LMFDB - Embedded newform 772.1.u.a.147.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [772,1,Mod(59,772)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(772, base_ring=CyclotomicField(48))

chi = DirichletCharacter(H, H._module([24, 37]))

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

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

chi := DirichletCharacter("772.59");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 772 = 2^{2} \cdot 193 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	772.u (of order $48$, degree $16$, 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:	$0.385278189753$
Analytic rank:	$0$
Dimension:	$16$
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			147.1
Root			$-0.130526 - 0.991445i$ of defining polynomial
Character	$\chi$	$=$	772.147
Dual form			772.1.u.a.751.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.608761 + 0.793353i) q^{2} +(-0.258819 - 0.965926i) q^{4} +(0.0255190 - 0.389345i) q^{5} +(0.923880 + 0.382683i) q^{8} -1.00000i q^{9} +(0.293353 + 0.257264i) q^{10} +(-0.257264 - 1.29335i) q^{13} +(-0.866025 + 0.500000i) q^{16} +(-1.49144 - 0.735499i) q^{17} +(0.793353 + 0.608761i) q^{18} +(-0.382683 + 0.0761205i) q^{20} +(0.840506 + 0.110655i) q^{25} +(1.18270 + 0.583242i) q^{26} +(1.47479 + 0.293353i) q^{29} +(0.130526 - 0.991445i) q^{32} +(1.49144 - 0.735499i) q^{34} +(-0.965926 + 0.258819i) q^{36} +(0.641502 - 1.88981i) q^{37} +(0.172572 - 0.349942i) q^{40} +(-0.837633 + 1.69855i) q^{41} +(-0.389345 - 0.0255190i) q^{45} +(0.500000 + 0.866025i) q^{49} +(-0.599456 + 0.599456i) q^{50} +(-1.18270 + 0.583242i) q^{52} +(0.837633 + 0.284338i) q^{53} +(-1.13053 + 0.991445i) q^{58} +(0.0420463 + 0.641502i) q^{61} +(0.707107 + 0.707107i) q^{64} +(-0.510126 + 0.0671594i) q^{65} +(-0.324423 + 1.63099i) q^{68} +(0.382683 - 0.923880i) q^{72} +(-0.576581 - 1.69855i) q^{73} +(1.10876 + 1.65938i) q^{74} +(0.172572 + 0.349942i) q^{80} -1.00000 q^{81} +(-0.837633 - 1.69855i) q^{82} +(-0.324423 + 0.561918i) q^{85} +(-0.382683 + 1.92388i) q^{89} +(0.257264 - 0.293353i) q^{90} +(1.17604 + 1.53264i) q^{97} +(-0.991445 - 0.130526i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{10} - 8 q^{17} + 8 q^{34} + 8 q^{49} - 16 q^{58} + 8 q^{65} + 8 q^{74} - 16 q^{81}+O(q^{100}) $ 16 * q - 8 * q^10 - 8 * q^17 + 8 * q^34 + 8 * q^49 - 16 * q^58 + 8 * q^65 + 8 * q^74 - 16 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/772\mathbb{Z}\right)^\times$.

$n$	$5$	$387$
$\chi(n)$	$e\left(\frac{25}{48}\right)$	$-1$

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			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$5$	0.0255190	−	0.389345i	0.0255190	−	0.389345i	−0.965926	−	0.258819i	$-0.916667\pi$
$5$	0.0255190	−	0.389345i	0.0255190	−	0.389345i	0.991445	−	0.130526i	$-0.0416667\pi$
$6$		0			0
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$	0.923880	+	0.382683i	0.923880	+	0.382683i
$9$		−	1.00000i		−	1.00000i
$10$	0.293353	+	0.257264i	0.293353	+	0.257264i
$11$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$11$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$12$		0			0
$13$	−0.257264	−	1.29335i	−0.257264	−	1.29335i	−0.866025	−	0.500000i	$-0.833333\pi$
$13$	−0.257264	−	1.29335i	−0.257264	−	1.29335i	0.608761	−	0.793353i	$-0.291667\pi$
$14$		0			0
$15$		0			0
$16$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$17$	−1.49144	−	0.735499i	−1.49144	−	0.735499i	−0.500000	−	0.866025i	$-0.666667\pi$
$17$	−1.49144	−	0.735499i	−1.49144	−	0.735499i	−0.991445	+	0.130526i	$0.958333\pi$
$18$	0.793353	+	0.608761i	0.793353	+	0.608761i
$19$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$19$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$20$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i
$21$		0			0
$22$		0			0
$23$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$23$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$24$		0			0
$25$	0.840506	+	0.110655i	0.840506	+	0.110655i
$26$	1.18270	+	0.583242i	1.18270	+	0.583242i
$27$		0			0
$28$		0			0
$29$	1.47479	+	0.293353i	1.47479	+	0.293353i	0.866025	−	0.500000i	$-0.166667\pi$
$29$	1.47479	+	0.293353i	1.47479	+	0.293353i	0.608761	+	0.793353i	$0.291667\pi$
$30$		0			0
$31$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$31$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$32$	0.130526	−	0.991445i	0.130526	−	0.991445i
$33$		0			0
$34$	1.49144	−	0.735499i	1.49144	−	0.735499i
$35$		0			0
$36$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$37$	0.641502	−	1.88981i	0.641502	−	1.88981i	0.258819	−	0.965926i	$-0.416667\pi$
$37$	0.641502	−	1.88981i	0.641502	−	1.88981i	0.382683	−	0.923880i	$-0.375000\pi$
$38$		0			0
$39$		0			0
$40$	0.172572	−	0.349942i	0.172572	−	0.349942i
$41$	−0.837633	+	1.69855i	−0.837633	+	1.69855i	−0.130526	+	0.991445i	$0.541667\pi$
$41$	−0.837633	+	1.69855i	−0.837633	+	1.69855i	−0.707107	+	0.707107i	$0.750000\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	−0.389345	−	0.0255190i	−0.389345	−	0.0255190i
$46$		0			0
$47$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$47$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$48$		0			0
$49$	0.500000	+	0.866025i	0.500000	+	0.866025i
$50$	−0.599456	+	0.599456i	−0.599456	+	0.599456i
$51$		0			0
$52$	−1.18270	+	0.583242i	−1.18270	+	0.583242i
$53$	0.837633	+	0.284338i	0.837633	+	0.284338i	0.707107	−	0.707107i	$-0.250000\pi$
$53$	0.837633	+	0.284338i	0.837633	+	0.284338i	0.130526	+	0.991445i	$0.458333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−1.13053	+	0.991445i	−1.13053	+	0.991445i
$59$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$59$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$60$		0			0
$61$	0.0420463	+	0.641502i	0.0420463	+	0.641502i	0.965926	+	0.258819i	$0.0833333\pi$
$61$	0.0420463	+	0.641502i	0.0420463	+	0.641502i	−0.923880	+	0.382683i	$0.875000\pi$
$62$		0			0
$63$		0			0
$64$	0.707107	+	0.707107i	0.707107	+	0.707107i
$65$	−0.510126	+	0.0671594i	−0.510126	+	0.0671594i
$66$		0			0
$67$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$67$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$68$	−0.324423	+	1.63099i	−0.324423	+	1.63099i
$69$		0			0
$70$		0			0
$71$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$71$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$72$	0.382683	−	0.923880i	0.382683	−	0.923880i
$73$	−0.576581	−	1.69855i	−0.576581	−	1.69855i	−0.707107	−	0.707107i	$-0.750000\pi$
$73$	−0.576581	−	1.69855i	−0.576581	−	1.69855i	0.130526	−	0.991445i	$-0.458333\pi$
$74$	1.10876	+	1.65938i	1.10876	+	1.65938i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$79$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$80$	0.172572	+	0.349942i	0.172572	+	0.349942i
$81$	−1.00000			−1.00000
$82$	−0.837633	−	1.69855i	−0.837633	−	1.69855i
$83$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$83$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$84$		0			0
$85$	−0.324423	+	0.561918i	−0.324423	+	0.561918i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.382683	+	1.92388i	−0.382683	+	1.92388i			1.00000i	$0.5\pi$
$89$	−0.382683	+	1.92388i	−0.382683	+	1.92388i	−0.382683	+	0.923880i	$0.625000\pi$
$90$	0.257264	−	0.293353i	0.257264	−	0.293353i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.17604	+	1.53264i	1.17604	+	1.53264i	0.793353	+	0.608761i	$0.208333\pi$
$97$	1.17604	+	1.53264i	1.17604	+	1.53264i	0.382683	+	0.923880i	$0.375000\pi$
$98$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$99$		0			0
$100$	−0.110655	−	0.840506i	−0.110655	−	0.840506i
$101$	−0.410670	+	0.315118i	−0.410670	+	0.315118i	−0.793353	−	0.608761i	$-0.791667\pi$
$101$	−0.410670	+	0.315118i	−0.410670	+	0.315118i	0.382683	+	0.923880i	$0.375000\pi$
$102$		0			0
$103$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$103$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$104$	0.257264	−	1.29335i	0.257264	−	1.29335i
$105$		0			0
$106$	−0.735499	+	0.491445i	−0.735499	+	0.491445i
$107$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$107$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$108$		0			0
$109$	−0.991445	−	1.71723i	−0.991445	−	1.71723i	−0.608761	−	0.793353i	$-0.708333\pi$
$109$	−0.991445	−	1.71723i	−0.991445	−	1.71723i	−0.382683	−	0.923880i	$-0.625000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.0578541	+	0.117317i	0.0578541	+	0.117317i	0.923880	−	0.382683i	$-0.125000\pi$
$113$	0.0578541	+	0.117317i	0.0578541	+	0.117317i	−0.866025	+	0.500000i	$0.833333\pi$
$114$		0			0
$115$		0			0
$116$	−0.0983454	−	1.50046i	−0.0983454	−	1.50046i
$117$	−1.29335	+	0.257264i	−1.29335	+	0.257264i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$122$	−0.534534	−	0.357164i	−0.534534	−	0.357164i
$123$		0			0
$124$		0			0
$125$	0.140652	−	0.707107i	0.140652	−	0.707107i
$126$		0			0
$127$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$127$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$128$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$129$		0			0
$130$	0.257264	−	0.445594i	0.257264	−	0.445594i
$131$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$131$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−1.09645	−	1.25026i	−1.09645	−	1.25026i
$137$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$137$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$138$		0			0
$139$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$139$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.500000	+	0.866025i	0.500000	+	0.866025i
$145$	0.151851	−	0.566715i	0.151851	−	0.566715i
$146$	1.69855	+	0.576581i	1.69855	+	0.576581i
$147$		0			0
$148$	−1.99144	−	0.130526i	−1.99144	−	0.130526i
$149$	0.732626	−	0.835400i	0.732626	−	0.835400i	−0.258819	−	0.965926i	$-0.583333\pi$
$149$	0.732626	−	0.835400i	0.732626	−	0.835400i	0.991445	+	0.130526i	$0.0416667\pi$
$150$		0			0
$151$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$151$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$152$		0			0
$153$	−0.735499	+	1.49144i	−0.735499	+	1.49144i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.53264	+	0.410670i	−1.53264	+	0.410670i	−0.923880	−	0.382683i	$-0.875000\pi$
$157$	−1.53264	+	0.410670i	−1.53264	+	0.410670i	−0.608761	+	0.793353i	$0.708333\pi$
$158$		0			0
$159$		0			0
$160$	−0.382683	−	0.0761205i	−0.382683	−	0.0761205i
$161$		0			0
$162$	0.608761	−	0.793353i	0.608761	−	0.793353i
$163$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$163$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$164$	1.85747	+	0.369474i	1.85747	+	0.369474i
$165$		0			0
$166$		0			0
$167$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$167$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$168$		0			0
$169$	−0.682699	+	0.282783i	−0.682699	+	0.282783i
$170$	−0.248303	−	0.599456i	−0.248303	−	0.599456i
$171$		0			0
$172$		0			0
$173$	0.128293	−	0.0255190i	0.128293	−	0.0255190i	−0.130526	−	0.991445i	$-0.541667\pi$
$173$	0.128293	−	0.0255190i	0.128293	−	0.0255190i	0.258819	+	0.965926i	$0.416667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−1.29335	−	1.47479i	−1.29335	−	1.47479i
$179$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$179$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$180$	0.0761205	+	0.382683i	0.0761205	+	0.382683i
$181$	1.71723	+	0.991445i	1.71723	+	0.991445i	0.923880	+	0.382683i	$0.125000\pi$
$181$	1.71723	+	0.991445i	1.71723	+	0.991445i	0.793353	+	0.608761i	$0.208333\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.719416	−	0.297992i	−0.719416	−	0.297992i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$191$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$192$		0			0
$193$	0.707107	−	0.707107i	0.707107	−	0.707107i
$193$	0.707107	−	0.707107i	0.707107	−	0.707107i
$194$	−1.93185			−1.93185
$195$		0			0
$196$	0.707107	−	0.707107i	0.707107	−	0.707107i
$197$	0.258819	+	0.965926i	0.258819	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$197$	0.258819	+	0.965926i	0.258819	+	0.965926i	−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$199$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$200$	0.734181	+	0.423880i	0.734181	+	0.423880i
$201$		0			0
$202$		−	0.517638i		−	0.517638i
$203$		0			0
$204$		0			0
$205$	0.639947	+	0.369474i	0.639947	+	0.369474i
$206$		0			0
$207$		0			0
$208$	0.869474	+	0.991445i	0.869474	+	0.991445i
$209$		0			0
$210$		0			0
$211$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$211$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$212$	0.0578541	−	0.882683i	0.0578541	−	0.882683i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.96593	+	0.258819i	1.96593	+	0.258819i
$219$		0			0
$220$		0			0
$221$	−0.567565	+	2.11818i	−0.567565	+	2.11818i
$222$		0			0
$223$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$223$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$224$		0			0
$225$	0.110655	−	0.840506i	0.110655	−	0.840506i
$226$	−0.128293	−	0.0255190i	−0.128293	−	0.0255190i
$227$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$227$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$228$		0			0
$229$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$229$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$230$		0			0
$231$		0			0
$232$	1.25026	+	0.835400i	1.25026	+	0.835400i
$233$	−0.867580	+	1.75928i	−0.867580	+	1.75928i	−0.258819	+	0.965926i	$0.583333\pi$
$233$	−0.867580	+	1.75928i	−0.867580	+	1.75928i	−0.608761	+	0.793353i	$0.708333\pi$
$234$	0.583242	−	1.18270i	0.583242	−	1.18270i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$239$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$240$		0			0
$241$	−0.198092	+	0.739288i	−0.198092	+	0.739288i	0.793353	+	0.608761i	$0.208333\pi$
$241$	−0.198092	+	0.739288i	−0.198092	+	0.739288i	−0.991445	+	0.130526i	$0.958333\pi$
$242$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$243$		0			0
$244$	0.608761	−	0.206647i	0.608761	−	0.206647i
$245$	0.349942	−	0.172572i	0.349942	−	0.172572i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.475362	+	0.542046i	0.475362	+	0.542046i
$251$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$251$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.500000	−	0.866025i	0.500000	−	0.866025i
$257$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$257$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$258$		0			0
$259$		0			0
$260$	0.196901	+	0.475362i	0.196901	+	0.475362i
$261$	0.293353	−	1.47479i	0.293353	−	1.47479i
$262$		0			0
$263$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$263$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$264$		0			0
$265$	0.132081	−	0.318872i	0.132081	−	0.318872i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.75928	−	0.349942i	1.75928	−	0.349942i	0.793353	−	0.608761i	$-0.208333\pi$
$269$	1.75928	−	0.349942i	1.75928	−	0.349942i	0.965926	+	0.258819i	$0.0833333\pi$
$270$		0			0
$271$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$271$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$272$	1.65938	−	0.108761i	1.65938	−	0.108761i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.793353	+	1.37413i	0.793353	+	1.37413i	0.923880	+	0.382683i	$0.125000\pi$
$277$	0.793353	+	1.37413i	0.793353	+	1.37413i	−0.130526	+	0.991445i	$0.541667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.382683	−	1.92388i	−0.382683	−	1.92388i	−0.382683	−	0.923880i	$-0.625000\pi$
$281$	−0.382683	−	1.92388i	−0.382683	−	1.92388i		−	1.00000i	$-0.5\pi$
$282$		0			0
$283$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$283$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$289$	1.07469	+	1.40056i	1.07469	+	1.40056i
$290$	0.357164	+	0.465466i	0.357164	+	0.465466i
$291$		0			0
$292$	−1.49144	+	0.996552i	−1.49144	+	0.996552i
$293$	−0.158919	−	1.20711i	−0.158919	−	1.20711i	−0.866025	−	0.500000i	$-0.833333\pi$
$293$	−0.158919	−	1.20711i	−0.158919	−	1.20711i	0.707107	−	0.707107i	$-0.250000\pi$
$294$		0			0
$295$		0			0
$296$	1.31587	−	1.50046i	1.31587	−	1.50046i
$297$		0			0
$298$	0.216773	+	1.08979i	0.216773	+	1.08979i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.250839			0.250839
$306$	−0.735499	−	1.49144i	−0.735499	−	1.49144i
$307$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$307$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$311$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$312$		0			0
$313$	−0.0420463	−	0.123864i	−0.0420463	−	0.123864i	0.923880	−	0.382683i	$-0.125000\pi$
$313$	−0.0420463	−	0.123864i	−0.0420463	−	0.123864i	−0.965926	+	0.258819i	$0.916667\pi$
$314$	0.607206	−	1.46593i	0.607206	−	1.46593i
$315$		0			0
$316$		0			0
$317$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$317$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$318$		0			0
$319$		0			0
$320$	0.293353	−	0.257264i	0.293353	−	0.257264i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.258819	+	0.965926i	0.258819	+	0.965926i
$325$	−0.0731163	−	1.11554i	−0.0731163	−	1.11554i
$326$		0			0
$327$		0			0
$328$	−1.42388	+	1.24871i	−1.42388	+	1.24871i
$329$		0			0
$330$		0			0
$331$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$331$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$332$		0			0
$333$	−1.88981	−	0.641502i	−1.88981	−	0.641502i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.866025	+	1.50000i	0.866025	+	1.50000i	0.866025	+	0.500000i	$0.166667\pi$
$337$	0.866025	+	1.50000i	0.866025	+	1.50000i			1.00000i	$0.5\pi$
$338$	0.191254	−	0.713769i	0.191254	−	0.713769i
$339$		0			0
$340$	0.626738	+	0.167934i	0.626738	+	0.167934i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.0578541	+	0.117317i	−0.0578541	+	0.117317i
$347$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$347$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$348$		0			0
$349$	0.206647	−	0.608761i	0.206647	−	0.608761i	−0.793353	−	0.608761i	$-0.791667\pi$
$349$	0.206647	−	0.608761i	0.206647	−	0.608761i	1.00000			$0$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.867580	+	0.172572i	0.867580	+	0.172572i	0.608761	−	0.793353i	$-0.291667\pi$
$353$	0.867580	+	0.172572i	0.867580	+	0.172572i	0.258819	+	0.965926i	$0.416667\pi$
$354$		0			0
$355$		0			0
$356$	1.95737	−	0.128293i	1.95737	−	0.128293i
$357$		0			0
$358$		0			0
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$	−0.349942	−	0.172572i	−0.349942	−	0.172572i
$361$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$362$	−1.83195	+	0.758819i	−1.83195	+	0.758819i
$363$		0			0
$364$		0			0
$365$	−0.676037	+	0.181144i	−0.676037	+	0.181144i
$366$		0			0
$367$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$367$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$368$		0			0
$369$	1.69855	+	0.837633i	1.69855	+	0.837633i
$370$	0.674366	−	0.389345i	0.674366	−	0.389345i
$371$		0			0
$372$		0			0
$373$	0.0255190	+	0.128293i	0.0255190	+	0.128293i	0.991445	−	0.130526i	$-0.0416667\pi$
$373$	0.0255190	+	0.128293i	0.0255190	+	0.128293i	−0.965926	+	0.258819i	$0.916667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.98289i		−	1.98289i
$378$		0			0
$379$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$379$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$	0.130526	+	0.991445i	0.130526	+	0.991445i
$387$		0			0
$388$	1.17604	−	1.53264i	1.17604	−	1.53264i
$389$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$389$	1.36603	−	1.36603i	1.36603	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$390$		0			0
$391$		0			0
$392$	0.130526	+	0.991445i	0.130526	+	0.991445i
$393$		0			0
$394$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$395$		0			0
$396$		0			0
$397$	0.617317	+	0.923880i	0.617317	+	0.923880i	1.00000			$0$
$397$	0.617317	+	0.923880i	0.617317	+	0.923880i	−0.382683	+	0.923880i	$0.625000\pi$
$398$		0			0
$399$		0			0
$400$	−0.783227	+	0.324423i	−0.783227	+	0.324423i
$401$	−0.732626	−	0.835400i	−0.732626	−	0.835400i	0.258819	−	0.965926i	$-0.416667\pi$
$401$	−0.732626	−	0.835400i	−0.732626	−	0.835400i	−0.991445	+	0.130526i	$0.958333\pi$
$402$		0			0
$403$		0			0
$404$	0.410670	+	0.315118i	0.410670	+	0.315118i
$405$	−0.0255190	+	0.389345i	−0.0255190	+	0.389345i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.198092	+	0.478235i	0.198092	+	0.478235i	0.991445	−	0.130526i	$-0.0416667\pi$
$409$	0.198092	+	0.478235i	0.198092	+	0.478235i	−0.793353	+	0.608761i	$0.791667\pi$
$410$	−0.682699	+	0.282783i	−0.682699	+	0.282783i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−1.31587	+	0.0862466i	−1.31587	+	0.0862466i
$417$		0			0
$418$		0			0
$419$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$419$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$420$		0			0
$421$	−0.617317	+	0.923880i	−0.617317	+	0.923880i	0.382683	+	0.923880i	$0.375000\pi$
$421$	−0.617317	+	0.923880i	−0.617317	+	0.923880i	−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	0.665060	+	0.583242i	0.665060	+	0.583242i
$425$	−1.17218	−	0.783227i	−1.17218	−	0.783227i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$431$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$432$		0			0
$433$	0.837633	+	0.284338i	0.837633	+	0.284338i	0.707107	−	0.707107i	$-0.250000\pi$
$433$	0.837633	+	0.284338i	0.837633	+	0.284338i	0.130526	+	0.991445i	$0.458333\pi$
$434$		0			0
$435$		0			0
$436$	−1.40211	+	1.40211i	−1.40211	+	1.40211i
$437$		0			0
$438$		0			0
$439$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$439$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$440$		0			0
$441$	0.866025	−	0.500000i	0.866025	−	0.500000i
$442$	−1.33496	−	1.73975i	−1.33496	−	1.73975i
$443$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$443$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$444$		0			0
$445$	0.739288	+	0.198092i	0.739288	+	0.198092i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	−0.965926	−	0.258819i	$-0.916667\pi$
$449$	−0.258819	+	0.448288i	−0.258819	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$450$	0.599456	+	0.599456i	0.599456	+	0.599456i
$451$		0			0
$452$	0.0983454	−	0.0862466i	0.0983454	−	0.0862466i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.57469	−	1.05217i	−1.57469	−	1.05217i	−0.965926	−	0.258819i	$-0.916667\pi$
$457$	−1.57469	−	1.05217i	−1.57469	−	1.05217i	−0.608761	−	0.793353i	$-0.708333\pi$
$458$	−0.541196	+	1.30656i	−0.541196	+	1.30656i
$459$		0			0
$460$		0			0
$461$	−1.12197	+	0.860919i	−1.12197	+	0.860919i	−0.991445	−	0.130526i	$-0.958333\pi$
$461$	−1.12197	+	0.860919i	−1.12197	+	0.860919i	−0.130526	+	0.991445i	$0.541667\pi$
$462$		0			0
$463$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$463$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$464$	−1.42388	+	0.483342i	−1.42388	+	0.483342i
$465$		0			0
$466$	−0.867580	−	1.75928i	−0.867580	−	1.75928i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	0.583242	+	1.18270i	0.583242	+	1.18270i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.284338	−	0.837633i	0.284338	−	0.837633i
$478$		0			0
$479$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$479$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$480$		0			0
$481$	−2.60922	−	0.343511i	−2.60922	−	0.343511i
$482$	−0.465926	−	0.607206i	−0.465926	−	0.607206i
$483$		0			0
$484$	0.991445	+	0.130526i	0.991445	+	0.130526i
$485$	0.626738	−	0.418773i	0.626738	−	0.418773i
$486$		0			0
$487$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$487$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$488$	−0.206647	+	0.608761i	−0.206647	+	0.608761i
$489$		0			0
$490$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$491$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$491$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$492$		0			0
$493$	−1.98380	−	1.52222i	−1.98380	−	1.52222i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$499$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$500$	−0.719416	+	0.0471530i	−0.719416	+	0.0471530i
$501$		0			0
$502$		0			0
$503$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$503$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$504$		0			0
$505$	0.112210	+	0.167934i	0.112210	+	0.167934i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.608761	−	1.79335i	−0.608761	−	1.79335i	−0.608761	−	0.793353i	$-0.708333\pi$
$509$	−0.608761	−	1.79335i	−0.608761	−	1.79335i		−	1.00000i	$-0.5\pi$
$510$		0			0
$511$		0			0
$512$	0.382683	+	0.923880i	0.382683	+	0.923880i
$513$		0			0
$514$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.496996	−	0.133170i	−0.496996	−	0.133170i
$521$	1.34861	−	1.18270i	1.34861	−	1.18270i	0.382683	−	0.923880i	$-0.375000\pi$
$521$	1.34861	−	1.18270i	1.34861	−	1.18270i	0.965926	−	0.258819i	$-0.0833333\pi$
$522$	0.991445	+	1.13053i	0.991445	+	1.13053i
$523$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$523$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$530$	0.172572	+	0.298904i	0.172572	+	0.298904i
$531$		0			0
$532$		0			0
$533$	2.41232	+	0.646379i	2.41232	+	0.646379i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.793353	+	1.60876i	−0.793353	+	1.60876i
$539$		0			0
$540$		0			0
$541$	1.47479	+	1.29335i	1.47479	+	1.29335i	0.866025	+	0.500000i	$0.166667\pi$
$541$	1.47479	+	1.29335i	1.47479	+	1.29335i	0.608761	+	0.793353i	$0.291667\pi$
$542$		0			0
$543$		0			0
$544$	−0.923880	+	1.38268i	−0.923880	+	1.38268i
$545$	−0.693897	+	0.342192i	−0.693897	+	0.342192i
$546$		0			0
$547$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$547$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$548$		0			0
$549$	0.641502	−	0.0420463i	0.641502	−	0.0420463i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.57313	−	0.207107i	−1.57313	−	0.207107i
$555$		0			0
$556$		0			0
$557$	−1.99144	−	0.130526i	−1.99144	−	0.130526i	−0.991445	−	0.130526i	$-0.958333\pi$
$557$	−1.99144	−	0.130526i	−1.99144	−	0.130526i	−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.75928	+	0.867580i	1.75928	+	0.867580i
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$	0.0471530	−	0.0195314i	0.0471530	−	0.0195314i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.42388	−	1.24871i	−1.42388	−	1.24871i	−0.923880	−	0.382683i	$-0.875000\pi$
$569$	−1.42388	−	1.24871i	−1.42388	−	1.24871i	−0.500000	−	0.866025i	$-0.666667\pi$
$570$		0			0
$571$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$571$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.707107	−	0.707107i	0.707107	−	0.707107i
$577$	−1.05441	+	1.37413i	−1.05441	+	1.37413i	−0.130526	+	0.991445i	$0.541667\pi$
$577$	−1.05441	+	1.37413i	−1.05441	+	1.37413i	−0.923880	+	0.382683i	$0.875000\pi$
$578$	−1.76537			−1.76537
$579$		0			0
$580$	−0.586707			−0.586707
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.117317	−	1.78990i	0.117317	−	1.78990i
$585$	0.0671594	+	0.510126i	0.0671594	+	0.510126i
$586$	1.05441	+	0.608761i	1.05441	+	0.608761i
$587$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$587$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.389345	+	1.95737i	0.389345	+	1.95737i
$593$	1.12484	−	0.465926i	1.12484	−	0.465926i	0.258819	−	0.965926i	$-0.416667\pi$
$593$	1.12484	−	0.465926i	1.12484	−	0.465926i	0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$	−0.996552	−	0.491445i	−0.996552	−	0.491445i
$597$		0			0
$598$		0			0
$599$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$599$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$600$		0			0
$601$	−1.99144	−	0.130526i	−1.99144	−	0.130526i	−0.991445	−	0.130526i	$-0.958333\pi$
$601$	−1.99144	−	0.130526i	−1.99144	−	0.130526i	−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.349942	+	0.172572i	0.349942	+	0.172572i
$606$		0			0
$607$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$607$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$608$		0			0
$609$		0			0
$610$	−0.152701	+	0.199004i	−0.152701	+	0.199004i
$611$		0			0
$612$	1.63099	+	0.324423i	1.63099	+	0.324423i
$613$	0.576581	−	0.284338i	0.576581	−	0.284338i	−0.130526	−	0.991445i	$-0.541667\pi$
$613$	0.576581	−	0.284338i	0.576581	−	0.284338i	0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.483342	+	0.423880i	0.483342	+	0.423880i	0.866025	−	0.500000i	$-0.166667\pi$
$617$	0.483342	+	0.423880i	0.483342	+	0.423880i	−0.382683	+	0.923880i	$0.625000\pi$
$618$		0			0
$619$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$619$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.547153	+	0.146609i	0.547153	+	0.146609i
$626$	0.123864	+	0.0420463i	0.123864	+	0.0420463i
$627$		0			0
$628$	0.793353	+	1.37413i	0.793353	+	1.37413i
$629$	−2.34672	+	2.34672i	−2.34672	+	2.34672i
$630$		0			0
$631$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$631$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$632$		0			0
$633$		0			0
$634$	1.22474	−	0.707107i	1.22474	−	0.707107i
$635$		0			0
$636$		0			0
$637$	0.991445	−	0.869474i	0.991445	−	0.869474i
$638$		0			0
$639$		0			0
$640$	0.0255190	+	0.389345i	0.0255190	+	0.389345i
$641$	−0.410670	−	1.53264i	−0.410670	−	1.53264i	−0.793353	−	0.608761i	$-0.791667\pi$
$641$	−0.410670	−	1.53264i	−0.410670	−	1.53264i	0.382683	−	0.923880i	$-0.375000\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$647$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$648$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$649$		0			0
$650$	0.929527	+	0.621090i	0.929527	+	0.621090i
$651$		0			0
$652$		0			0
$653$	−0.835400	−	1.25026i	−0.835400	−	1.25026i	−0.965926	−	0.258819i	$-0.916667\pi$
$653$	−0.835400	−	1.25026i	−0.835400	−	1.25026i	0.130526	−	0.991445i	$-0.458333\pi$
$654$		0			0
$655$		0			0
$656$	−0.123864	−	1.88981i	−0.123864	−	1.88981i
$657$	−1.69855	+	0.576581i	−1.69855	+	0.576581i
$658$		0			0
$659$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$659$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$660$		0			0
$661$	−0.665060	−	1.34861i	−0.665060	−	1.34861i	−0.923880	−	0.382683i	$-0.875000\pi$
$661$	−0.665060	−	1.34861i	−0.665060	−	1.34861i	0.258819	−	0.965926i	$-0.416667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.65938	−	1.10876i	1.65938	−	1.10876i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.49144	−	0.996552i	1.49144	−	0.996552i	0.500000	−	0.866025i	$-0.333333\pi$
$673$	1.49144	−	0.996552i	1.49144	−	0.996552i	0.991445	−	0.130526i	$-0.0416667\pi$
$674$	−1.71723	−	0.226078i	−1.71723	−	0.226078i
$675$		0			0
$676$	0.449843	+	0.586247i	0.449843	+	0.586247i
$677$	0.758819	+	0.0999004i	0.758819	+	0.0999004i	0.500000	−	0.866025i	$-0.333333\pi$
$677$	0.758819	+	0.0999004i	0.758819	+	0.0999004i	0.258819	+	0.965926i	$0.416667\pi$
$678$		0			0
$679$		0			0
$680$	−0.514765	+	0.394993i	−0.514765	+	0.394993i
$681$		0			0
$682$		0			0
$683$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$683$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.152257	−	1.15651i	0.152257	−	1.15651i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	−0.0578541	−	0.117317i	−0.0578541	−	0.117317i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.49857	−	1.91722i	2.49857	−	1.91722i
$698$	0.357164	+	0.534534i	0.357164	+	0.534534i
$699$		0			0
$700$		0			0
$701$	1.49144	+	0.996552i	1.49144	+	0.996552i	0.991445	+	0.130526i	$0.0416667\pi$
$701$	1.49144	+	0.996552i	1.49144	+	0.996552i	0.500000	+	0.866025i	$0.333333\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.665060	+	0.583242i	−0.665060	+	0.583242i
$707$		0			0
$708$		0			0
$709$	0.608761	−	1.05441i	0.608761	−	1.05441i	−0.382683	−	0.923880i	$-0.625000\pi$
$709$	0.608761	−	1.05441i	0.608761	−	1.05441i	0.991445	−	0.130526i	$-0.0416667\pi$
$710$		0			0
$711$		0			0
$712$	−1.08979	+	1.63099i	−1.08979	+	1.63099i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$719$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$720$	0.349942	−	0.172572i	0.349942	−	0.172572i
$721$		0			0
$722$	0.707107	−	0.707107i	0.707107	−	0.707107i
$723$		0			0
$724$	0.513210	−	1.91532i	0.513210	−	1.91532i
$725$	1.20711	+	0.409758i	1.20711	+	0.409758i
$726$		0			0
$727$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$727$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$	0.267834	−	0.646609i	0.267834	−	0.646609i
$731$		0			0
$732$		0			0
$733$	−0.735499	−	0.491445i	−0.735499	−	0.491445i	0.130526	−	0.991445i	$-0.458333\pi$
$733$	−0.735499	−	0.491445i	−0.735499	−	0.491445i	−0.866025	+	0.500000i	$0.833333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−1.69855	+	0.837633i	−1.69855	+	0.837633i
$739$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$739$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$740$	−0.101640	+	0.772029i	−0.101640	+	0.772029i
$741$		0			0
$742$		0			0
$743$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$743$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$744$		0			0
$745$	−0.306563	−	0.306563i	−0.306563	−	0.306563i
$746$	−0.117317	−	0.0578541i	−0.117317	−	0.0578541i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$751$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$752$		0			0
$753$		0			0
$754$	1.57313	+	1.20711i	1.57313	+	1.20711i
$755$		0			0
$756$		0			0
$757$	1.31587	+	1.50046i	1.31587	+	1.50046i	0.707107	+	0.707107i	$0.250000\pi$
$757$	1.31587	+	1.50046i	1.31587	+	1.50046i	0.608761	+	0.793353i	$0.291667\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.835400	+	1.25026i	0.835400	+	1.25026i	0.965926	+	0.258819i	$0.0833333\pi$
$761$	0.835400	+	1.25026i	0.835400	+	1.25026i	−0.130526	+	0.991445i	$0.541667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.561918	+	0.324423i	0.561918	+	0.324423i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	−0.965926	−	0.258819i	$-0.916667\pi$
$769$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$770$		0			0
$771$		0			0
$772$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$773$	−1.73205			−1.73205			−0.866025	−	0.500000i	$-0.833333\pi$
$773$	−1.73205			−1.73205			−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$		0			0
$776$	0.500000	+	1.86603i	0.500000	+	1.86603i
$777$		0			0
$778$	0.252157	+	1.91532i	0.252157	+	1.91532i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$785$	0.120781	+	0.607206i	0.120781	+	0.607206i
$786$		0			0
$787$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$787$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$788$	0.866025	−	0.500000i	0.866025	−	0.500000i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.818872	−	0.219416i	0.818872	−	0.219416i
$794$	−1.10876	−	0.0726721i	−1.10876	−	0.0726721i
$795$		0			0
$796$		0			0
$797$	−1.83195	−	0.241181i	−1.83195	−	0.241181i	−0.866025	−	0.500000i	$-0.833333\pi$
$797$	−1.83195	−	0.241181i	−1.83195	−	0.241181i	−0.965926	+	0.258819i	$0.916667\pi$
$798$		0			0
$799$		0			0
$800$	0.219416	−	0.818872i	0.219416	−	0.818872i
$801$	1.92388	+	0.382683i	1.92388	+	0.382683i
$802$	1.10876	−	0.0726721i	1.10876	−	0.0726721i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.500000	+	0.133975i	−0.500000	+	0.133975i
$809$	−0.206647	+	0.608761i	−0.206647	+	0.608761i	0.793353	+	0.608761i	$0.208333\pi$
$809$	−0.206647	+	0.608761i	−0.206647	+	0.608761i	−1.00000			$1.00000\pi$
$810$	−0.293353	−	0.257264i	−0.293353	−	0.257264i
$811$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$811$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.500000	−	0.133975i	−0.500000	−	0.133975i
$819$		0			0
$820$	0.191254	−	0.713769i	0.191254	−	0.713769i
$821$	0.500000	+	0.866025i	0.500000	+	0.866025i	1.00000			$0$
$821$	0.500000	+	0.866025i	0.500000	+	0.866025i	−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$823$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$827$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$828$		0			0
$829$	0.583242	+	0.665060i	0.583242	+	0.665060i	0.965926	−	0.258819i	$-0.0833333\pi$
$829$	0.583242	+	0.665060i	0.583242	+	0.665060i	−0.382683	+	0.923880i	$0.625000\pi$
$830$		0			0
$831$		0			0
$832$	0.732626	−	1.09645i	0.732626	−	1.09645i
$833$	−0.108761	−	1.65938i	−0.108761	−	1.65938i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		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
$841$	1.16506	+	0.482584i	1.16506	+	0.482584i
$842$	−0.357164	−	1.05217i	−0.357164	−	1.05217i
$843$		0			0
$844$		0			0
$845$	0.0926784	+	0.273022i	0.0926784	+	0.273022i
$846$		0			0
$847$		0			0
$848$	−0.867580	+	0.172572i	−0.867580	+	0.172572i
$849$		0			0
$850$	1.33496	−	0.453156i	1.33496	−	0.453156i
$851$		0			0
$852$		0			0
$853$	−1.00000			−1.00000			−0.500000	−	0.866025i	$-0.666667\pi$
$853$	−1.00000			−1.00000			−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.130526	−	0.226078i	0.130526	−	0.226078i	−0.793353	−	0.608761i	$-0.791667\pi$
$857$	0.130526	−	0.226078i	0.130526	−	0.226078i	0.923880	+	0.382683i	$0.125000\pi$
$858$		0			0
$859$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$859$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$863$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$864$		0			0
$865$	−0.00666180	−	0.0506014i	−0.00666180	−	0.0506014i
$866$	−0.735499	+	0.491445i	−0.735499	+	0.491445i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.258819	−	1.96593i	−0.258819	−	1.96593i
$873$	1.53264	−	1.17604i	1.53264	−	1.17604i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.382683	+	1.92388i	0.382683	+	1.92388i	0.382683	+	0.923880i	$0.375000\pi$
$877$	0.382683	+	1.92388i	0.382683	+	1.92388i			1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.382683	+	0.662827i	0.382683	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$881$	0.382683	+	0.662827i	0.382683	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$882$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$883$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$883$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$884$	2.19290			2.19290
$885$		0			0
$886$		0			0
$887$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$887$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$888$		0			0
$889$		0			0
$890$	−0.607206	+	0.465926i	−0.607206	+	0.465926i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.198092	−	0.478235i	−0.198092	−	0.478235i
$899$		0			0
$900$	−0.840506	+	0.110655i	−0.840506	+	0.110655i
$901$	−1.04015	−	1.04015i	−1.04015	−	1.04015i
$902$		0			0
$903$		0			0
$904$	0.00855514	+	0.130526i	0.00855514	+	0.130526i
$905$	0.429836	−	0.643296i	0.429836	−	0.643296i
$906$		0			0
$907$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$907$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$908$		0			0
$909$	0.315118	+	0.410670i	0.315118	+	0.410670i
$910$		0			0
$911$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$911$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$912$		0			0
$913$		0			0
$914$	1.79335	−	0.608761i	1.79335	−	0.608761i
$915$		0			0
$916$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$917$		0			0
$918$		0			0
$919$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$919$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$920$		0			0
$921$		0			0
$922$		−	1.41421i		−	1.41421i
$923$		0			0
$924$		0			0
$925$	0.748303	−	1.51741i	0.748303	−	1.51741i
$926$		0			0
$927$		0			0
$928$	0.483342	−	1.42388i	0.483342	−	1.42388i
$929$	1.17604	−	0.315118i	1.17604	−	0.315118i	0.382683	−	0.923880i	$-0.375000\pi$
$929$	1.17604	−	0.315118i	1.17604	−	0.315118i	0.793353	+	0.608761i	$0.208333\pi$
$930$		0			0
$931$		0			0
$932$	1.92388	+	0.382683i	1.92388	+	0.382683i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−1.29335	−	0.257264i	−1.29335	−	0.257264i
$937$	−0.0675653	+	0.252157i	−0.0675653	+	0.252157i	−0.991445	−	0.130526i	$-0.958333\pi$
$937$	−0.0675653	+	0.252157i	−0.0675653	+	0.252157i	0.923880	+	0.382683i	$0.125000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$941$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$947$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$948$		0			0
$949$	−2.04849	+	1.18270i	−2.04849	+	1.18270i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.37413	−	0.793353i	−1.37413	−	0.793353i	−0.382683	−	0.923880i	$-0.625000\pi$
$953$	−1.37413	−	0.793353i	−1.37413	−	0.793353i	−0.991445	+	0.130526i	$0.958333\pi$
$954$	0.491445	+	0.735499i	0.491445	+	0.735499i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$962$	1.86092	−	1.86092i	1.86092	−	1.86092i
$963$		0			0
$964$	0.765367			0.765367
$965$	−0.257264	−	0.293353i	−0.257264	−	0.293353i
$966$		0			0
$967$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$967$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$968$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$969$		0			0
$970$	−0.0492990	+	0.752157i	−0.0492990	+	0.752157i
$971$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$971$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.357164	−	0.534534i	−0.357164	−	0.534534i
$977$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.258819	−	0.965926i	$-0.583333\pi$
$977$	−1.22474	−	0.707107i	−1.22474	−	0.707107i	−0.965926	+	0.258819i	$0.916667\pi$
$978$		0			0
$979$		0			0
$980$	−0.257264	−	0.293353i	−0.257264	−	0.293353i
$981$	−1.71723	+	0.991445i	−1.71723	+	0.991445i
$982$		0			0
$983$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$983$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$984$		0			0
$985$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$986$	2.41532	−	0.647184i	2.41532	−	0.647184i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$991$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.158919	+	1.20711i	−0.158919	+	1.20711i	0.707107	+	0.707107i	$0.250000\pi$
$997$	−0.158919	+	1.20711i	−0.158919	+	1.20711i	−0.866025	+	0.500000i	$0.833333\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	772.1.u.a.147.1	✓	16
4.3	odd	2	CM	772.1.u.a.147.1	✓	16
193.172	even	48	inner	772.1.u.a.751.1	yes	16
772.751	odd	48	inner	772.1.u.a.751.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
772.1.u.a.147.1	✓	16	1.1	even	1	trivial
772.1.u.a.147.1	✓	16	4.3	odd	2	CM
772.1.u.a.751.1	yes	16	193.172	even	48	inner
772.1.u.a.751.1	yes	16	772.751	odd	48	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 772 = 2^{2} \cdot 193 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	772.u (of order \(48\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.608761 + 0.793353i) q^{2} +(-0.258819 - 0.965926i) q^{4} +(0.0255190 - 0.389345i) q^{5} +(0.923880 + 0.382683i) q^{8} -1.00000i q^{9} +(0.293353 + 0.257264i) q^{10} +(-0.257264 - 1.29335i) q^{13} +(-0.866025 + 0.500000i) q^{16} +(-1.49144 - 0.735499i) q^{17} +(0.793353 + 0.608761i) q^{18} +(-0.382683 + 0.0761205i) q^{20} +(0.840506 + 0.110655i) q^{25} +(1.18270 + 0.583242i) q^{26} +(1.47479 + 0.293353i) q^{29} +(0.130526 - 0.991445i) q^{32} +(1.49144 - 0.735499i) q^{34} +(-0.965926 + 0.258819i) q^{36} +(0.641502 - 1.88981i) q^{37} +(0.172572 - 0.349942i) q^{40} +(-0.837633 + 1.69855i) q^{41} +(-0.389345 - 0.0255190i) q^{45} +(0.500000 + 0.866025i) q^{49} +(-0.599456 + 0.599456i) q^{50} +(-1.18270 + 0.583242i) q^{52} +(0.837633 + 0.284338i) q^{53} +(-1.13053 + 0.991445i) q^{58} +(0.0420463 + 0.641502i) q^{61} +(0.707107 + 0.707107i) q^{64} +(-0.510126 + 0.0671594i) q^{65} +(-0.324423 + 1.63099i) q^{68} +(0.382683 - 0.923880i) q^{72} +(-0.576581 - 1.69855i) q^{73} +(1.10876 + 1.65938i) q^{74} +(0.172572 + 0.349942i) q^{80} -1.00000 q^{81} +(-0.837633 - 1.69855i) q^{82} +(-0.324423 + 0.561918i) q^{85} +(-0.382683 + 1.92388i) q^{89} +(0.257264 - 0.293353i) q^{90} +(1.17604 + 1.53264i) q^{97} +(-0.991445 - 0.130526i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{10} - 8 q^{17} + 8 q^{34} + 8 q^{49} - 16 q^{58} + 8 q^{65} + 8 q^{74} - 16 q^{81}+O(q^{100}) \) 16 * q - 8 * q^10 - 8 * q^17 + 8 * q^34 + 8 * q^49 - 16 * q^58 + 8 * q^65 + 8 * q^74 - 16 * q^81

Introduction

L-functions

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