	$0.166667\pi$
$3$	0.996552	+	1.49144i	0.996552	+	1.49144i	0.130526	+	0.991445i	$0.458333\pi$
$4$	0.258819	−	0.965926i	0.258819	−	0.965926i
$5$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$5$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$6$	1.69855	+	0.576581i	1.69855	+	0.576581i
$7$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$7$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$8$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$9$	−0.848609	+	2.04872i	−0.848609	+	2.04872i
$10$		0			0
$11$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$11$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$12$	1.69855	−	0.576581i	1.69855	−	0.576581i
$13$	−0.0255190	+	0.128293i	−0.0255190	+	0.128293i	−0.991445	−	0.130526i	$-0.958333\pi$
$13$	−0.0255190	+	0.128293i	−0.0255190	+	0.128293i	0.965926	+	0.258819i	$0.0833333\pi$
$14$		0			0
$15$		0			0
$16$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$	0.573937	+	2.14196i	0.573937	+	2.14196i
$19$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$19$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.923880	+	0.382683i	0.923880	+	0.382683i
$23$	0.923880	+	0.382683i	0.923880	+	0.382683i
$24$	0.996552	−	1.49144i	0.996552	−	1.49144i
$25$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$26$	0.0578541	+	0.117317i	0.0578541	+	0.117317i
$27$	−2.14196	+	0.426063i	−2.14196	+	0.426063i
$28$		0			0
$29$	0.534534	−	0.357164i	0.534534	−	0.357164i	−0.258819	−	0.965926i	$-0.583333\pi$
$29$	0.534534	−	0.357164i	0.534534	−	0.357164i	0.793353	+	0.608761i	$0.208333\pi$
$30$		0			0
$31$		−	1.58671i		−	1.58671i	−0.608761	−	0.793353i	$-0.708333\pi$
$31$		−	1.58671i		−	1.58671i	0.608761	−	0.793353i	$-0.291667\pi$
$32$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.75928	+	1.34994i	1.75928	+	1.34994i
$37$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$37$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$38$		0			0
$39$	−0.216773	+	0.0897902i	−0.216773	+	0.0897902i
$40$		0			0
$41$	−0.923880	−	0.382683i	−0.923880	−	0.382683i	−0.130526	−	0.991445i	$-0.541667\pi$
$41$	−0.923880	−	0.382683i	−0.923880	−	0.382683i	−0.793353	+	0.608761i	$0.791667\pi$
$42$		0			0
$43$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$43$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$44$		0			0
$45$		0			0
$46$	0.965926	−	0.258819i	0.965926	−	0.258819i
$47$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$47$	−0.366025	−	0.366025i	−0.366025	−	0.366025i	−0.866025	+	0.500000i	$0.833333\pi$
$48$	−0.117317	−	1.78990i	−0.117317	−	1.78990i
$49$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$50$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$51$		0			0
$52$	0.117317	+	0.0578541i	0.117317	+	0.0578541i
$53$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$53$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$54$	−1.43996	+	1.64196i	−1.43996	+	1.64196i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.206647	−	0.608761i	0.206647	−	0.608761i
$59$	−0.216773	−	1.08979i	−0.216773	−	1.08979i	−0.923880	−	0.382683i	$-0.875000\pi$
$59$	−0.216773	−	1.08979i	−0.216773	−	1.08979i	0.707107	−	0.707107i	$-0.250000\pi$
$60$		0			0
$61$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$61$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$62$	−0.965926	−	1.25882i	−0.965926	−	1.25882i
$63$		0			0
$64$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$65$		0			0
$66$		0			0
$67$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$67$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$68$		0			0
$69$	0.349942	+	1.75928i	0.349942	+	1.75928i
$70$		0			0
$71$	−0.758819	−	1.83195i	−0.758819	−	1.83195i	−0.500000	−	0.866025i	$-0.666667\pi$
$71$	−0.758819	−	1.83195i	−0.758819	−	1.83195i	−0.258819	−	0.965926i	$-0.583333\pi$
$72$	2.21752			2.21752
$73$	−0.739288	+	1.78480i	−0.739288	+	1.78480i	−0.130526	+	0.991445i	$0.541667\pi$
$73$	−0.739288	+	1.78480i	−0.739288	+	1.78480i	−0.608761	+	0.793353i	$0.708333\pi$
$74$		0			0
$75$	−1.49144	−	0.996552i	−1.49144	−	0.996552i
$76$		0			0
$77$		0			0
$78$	−0.117317	+	0.203198i	−0.117317	+	0.203198i
$79$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$79$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$80$		0			0
$81$	−1.20200	−	1.20200i	−1.20200	−	1.20200i
$82$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$83$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$83$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	1.06538	+	0.441296i	1.06538	+	0.441296i
$88$		0			0
$89$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$89$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$90$		0			0
$91$		0			0
$92$	0.608761	−	0.793353i	0.608761	−	0.793353i
$93$	2.36649	−	1.58124i	2.36649	−	1.58124i
$94$	−0.513210	−	0.0675653i	−0.513210	−	0.0675653i
$95$		0			0
$96$	−1.18270	−	1.34861i	−1.18270	−	1.34861i
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$99$		0			0
$100$	0.130526	+	0.991445i	0.130526	+	0.991445i
$101$	1.08979	−	0.216773i	1.08979	−	0.216773i	0.382683	−	0.923880i	$-0.375000\pi$
$101$	1.08979	−	0.216773i	1.08979	−	0.216773i	0.707107	+	0.707107i	$0.250000\pi$
$102$		0			0
$103$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$103$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$104$	0.128293	−	0.0255190i	0.128293	−	0.0255190i
$105$		0			0
$106$		0			0
$107$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$107$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$108$	−0.142836	+	2.17925i	−0.142836	+	2.17925i
$109$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$109$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$113$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$114$		0			0
$115$		0			0
$116$	−0.206647	−	0.608761i	−0.206647	−	0.608761i
$117$	−0.241181	−	0.161152i	−0.241181	−	0.161152i
$118$	−0.835400	−	0.732626i	−0.835400	−	0.732626i
$119$		0			0
$120$		0			0
$121$	0.382683	+	0.923880i	0.382683	+	0.923880i
$122$		0			0
$123$	−0.349942	−	1.75928i	−0.349942	−	1.75928i
$124$	−1.53264	−	0.410670i	−1.53264	−	0.410670i
$125$		0			0
$126$		0			0
$127$	−1.93185			−1.93185			−0.965926	−	0.258819i	$-0.916667\pi$
$127$	−1.93185			−1.93185			−0.965926	+	0.258819i	$0.916667\pi$
$128$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$129$		0			0
$130$		0			0
$131$	0.732626	+	1.09645i	0.732626	+	1.09645i	0.991445	+	0.130526i	$0.0416667\pi$
$131$	0.732626	+	1.09645i	0.732626	+	1.09645i	−0.258819	+	0.965926i	$0.583333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$137$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$138$	1.34861	+	1.18270i	1.34861	+	1.18270i
$139$	1.25026	+	0.835400i	1.25026	+	0.835400i	0.991445	−	0.130526i	$-0.0416667\pi$
$139$	1.25026	+	0.835400i	1.25026	+	0.835400i	0.258819	+	0.965926i	$0.416667\pi$
$140$		0			0
$141$	0.181144	−	0.910670i	0.181144	−	0.910670i
$142$	−1.71723	−	0.991445i	−1.71723	−	0.991445i
$143$		0			0
$144$	1.75928	−	1.34994i	1.75928	−	1.34994i
$145$		0			0
$146$	0.500000	+	1.86603i	0.500000	+	1.86603i
$147$	−1.75928	−	0.349942i	−1.75928	−	0.349942i
$148$		0			0
$149$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$149$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$150$	−1.78990	+	0.117317i	−1.78990	+	0.117317i
$151$	1.60021	+	0.662827i	1.60021	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$151$	1.60021	+	0.662827i	1.60021	+	0.662827i	0.608761	+	0.793353i	$0.291667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.0306258	+	0.232626i	0.0306258	+	0.232626i
$157$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$157$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−1.68534	−	0.221879i	−1.68534	−	0.221879i
$163$	1.65938	−	1.10876i	1.65938	−	1.10876i	0.793353	−	0.608761i	$-0.208333\pi$
$163$	1.65938	−	1.10876i	1.65938	−	1.10876i	0.866025	−	0.500000i	$-0.166667\pi$
$164$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$165$		0			0
$166$		0			0
$167$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$167$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$168$		0			0
$169$	0.908072	+	0.376136i	0.908072	+	0.376136i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.382683	+	0.0761205i	0.382683	+	0.0761205i	0.382683	−	0.923880i	$-0.375000\pi$
$173$	0.382683	+	0.0761205i	0.382683	+	0.0761205i			1.00000i	$0.5\pi$
$174$	1.11387	−	0.298460i	1.11387	−	0.298460i
$175$		0			0
$176$		0			0
$177$	1.40934	−	1.40934i	1.40934	−	1.40934i
$178$		0			0
$179$	0.389345	−	1.95737i	0.389345	−	1.95737i	0.130526	−	0.991445i	$-0.458333\pi$
$179$	0.389345	−	1.95737i	0.389345	−	1.95737i	0.258819	−	0.965926i	$-0.416667\pi$
$180$		0			0
$181$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$181$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$182$		0			0
$183$		0			0
$184$		−	1.00000i		−	1.00000i
$185$		0			0
$186$	0.914864	−	2.69510i	0.914864	−	2.69510i
$187$		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			$\pi$
$192$	−1.75928	−	0.349942i	−1.75928	−	0.349942i
$193$	−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
$197$	−0.257264	−	1.29335i	−0.257264	−	1.29335i	−0.866025	−	0.500000i	$-0.833333\pi$
$197$	−0.257264	−	1.29335i	−0.257264	−	1.29335i	0.608761	−	0.793353i	$-0.291667\pi$
$198$		0			0
$199$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$199$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$200$	0.707107	+	0.707107i	0.707107	+	0.707107i
$201$		0			0
$202$	0.732626	−	0.835400i	0.732626	−	0.835400i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.56803	+	1.56803i	−1.56803	+	1.56803i
$208$	0.0862466	−	0.0983454i	0.0862466	−	0.0983454i
$209$		0			0
$210$		0			0
$211$	−1.63099	−	0.324423i	−1.63099	−	0.324423i	−0.707107	−	0.707107i	$-0.750000\pi$
$211$	−1.63099	−	0.324423i	−1.63099	−	0.324423i	−0.923880	+	0.382683i	$0.875000\pi$
$212$		0			0
$213$	1.97605	−	2.95737i	1.97605	−	2.95737i
$214$		0			0
$215$		0			0
$216$	1.21332	+	1.81587i	1.21332	+	1.81587i
$217$		0			0
$218$		0			0
$219$	−3.39867	+	0.676037i	−3.39867	+	0.676037i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			1.84776i			1.84776i	0.382683	+	0.923880i	$0.375000\pi$
$223$			1.84776i			1.84776i	−0.382683	+	0.923880i	$0.625000\pi$
$224$		0			0
$225$		−	2.21752i		−	2.21752i
$226$		0			0
$227$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$227$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$228$		0			0
$229$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$229$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$230$		0			0
$231$		0			0
$232$	−0.534534	−	0.357164i	−0.534534	−	0.357164i
$233$	0.241181	+	0.0999004i	0.241181	+	0.0999004i	0.500000	−	0.866025i	$-0.333333\pi$
$233$	0.241181	+	0.0999004i	0.241181	+	0.0999004i	−0.258819	+	0.965926i	$0.583333\pi$
$234$	−0.289445	+	0.0189712i	−0.289445	+	0.0189712i
$235$		0			0
$236$	−1.10876	−	0.0726721i	−1.10876	−	0.0726721i
$237$		0			0
$238$		0			0
$239$	−0.860919	−	0.860919i	−0.860919	−	0.860919i	0.130526	−	0.991445i	$-0.458333\pi$
$239$	−0.860919	−	0.860919i	−0.860919	−	0.860919i	−0.991445	+	0.130526i	$0.958333\pi$
$240$		0			0
$241$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$241$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$242$	0.866025	+	0.500000i	0.866025	+	0.500000i
$243$	0.168799	−	0.848609i	0.168799	−	0.848609i
$244$		0			0
$245$		0			0
$246$	−1.34861	−	1.18270i	−1.34861	−	1.18270i
$247$		0			0
$248$	−1.46593	+	0.607206i	−1.46593	+	0.607206i
$249$		0			0
$250$		0			0
$251$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$251$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$252$		0			0
$253$		0			0
$254$	−1.53264	+	1.17604i	−1.53264	+	1.17604i
$255$		0			0
$256$	0.500000	+	0.866025i	0.500000	+	0.866025i
$257$	1.98289			1.98289			0.991445	−	0.130526i	$-0.0416667\pi$
$257$	1.98289			1.98289			0.991445	+	0.130526i	$0.0416667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	0.278121	+	1.39821i	0.278121	+	1.39821i
$262$	1.24871	+	0.423880i	1.24871	+	0.423880i
$263$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$263$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.293353	+	1.47479i	−0.293353	+	1.47479i	0.500000	+	0.866025i	$0.333333\pi$
$269$	−0.293353	+	1.47479i	−0.293353	+	1.47479i	−0.793353	+	0.608761i	$0.791667\pi$
$270$		0			0
$271$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$271$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	1.78990	+	0.117317i	1.78990	+	0.117317i
$277$	0.491445	−	0.735499i	0.491445	−	0.735499i	−0.500000	−	0.866025i	$-0.666667\pi$
$277$	0.491445	−	0.735499i	0.491445	−	0.735499i	0.991445	+	0.130526i	$0.0416667\pi$
$278$	1.50046	−	0.0983454i	1.50046	−	0.0983454i
$279$	3.25072	+	1.34649i	3.25072	+	1.34649i
$280$		0			0
$281$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$281$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$282$	−0.410670	−	0.832756i	−0.410670	−	0.832756i
$283$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$283$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$284$	−1.96593	+	0.258819i	−1.96593	+	0.258819i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.573937	−	2.14196i	0.573937	−	2.14196i
$289$			1.00000i			1.00000i
$290$		0			0
$291$		0			0
$292$	1.53264	+	1.17604i	1.53264	+	1.17604i
$293$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$293$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$294$	−1.60876	+	0.793353i	−1.60876	+	0.793353i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.0726721	+	0.108761i	−0.0726721	+	0.108761i
$300$	−1.34861	+	1.18270i	−1.34861	+	1.18270i
$301$		0			0
$302$	1.67303	−	0.448288i	1.67303	−	0.448288i
$303$	1.40934	+	1.40934i	1.40934	+	1.40934i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−0.382683	+	1.92388i	−0.382683	+	1.92388i			1.00000i	$0.5\pi$
$307$	−0.382683	+	1.92388i	−0.382683	+	1.92388i	−0.382683	+	0.923880i	$0.625000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−0.662827	+	1.60021i	−0.662827	+	1.60021i	0.130526	+	0.991445i	$0.458333\pi$
$311$	−0.662827	+	1.60021i	−0.662827	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$312$	0.165911	+	0.165911i	0.165911	+	0.165911i
$313$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$313$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.08979	+	1.63099i	1.08979	+	1.63099i	0.707107	+	0.707107i	$0.250000\pi$
$317$	1.08979	+	1.63099i	1.08979	+	1.63099i	0.382683	+	0.923880i	$0.375000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.47214	+	0.849942i	−1.47214	+	0.849942i
$325$	−0.0255190	−	0.128293i	−0.0255190	−	0.128293i
$326$	0.641502	−	1.88981i	0.641502	−	1.88981i
$327$		0			0
$328$			1.00000i			1.00000i
$329$		0			0
$330$		0			0
$331$	0.108761	+	0.0726721i	0.108761	+	0.0726721i	0.608761	−	0.793353i	$-0.291667\pi$
$331$	0.108761	+	0.0726721i	0.108761	+	0.0726721i	−0.500000	+	0.866025i	$0.666667\pi$
$332$		0			0
$333$		0			0
$334$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$335$		0			0
$336$		0			0
$337$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$337$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$338$	0.949399	−	0.254391i	0.949399	−	0.254391i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.349942	−	0.172572i	0.349942	−	0.172572i
$347$	1.92388	−	0.382683i	1.92388	−	0.382683i	0.923880	−	0.382683i	$-0.125000\pi$
$347$	1.92388	−	0.382683i	1.92388	−	0.382683i	1.00000			$0$
$348$	0.702000	−	0.914864i	0.702000	−	0.914864i
$349$	−1.49144	+	0.996552i	−1.49144	+	0.996552i	−0.500000	+	0.866025i	$0.666667\pi$
$349$	−1.49144	+	0.996552i	−1.49144	+	0.996552i	−0.991445	+	0.130526i	$0.958333\pi$
$350$		0			0
$351$		−	0.285671i		−	0.285671i
$352$		0			0
$353$		−	0.517638i		−	0.517638i	−0.965926	−	0.258819i	$-0.916667\pi$
$353$		−	0.517638i		−	0.517638i	0.965926	−	0.258819i	$-0.0833333\pi$
$354$	0.260152	−	1.97605i	0.260152	−	1.97605i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−0.882683	−	1.78990i	−0.882683	−	1.78990i
$359$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$359$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$360$		0			0
$361$	0.923880	+	0.382683i	0.923880	+	0.382683i
$362$		0			0
$363$	−0.996552	+	1.49144i	−0.996552	+	1.49144i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$369$	1.56803	−	1.56803i	1.56803	−	1.56803i
$370$		0			0
$371$		0			0
$372$	−0.914864	−	2.69510i	−0.914864	−	2.69510i
$373$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$373$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$374$		0			0
$375$		0			0
$376$	−0.198092	+	0.478235i	−0.198092	+	0.478235i
$377$	0.0321808	+	0.0776914i	0.0321808	+	0.0776914i
$378$		0			0
$379$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$379$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$380$		0			0
$381$	−1.92519	−	2.88125i	−1.92519	−	2.88125i
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$	−1.60876	+	0.793353i	−1.60876	+	0.793353i
$385$		0			0
$386$	−1.37413	+	1.05441i	−1.37413	+	1.05441i
$387$		0			0
$388$		0			0
$389$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$389$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$390$		0			0
$391$		0			0
$392$	0.923880	+	0.382683i	0.923880	+	0.382683i
$393$	−0.905198	+	2.18534i	−0.905198	+	2.18534i
$394$	−0.991445	−	0.869474i	−0.991445	−	0.869474i
$395$		0			0
$396$		0			0
$397$	−0.369474	+	1.85747i	−0.369474	+	1.85747i	0.130526	+	0.991445i	$0.458333\pi$
$397$	−0.369474	+	1.85747i	−0.369474	+	1.85747i	−0.500000	+	0.866025i	$0.666667\pi$
$398$		0			0
$399$		0			0
$400$	0.991445	+	0.130526i	0.991445	+	0.130526i
$401$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$401$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$402$		0			0
$403$	0.203563	+	0.0404912i	0.203563	+	0.0404912i
$404$	0.0726721	−	1.10876i	0.0726721	−	1.10876i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.241181	+	0.0999004i	−0.241181	+	0.0999004i	−0.500000	−	0.866025i	$-0.666667\pi$
$409$	−0.241181	+	0.0999004i	−0.241181	+	0.0999004i	0.258819	+	0.965926i	$0.416667\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	−0.289445	+	2.19855i	−0.289445	+	2.19855i
$415$		0			0
$416$	0.00855514	−	0.130526i	0.00855514	−	0.130526i
$417$			2.69722i			2.69722i
$418$		0			0
$419$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$419$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$420$		0			0
$421$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$421$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$422$	−1.49144	+	0.735499i	−1.49144	+	0.735499i
$423$	1.06050	−	0.439272i	1.06050	−	0.439272i
$424$		0			0
$425$		0			0
$426$	−0.232626	−	3.54918i	−0.232626	−	3.54918i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$	2.06803	+	0.702000i	2.06803	+	0.702000i
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−2.28480	+	2.60531i	−2.28480	+	2.60531i
$439$	−0.0999004	+	0.241181i	−0.0999004	+	0.241181i	−0.965926	−	0.258819i	$-0.916667\pi$
$439$	−0.0999004	+	0.241181i	−0.0999004	+	0.241181i	0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$	−0.848609	−	2.04872i	−0.848609	−	2.04872i
$442$		0			0
$443$	−0.293353	−	1.47479i	−0.293353	−	1.47479i	−0.793353	−	0.608761i	$-0.791667\pi$
$443$	−0.293353	−	1.47479i	−0.293353	−	1.47479i	0.500000	−	0.866025i	$-0.333333\pi$
$444$		0			0
$445$		0			0
$446$	1.12484	+	1.46593i	1.12484	+	1.46593i
$447$		0			0
$448$		0			0
$449$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$449$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$450$	−1.34994	−	1.75928i	−1.34994	−	1.75928i
$451$		0			0
$452$		0			0
$453$	0.606118	+	3.04716i	0.606118	+	3.04716i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$457$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	0.172572	−	0.867580i	0.172572	−	0.867580i	−0.793353	−	0.608761i	$-0.791667\pi$
$461$	0.172572	−	0.867580i	0.172572	−	0.867580i	0.965926	−	0.258819i	$-0.0833333\pi$
$462$		0			0
$463$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$463$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$464$	−0.641502	+	0.0420463i	−0.641502	+	0.0420463i
$465$		0			0
$466$	0.252157	−	0.0675653i	0.252157	−	0.0675653i
$467$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$467$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$468$	−0.218083	+	0.191254i	−0.218083	+	0.191254i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−0.923880	+	0.617317i	−0.923880	+	0.617317i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−1.20711	−	0.158919i	−1.20711	−	0.158919i
$479$		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
$484$	0.991445	−	0.130526i	0.991445	−	0.130526i
$485$		0			0
$486$	−0.382683	−	0.776005i	−0.382683	−	0.776005i
$487$	1.83195	−	0.758819i	1.83195	−	0.758819i	0.866025	−	0.500000i	$-0.166667\pi$
$487$	1.83195	−	0.758819i	1.83195	−	0.758819i	0.965926	−	0.258819i	$-0.0833333\pi$
$488$		0			0
$489$	3.30731	+	1.36993i	3.30731	+	1.36993i
$490$		0			0
$491$	0.357164	−	0.534534i	0.357164	−	0.534534i	−0.608761	−	0.793353i	$-0.708333\pi$
$491$	0.357164	−	0.534534i	0.357164	−	0.534534i	0.965926	+	0.258819i	$0.0833333\pi$
$492$	−1.78990	−	0.117317i	−1.78990	−	0.117317i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.793353	+	1.37413i	−0.793353	+	1.37413i
$497$		0			0
$498$		0			0
$499$	0.369474	−	1.85747i	0.369474	−	1.85747i	−0.130526	−	0.991445i	$-0.541667\pi$
$499$	0.369474	−	1.85747i	0.369474	−	1.85747i	0.500000	−	0.866025i	$-0.333333\pi$
$500$		0			0
$501$	−2.75583	−	1.84139i	−2.75583	−	1.84139i
$502$		0			0
$503$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$503$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.343955	+	1.72918i	0.343955	+	1.72918i
$508$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$509$	−1.10876	−	1.65938i	−1.10876	−	1.65938i	−0.608761	−	0.793353i	$-0.708333\pi$
$509$	−1.10876	−	1.65938i	−1.10876	−	1.65938i	−0.500000	−	0.866025i	$-0.666667\pi$
$510$		0			0
$511$		0			0
$512$	0.923880	+	0.382683i	0.923880	+	0.382683i
$513$		0			0
$514$	1.57313	−	1.20711i	1.57313	−	1.20711i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	0.267834	+	0.646609i	0.267834	+	0.646609i
$520$		0			0
$521$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$521$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$522$	1.07182	+	0.939963i	1.07182	+	0.939963i
$523$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$523$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$524$	1.24871	−	0.423880i	1.24871	−	0.423880i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.707107	+	0.707107i	0.707107	+	0.707107i
$530$		0			0
$531$	2.41663	+	0.480699i	2.41663	+	0.480699i
$532$		0			0
$533$	0.0726721	−	0.108761i	0.0726721	−	0.108761i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	3.30731	−	1.36993i	3.30731	−	1.36993i
$538$	0.665060	+	1.34861i	0.665060	+	1.34861i
$539$		0			0
$540$		0			0
$541$	−1.57469	+	1.05217i	−1.57469	+	1.05217i	−0.608761	+	0.793353i	$0.708333\pi$
$541$	−1.57469	+	1.05217i	−1.57469	+	1.05217i	−0.965926	+	0.258819i	$0.916667\pi$
$542$	0.184592	−	1.40211i	0.184592	−	1.40211i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.534534	−	0.357164i	0.534534	−	0.357164i	−0.258819	−	0.965926i	$-0.583333\pi$
$547$	0.534534	−	0.357164i	0.534534	−	0.357164i	0.793353	+	0.608761i	$0.208333\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	1.49144	−	0.996552i	1.49144	−	0.996552i
$553$		0			0
$554$	−0.0578541	−	0.882683i	−0.0578541	−	0.882683i
$555$		0			0
$556$	1.13053	−	0.991445i	1.13053	−	0.991445i
$557$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$557$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$558$	3.39867	−	0.910670i	3.39867	−	0.910670i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$563$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$564$	−0.832756	−	0.410670i	−0.832756	−	0.410670i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	−1.40211	+	1.40211i	−1.40211	+	1.40211i
$569$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$569$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$570$		0			0
$571$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$571$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.00000			−1.00000
$576$	−0.848609	−	2.04872i	−0.848609	−	2.04872i
$577$	0.261052			0.261052			0.130526	−	0.991445i	$-0.458333\pi$
$577$	0.261052			0.261052			0.130526	+	0.991445i	$0.458333\pi$
$578$	0.608761	+	0.793353i	0.608761	+	0.793353i
$579$	−1.72608	−	2.58326i	−1.72608	−	2.58326i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	1.93185			1.93185
$585$		0			0
$586$		0			0
$587$	1.57469	+	1.05217i	1.57469	+	1.05217i	0.965926	+	0.258819i	$0.0833333\pi$
$587$	1.57469	+	1.05217i	1.57469	+	1.05217i	0.608761	+	0.793353i	$0.291667\pi$
$588$	−0.793353	+	1.60876i	−0.793353	+	1.60876i
$589$		0			0
$590$		0			0
$591$	1.67259	−	1.67259i	1.67259	−	1.67259i
$592$		0			0
$593$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$593$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	0.00855514	+	0.130526i	0.00855514	+	0.130526i
$599$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$599$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$600$	−0.349942	+	1.75928i	−0.349942	+	1.75928i
$601$	−0.923880	+	0.382683i	−0.923880	+	0.382683i	−0.793353	−	0.608761i	$-0.791667\pi$
$601$	−0.923880	+	0.382683i	−0.923880	+	0.382683i	−0.130526	+	0.991445i	$0.541667\pi$
$602$		0			0
$603$		0			0
$604$	1.05441	−	1.37413i	1.05441	−	1.37413i
$605$		0			0
$606$	1.97605	+	0.260152i	1.97605	+	0.260152i
$607$		−	0.765367i		−	0.765367i	−0.923880	−	0.382683i	$-0.875000\pi$
$607$		−	0.765367i		−	0.765367i	0.923880	−	0.382683i	$-0.125000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	0.0562991	−	0.0376178i	0.0562991	−	0.0376178i
$612$		0			0
$613$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$613$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$614$	0.867580	+	1.75928i	0.867580	+	1.75928i
$615$		0			0
$616$		0			0
$617$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$617$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$618$		0			0
$619$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$619$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$620$		0			0
$621$	−2.14196	−	0.426063i	−2.14196	−	0.426063i
$622$	0.448288	+	1.67303i	0.448288	+	1.67303i
$623$		0			0
$624$	0.232626	+	0.0306258i	0.232626	+	0.0306258i
$625$	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		0.382683	−	0.923880i	$-0.375000\pi$
$631$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$632$		0			0
$633$	−1.14150	−	2.75583i	−1.14150	−	2.75583i
$634$	1.85747	+	0.630526i	1.85747	+	0.630526i
$635$		0			0
$636$		0			0
$637$	−0.0726721	−	0.108761i	−0.0726721	−	0.108761i
$638$		0			0
$639$	4.39710			4.39710
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$643$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.382683	+	0.923880i	0.382683	+	0.923880i	0.991445	+	0.130526i	$0.0416667\pi$
$647$	0.382683	+	0.923880i	0.382683	+	0.923880i	−0.608761	+	0.793353i	$0.708333\pi$
$648$	−0.650518	+	1.57049i	−0.650518	+	1.57049i
$649$		0			0
$650$	−0.0983454	−	0.0862466i	−0.0983454	−	0.0862466i
$651$		0			0
$652$	−0.641502	−	1.88981i	−0.641502	−	1.88981i
$653$	0.389345	−	1.95737i	0.389345	−	1.95737i	0.130526	−	0.991445i	$-0.458333\pi$
$653$	0.389345	−	1.95737i	0.389345	−	1.95737i	0.258819	−	0.965926i	$-0.416667\pi$
$654$		0			0
$655$		0			0
$656$	0.608761	+	0.793353i	0.608761	+	0.793353i
$657$	−3.02919	−	3.02919i	−3.02919	−	3.02919i
$658$		0			0
$659$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$659$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$660$		0			0
$661$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$661$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$662$	0.130526	−	0.00855514i	0.130526	−	0.00855514i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	0.630526	−	0.125419i	0.630526	−	0.125419i
$668$	0.241181	+	1.83195i	0.241181	+	1.83195i
$669$	−2.75583	+	1.84139i	−2.75583	+	1.84139i
$670$		0			0
$671$		0			0
$672$		0			0
$673$			1.58671i			1.58671i	0.608761	+	0.793353i	$0.291667\pi$
$673$			1.58671i			1.58671i	−0.608761	+	0.793353i	$0.708333\pi$
$674$		0			0
$675$	1.81587	−	1.21332i	1.81587	−	1.21332i
$676$	0.598345	−	0.779779i	0.598345	−	0.779779i
$677$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$677$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.835400	−	1.25026i	0.835400	−	1.25026i	−0.130526	−	0.991445i	$-0.541667\pi$
$683$	0.835400	−	1.25026i	0.835400	−	1.25026i	0.965926	−	0.258819i	$-0.0833333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.382683	−	1.92388i	0.382683	−	1.92388i		−	1.00000i	$-0.5\pi$
$691$	0.382683	−	1.92388i	0.382683	−	1.92388i	0.382683	−	0.923880i	$-0.375000\pi$
$692$	0.172572	−	0.349942i	0.172572	−	0.349942i
$693$		0			0
$694$	1.29335	−	1.47479i	1.29335	−	1.47479i
$695$		0			0
$696$		−	1.15316i		−	1.15316i
$697$		0			0
$698$	−0.576581	+	1.69855i	−0.576581	+	1.69855i
$699$	0.0913533	+	0.459264i	0.0913533	+	0.459264i
$700$		0			0
$701$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$701$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$702$	−0.173906	−	0.226638i	−0.173906	−	0.226638i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.315118	−	0.410670i	−0.315118	−	0.410670i
$707$		0			0
$708$	−0.996552	−	1.72608i	−0.996552	−	1.72608i
$709$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$709$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.607206	−	1.46593i	0.607206	−	1.46593i
$714$		0			0
$715$		0			0
$716$	−1.78990	−	0.882683i	−1.78990	−	0.882683i
$717$	0.426063	−	2.14196i	0.426063	−	2.14196i
$718$		0			0
$719$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$719$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$720$		0			0
$721$		0			0
$722$	0.965926	−	0.258819i	0.965926	−	0.258819i
$723$		0			0
$724$		0			0
$725$	−0.357164	+	0.534534i	−0.357164	+	0.534534i
$726$	0.117317	+	1.78990i	0.117317	+	1.78990i
$727$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$727$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$728$		0			0
$729$	−0.136618	+	0.0565890i	−0.136618	+	0.0565890i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$733$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$734$		0			0
$735$		0			0
$736$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$737$		0			0
$738$	0.289445	−	2.19855i	0.289445	−	2.19855i
$739$	−1.09645	+	0.732626i	−1.09645	+	0.732626i	−0.965926	−	0.258819i	$-0.916667\pi$
$739$	−1.09645	+	0.732626i	−1.09645	+	0.732626i	−0.130526	+	0.991445i	$0.541667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$743$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$744$	−2.36649	−	1.58124i	−2.36649	−	1.58124i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$751$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$752$	0.133975	+	0.500000i	0.133975	+	0.500000i
$753$		0			0
$754$	0.0728263	+	0.0420463i	0.0728263	+	0.0420463i
$755$		0			0
$756$		0			0
$757$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$757$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.607206	−	1.46593i	−0.607206	−	1.46593i	−0.866025	−	0.500000i	$-0.833333\pi$
$761$	−0.607206	−	1.46593i	−0.607206	−	1.46593i	0.258819	−	0.965926i	$-0.416667\pi$
$762$	−3.28135	−	1.11387i	−3.28135	−	1.11387i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.145344			0.145344
$768$	−0.793353	+	1.60876i	−0.793353	+	1.60876i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	1.97605	+	2.95737i	1.97605	+	2.95737i
$772$	−0.448288	+	1.67303i	−0.448288	+	1.67303i
$773$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$773$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$774$		0			0
$775$	0.607206	+	1.46593i	0.607206	+	1.46593i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−0.992778	+	0.992778i	−0.992778	+	0.992778i
$784$	0.965926	−	0.258819i	0.965926	−	0.258819i
$785$		0			0
$786$	0.612210	+	2.28480i	0.612210	+	2.28480i
$787$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$787$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$788$	−1.31587	−	0.0862466i	−1.31587	−	0.0862466i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	0.837633	+	1.69855i	0.837633	+	1.69855i
$795$		0			0
$796$		0			0
$797$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$797$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$798$		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
$806$	0.186147	−	0.0917975i	0.186147	−	0.0917975i
$807$	−2.49190	+	1.03218i	−2.49190	+	1.03218i
$808$	−0.617317	−	0.923880i	−0.617317	−	0.923880i
$809$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$809$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$810$		0			0
$811$	−1.05217	+	1.57469i	−1.05217	+	1.57469i	−0.258819	+	0.965926i	$0.583333\pi$
$811$	−1.05217	+	1.57469i	−1.05217	+	1.57469i	−0.793353	+	0.608761i	$0.791667\pi$
$812$		0			0
$813$	2.48800	+	0.494893i	2.48800	+	0.494893i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.130526	+	0.226078i	−0.130526	+	0.226078i
$819$		0			0
$820$		0			0
$821$	1.38268	+	0.923880i	1.38268	+	0.923880i	1.00000			$0$
$821$	1.38268	+	0.923880i	1.38268	+	0.923880i	0.382683	+	0.923880i	$0.375000\pi$
$822$		0			0
$823$	−0.198092	+	0.478235i	−0.198092	+	0.478235i	−0.991445	−	0.130526i	$-0.958333\pi$
$823$	−0.198092	+	0.478235i	−0.198092	+	0.478235i	0.793353	+	0.608761i	$0.208333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$827$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$828$	1.10876	+	1.92043i	1.10876	+	1.92043i
$829$	−0.923880	−	1.38268i	−0.923880	−	1.38268i	−0.923880	−	0.382683i	$-0.875000\pi$
$829$	−0.923880	−	1.38268i	−0.923880	−	1.38268i		−	1.00000i	$-0.5\pi$
$830$		0			0
$831$	1.58671			1.58671
$832$	−0.0726721	−	0.108761i	−0.0726721	−	0.108761i
$833$		0			0
$834$	1.64196	+	2.13985i	1.64196	+	2.13985i
$835$		0			0
$836$		0			0
$837$	0.676037	+	3.39867i	0.676037	+	3.39867i
$838$		0			0
$839$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$839$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$840$		0			0
$841$	−0.224523	+	0.542046i	−0.224523	+	0.542046i
$842$		0			0
$843$		0			0
$844$	−0.735499	+	1.49144i	−0.735499	+	1.49144i
$845$		0			0
$846$	0.573937	−	0.994088i	0.573937	−	0.994088i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−2.34516	−	2.67414i	−2.34516	−	2.67414i
$853$	0.216773	−	0.324423i	0.216773	−	0.324423i	−0.707107	−	0.707107i	$-0.750000\pi$
$853$	0.216773	−	0.324423i	0.216773	−	0.324423i	0.923880	+	0.382683i	$0.125000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.83195	+	0.758819i	−1.83195	+	0.758819i	−0.866025	+	0.500000i	$0.833333\pi$
$857$	−1.83195	+	0.758819i	−1.83195	+	0.758819i	−0.965926	+	0.258819i	$0.916667\pi$
$858$		0			0
$859$	0.128293	−	0.0255190i	0.128293	−	0.0255190i	−0.130526	−	0.991445i	$-0.541667\pi$
$859$	0.128293	−	0.0255190i	0.128293	−	0.0255190i	0.258819	+	0.965926i	$0.416667\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	0.261052i		−	0.261052i	−0.991445	−	0.130526i	$-0.958333\pi$
$863$		−	0.261052i		−	0.261052i	0.991445	−	0.130526i	$-0.0416667\pi$
$864$	2.06803	−	0.702000i	2.06803	−	0.702000i
$865$		0			0
$866$		0			0
$867$	−1.49144	+	0.996552i	−1.49144	+	0.996552i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−0.226638	+	3.45783i	−0.226638	+	3.45783i
$877$	−1.08979	−	0.216773i	−1.08979	−	0.216773i	−0.382683	−	0.923880i	$-0.625000\pi$
$877$	−1.08979	−	0.216773i	−1.08979	−	0.216773i	−0.707107	+	0.707107i	$0.750000\pi$
$878$	0.0675653	+	0.252157i	0.0675653	+	0.252157i
$879$		0			0
$880$		0			0
$881$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$881$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$882$	−1.92043	−	1.10876i	−1.92043	−	1.10876i
$883$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$883$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i	−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$	−1.13053	−	0.991445i	−1.13053	−	0.991445i
$887$	−0.465926	+	1.12484i	−0.465926	+	1.12484i	0.500000	+	0.866025i	$0.333333\pi$
$887$	−0.465926	+	1.12484i	−0.465926	+	1.12484i	−0.965926	+	0.258819i	$0.916667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	1.78480	+	0.478235i	1.78480	+	0.478235i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−0.234633			−0.234633
$898$	−0.607206	+	0.465926i	−0.607206	+	0.465926i
$899$	−0.566715	−	0.848149i	−0.566715	−	0.848149i
$900$	−2.14196	−	0.573937i	−2.14196	−	0.573937i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	2.33586	+	2.04849i	2.33586	+	2.04849i
$907$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$907$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$908$		0			0
$909$	−0.480699	+	2.41663i	−0.480699	+	2.41663i
$910$		0			0
$911$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$911$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$919$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$920$		0			0
$921$	−3.25072	+	1.34649i	−3.25072	+	1.34649i
$922$	−0.391239	−	0.793353i	−0.391239	−	0.793353i
$923$	0.254391	−	0.0506014i	0.254391	−	0.0506014i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$	−0.483342	+	0.423880i	−0.483342	+	0.423880i
$929$		−	0.261052i		−	0.261052i	−0.991445	−	0.130526i	$-0.958333\pi$
$929$		−	0.261052i		−	0.261052i	0.991445	−	0.130526i	$-0.0416667\pi$
$930$		0			0
$931$		0			0
$932$	0.158919	−	0.207107i	0.158919	−	0.207107i
$933$	−3.04716	+	0.606118i	−3.04716	+	0.606118i
$934$		0			0
$935$		0			0
$936$	−0.0565890	+	0.284492i	−0.0565890	+	0.284492i
$937$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$937$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$941$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$942$		0			0
$943$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$944$	−0.357164	+	1.05217i	−0.357164	+	1.05217i
$945$		0			0
$946$		0			0
$947$	−0.125419	+	0.630526i	−0.125419	+	0.630526i	0.866025	+	0.500000i	$0.166667\pi$
$947$	−0.125419	+	0.630526i	−0.125419	+	0.630526i	−0.991445	+	0.130526i	$0.958333\pi$
$948$		0			0
$949$	−0.210111	−	0.140392i	−0.210111	−	0.140392i
$950$		0			0
$951$	−1.34649	+	3.25072i	−1.34649	+	3.25072i
$952$		0			0
$953$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$953$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$954$		0			0
$955$		0			0
$956$	−1.05441	+	0.608761i	−1.05441	+	0.608761i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.51764			−1.51764
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.382683	−	0.923880i	−0.382683	−	0.923880i	−0.991445	−	0.130526i	$-0.958333\pi$
$967$	−0.382683	−	0.923880i	−0.382683	−	0.923880i	0.608761	−	0.793353i	$-0.291667\pi$
$968$	0.707107	−	0.707107i	0.707107	−	0.707107i
$969$		0			0
$970$		0			0
$971$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$971$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$972$	−0.776005	−	0.382683i	−0.776005	−	0.382683i
$973$		0			0
$974$	0.991445	−	1.71723i	0.991445	−	1.71723i
$975$	0.165911	−	0.165911i	0.165911	−	0.165911i
$976$		0			0
$977$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$977$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$978$	3.45783	−	0.926523i	3.45783	−	0.926523i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−0.0420463	−	0.641502i	−0.0420463	−	0.641502i
$983$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$983$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$984$	−1.49144	+	0.996552i	−1.49144	+	0.996552i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	1.84776i		−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$991$		−	1.84776i		−	1.84776i	0.382683	−	0.923880i	$-0.375000\pi$
$992$	0.207107	+	1.57313i	0.207107	+	1.57313i
$993$			0.234633i			0.234633i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.92388	−	0.382683i	1.92388	−	0.382683i	0.923880	−	0.382683i	$-0.125000\pi$
$997$	1.92388	−	0.382683i	1.92388	−	0.382683i	1.00000			$0$
$998$	−0.837633	−	1.69855i	−0.837633	−	1.69855i
$999$		0			0

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

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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.45.2	✓	16
23.22	odd	2	CM	1472.1.s.b.45.2	✓	16
64.37	even	16	inner	1472.1.s.b.229.2	yes	16
1472.229	odd	16	inner	1472.1.s.b.229.2	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1472.1.s.b.45.2	✓	16	1.1	even	1	trivial
1472.1.s.b.45.2	✓	16	23.22	odd	2	CM
1472.1.s.b.229.2	yes	16	64.37	even	16	inner
1472.1.s.b.229.2	yes	16	1472.229	odd	16	inner

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Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
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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.45.2

Level	$1472$
Weight	$1$
Character	1472.45
Analytic conductor	$0.735$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{48}$
CM discriminant	-23
Inner twists	$4$