LMFDB - Embedded newform 772.1.u.a.615.1

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			615.1
Root			$-0.991445 - 0.130526i$ of defining polynomial
Character	$\chi$	$=$	772.615
Dual form			772.1.u.a.59.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.793353 + 0.608761i) q^{2} +(0.258819 - 0.965926i) q^{4} +(0.835400 + 0.732626i) q^{5} +(0.382683 + 0.923880i) q^{8} +1.00000i q^{9} +(-1.10876 - 0.0726721i) q^{10} +(-0.0726721 - 0.108761i) q^{13} +(-0.866025 - 0.500000i) q^{16} +(-0.369474 - 0.125419i) q^{17} +(-0.608761 - 0.793353i) q^{18} +(0.923880 - 0.617317i) q^{20} +(0.0306258 + 0.232626i) q^{25} +(0.123864 + 0.0420463i) q^{26} +(1.65938 + 1.10876i) q^{29} +(0.991445 - 0.130526i) q^{32} +(0.369474 - 0.125419i) q^{34} +(0.965926 + 0.258819i) q^{36} +(-1.18270 - 0.583242i) q^{37} +(-0.357164 + 1.05217i) q^{40} +(-0.284338 + 0.837633i) q^{41} +(-0.732626 + 0.835400i) q^{45} +(0.500000 - 0.866025i) q^{49} +(-0.165911 - 0.165911i) q^{50} +(-0.123864 + 0.0420463i) q^{52} +(0.284338 - 0.576581i) q^{53} +(-1.99144 + 0.130526i) q^{58} +(-1.34861 + 1.18270i) q^{61} +(-0.707107 + 0.707107i) q^{64} +(0.0189712 - 0.144101i) q^{65} +(-0.216773 + 0.324423i) q^{68} +(-0.923880 + 0.382683i) q^{72} +(1.69855 - 0.837633i) q^{73} +(1.29335 - 0.257264i) q^{74} +(-0.357164 - 1.05217i) q^{80} -1.00000 q^{81} +(-0.284338 - 0.837633i) q^{82} +(-0.216773 - 0.375461i) q^{85} +(0.923880 - 1.38268i) q^{89} +(0.0726721 - 1.10876i) q^{90} +(-1.53264 - 1.17604i) q^{97} +(0.130526 + 0.991445i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/772\mathbb{Z}\right)^\times$.

$n$	$5$	$387$
$\chi(n)$	$e\left(\frac{11}{48}\right)$	$-1$

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 $

$2$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$2$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$3$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$3$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$4$	0.258819	−	0.965926i	0.258819	−	0.965926i
$5$	0.835400	+	0.732626i	0.835400	+	0.732626i	0.965926	−	0.258819i	$-0.0833333\pi$
$5$	0.835400	+	0.732626i	0.835400	+	0.732626i	−0.130526	+	0.991445i	$0.541667\pi$
$6$		0			0
$7$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$7$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$8$	0.382683	+	0.923880i	0.382683	+	0.923880i
$9$			1.00000i			1.00000i
$10$	−1.10876	−	0.0726721i	−1.10876	−	0.0726721i
$11$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$11$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$12$		0			0
$13$	−0.0726721	−	0.108761i	−0.0726721	−	0.108761i	0.793353	−	0.608761i	$-0.208333\pi$
$13$	−0.0726721	−	0.108761i	−0.0726721	−	0.108761i	−0.866025	+	0.500000i	$0.833333\pi$
$14$		0			0
$15$		0			0
$16$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$17$	−0.369474	−	0.125419i	−0.369474	−	0.125419i	0.130526	−	0.991445i	$-0.458333\pi$
$17$	−0.369474	−	0.125419i	−0.369474	−	0.125419i	−0.500000	+	0.866025i	$0.666667\pi$
$18$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$19$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$19$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$20$	0.923880	−	0.617317i	0.923880	−	0.617317i
$21$		0			0
$22$		0			0
$23$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$23$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$24$		0			0
$25$	0.0306258	+	0.232626i	0.0306258	+	0.232626i
$26$	0.123864	+	0.0420463i	0.123864	+	0.0420463i
$27$		0			0
$28$		0			0
$29$	1.65938	+	1.10876i	1.65938	+	1.10876i	0.866025	+	0.500000i	$0.166667\pi$
$29$	1.65938	+	1.10876i	1.65938	+	1.10876i	0.793353	+	0.608761i	$0.208333\pi$
$30$		0			0
$31$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$31$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$32$	0.991445	−	0.130526i	0.991445	−	0.130526i
$33$		0			0
$34$	0.369474	−	0.125419i	0.369474	−	0.125419i
$35$		0			0
$36$	0.965926	+	0.258819i	0.965926	+	0.258819i
$37$	−1.18270	−	0.583242i	−1.18270	−	0.583242i	−0.258819	−	0.965926i	$-0.583333\pi$
$37$	−1.18270	−	0.583242i	−1.18270	−	0.583242i	−0.923880	+	0.382683i	$0.875000\pi$
$38$		0			0
$39$		0			0
$40$	−0.357164	+	1.05217i	−0.357164	+	1.05217i
$41$	−0.284338	+	0.837633i	−0.284338	+	0.837633i	0.707107	+	0.707107i	$0.250000\pi$
$41$	−0.284338	+	0.837633i	−0.284338	+	0.837633i	−0.991445	+	0.130526i	$0.958333\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	−0.732626	+	0.835400i	−0.732626	+	0.835400i
$46$		0			0
$47$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$47$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$48$		0			0
$49$	0.500000	−	0.866025i	0.500000	−	0.866025i
$50$	−0.165911	−	0.165911i	−0.165911	−	0.165911i
$51$		0			0
$52$	−0.123864	+	0.0420463i	−0.123864	+	0.0420463i
$53$	0.284338	−	0.576581i	0.284338	−	0.576581i	−0.707107	−	0.707107i	$-0.750000\pi$
$53$	0.284338	−	0.576581i	0.284338	−	0.576581i	0.991445	+	0.130526i	$0.0416667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−1.99144	+	0.130526i	−1.99144	+	0.130526i
$59$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$59$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$60$		0			0
$61$	−1.34861	+	1.18270i	−1.34861	+	1.18270i	−0.382683	+	0.923880i	$0.625000\pi$
$61$	−1.34861	+	1.18270i	−1.34861	+	1.18270i	−0.965926	+	0.258819i	$0.916667\pi$
$62$		0			0
$63$		0			0
$64$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$65$	0.0189712	−	0.144101i	0.0189712	−	0.144101i
$66$		0			0
$67$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$67$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$68$	−0.216773	+	0.324423i	−0.216773	+	0.324423i
$69$		0			0
$70$		0			0
$71$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$71$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$72$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$73$	1.69855	−	0.837633i	1.69855	−	0.837633i	0.707107	−	0.707107i	$-0.250000\pi$
$73$	1.69855	−	0.837633i	1.69855	−	0.837633i	0.991445	−	0.130526i	$-0.0416667\pi$
$74$	1.29335	−	0.257264i	1.29335	−	0.257264i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$79$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$80$	−0.357164	−	1.05217i	−0.357164	−	1.05217i
$81$	−1.00000			−1.00000
$82$	−0.284338	−	0.837633i	−0.284338	−	0.837633i
$83$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$83$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$84$		0			0
$85$	−0.216773	−	0.375461i	−0.216773	−	0.375461i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.923880	−	1.38268i	0.923880	−	1.38268i		−	1.00000i	$-0.5\pi$
$89$	0.923880	−	1.38268i	0.923880	−	1.38268i	0.923880	−	0.382683i	$-0.125000\pi$
$90$	0.0726721	−	1.10876i	0.0726721	−	1.10876i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.53264	−	1.17604i	−1.53264	−	1.17604i	−0.923880	−	0.382683i	$-0.875000\pi$
$97$	−1.53264	−	1.17604i	−1.53264	−	1.17604i	−0.608761	−	0.793353i	$-0.708333\pi$
$98$	0.130526	+	0.991445i	0.130526	+	0.991445i
$99$		0			0
$100$	0.232626	+	0.0306258i	0.232626	+	0.0306258i
$101$	−0.315118	+	0.410670i	−0.315118	+	0.410670i	−0.923880	−	0.382683i	$-0.875000\pi$
$101$	−0.315118	+	0.410670i	−0.315118	+	0.410670i	0.608761	+	0.793353i	$0.291667\pi$
$102$		0			0
$103$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$103$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$104$	0.0726721	−	0.108761i	0.0726721	−	0.108761i
$105$		0			0
$106$	0.125419	+	0.630526i	0.125419	+	0.630526i
$107$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$107$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$108$		0			0
$109$	0.130526	−	0.226078i	0.130526	−	0.226078i	−0.793353	−	0.608761i	$-0.791667\pi$
$109$	0.130526	−	0.226078i	0.130526	−	0.226078i	0.923880	+	0.382683i	$0.125000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.483342	−	1.42388i	−0.483342	−	1.42388i	−0.866025	−	0.500000i	$-0.833333\pi$
$113$	−0.483342	−	1.42388i	−0.483342	−	1.42388i	0.382683	−	0.923880i	$-0.375000\pi$
$114$		0			0
$115$		0			0
$116$	1.50046	−	1.31587i	1.50046	−	1.31587i
$117$	0.108761	−	0.0726721i	0.108761	−	0.0726721i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.923880	−	0.382683i	0.923880	−	0.382683i
$122$	0.349942	−	1.75928i	0.349942	−	1.75928i
$123$		0			0
$124$		0			0
$125$	0.472474	−	0.707107i	0.472474	−	0.707107i
$126$		0			0
$127$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$127$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$128$	0.130526	−	0.991445i	0.130526	−	0.991445i
$129$		0			0
$130$	0.0726721	+	0.125872i	0.0726721	+	0.125872i
$131$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$131$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.0255190	−	0.389345i	−0.0255190	−	0.389345i
$137$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$137$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$138$		0			0
$139$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$139$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.500000	−	0.866025i	0.500000	−	0.866025i
$145$	0.573937	+	2.14196i	0.573937	+	2.14196i
$146$	−0.837633	+	1.69855i	−0.837633	+	1.69855i
$147$		0			0
$148$	−0.869474	+	0.991445i	−0.869474	+	0.991445i
$149$	0.128293	−	1.95737i	0.128293	−	1.95737i	−0.130526	−	0.991445i	$-0.541667\pi$
$149$	0.128293	−	1.95737i	0.128293	−	1.95737i	0.258819	−	0.965926i	$-0.416667\pi$
$150$		0			0
$151$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$151$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$152$		0			0
$153$	0.125419	−	0.369474i	0.125419	−	0.369474i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.17604	−	0.315118i	−1.17604	−	0.315118i	−0.382683	−	0.923880i	$-0.625000\pi$
$157$	−1.17604	−	0.315118i	−1.17604	−	0.315118i	−0.793353	+	0.608761i	$0.791667\pi$
$158$		0			0
$159$		0			0
$160$	0.923880	+	0.617317i	0.923880	+	0.617317i
$161$		0			0
$162$	0.793353	−	0.608761i	0.793353	−	0.608761i
$163$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$163$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$164$	0.735499	+	0.491445i	0.735499	+	0.491445i
$165$		0			0
$166$		0			0
$167$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$167$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$168$		0			0
$169$	0.376136	−	0.908072i	0.376136	−	0.908072i
$170$	0.400544	+	0.165911i	0.400544	+	0.165911i
$171$		0			0
$172$		0			0
$173$	−1.25026	+	0.835400i	−1.25026	+	0.835400i	−0.991445	−	0.130526i	$-0.958333\pi$
$173$	−1.25026	+	0.835400i	−1.25026	+	0.835400i	−0.258819	+	0.965926i	$0.583333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.108761	+	1.65938i	0.108761	+	1.65938i
$179$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$179$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$180$	0.617317	+	0.923880i	0.617317	+	0.923880i
$181$	−0.226078	+	0.130526i	−0.226078	+	0.130526i	−0.608761	−	0.793353i	$-0.708333\pi$
$181$	−0.226078	+	0.130526i	−0.226078	+	0.130526i	0.382683	+	0.923880i	$0.375000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.560728	−	1.35372i	−0.560728	−	1.35372i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$191$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$192$		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	0.707107	+	0.707107i	$0.250000\pi$
$197$	−0.258819	+	0.965926i	−0.258819	+	0.965926i	−0.965926	+	0.258819i	$0.916667\pi$
$198$		0			0
$199$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$199$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$200$	−0.203198	+	0.117317i	−0.203198	+	0.117317i
$201$		0			0
$202$		−	0.517638i		−	0.517638i
$203$		0			0
$204$		0			0
$205$	−0.851207	+	0.491445i	−0.851207	+	0.491445i
$206$		0			0
$207$		0			0
$208$	0.00855514	+	0.130526i	0.00855514	+	0.130526i
$209$		0			0
$210$		0			0
$211$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$211$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$212$	−0.483342	−	0.423880i	−0.483342	−	0.423880i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.0340742	+	0.258819i	0.0340742	+	0.258819i
$219$		0			0
$220$		0			0
$221$	0.0132096	+	0.0492990i	0.0132096	+	0.0492990i
$222$		0			0
$223$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$223$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$224$		0			0
$225$	−0.232626	+	0.0306258i	−0.232626	+	0.0306258i
$226$	1.25026	+	0.835400i	1.25026	+	0.835400i
$227$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$227$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$228$		0			0
$229$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.866025	−	0.500000i	$-0.166667\pi$
$229$	1.36603	+	0.366025i	1.36603	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$	−0.389345	+	1.95737i	−0.389345	+	1.95737i
$233$	−0.534534	+	1.57469i	−0.534534	+	1.57469i	0.258819	+	0.965926i	$0.416667\pi$
$233$	−0.534534	+	1.57469i	−0.534534	+	1.57469i	−0.793353	+	0.608761i	$0.791667\pi$
$234$	−0.0420463	+	0.123864i	−0.0420463	+	0.123864i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$239$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$240$		0			0
$241$	−0.478235	−	1.78480i	−0.478235	−	1.78480i	−0.608761	−	0.793353i	$-0.708333\pi$
$241$	−0.478235	−	1.78480i	−0.478235	−	1.78480i	0.130526	−	0.991445i	$-0.458333\pi$
$242$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$243$		0			0
$244$	0.793353	+	1.60876i	0.793353	+	1.60876i
$245$	1.05217	−	0.357164i	1.05217	−	0.357164i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.0556208	+	0.848609i	0.0556208	+	0.848609i
$251$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$251$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$252$		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	$-0.916667\pi$
$257$	−0.707107	+	0.707107i	−0.707107	+	0.707107i	0.258819	+	0.965926i	$0.416667\pi$
$258$		0			0
$259$		0			0
$260$	−0.134280	−	0.0556208i	−0.134280	−	0.0556208i
$261$	−1.10876	+	1.65938i	−1.10876	+	1.65938i
$262$		0			0
$263$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$263$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$264$		0			0
$265$	0.659954	−	0.273362i	0.659954	−	0.273362i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.57469	+	1.05217i	−1.57469	+	1.05217i	−0.608761	+	0.793353i	$0.708333\pi$
$269$	−1.57469	+	1.05217i	−1.57469	+	1.05217i	−0.965926	+	0.258819i	$0.916667\pi$
$270$		0			0
$271$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$271$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$272$	0.257264	+	0.293353i	0.257264	+	0.293353i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.608761	+	1.05441i	−0.608761	+	1.05441i	0.382683	+	0.923880i	$0.375000\pi$
$277$	−0.608761	+	1.05441i	−0.608761	+	1.05441i	−0.991445	+	0.130526i	$0.958333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.923880	+	1.38268i	0.923880	+	1.38268i	0.923880	+	0.382683i	$0.125000\pi$
$281$	0.923880	+	1.38268i	0.923880	+	1.38268i			1.00000i	$0.5\pi$
$282$		0			0
$283$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$283$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.130526	+	0.991445i	0.130526	+	0.991445i
$289$	−0.672572	−	0.516083i	−0.672572	−	0.516083i
$290$	−1.75928	−	1.34994i	−1.75928	−	1.34994i
$291$		0			0
$292$	−0.369474	−	1.85747i	−0.369474	−	1.85747i
$293$	−1.57313	−	0.207107i	−1.57313	−	0.207107i	−0.707107	−	0.707107i	$-0.750000\pi$
$293$	−1.57313	−	0.207107i	−1.57313	−	0.207107i	−0.866025	+	0.500000i	$0.833333\pi$
$294$		0			0
$295$		0			0
$296$	0.0862466	−	1.31587i	0.0862466	−	1.31587i
$297$		0			0
$298$	1.08979	+	1.63099i	1.08979	+	1.63099i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.99310			−1.99310
$306$	0.125419	+	0.369474i	0.125419	+	0.369474i
$307$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$307$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$311$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$312$		0			0
$313$	1.34861	−	0.665060i	1.34861	−	0.665060i	0.382683	−	0.923880i	$-0.375000\pi$
$313$	1.34861	−	0.665060i	1.34861	−	0.665060i	0.965926	+	0.258819i	$0.0833333\pi$
$314$	1.12484	−	0.465926i	1.12484	−	0.465926i
$315$		0			0
$316$		0			0
$317$	0.541196	+	1.30656i	0.541196	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$317$	0.541196	+	1.30656i	0.541196	+	1.30656i	−0.382683	+	0.923880i	$0.625000\pi$
$318$		0			0
$319$		0			0
$320$	−1.10876	+	0.0726721i	−1.10876	+	0.0726721i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$325$	0.0230751	−	0.0202363i	0.0230751	−	0.0202363i
$326$		0			0
$327$		0			0
$328$	−0.882683	+	0.0578541i	−0.882683	+	0.0578541i
$329$		0			0
$330$		0			0
$331$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$331$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$332$		0			0
$333$	0.583242	−	1.18270i	0.583242	−	1.18270i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.866025	−	1.50000i	0.866025	−	1.50000i		−	1.00000i	$-0.5\pi$
$337$	0.866025	−	1.50000i	0.866025	−	1.50000i	0.866025	−	0.500000i	$-0.166667\pi$
$338$	0.254391	+	0.949399i	0.254391	+	0.949399i
$339$		0			0
$340$	−0.418773	+	0.112210i	−0.418773	+	0.112210i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.483342	−	1.42388i	0.483342	−	1.42388i
$347$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$347$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$348$		0			0
$349$	1.60876	+	0.793353i	1.60876	+	0.793353i	1.00000			$0$
$349$	1.60876	+	0.793353i	1.60876	+	0.793353i	0.608761	+	0.793353i	$0.291667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.534534	+	0.357164i	0.534534	+	0.357164i	0.793353	−	0.608761i	$-0.208333\pi$
$353$	0.534534	+	0.357164i	0.534534	+	0.357164i	−0.258819	+	0.965926i	$0.583333\pi$
$354$		0			0
$355$		0			0
$356$	−1.09645	−	1.25026i	−1.09645	−	1.25026i
$357$		0			0
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$	−1.05217	−	0.357164i	−1.05217	−	0.357164i
$361$	0.130526	+	0.991445i	0.130526	+	0.991445i
$362$	0.0999004	−	0.241181i	0.0999004	−	0.241181i
$363$		0			0
$364$		0			0
$365$	2.03264	+	0.544645i	2.03264	+	0.544645i
$366$		0			0
$367$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$367$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$368$		0			0
$369$	−0.837633	−	0.284338i	−0.837633	−	0.284338i
$370$	1.26895	+	0.732626i	1.26895	+	0.732626i
$371$		0			0
$372$		0			0
$373$	0.835400	+	1.25026i	0.835400	+	1.25026i	0.965926	+	0.258819i	$0.0833333\pi$
$373$	0.835400	+	1.25026i	0.835400	+	1.25026i	−0.130526	+	0.991445i	$0.541667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	0.261052i		−	0.261052i
$378$		0			0
$379$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$379$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$383$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$384$		0			0
$385$		0			0
$386$	0.991445	+	0.130526i	0.991445	+	0.130526i
$387$		0			0
$388$	−1.53264	+	1.17604i	−1.53264	+	1.17604i
$389$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$389$	1.36603	+	1.36603i	1.36603	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	0.991445	+	0.130526i	0.991445	+	0.130526i
$393$		0			0
$394$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$395$		0			0
$396$		0			0
$397$	1.92388	−	0.382683i	1.92388	−	0.382683i	0.923880	−	0.382683i	$-0.125000\pi$
$397$	1.92388	−	0.382683i	1.92388	−	0.382683i	1.00000			$0$
$398$		0			0
$399$		0			0
$400$	0.0897902	−	0.216773i	0.0897902	−	0.216773i
$401$	−0.128293	−	1.95737i	−0.128293	−	1.95737i	−0.258819	−	0.965926i	$-0.583333\pi$
$401$	−0.128293	−	1.95737i	−0.128293	−	1.95737i	0.130526	−	0.991445i	$-0.458333\pi$
$402$		0			0
$403$		0			0
$404$	0.315118	+	0.410670i	0.315118	+	0.410670i
$405$	−0.835400	−	0.732626i	−0.835400	−	0.732626i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.478235	+	0.198092i	0.478235	+	0.198092i	0.608761	−	0.793353i	$-0.291667\pi$
$409$	0.478235	+	0.198092i	0.478235	+	0.198092i	−0.130526	+	0.991445i	$0.541667\pi$
$410$	0.376136	−	0.908072i	0.376136	−	0.908072i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.0862466	−	0.0983454i	−0.0862466	−	0.0983454i
$417$		0			0
$418$		0			0
$419$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$419$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$420$		0			0
$421$	−1.92388	−	0.382683i	−1.92388	−	0.382683i	−0.923880	−	0.382683i	$-0.875000\pi$
$421$	−1.92388	−	0.382683i	−1.92388	−	0.382683i	−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	0.641502	+	0.0420463i	0.641502	+	0.0420463i
$425$	0.0178604	−	0.0897902i	0.0178604	−	0.0897902i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$431$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$432$		0			0
$433$	0.284338	−	0.576581i	0.284338	−	0.576581i	−0.707107	−	0.707107i	$-0.750000\pi$
$433$	0.284338	−	0.576581i	0.284338	−	0.576581i	0.991445	+	0.130526i	$0.0416667\pi$
$434$		0			0
$435$		0			0
$436$	−0.184592	−	0.184592i	−0.184592	−	0.184592i
$437$		0			0
$438$		0			0
$439$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$439$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$440$		0			0
$441$	0.866025	+	0.500000i	0.866025	+	0.500000i
$442$	−0.0404912	−	0.0310700i	−0.0404912	−	0.0310700i
$443$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$443$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$444$		0			0
$445$	1.78480	−	0.478235i	1.78480	−	0.478235i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.258819	+	0.448288i	0.258819	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$449$	0.258819	+	0.448288i	0.258819	+	0.448288i	−0.707107	+	0.707107i	$0.750000\pi$
$450$	0.165911	−	0.165911i	0.165911	−	0.165911i
$451$		0			0
$452$	−1.50046	+	0.0983454i	−1.50046	+	0.0983454i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.172572	−	0.867580i	0.172572	−	0.867580i	−0.793353	−	0.608761i	$-0.791667\pi$
$457$	0.172572	−	0.867580i	0.172572	−	0.867580i	0.965926	−	0.258819i	$-0.0833333\pi$
$458$	−1.30656	+	0.541196i	−1.30656	+	0.541196i
$459$		0			0
$460$		0			0
$461$	−0.860919	+	1.12197i	−0.860919	+	1.12197i	0.130526	+	0.991445i	$0.458333\pi$
$461$	−0.860919	+	1.12197i	−0.860919	+	1.12197i	−0.991445	+	0.130526i	$0.958333\pi$
$462$		0			0
$463$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$463$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$464$	−0.882683	−	1.78990i	−0.882683	−	1.78990i
$465$		0			0
$466$	−0.534534	−	1.57469i	−0.534534	−	1.57469i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	−0.0420463	−	0.123864i	−0.0420463	−	0.123864i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.576581	+	0.284338i	0.576581	+	0.284338i
$478$		0			0
$479$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$479$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$480$		0			0
$481$	0.0225149	+	0.171017i	0.0225149	+	0.171017i
$482$	1.46593	+	1.12484i	1.46593	+	1.12484i
$483$		0			0
$484$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$485$	−0.418773	−	2.10531i	−0.418773	−	2.10531i
$486$		0			0
$487$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$487$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$488$	−1.60876	−	0.793353i	−1.60876	−	0.793353i
$489$		0			0
$490$	−0.617317	+	0.923880i	−0.617317	+	0.923880i
$491$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$491$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$492$		0			0
$493$	−0.474037	−	0.617777i	−0.474037	−	0.617777i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$499$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$500$	−0.560728	−	0.639387i	−0.560728	−	0.639387i
$501$		0			0
$502$		0			0
$503$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$503$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$504$		0			0
$505$	−0.564117	+	0.112210i	−0.564117	+	0.112210i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.793353	+	0.391239i	−0.793353	+	0.391239i	−0.793353	−	0.608761i	$-0.791667\pi$
$509$	−0.793353	+	0.391239i	−0.793353	+	0.391239i			1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$513$		0			0
$514$	0.130526	−	0.991445i	0.130526	−	0.991445i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.140392	−	0.0376178i	0.140392	−	0.0376178i
$521$	−1.88981	+	0.123864i	−1.88981	+	0.123864i	−0.965926	−	0.258819i	$-0.916667\pi$
$521$	−1.88981	+	0.123864i	−1.88981	+	0.123864i	−0.923880	+	0.382683i	$0.875000\pi$
$522$	−0.130526	−	1.99144i	−0.130526	−	1.99144i
$523$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$523$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.707107	+	0.707107i	0.707107	+	0.707107i
$530$	−0.357164	+	0.618627i	−0.357164	+	0.618627i
$531$		0			0
$532$		0			0
$533$	0.111766	−	0.0299475i	0.111766	−	0.0299475i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	0.608761	−	1.79335i	0.608761	−	1.79335i
$539$		0			0
$540$		0			0
$541$	1.65938	+	0.108761i	1.65938	+	0.108761i	0.866025	−	0.500000i	$-0.166667\pi$
$541$	1.65938	+	0.108761i	1.65938	+	0.108761i	0.793353	+	0.608761i	$0.208333\pi$
$542$		0			0
$543$		0			0
$544$	−0.382683	−	0.0761205i	−0.382683	−	0.0761205i
$545$	0.274672	−	0.0932386i	0.274672	−	0.0932386i
$546$		0			0
$547$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$547$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$548$		0			0
$549$	−1.18270	−	1.34861i	−1.18270	−	1.34861i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.158919	−	1.20711i	−0.158919	−	1.20711i
$555$		0			0
$556$		0			0
$557$	−0.869474	+	0.991445i	−0.869474	+	0.991445i	0.130526	+	0.991445i	$0.458333\pi$
$557$	−0.869474	+	0.991445i	−0.869474	+	0.991445i	−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.57469	−	0.534534i	−1.57469	−	0.534534i
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$	0.639387	−	1.54362i	0.639387	−	1.54362i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.882683	−	0.0578541i	−0.882683	−	0.0578541i	−0.382683	−	0.923880i	$-0.625000\pi$
$569$	−0.882683	−	0.0578541i	−0.882683	−	0.0578541i	−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$571$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$577$	−1.37413	+	1.05441i	−1.37413	+	1.05441i	−0.382683	+	0.923880i	$0.625000\pi$
$577$	−1.37413	+	1.05441i	−1.37413	+	1.05441i	−0.991445	+	0.130526i	$0.958333\pi$
$578$	0.847759			0.847759
$579$		0			0
$580$	2.21752			2.21752
$581$		0			0
$582$		0			0
$583$		0			0
$584$	1.42388	+	1.24871i	1.42388	+	1.24871i
$585$	0.144101	+	0.0189712i	0.144101	+	0.0189712i
$586$	1.37413	−	0.793353i	1.37413	−	0.793353i
$587$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$587$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.732626	+	1.09645i	0.732626	+	1.09645i
$593$	0.607206	−	1.46593i	0.607206	−	1.46593i	−0.258819	−	0.965926i	$-0.583333\pi$
$593$	0.607206	−	1.46593i	0.607206	−	1.46593i	0.866025	−	0.500000i	$-0.166667\pi$
$594$		0			0
$595$		0			0
$596$	−1.85747	−	0.630526i	−1.85747	−	0.630526i
$597$		0			0
$598$		0			0
$599$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$599$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$600$		0			0
$601$	−0.869474	+	0.991445i	−0.869474	+	0.991445i	0.130526	+	0.991445i	$0.458333\pi$
$601$	−0.869474	+	0.991445i	−0.869474	+	0.991445i	−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.05217	+	0.357164i	1.05217	+	0.357164i
$606$		0			0
$607$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$607$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$608$		0			0
$609$		0			0
$610$	1.58124	−	1.21332i	1.58124	−	1.21332i
$611$		0			0
$612$	−0.324423	−	0.216773i	−0.324423	−	0.216773i
$613$	−1.69855	+	0.576581i	−1.69855	+	0.576581i	−0.991445	−	0.130526i	$-0.958333\pi$
$613$	−1.69855	+	0.576581i	−1.69855	+	0.576581i	−0.707107	+	0.707107i	$0.750000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.78990	+	0.117317i	1.78990	+	0.117317i	0.923880	−	0.382683i	$-0.125000\pi$
$617$	1.78990	+	0.117317i	1.78990	+	0.117317i	0.866025	+	0.500000i	$0.166667\pi$
$618$		0			0
$619$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$619$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.13939	−	0.305298i	1.13939	−	0.305298i
$626$	−0.665060	+	1.34861i	−0.665060	+	1.34861i
$627$		0			0
$628$	−0.608761	+	1.05441i	−0.608761	+	1.05441i
$629$	0.363826	+	0.363826i	0.363826	+	0.363826i
$630$		0			0
$631$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$631$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$632$		0			0
$633$		0			0
$634$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$635$		0			0
$636$		0			0
$637$	−0.130526	+	0.00855514i	−0.130526	+	0.00855514i
$638$		0			0
$639$		0			0
$640$	0.835400	−	0.732626i	0.835400	−	0.732626i
$641$	−0.315118	+	1.17604i	−0.315118	+	1.17604i	0.608761	+	0.793353i	$0.291667\pi$
$641$	−0.315118	+	1.17604i	−0.315118	+	1.17604i	−0.923880	+	0.382683i	$0.875000\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$647$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$648$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$649$		0			0
$650$	−0.00598761	+	0.0301018i	−0.00598761	+	0.0301018i
$651$		0			0
$652$		0			0
$653$	1.95737	−	0.389345i	1.95737	−	0.389345i	0.965926	−	0.258819i	$-0.0833333\pi$
$653$	1.95737	−	0.389345i	1.95737	−	0.389345i	0.991445	−	0.130526i	$-0.0416667\pi$
$654$		0			0
$655$		0			0
$656$	0.665060	−	0.583242i	0.665060	−	0.583242i
$657$	0.837633	+	1.69855i	0.837633	+	1.69855i
$658$		0			0
$659$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$659$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$660$		0			0
$661$	−0.641502	−	1.88981i	−0.641502	−	1.88981i	−0.382683	−	0.923880i	$-0.625000\pi$
$661$	−0.641502	−	1.88981i	−0.641502	−	1.88981i	−0.258819	−	0.965926i	$-0.583333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	0.257264	+	1.29335i	0.257264	+	1.29335i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.369474	+	1.85747i	0.369474	+	1.85747i	0.500000	+	0.866025i	$0.333333\pi$
$673$	0.369474	+	1.85747i	0.369474	+	1.85747i	−0.130526	+	0.991445i	$0.541667\pi$
$674$	0.226078	+	1.71723i	0.226078	+	1.71723i
$675$		0			0
$676$	−0.779779	−	0.598345i	−0.779779	−	0.598345i
$677$	0.241181	+	1.83195i	0.241181	+	1.83195i	0.500000	+	0.866025i	$0.333333\pi$
$677$	0.241181	+	1.83195i	0.241181	+	1.83195i	−0.258819	+	0.965926i	$0.583333\pi$
$678$		0			0
$679$		0			0
$680$	0.263926	−	0.343955i	0.263926	−	0.343955i
$681$		0			0
$682$		0			0
$683$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$683$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.0833732	+	0.0109763i	−0.0833732	+	0.0109763i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	0.483342	+	1.42388i	0.483342	+	1.42388i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.210111	−	0.273822i	0.210111	−	0.273822i
$698$	−1.75928	+	0.349942i	−1.75928	+	0.349942i
$699$		0			0
$700$		0			0
$701$	0.369474	−	1.85747i	0.369474	−	1.85747i	−0.130526	−	0.991445i	$-0.541667\pi$
$701$	0.369474	−	1.85747i	0.369474	−	1.85747i	0.500000	−	0.866025i	$-0.333333\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.641502	+	0.0420463i	−0.641502	+	0.0420463i
$707$		0			0
$708$		0			0
$709$	0.793353	+	1.37413i	0.793353	+	1.37413i	0.923880	+	0.382683i	$0.125000\pi$
$709$	0.793353	+	1.37413i	0.793353	+	1.37413i	−0.130526	+	0.991445i	$0.541667\pi$
$710$		0			0
$711$		0			0
$712$	1.63099	+	0.324423i	1.63099	+	0.324423i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$719$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$720$	1.05217	−	0.357164i	1.05217	−	0.357164i
$721$		0			0
$722$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$723$		0			0
$724$	0.0675653	+	0.252157i	0.0675653	+	0.252157i
$725$	−0.207107	+	0.419971i	−0.207107	+	0.419971i
$726$		0			0
$727$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$727$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$728$		0			0
$729$		−	1.00000i		−	1.00000i
$730$	−1.94416	+	0.805298i	−1.94416	+	0.805298i
$731$		0			0
$732$		0			0
$733$	0.125419	−	0.630526i	0.125419	−	0.630526i	−0.866025	−	0.500000i	$-0.833333\pi$
$733$	0.125419	−	0.630526i	0.125419	−	0.630526i	0.991445	−	0.130526i	$-0.0416667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	0.837633	−	0.284338i	0.837633	−	0.284338i
$739$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$739$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$740$	−1.45272	+	0.191254i	−1.45272	+	0.191254i
$741$		0			0
$742$		0			0
$743$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$743$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$744$		0			0
$745$	1.54120	−	1.54120i	1.54120	−	1.54120i
$746$	−1.42388	−	0.483342i	−1.42388	−	0.483342i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$751$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$752$		0			0
$753$		0			0
$754$	0.158919	+	0.207107i	0.158919	+	0.207107i
$755$		0			0
$756$		0			0
$757$	0.0862466	+	1.31587i	0.0862466	+	1.31587i	0.793353	+	0.608761i	$0.208333\pi$
$757$	0.0862466	+	1.31587i	0.0862466	+	1.31587i	−0.707107	+	0.707107i	$0.750000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.95737	+	0.389345i	−1.95737	+	0.389345i	−0.965926	+	0.258819i	$0.916667\pi$
$761$	−1.95737	+	0.389345i	−1.95737	+	0.389345i	−0.991445	+	0.130526i	$0.958333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.375461	−	0.216773i	0.375461	−	0.216773i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.707107	+	0.707107i	0.707107	+	0.707107i	0.965926	−	0.258819i	$-0.0833333\pi$
$769$	0.707107	+	0.707107i	0.707107	+	0.707107i	−0.258819	+	0.965926i	$0.583333\pi$
$770$		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
$778$	−1.91532	−	0.252157i	−1.91532	−	0.252157i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.866025	+	0.500000i	−0.866025	+	0.500000i
$785$	−0.751597	−	1.12484i	−0.751597	−	1.12484i
$786$		0			0
$787$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$787$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$788$	0.866025	+	0.500000i	0.866025	+	0.500000i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.226638	+	0.0607275i	0.226638	+	0.0607275i
$794$	−1.29335	+	1.47479i	−1.29335	+	1.47479i
$795$		0			0
$796$		0			0
$797$	0.0999004	+	0.758819i	0.0999004	+	0.758819i	0.965926	+	0.258819i	$0.0833333\pi$
$797$	0.0999004	+	0.758819i	0.0999004	+	0.758819i	−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$	0.0607275	+	0.226638i	0.0607275	+	0.226638i
$801$	1.38268	+	0.923880i	1.38268	+	0.923880i
$802$	1.29335	+	1.47479i	1.29335	+	1.47479i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.500000	−	0.133975i	−0.500000	−	0.133975i
$809$	−1.60876	−	0.793353i	−1.60876	−	0.793353i	−0.608761	−	0.793353i	$-0.708333\pi$
$809$	−1.60876	−	0.793353i	−1.60876	−	0.793353i	−1.00000			$\pi$
$810$	1.10876	+	0.0726721i	1.10876	+	0.0726721i
$811$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$811$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$812$		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
$820$	0.254391	+	0.949399i	0.254391	+	0.949399i
$821$	0.500000	−	0.866025i	0.500000	−	0.866025i	−0.500000	−	0.866025i	$-0.666667\pi$
$821$	0.500000	−	0.866025i	0.500000	−	0.866025i	1.00000			$0$
$822$		0			0
$823$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$823$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$827$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$828$		0			0
$829$	−0.0420463	−	0.641502i	−0.0420463	−	0.641502i	−0.965926	−	0.258819i	$-0.916667\pi$
$829$	−0.0420463	−	0.641502i	−0.0420463	−	0.641502i	0.923880	−	0.382683i	$-0.125000\pi$
$830$		0			0
$831$		0			0
$832$	0.128293	+	0.0255190i	0.128293	+	0.0255190i
$833$	−0.293353	+	0.257264i	−0.293353	+	0.257264i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$839$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$840$		0			0
$841$	1.14150	+	2.75583i	1.14150	+	2.75583i
$842$	1.75928	−	0.867580i	1.75928	−	0.867580i
$843$		0			0
$844$		0			0
$845$	0.979500	−	0.483036i	0.979500	−	0.483036i
$846$		0			0
$847$		0			0
$848$	−0.534534	+	0.357164i	−0.534534	+	0.357164i
$849$		0			0
$850$	0.0404912	+	0.0821081i	0.0404912	+	0.0821081i
$851$		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
$857$	0.991445	+	1.71723i	0.991445	+	1.71723i	0.608761	+	0.793353i	$0.291667\pi$
$857$	0.991445	+	1.71723i	0.991445	+	1.71723i	0.382683	+	0.923880i	$0.375000\pi$
$858$		0			0
$859$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$859$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$863$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$864$		0			0
$865$	−1.65651	−	0.218083i	−1.65651	−	0.218083i
$866$	0.125419	+	0.630526i	0.125419	+	0.630526i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.258819	+	0.0340742i	0.258819	+	0.0340742i
$873$	1.17604	−	1.53264i	1.17604	−	1.53264i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.923880	−	1.38268i	−0.923880	−	1.38268i	−0.923880	−	0.382683i	$-0.875000\pi$
$877$	−0.923880	−	1.38268i	−0.923880	−	1.38268i		−	1.00000i	$-0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$881$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$882$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$883$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$883$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$884$	0.0510381			0.0510381
$885$		0			0
$886$		0			0
$887$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$887$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$888$		0			0
$889$		0			0
$890$	−1.12484	+	1.46593i	−1.12484	+	1.46593i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.478235	−	0.198092i	−0.478235	−	0.198092i
$899$		0			0
$900$	−0.0306258	+	0.232626i	−0.0306258	+	0.232626i
$901$	−0.177370	+	0.177370i	−0.177370	+	0.177370i
$902$		0			0
$903$		0			0
$904$	1.13053	−	0.991445i	1.13053	−	0.991445i
$905$	−0.284492	−	0.0565890i	−0.284492	−	0.0565890i
$906$		0			0
$907$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$907$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$908$		0			0
$909$	−0.410670	−	0.315118i	−0.410670	−	0.315118i
$910$		0			0
$911$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$911$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$912$		0			0
$913$		0			0
$914$	0.391239	+	0.793353i	0.391239	+	0.793353i
$915$		0			0
$916$	0.707107	−	1.22474i	0.707107	−	1.22474i
$917$		0			0
$918$		0			0
$919$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$919$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$920$		0			0
$921$		0			0
$922$		−	1.41421i		−	1.41421i
$923$		0			0
$924$		0			0
$925$	0.0994562	−	0.292988i	0.0994562	−	0.292988i
$926$		0			0
$927$		0			0
$928$	1.78990	+	0.882683i	1.78990	+	0.882683i
$929$	−1.53264	−	0.410670i	−1.53264	−	0.410670i	−0.608761	−	0.793353i	$-0.708333\pi$
$929$	−1.53264	−	0.410670i	−1.53264	−	0.410670i	−0.923880	+	0.382683i	$0.875000\pi$
$930$		0			0
$931$		0			0
$932$	1.38268	+	0.923880i	1.38268	+	0.923880i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.108761	+	0.0726721i	0.108761	+	0.0726721i
$937$	0.513210	+	1.91532i	0.513210	+	1.91532i	0.382683	+	0.923880i	$0.375000\pi$
$937$	0.513210	+	1.91532i	0.513210	+	1.91532i	0.130526	+	0.991445i	$0.458333\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	0.707107	+	0.707107i	$0.250000\pi$
$941$	−0.292893	+	0.707107i	−0.292893	+	0.707107i	−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$947$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$948$		0			0
$949$	−0.214539	−	0.123864i	−0.214539	−	0.123864i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.05441	−	0.608761i	1.05441	−	0.608761i	0.130526	−	0.991445i	$-0.458333\pi$
$953$	1.05441	−	0.608761i	1.05441	−	0.608761i	0.923880	+	0.382683i	$0.125000\pi$
$954$	−0.630526	+	0.125419i	−0.630526	+	0.125419i
$955$		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
$962$	−0.121971	−	0.121971i	−0.121971	−	0.121971i
$963$		0			0
$964$	−1.84776			−1.84776
$965$	−0.0726721	−	1.10876i	−0.0726721	−	1.10876i
$966$		0			0
$967$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$967$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$968$	0.707107	+	0.707107i	0.707107	+	0.707107i
$969$		0			0
$970$	1.61387	+	1.41532i	1.61387	+	1.41532i
$971$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$971$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.75928	−	0.349942i	1.75928	−	0.349942i
$977$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.258819	−	0.965926i	$-0.416667\pi$
$977$	1.22474	−	0.707107i	1.22474	−	0.707107i	0.965926	+	0.258819i	$0.0833333\pi$
$978$		0			0
$979$		0			0
$980$	−0.0726721	−	1.10876i	−0.0726721	−	1.10876i
$981$	0.226078	+	0.130526i	0.226078	+	0.130526i
$982$		0			0
$983$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$983$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$984$		0			0
$985$	−0.923880	+	0.617317i	−0.923880	+	0.617317i
$986$	0.752157	+	0.201540i	0.752157	+	0.201540i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$991$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.57313	+	0.207107i	−1.57313	+	0.207107i	−0.866025	−	0.500000i	$-0.833333\pi$
$997$	−1.57313	+	0.207107i	−1.57313	+	0.207107i	−0.707107	+	0.707107i	$0.750000\pi$
$998$		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.615.1	yes	16
4.3	odd	2	CM	772.1.u.a.615.1	yes	16
193.59	even	48	inner	772.1.u.a.59.1	✓	16
772.59	odd	48	inner	772.1.u.a.59.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
772.1.u.a.59.1	✓	16	193.59	even	48	inner
772.1.u.a.59.1	✓	16	772.59	odd	48	inner
772.1.u.a.615.1	yes	16	1.1	even	1	trivial
772.1.u.a.615.1	yes	16	4.3	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(-0.793353 + 0.608761i) q^{2} +(0.258819 - 0.965926i) q^{4} +(0.835400 + 0.732626i) q^{5} +(0.382683 + 0.923880i) q^{8} +1.00000i q^{9} +(-1.10876 - 0.0726721i) q^{10} +(-0.0726721 - 0.108761i) q^{13} +(-0.866025 - 0.500000i) q^{16} +(-0.369474 - 0.125419i) q^{17} +(-0.608761 - 0.793353i) q^{18} +(0.923880 - 0.617317i) q^{20} +(0.0306258 + 0.232626i) q^{25} +(0.123864 + 0.0420463i) q^{26} +(1.65938 + 1.10876i) q^{29} +(0.991445 - 0.130526i) q^{32} +(0.369474 - 0.125419i) q^{34} +(0.965926 + 0.258819i) q^{36} +(-1.18270 - 0.583242i) q^{37} +(-0.357164 + 1.05217i) q^{40} +(-0.284338 + 0.837633i) q^{41} +(-0.732626 + 0.835400i) q^{45} +(0.500000 - 0.866025i) q^{49} +(-0.165911 - 0.165911i) q^{50} +(-0.123864 + 0.0420463i) q^{52} +(0.284338 - 0.576581i) q^{53} +(-1.99144 + 0.130526i) q^{58} +(-1.34861 + 1.18270i) q^{61} +(-0.707107 + 0.707107i) q^{64} +(0.0189712 - 0.144101i) q^{65} +(-0.216773 + 0.324423i) q^{68} +(-0.923880 + 0.382683i) q^{72} +(1.69855 - 0.837633i) q^{73} +(1.29335 - 0.257264i) q^{74} +(-0.357164 - 1.05217i) q^{80} -1.00000 q^{81} +(-0.284338 - 0.837633i) q^{82} +(-0.216773 - 0.375461i) q^{85} +(0.923880 - 1.38268i) q^{89} +(0.0726721 - 1.10876i) q^{90} +(-1.53264 - 1.17604i) q^{97} +(0.130526 + 0.991445i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{10} - 8 q^{17} + 8 q^{34} + 8 q^{49} - 16 q^{58} + 8 q^{65} + 8 q^{74} - 16 q^{81}+O(q^{100}) \) 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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