LMFDB - Embedded newform 772.1.u.a.583.1

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			583.1
Root			$0.793353 - 0.608761i$ of defining polynomial
Character	$\chi$	$=$	772.583
Dual form			772.1.u.a.531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.991445 - 0.130526i) q^{2} +(0.965926 + 0.258819i) q^{4} +(0.867580 + 1.75928i) q^{5} +(-0.923880 - 0.382683i) q^{8} -1.00000i q^{9} +O(q^{10})$	\(q+(-0.991445 - 0.130526i) q^{2} +(0.965926 + 0.258819i) q^{4} +(0.867580 + 1.75928i) q^{5} +(-0.923880 - 0.382683i) q^{8} -1.00000i q^{9} +(-0.630526 - 1.85747i) q^{10} +(1.85747 - 0.369474i) q^{13} +(0.866025 + 0.500000i) q^{16} +(-1.10876 + 0.0726721i) q^{17} +(-0.130526 + 0.991445i) q^{18} +(0.382683 + 1.92388i) q^{20} +(-1.73361 + 2.25928i) q^{25} +(-1.88981 + 0.123864i) q^{26} +(0.125419 - 0.630526i) q^{29} +(-0.793353 - 0.608761i) q^{32} +(1.10876 + 0.0726721i) q^{34} +(0.258819 - 0.965926i) q^{36} +(-1.34861 + 1.18270i) q^{37} +(-0.128293 - 1.95737i) q^{40} +(0.0862466 + 1.31587i) q^{41} +(1.75928 - 0.867580i) q^{45} +(0.500000 - 0.866025i) q^{49} +(2.01367 - 2.01367i) q^{50} +(1.88981 + 0.123864i) q^{52} +(-0.0862466 - 0.0983454i) q^{53} +(-0.206647 + 0.608761i) q^{58} +(0.665060 - 1.34861i) q^{61} +(0.707107 + 0.707107i) q^{64} +(2.26151 + 2.94726i) q^{65} +(-1.08979 - 0.216773i) q^{68} +(-0.382683 + 0.923880i) q^{72} +(-1.50046 - 1.31587i) q^{73} +(1.49144 - 0.996552i) q^{74} +(-0.128293 + 1.95737i) q^{80} -1.00000 q^{81} +(0.0862466 - 1.31587i) q^{82} +(-1.08979 - 1.88757i) q^{85} +(0.382683 + 0.0761205i) q^{89} +(-1.85747 + 0.630526i) q^{90} +(-0.513210 + 0.0675653i) q^{97} +(-0.608761 + 0.793353i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * 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

Display 100 coefficients 10 coefficients

Character values

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

$n$	$5$	$387$
$\chi(n)$	$e\left(\frac{17}{48}\right)$	$-1$

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 $

$2$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$2$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$	0.965926	+	0.258819i	0.965926	+	0.258819i
$5$	0.867580	+	1.75928i	0.867580	+	1.75928i	0.608761	+	0.793353i	$0.291667\pi$
$5$	0.867580	+	1.75928i	0.867580	+	1.75928i	0.258819	+	0.965926i	$0.416667\pi$
$6$		0			0
$7$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$7$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$8$	−0.923880	−	0.382683i	−0.923880	−	0.382683i
$9$		−	1.00000i		−	1.00000i
$10$	−0.630526	−	1.85747i	−0.630526	−	1.85747i
$11$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$11$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$12$		0			0
$13$	1.85747	−	0.369474i	1.85747	−	0.369474i	0.866025	−	0.500000i	$-0.166667\pi$
$13$	1.85747	−	0.369474i	1.85747	−	0.369474i	0.991445	+	0.130526i	$0.0416667\pi$
$14$		0			0
$15$		0			0
$16$	0.866025	+	0.500000i	0.866025	+	0.500000i
$17$	−1.10876	+	0.0726721i	−1.10876	+	0.0726721i	−0.608761	−	0.793353i	$-0.708333\pi$
$17$	−1.10876	+	0.0726721i	−1.10876	+	0.0726721i	−0.500000	+	0.866025i	$0.666667\pi$
$18$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$19$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$19$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$20$	0.382683	+	1.92388i	0.382683	+	1.92388i
$21$		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
$25$	−1.73361	+	2.25928i	−1.73361	+	2.25928i
$26$	−1.88981	+	0.123864i	−1.88981	+	0.123864i
$27$		0			0
$28$		0			0
$29$	0.125419	−	0.630526i	0.125419	−	0.630526i	−0.866025	−	0.500000i	$-0.833333\pi$
$29$	0.125419	−	0.630526i	0.125419	−	0.630526i	0.991445	−	0.130526i	$-0.0416667\pi$
$30$		0			0
$31$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$31$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$32$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$33$		0			0
$34$	1.10876	+	0.0726721i	1.10876	+	0.0726721i
$35$		0			0
$36$	0.258819	−	0.965926i	0.258819	−	0.965926i
$37$	−1.34861	+	1.18270i	−1.34861	+	1.18270i	−0.382683	+	0.923880i	$0.625000\pi$
$37$	−1.34861	+	1.18270i	−1.34861	+	1.18270i	−0.965926	+	0.258819i	$0.916667\pi$
$38$		0			0
$39$		0			0
$40$	−0.128293	−	1.95737i	−0.128293	−	1.95737i
$41$	0.0862466	+	1.31587i	0.0862466	+	1.31587i	0.793353	+	0.608761i	$0.208333\pi$
$41$	0.0862466	+	1.31587i	0.0862466	+	1.31587i	−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
$45$	1.75928	−	0.867580i	1.75928	−	0.867580i
$46$		0			0
$47$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$47$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$48$		0			0
$49$	0.500000	−	0.866025i	0.500000	−	0.866025i
$50$	2.01367	−	2.01367i	2.01367	−	2.01367i
$51$		0			0
$52$	1.88981	+	0.123864i	1.88981	+	0.123864i
$53$	−0.0862466	−	0.0983454i	−0.0862466	−	0.0983454i	0.707107	−	0.707107i	$-0.250000\pi$
$53$	−0.0862466	−	0.0983454i	−0.0862466	−	0.0983454i	−0.793353	+	0.608761i	$0.791667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.206647	+	0.608761i	−0.206647	+	0.608761i
$59$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$59$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$60$		0			0
$61$	0.665060	−	1.34861i	0.665060	−	1.34861i	−0.258819	−	0.965926i	$-0.583333\pi$
$61$	0.665060	−	1.34861i	0.665060	−	1.34861i	0.923880	−	0.382683i	$-0.125000\pi$
$62$		0			0
$63$		0			0
$64$	0.707107	+	0.707107i	0.707107	+	0.707107i
$65$	2.26151	+	2.94726i	2.26151	+	2.94726i
$66$		0			0
$67$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$67$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$68$	−1.08979	−	0.216773i	−1.08979	−	0.216773i
$69$		0			0
$70$		0			0
$71$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$71$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$72$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$73$	−1.50046	−	1.31587i	−1.50046	−	1.31587i	−0.793353	−	0.608761i	$-0.791667\pi$
$73$	−1.50046	−	1.31587i	−1.50046	−	1.31587i	−0.707107	−	0.707107i	$-0.750000\pi$
$74$	1.49144	−	0.996552i	1.49144	−	0.996552i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$79$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$80$	−0.128293	+	1.95737i	−0.128293	+	1.95737i
$81$	−1.00000			−1.00000
$82$	0.0862466	−	1.31587i	0.0862466	−	1.31587i
$83$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$83$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$84$		0			0
$85$	−1.08979	−	1.88757i	−1.08979	−	1.88757i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.382683	+	0.0761205i	0.382683	+	0.0761205i	0.382683	−	0.923880i	$-0.375000\pi$
$89$	0.382683	+	0.0761205i	0.382683	+	0.0761205i			1.00000i	$0.5\pi$
$90$	−1.85747	+	0.630526i	−1.85747	+	0.630526i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.513210	+	0.0675653i	−0.513210	+	0.0675653i	−0.382683	−	0.923880i	$-0.625000\pi$
$97$	−0.513210	+	0.0675653i	−0.513210	+	0.0675653i	−0.130526	+	0.991445i	$0.541667\pi$
$98$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$99$		0			0
$100$	−2.25928	+	1.73361i	−2.25928	+	1.73361i
$101$	−0.252157	−	1.91532i	−0.252157	−	1.91532i	−0.382683	−	0.923880i	$-0.625000\pi$
$101$	−0.252157	−	1.91532i	−0.252157	−	1.91532i	0.130526	−	0.991445i	$-0.458333\pi$
$102$		0			0
$103$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$103$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$104$	−1.85747	−	0.369474i	−1.85747	−	0.369474i
$105$		0			0
$106$	0.0726721	+	0.108761i	0.0726721	+	0.108761i
$107$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$107$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$108$		0			0
$109$	−0.608761	+	1.05441i	−0.608761	+	1.05441i	0.382683	+	0.923880i	$0.375000\pi$
$109$	−0.608761	+	1.05441i	−0.608761	+	1.05441i	−0.991445	+	0.130526i	$0.958333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.0578541	+	0.882683i	−0.0578541	+	0.882683i	0.866025	+	0.500000i	$0.166667\pi$
$113$	−0.0578541	+	0.882683i	−0.0578541	+	0.882683i	−0.923880	+	0.382683i	$0.875000\pi$
$114$		0			0
$115$		0			0
$116$	0.284338	−	0.576581i	0.284338	−	0.576581i
$117$	−0.369474	−	1.85747i	−0.369474	−	1.85747i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.382683	−	0.923880i	0.382683	−	0.923880i
$122$	−0.835400	+	1.25026i	−0.835400	+	1.25026i
$123$		0			0
$124$		0			0
$125$	−3.55487	−	0.707107i	−3.55487	−	0.707107i
$126$		0			0
$127$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$127$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$128$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$129$		0			0
$130$	−1.85747	−	3.21723i	−1.85747	−	3.21723i
$131$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$131$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.05217	+	0.357164i	1.05217	+	0.357164i
$137$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$137$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$138$		0			0
$139$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$139$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.500000	−	0.866025i	0.500000	−	0.866025i
$145$	1.21808	−	0.326384i	1.21808	−	0.326384i
$146$	1.31587	+	1.50046i	1.31587	+	1.50046i
$147$		0			0
$148$	−1.60876	+	0.793353i	−1.60876	+	0.793353i
$149$	1.57469	−	0.534534i	1.57469	−	0.534534i	0.608761	−	0.793353i	$-0.291667\pi$
$149$	1.57469	−	0.534534i	1.57469	−	0.534534i	0.965926	+	0.258819i	$0.0833333\pi$
$150$		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
$153$	0.0726721	+	1.10876i	0.0726721	+	1.10876i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.0675653	+	0.252157i	−0.0675653	+	0.252157i	−0.991445	−	0.130526i	$-0.958333\pi$
$157$	−0.0675653	+	0.252157i	−0.0675653	+	0.252157i	0.923880	+	0.382683i	$0.125000\pi$
$158$		0			0
$159$		0			0
$160$	0.382683	−	1.92388i	0.382683	−	1.92388i
$161$		0			0
$162$	0.991445	+	0.130526i	0.991445	+	0.130526i
$163$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$163$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$164$	−0.257264	+	1.29335i	−0.257264	+	1.29335i
$165$		0			0
$166$		0			0
$167$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$167$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$168$		0			0
$169$	2.38981	−	0.989890i	2.38981	−	0.989890i
$170$	0.834089	+	2.01367i	0.834089	+	2.01367i
$171$		0			0
$172$		0			0
$173$	−0.172572	−	0.867580i	−0.172572	−	0.867580i	−0.965926	−	0.258819i	$-0.916667\pi$
$173$	−0.172572	−	0.867580i	−0.172572	−	0.867580i	0.793353	−	0.608761i	$-0.208333\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.369474	−	0.125419i	−0.369474	−	0.125419i
$179$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$179$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$180$	1.92388	−	0.382683i	1.92388	−	0.382683i
$181$	−1.05441	+	0.608761i	−1.05441	+	0.608761i	−0.923880	−	0.382683i	$-0.875000\pi$
$181$	−1.05441	+	0.608761i	−1.05441	+	0.608761i	−0.130526	+	0.991445i	$0.541667\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.25072	−	1.34649i	−3.25072	−	1.34649i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$191$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$192$		0			0
$193$	0.707107	−	0.707107i	0.707107	−	0.707107i
$193$	0.707107	−	0.707107i	0.707107	−	0.707107i
$194$	0.517638			0.517638
$195$		0			0
$196$	0.707107	−	0.707107i	0.707107	−	0.707107i
$197$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.258819	−	0.965926i	$-0.583333\pi$
$197$	−0.965926	−	0.258819i	−0.965926	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$199$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$200$	2.46623	−	1.42388i	2.46623	−	1.42388i
$201$		0			0
$202$			1.93185i			1.93185i
$203$		0			0
$204$		0			0
$205$	−2.24015	+	1.29335i	−2.24015	+	1.29335i
$206$		0			0
$207$		0			0
$208$	1.79335	+	0.608761i	1.79335	+	0.608761i
$209$		0			0
$210$		0			0
$211$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$211$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$212$	−0.0578541	−	0.117317i	−0.0578541	−	0.117317i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.741181	−	0.965926i	0.741181	−	0.965926i
$219$		0			0
$220$		0			0
$221$	−2.03264	+	0.544645i	−2.03264	+	0.544645i
$222$		0			0
$223$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$223$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$224$		0			0
$225$	2.25928	+	1.73361i	2.25928	+	1.73361i
$226$	0.172572	−	0.867580i	0.172572	−	0.867580i
$227$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$227$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$228$		0			0
$229$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	0.500000	+	0.866025i	$0.333333\pi$
$229$	−0.366025	+	1.36603i	−0.366025	+	1.36603i	−0.866025	+	0.500000i	$0.833333\pi$
$230$		0			0
$231$		0			0
$232$	−0.357164	+	0.534534i	−0.357164	+	0.534534i
$233$	−0.0255190	−	0.389345i	−0.0255190	−	0.389345i	−0.991445	−	0.130526i	$-0.958333\pi$
$233$	−0.0255190	−	0.389345i	−0.0255190	−	0.389345i	0.965926	−	0.258819i	$-0.0833333\pi$
$234$	0.123864	+	1.88981i	0.123864	+	1.88981i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$239$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$240$		0			0
$241$	−0.739288	+	0.198092i	−0.739288	+	0.198092i	−0.608761	−	0.793353i	$-0.708333\pi$
$241$	−0.739288	+	0.198092i	−0.739288	+	0.198092i	−0.130526	+	0.991445i	$0.541667\pi$
$242$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$243$		0			0
$244$	0.991445	−	1.13053i	0.991445	−	1.13053i
$245$	1.95737	+	0.128293i	1.95737	+	0.128293i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	3.43216	+	1.16506i	3.43216	+	1.16506i
$251$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$251$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$252$		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
$260$	1.42165	+	3.43216i	1.42165	+	3.43216i
$261$	−0.630526	−	0.125419i	−0.630526	−	0.125419i
$262$		0			0
$263$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$263$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$264$		0			0
$265$	0.0981911	−	0.237054i	0.0981911	−	0.237054i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.389345	−	1.95737i	−0.389345	−	1.95737i	−0.258819	−	0.965926i	$-0.583333\pi$
$269$	−0.389345	−	1.95737i	−0.389345	−	1.95737i	−0.130526	−	0.991445i	$-0.541667\pi$
$270$		0			0
$271$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$271$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$272$	−0.996552	−	0.491445i	−0.996552	−	0.491445i
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.130526	+	0.226078i	−0.130526	+	0.226078i	−0.923880	−	0.382683i	$-0.875000\pi$
$277$	−0.130526	+	0.226078i	−0.130526	+	0.226078i	0.793353	+	0.608761i	$0.208333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.382683	−	0.0761205i	0.382683	−	0.0761205i		−	1.00000i	$-0.5\pi$
$281$	0.382683	−	0.0761205i	0.382683	−	0.0761205i	0.382683	+	0.923880i	$0.375000\pi$
$282$		0			0
$283$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$283$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$289$	0.232626	−	0.0306258i	0.232626	−	0.0306258i
$290$	−1.25026	+	0.164600i	−1.25026	+	0.164600i
$291$		0			0
$292$	−1.10876	−	1.65938i	−1.10876	−	1.65938i
$293$	1.57313	−	1.20711i	1.57313	−	1.20711i	0.707107	−	0.707107i	$-0.250000\pi$
$293$	1.57313	−	1.20711i	1.57313	−	1.20711i	0.866025	−	0.500000i	$-0.166667\pi$
$294$		0			0
$295$		0			0
$296$	1.69855	−	0.576581i	1.69855	−	0.576581i
$297$		0			0
$298$	−1.63099	+	0.324423i	−1.63099	+	0.324423i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.94957			2.94957
$306$	0.0726721	−	1.10876i	0.0726721	−	1.10876i
$307$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$307$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$311$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$312$		0			0
$313$	−0.665060	−	0.583242i	−0.665060	−	0.583242i	0.258819	−	0.965926i	$-0.416667\pi$
$313$	−0.665060	−	0.583242i	−0.665060	−	0.583242i	−0.923880	+	0.382683i	$0.875000\pi$
$314$	0.0999004	−	0.241181i	0.0999004	−	0.241181i
$315$		0			0
$316$		0			0
$317$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$317$	1.30656	+	0.541196i	1.30656	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$318$		0			0
$319$		0			0
$320$	−0.630526	+	1.85747i	−0.630526	+	1.85747i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$325$	−2.38538	+	4.83707i	−2.38538	+	4.83707i
$326$		0			0
$327$		0			0
$328$	0.423880	−	1.24871i	0.423880	−	1.24871i
$329$		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
$333$	1.18270	+	1.34861i	1.18270	+	1.34861i
$334$		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	$0.833333\pi$
$338$	−2.49857	+	0.669489i	−2.49857	+	0.669489i
$339$		0			0
$340$	−0.564117	−	2.10531i	−0.564117	−	2.10531i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.0578541	+	0.882683i	0.0578541	+	0.882683i
$347$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$347$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$348$		0			0
$349$	1.13053	−	0.991445i	1.13053	−	0.991445i	0.130526	−	0.991445i	$-0.458333\pi$
$349$	1.13053	−	0.991445i	1.13053	−	0.991445i	1.00000			$0$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.0255190	−	0.128293i	0.0255190	−	0.128293i	−0.965926	−	0.258819i	$-0.916667\pi$
$353$	0.0255190	−	0.128293i	0.0255190	−	0.128293i	0.991445	+	0.130526i	$0.0416667\pi$
$354$		0			0
$355$		0			0
$356$	0.349942	+	0.172572i	0.349942	+	0.172572i
$357$		0			0
$358$		0			0
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$	−1.95737	+	0.128293i	−1.95737	+	0.128293i
$361$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$362$	1.12484	−	0.465926i	1.12484	−	0.465926i
$363$		0			0
$364$		0			0
$365$	1.01321	−	3.78135i	1.01321	−	3.78135i
$366$		0			0
$367$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$367$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$368$		0			0
$369$	1.31587	−	0.0862466i	1.31587	−	0.0862466i
$370$	3.04716	+	1.75928i	3.04716	+	1.75928i
$371$		0			0
$372$		0			0
$373$	0.867580	−	0.172572i	0.867580	−	0.172572i	0.258819	−	0.965926i	$-0.416667\pi$
$373$	0.867580	−	0.172572i	0.867580	−	0.172572i	0.608761	+	0.793353i	$0.291667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	1.21752i		−	1.21752i
$378$		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	$-0.250000\pi$
$383$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$384$		0			0
$385$		0			0
$386$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$387$		0			0
$388$	−0.513210	−	0.0675653i	−0.513210	−	0.0675653i
$389$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$389$	−0.366025	+	0.366025i	−0.366025	+	0.366025i	0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$393$		0			0
$394$	0.923880	+	0.382683i	0.923880	+	0.382683i
$395$		0			0
$396$		0			0
$397$	1.38268	−	0.923880i	1.38268	−	0.923880i	0.382683	−	0.923880i	$-0.375000\pi$
$397$	1.38268	−	0.923880i	1.38268	−	0.923880i	1.00000			$0$
$398$		0			0
$399$		0			0
$400$	−2.63099	+	1.08979i	−2.63099	+	1.08979i
$401$	−1.57469	−	0.534534i	−1.57469	−	0.534534i	−0.608761	−	0.793353i	$-0.708333\pi$
$401$	−1.57469	−	0.534534i	−1.57469	−	0.534534i	−0.965926	+	0.258819i	$0.916667\pi$
$402$		0			0
$403$		0			0
$404$	0.252157	−	1.91532i	0.252157	−	1.91532i
$405$	−0.867580	−	1.75928i	−0.867580	−	1.75928i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.739288	+	1.78480i	0.739288	+	1.78480i	0.608761	+	0.793353i	$0.291667\pi$
$409$	0.739288	+	1.78480i	0.739288	+	1.78480i	0.130526	+	0.991445i	$0.458333\pi$
$410$	2.38981	−	0.989890i	2.38981	−	0.989890i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−1.69855	−	0.837633i	−1.69855	−	0.837633i
$417$		0			0
$418$		0			0
$419$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$419$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$420$		0			0
$421$	−1.38268	−	0.923880i	−1.38268	−	0.923880i	−0.382683	−	0.923880i	$-0.625000\pi$
$421$	−1.38268	−	0.923880i	−1.38268	−	0.923880i	−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$	0.0420463	+	0.123864i	0.0420463	+	0.123864i
$425$	1.75797	−	2.63099i	1.75797	−	2.63099i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$431$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$432$		0			0
$433$	−0.0862466	−	0.0983454i	−0.0862466	−	0.0983454i	0.707107	−	0.707107i	$-0.250000\pi$
$433$	−0.0862466	−	0.0983454i	−0.0862466	−	0.0983454i	−0.793353	+	0.608761i	$0.791667\pi$
$434$		0			0
$435$		0			0
$436$	−0.860919	+	0.860919i	−0.860919	+	0.860919i
$437$		0			0
$438$		0			0
$439$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$439$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$440$		0			0
$441$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$442$	2.08634	−	0.274672i	2.08634	−	0.274672i
$443$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$443$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$444$		0			0
$445$	0.198092	+	0.739288i	0.198092	+	0.739288i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.707107	+	0.707107i	$0.250000\pi$
$449$	0.965926	+	1.67303i	0.965926	+	1.67303i	0.258819	+	0.965926i	$0.416667\pi$
$450$	−2.01367	−	2.01367i	−2.01367	−	2.01367i
$451$		0			0
$452$	−0.284338	+	0.837633i	−0.284338	+	0.837633i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.732626	+	1.09645i	−0.732626	+	1.09645i	0.258819	+	0.965926i	$0.416667\pi$
$457$	−0.732626	+	1.09645i	−0.732626	+	1.09645i	−0.991445	+	0.130526i	$0.958333\pi$
$458$	0.541196	−	1.30656i	0.541196	−	1.30656i
$459$		0			0
$460$		0			0
$461$	0.184592	+	1.40211i	0.184592	+	1.40211i	0.793353	+	0.608761i	$0.208333\pi$
$461$	0.184592	+	1.40211i	0.184592	+	1.40211i	−0.608761	+	0.793353i	$0.708333\pi$
$462$		0			0
$463$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$463$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$464$	0.423880	−	0.483342i	0.423880	−	0.483342i
$465$		0			0
$466$	−0.0255190	+	0.389345i	−0.0255190	+	0.389345i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	0.123864	−	1.88981i	0.123864	−	1.88981i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.0983454	+	0.0862466i	−0.0983454	+	0.0862466i
$478$		0			0
$479$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$479$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$480$		0			0
$481$	−2.06803	+	2.69510i	−2.06803	+	2.69510i
$482$	0.758819	−	0.0999004i	0.758819	−	0.0999004i
$483$		0			0
$484$	0.608761	−	0.793353i	0.608761	−	0.793353i
$485$	−0.564117	−	0.844261i	−0.564117	−	0.844261i
$486$		0			0
$487$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$487$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$488$	−1.13053	+	0.991445i	−1.13053	+	0.991445i
$489$		0			0
$490$	−1.92388	−	0.382683i	−1.92388	−	0.382683i
$491$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$491$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$492$		0			0
$493$	−0.0932386	+	0.708218i	−0.0932386	+	0.708218i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$499$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$500$	−3.25072	−	1.60308i	−3.25072	−	1.60308i
$501$		0			0
$502$		0			0
$503$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$503$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$504$		0			0
$505$	3.15082	−	2.10531i	3.15082	−	2.10531i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.991445	−	0.869474i	−0.991445	−	0.869474i		−	1.00000i	$-0.5\pi$
$509$	−0.991445	−	0.869474i	−0.991445	−	0.869474i	−0.991445	+	0.130526i	$0.958333\pi$
$510$		0			0
$511$		0			0
$512$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$513$		0			0
$514$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.961497	−	3.58836i	−0.961497	−	3.58836i
$521$	−0.641502	+	1.88981i	−0.641502	+	1.88981i	−0.258819	+	0.965926i	$0.583333\pi$
$521$	−0.641502	+	1.88981i	−0.641502	+	1.88981i	−0.382683	+	0.923880i	$0.625000\pi$
$522$	0.608761	+	0.206647i	0.608761	+	0.206647i
$523$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$523$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$530$	−0.128293	+	0.222210i	−0.128293	+	0.222210i
$531$		0			0
$532$		0			0
$533$	0.646379	+	2.41232i	0.646379	+	2.41232i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	0.130526	+	1.99144i	0.130526	+	1.99144i
$539$		0			0
$540$		0			0
$541$	0.125419	+	0.369474i	0.125419	+	0.369474i	0.991445	−	0.130526i	$-0.0416667\pi$
$541$	0.125419	+	0.369474i	0.125419	+	0.369474i	−0.866025	+	0.500000i	$0.833333\pi$
$542$		0			0
$543$		0			0
$544$	0.923880	+	0.617317i	0.923880	+	0.617317i
$545$	−2.38314	−	0.156199i	−2.38314	−	0.156199i
$546$		0			0
$547$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$547$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$548$		0			0
$549$	−1.34861	−	0.665060i	−1.34861	−	0.665060i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.158919	−	0.207107i	0.158919	−	0.207107i
$555$		0			0
$556$		0			0
$557$	−1.60876	+	0.793353i	−1.60876	+	0.793353i	−0.608761	+	0.793353i	$0.708333\pi$
$557$	−1.60876	+	0.793353i	−1.60876	+	0.793353i	−1.00000			$1.00000\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.389345	+	0.0255190i	−0.389345	+	0.0255190i
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$	−1.60308	+	0.664017i	−1.60308	+	0.664017i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.423880	+	1.24871i	0.423880	+	1.24871i	0.923880	+	0.382683i	$0.125000\pi$
$569$	0.423880	+	1.24871i	0.423880	+	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
$577$	1.71723	+	0.226078i	1.71723	+	0.226078i	0.923880	−	0.382683i	$-0.125000\pi$
$577$	1.71723	+	0.226078i	1.71723	+	0.226078i	0.793353	+	0.608761i	$0.208333\pi$
$578$	−0.234633			−0.234633
$579$		0			0
$580$	1.26105			1.26105
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.882683	+	1.78990i	0.882683	+	1.78990i
$585$	2.94726	−	2.26151i	2.94726	−	2.26151i
$586$	−1.71723	+	0.991445i	−1.71723	+	0.991445i
$587$		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
$592$	−1.75928	+	0.349942i	−1.75928	+	0.349942i
$593$	−1.83195	+	0.758819i	−1.83195	+	0.758819i	−0.866025	+	0.500000i	$0.833333\pi$
$593$	−1.83195	+	0.758819i	−1.83195	+	0.758819i	−0.965926	+	0.258819i	$0.916667\pi$
$594$		0			0
$595$		0			0
$596$	1.65938	−	0.108761i	1.65938	−	0.108761i
$597$		0			0
$598$		0			0
$599$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$599$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$600$		0			0
$601$	−1.60876	+	0.793353i	−1.60876	+	0.793353i	−0.608761	+	0.793353i	$0.708333\pi$
$601$	−1.60876	+	0.793353i	−1.60876	+	0.793353i	−1.00000			$1.00000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.95737	−	0.128293i	1.95737	−	0.128293i
$606$		0			0
$607$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$607$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$608$		0			0
$609$		0			0
$610$	−2.92434	−	0.384997i	−2.92434	−	0.384997i
$611$		0			0
$612$	−0.216773	+	1.08979i	−0.216773	+	1.08979i
$613$	1.50046	+	0.0983454i	1.50046	+	0.0983454i	0.793353	−	0.608761i	$-0.208333\pi$
$613$	1.50046	+	0.0983454i	1.50046	+	0.0983454i	0.707107	+	0.707107i	$0.250000\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.483342	−	1.42388i	−0.483342	−	1.42388i	−0.866025	−	0.500000i	$-0.833333\pi$
$617$	−0.483342	−	1.42388i	−0.483342	−	1.42388i	0.382683	−	0.923880i	$-0.375000\pi$
$618$		0			0
$619$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$619$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.10308	−	4.11675i	−1.10308	−	4.11675i
$626$	0.583242	+	0.665060i	0.583242	+	0.665060i
$627$		0			0
$628$	−0.130526	+	0.226078i	−0.130526	+	0.226078i
$629$	1.40934	−	1.40934i	1.40934	−	1.40934i
$630$		0			0
$631$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$631$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$632$		0			0
$633$		0			0
$634$	−1.22474	−	0.707107i	−1.22474	−	0.707107i
$635$		0			0
$636$		0			0
$637$	0.608761	−	1.79335i	0.608761	−	1.79335i
$638$		0			0
$639$		0			0
$640$	0.867580	−	1.75928i	0.867580	−	1.75928i
$641$	−0.252157	−	0.0675653i	−0.252157	−	0.0675653i	0.130526	−	0.991445i	$-0.458333\pi$
$641$	−0.252157	−	0.0675653i	−0.252157	−	0.0675653i	−0.382683	+	0.923880i	$0.625000\pi$
$642$		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		−0.980785	−	0.195090i	$-0.937500\pi$
$647$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$648$	0.923880	+	0.382683i	0.923880	+	0.382683i
$649$		0			0
$650$	2.99633	−	4.48433i	2.99633	−	4.48433i
$651$		0			0
$652$		0			0
$653$	−0.534534	+	0.357164i	−0.534534	+	0.357164i	−0.793353	−	0.608761i	$-0.791667\pi$
$653$	−0.534534	+	0.357164i	−0.534534	+	0.357164i	0.258819	+	0.965926i	$0.416667\pi$
$654$		0			0
$655$		0			0
$656$	−0.583242	+	1.18270i	−0.583242	+	1.18270i
$657$	−1.31587	+	1.50046i	−1.31587	+	1.50046i
$658$		0			0
$659$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$659$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$660$		0			0
$661$	−0.0420463	+	0.641502i	−0.0420463	+	0.641502i	0.923880	+	0.382683i	$0.125000\pi$
$661$	−0.0420463	+	0.641502i	−0.0420463	+	0.641502i	−0.965926	+	0.258819i	$0.916667\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.996552	−	1.49144i	−0.996552	−	1.49144i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.10876	+	1.65938i	1.10876	+	1.65938i	0.608761	+	0.793353i	$0.291667\pi$
$673$	1.10876	+	1.65938i	1.10876	+	1.65938i	0.500000	+	0.866025i	$0.333333\pi$
$674$	1.05441	−	1.37413i	1.05441	−	1.37413i
$675$		0			0
$676$	2.56458	−	0.337633i	2.56458	−	0.337633i
$677$	−0.465926	+	0.607206i	−0.465926	+	0.607206i	−0.965926	−	0.258819i	$-0.916667\pi$
$677$	−0.465926	+	0.607206i	−0.465926	+	0.607206i	0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$		0			0
$680$	0.284492	+	2.16093i	0.284492	+	2.16093i
$681$		0			0
$682$		0			0
$683$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$683$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.196536	−	0.150808i	−0.196536	−	0.150808i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$	0.0578541	−	0.882683i	0.0578541	−	0.882683i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.191254	−	1.45272i	−0.191254	−	1.45272i
$698$	−1.25026	+	0.835400i	−1.25026	+	0.835400i
$699$		0			0
$700$		0			0
$701$	1.10876	−	1.65938i	1.10876	−	1.65938i	0.500000	−	0.866025i	$-0.333333\pi$
$701$	1.10876	−	1.65938i	1.10876	−	1.65938i	0.608761	−	0.793353i	$-0.291667\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.0420463	+	0.123864i	−0.0420463	+	0.123864i
$707$		0			0
$708$		0			0
$709$	0.991445	+	1.71723i	0.991445	+	1.71723i	0.608761	+	0.793353i	$0.291667\pi$
$709$	0.991445	+	1.71723i	0.991445	+	1.71723i	0.382683	+	0.923880i	$0.375000\pi$
$710$		0			0
$711$		0			0
$712$	−0.324423	−	0.216773i	−0.324423	−	0.216773i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$719$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$720$	1.95737	+	0.128293i	1.95737	+	0.128293i
$721$		0			0
$722$	0.707107	−	0.707107i	0.707107	−	0.707107i
$723$		0			0
$724$	−1.17604	+	0.315118i	−1.17604	+	0.315118i
$725$	1.20711	+	1.37644i	1.20711	+	1.37644i
$726$		0			0
$727$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$727$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$728$		0			0
$729$			1.00000i			1.00000i
$730$	−1.49811	+	3.61675i	−1.49811	+	3.61675i
$731$		0			0
$732$		0			0
$733$	0.0726721	−	0.108761i	0.0726721	−	0.108761i	−0.793353	−	0.608761i	$-0.791667\pi$
$733$	0.0726721	−	0.108761i	0.0726721	−	0.108761i	0.866025	+	0.500000i	$0.166667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−1.31587	−	0.0862466i	−1.31587	−	0.0862466i
$739$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$739$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$740$	−2.79146	−	2.14196i	−2.79146	−	2.14196i
$741$		0			0
$742$		0			0
$743$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$743$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$744$		0			0
$745$	2.30656	+	2.30656i	2.30656	+	2.30656i
$746$	−0.882683	+	0.0578541i	−0.882683	+	0.0578541i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$751$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$752$		0			0
$753$		0			0
$754$	−0.158919	+	1.20711i	−0.158919	+	1.20711i
$755$		0			0
$756$		0			0
$757$	1.69855	+	0.576581i	1.69855	+	0.576581i	0.991445	−	0.130526i	$-0.0416667\pi$
$757$	1.69855	+	0.576581i	1.69855	+	0.576581i	0.707107	+	0.707107i	$0.250000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.534534	−	0.357164i	0.534534	−	0.357164i	−0.258819	−	0.965926i	$-0.583333\pi$
$761$	0.534534	−	0.357164i	0.534534	−	0.357164i	0.793353	+	0.608761i	$0.208333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1.88757	+	1.08979i	−1.88757	+	1.08979i
$766$		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	$-0.166667\pi$
$773$	1.73205			1.73205			0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$		0			0
$776$	0.500000	+	0.133975i	0.500000	+	0.133975i
$777$		0			0
$778$	0.410670	−	0.315118i	0.410670	−	0.315118i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.866025	−	0.500000i	0.866025	−	0.500000i
$785$	−0.502233	+	0.0999004i	−0.502233	+	0.0999004i
$786$		0			0
$787$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$787$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$788$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	0.737054	−	2.75072i	0.737054	−	2.75072i
$794$	−1.49144	+	0.735499i	−1.49144	+	0.735499i
$795$		0			0
$796$		0			0
$797$	1.12484	−	1.46593i	1.12484	−	1.46593i	0.258819	−	0.965926i	$-0.416667\pi$
$797$	1.12484	−	1.46593i	1.12484	−	1.46593i	0.866025	−	0.500000i	$-0.166667\pi$
$798$		0			0
$799$		0			0
$800$	2.75072	−	0.737054i	2.75072	−	0.737054i
$801$	0.0761205	−	0.382683i	0.0761205	−	0.382683i
$802$	1.49144	+	0.735499i	1.49144	+	0.735499i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.500000	+	1.86603i	−0.500000	+	1.86603i
$809$	−1.13053	+	0.991445i	−1.13053	+	0.991445i	−0.130526	+	0.991445i	$0.541667\pi$
$809$	−1.13053	+	0.991445i	−1.13053	+	0.991445i	−1.00000			$\pi$
$810$	0.630526	+	1.85747i	0.630526	+	1.85747i
$811$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$811$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.500000	−	1.86603i	−0.500000	−	1.86603i
$819$		0			0
$820$	−2.49857	+	0.669489i	−2.49857	+	0.669489i
$821$	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		0.659346	−	0.751840i	$-0.270833\pi$
$823$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$824$		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
$829$	0.123864	+	0.0420463i	0.123864	+	0.0420463i	0.382683	−	0.923880i	$-0.375000\pi$
$829$	0.123864	+	0.0420463i	0.123864	+	0.0420463i	−0.258819	+	0.965926i	$0.583333\pi$
$830$		0			0
$831$		0			0
$832$	1.57469	+	1.05217i	1.57469	+	1.05217i
$833$	−0.491445	+	0.996552i	−0.491445	+	0.996552i
$834$		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
$841$	0.542046	+	0.224523i	0.542046	+	0.224523i
$842$	1.25026	+	1.09645i	1.25026	+	1.09645i
$843$		0			0
$844$		0			0
$845$	3.81484	+	3.34553i	3.81484	+	3.34553i
$846$		0			0
$847$		0			0
$848$	−0.0255190	−	0.128293i	−0.0255190	−	0.128293i
$849$		0			0
$850$	−2.08634	+	2.37902i	−2.08634	+	2.37902i
$851$		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
$857$	−0.793353	−	1.37413i	−0.793353	−	1.37413i	−0.923880	−	0.382683i	$-0.875000\pi$
$857$	−0.793353	−	1.37413i	−0.793353	−	1.37413i	0.130526	−	0.991445i	$-0.458333\pi$
$858$		0			0
$859$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$859$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$863$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$864$		0			0
$865$	1.37660	−	1.05630i	1.37660	−	1.05630i
$866$	0.0726721	+	0.108761i	0.0726721	+	0.108761i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.965926	−	0.741181i	0.965926	−	0.741181i
$873$	0.0675653	+	0.513210i	0.0675653	+	0.513210i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i	−0.382683	−	0.923880i	$-0.625000\pi$
$877$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i			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	$-0.958333\pi$
$881$	−0.382683	+	0.662827i	−0.382683	+	0.662827i	0.608761	+	0.793353i	$0.291667\pi$
$882$	0.793353	+	0.608761i	0.793353	+	0.608761i
$883$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$883$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$884$	−2.10434			−2.10434
$885$		0			0
$886$		0			0
$887$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$887$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$888$		0			0
$889$		0			0
$890$	−0.0999004	−	0.758819i	−0.0999004	−	0.758819i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.739288	−	1.78480i	−0.739288	−	1.78480i
$899$		0			0
$900$	1.73361	+	2.25928i	1.73361	+	2.25928i
$901$	0.102774	+	0.102774i	0.102774	+	0.102774i
$902$		0			0
$903$		0			0
$904$	0.391239	−	0.793353i	0.391239	−	0.793353i
$905$	−1.98576	−	1.32684i	−1.98576	−	1.32684i
$906$		0			0
$907$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$907$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$908$		0			0
$909$	−1.91532	+	0.252157i	−1.91532	+	0.252157i
$910$		0			0
$911$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$911$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$912$		0			0
$913$		0			0
$914$	0.869474	−	0.991445i	0.869474	−	0.991445i
$915$		0			0
$916$	−0.707107	+	1.22474i	−0.707107	+	1.22474i
$917$		0			0
$918$		0			0
$919$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$919$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$920$		0			0
$921$		0			0
$922$		−	1.41421i		−	1.41421i
$923$		0			0
$924$		0			0
$925$	−0.334089	−	5.09722i	−0.334089	−	5.09722i
$926$		0			0
$927$		0			0
$928$	−0.483342	+	0.423880i	−0.483342	+	0.423880i
$929$	−0.513210	+	1.91532i	−0.513210	+	1.91532i	−0.130526	+	0.991445i	$0.541667\pi$
$929$	−0.513210	+	1.91532i	−0.513210	+	1.91532i	−0.382683	+	0.923880i	$0.625000\pi$
$930$		0			0
$931$		0			0
$932$	0.0761205	−	0.382683i	0.0761205	−	0.382683i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−0.369474	+	1.85747i	−0.369474	+	1.85747i
$937$	−1.53264	+	0.410670i	−1.53264	+	0.410670i	−0.923880	−	0.382683i	$-0.875000\pi$
$937$	−1.53264	+	0.410670i	−1.53264	+	0.410670i	−0.608761	+	0.793353i	$0.708333\pi$
$938$		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		0.130526	−	0.991445i	$-0.458333\pi$
$947$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$948$		0			0
$949$	−3.27324	−	1.88981i	−3.27324	−	1.88981i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.226078	+	0.130526i	−0.226078	+	0.130526i	−0.608761	−	0.793353i	$-0.708333\pi$
$953$	−0.226078	+	0.130526i	−0.226078	+	0.130526i	0.382683	+	0.923880i	$0.375000\pi$
$954$	0.108761	−	0.0726721i	0.108761	−	0.0726721i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.965926	+	0.258819i	0.965926	+	0.258819i
$962$	2.40211	−	2.40211i	2.40211	−	2.40211i
$963$		0			0
$964$	−0.765367			−0.765367
$965$	1.85747	+	0.630526i	1.85747	+	0.630526i
$966$		0			0
$967$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$967$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$968$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$969$		0			0
$970$	0.449093	+	0.910670i	0.449093	+	0.910670i
$971$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$971$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.25026	−	0.835400i	1.25026	−	0.835400i
$977$	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
$980$	1.85747	+	0.630526i	1.85747	+	0.630526i
$981$	1.05441	+	0.608761i	1.05441	+	0.608761i
$982$		0			0
$983$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$983$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$984$		0			0
$985$	−0.382683	−	1.92388i	−0.382683	−	1.92388i
$986$	0.184882	−	0.689989i	0.184882	−	0.689989i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$991$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.57313	+	1.20711i	1.57313	+	1.20711i	0.866025	+	0.500000i	$0.166667\pi$
$997$	1.57313	+	1.20711i	1.57313	+	1.20711i	0.707107	+	0.707107i	$0.250000\pi$
$998$		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.583.1	yes	16
4.3	odd	2	CM	772.1.u.a.583.1	yes	16
193.145	even	48	inner	772.1.u.a.531.1	✓	16
772.531	odd	48	inner	772.1.u.a.531.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
772.1.u.a.531.1	✓	16	193.145	even	48	inner
772.1.u.a.531.1	✓	16	772.531	odd	48	inner
772.1.u.a.583.1	yes	16	1.1	even	1	trivial
772.1.u.a.583.1	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)

\(f(q)\)	\(=\)	\(q+(-0.991445 - 0.130526i) q^{2} +(0.965926 + 0.258819i) q^{4} +(0.867580 + 1.75928i) q^{5} +(-0.923880 - 0.382683i) q^{8} -1.00000i q^{9} +O(q^{10})\)	\(q+(-0.991445 - 0.130526i) q^{2} +(0.965926 + 0.258819i) q^{4} +(0.867580 + 1.75928i) q^{5} +(-0.923880 - 0.382683i) q^{8} -1.00000i q^{9} +(-0.630526 - 1.85747i) q^{10} +(1.85747 - 0.369474i) q^{13} +(0.866025 + 0.500000i) q^{16} +(-1.10876 + 0.0726721i) q^{17} +(-0.130526 + 0.991445i) q^{18} +(0.382683 + 1.92388i) q^{20} +(-1.73361 + 2.25928i) q^{25} +(-1.88981 + 0.123864i) q^{26} +(0.125419 - 0.630526i) q^{29} +(-0.793353 - 0.608761i) q^{32} +(1.10876 + 0.0726721i) q^{34} +(0.258819 - 0.965926i) q^{36} +(-1.34861 + 1.18270i) q^{37} +(-0.128293 - 1.95737i) q^{40} +(0.0862466 + 1.31587i) q^{41} +(1.75928 - 0.867580i) q^{45} +(0.500000 - 0.866025i) q^{49} +(2.01367 - 2.01367i) q^{50} +(1.88981 + 0.123864i) q^{52} +(-0.0862466 - 0.0983454i) q^{53} +(-0.206647 + 0.608761i) q^{58} +(0.665060 - 1.34861i) q^{61} +(0.707107 + 0.707107i) q^{64} +(2.26151 + 2.94726i) q^{65} +(-1.08979 - 0.216773i) q^{68} +(-0.382683 + 0.923880i) q^{72} +(-1.50046 - 1.31587i) q^{73} +(1.49144 - 0.996552i) q^{74} +(-0.128293 + 1.95737i) q^{80} -1.00000 q^{81} +(0.0862466 - 1.31587i) q^{82} +(-1.08979 - 1.88757i) q^{85} +(0.382683 + 0.0761205i) q^{89} +(-1.85747 + 0.630526i) q^{90} +(-0.513210 + 0.0675653i) q^{97} +(-0.608761 + 0.793353i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more