LMFDB - Embedded newform 776.1.bp.a.723.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [776,1,Mod(3,776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(776, base_ring=CyclotomicField(48))

chi = DirichletCharacter(H, H._module([24, 24, 35]))

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

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

chi := DirichletCharacter("776.3");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 776 = 2^{3} \cdot 97 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	776.bp (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:	$0.387274449803$
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			723.1
Root			$-0.608761 - 0.793353i$ of defining polynomial
Character	$\chi$	$=$	776.723
Dual form			776.1.bp.a.571.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.130526 - 0.991445i) q^{2} +(1.20711 + 1.57313i) q^{3} +(-0.965926 - 0.258819i) q^{4} +(1.71723 - 0.991445i) q^{6} +(-0.382683 + 0.923880i) q^{8} +(-0.758819 + 2.83195i) q^{9} +O(q^{10})$	\(q+(0.130526 - 0.991445i) q^{2} +(1.20711 + 1.57313i) q^{3} +(-0.965926 - 0.258819i) q^{4} +(1.71723 - 0.991445i) q^{6} +(-0.382683 + 0.923880i) q^{8} +(-0.758819 + 2.83195i) q^{9} +(-0.513210 + 0.0675653i) q^{11} +(-0.758819 - 1.83195i) q^{12} +(0.866025 + 0.500000i) q^{16} +(1.25026 - 1.09645i) q^{17} +(2.70868 + 1.12197i) q^{18} +(0.382683 - 0.0761205i) q^{19} +0.517638i q^{22} +(-1.91532 + 0.513210i) q^{24} +(-0.793353 - 0.608761i) q^{25} +(-3.53906 + 1.46593i) q^{27} +(0.608761 - 0.793353i) q^{32} +(-0.725788 - 0.725788i) q^{33} +(-0.923880 - 1.38268i) q^{34} +(1.46593 - 2.53906i) q^{36} +(-0.0255190 - 0.389345i) q^{38} +(-0.867580 - 1.75928i) q^{41} +(0.315118 + 1.17604i) q^{43} +(0.513210 + 0.0675653i) q^{44} +(0.258819 + 1.96593i) q^{48} +(-0.130526 - 0.991445i) q^{49} +(-0.707107 + 0.707107i) q^{50} +(3.23407 + 0.643296i) q^{51} +(0.991445 + 3.70012i) q^{54} +(0.581687 + 0.510126i) q^{57} +(-0.0726721 - 1.10876i) q^{59} +(-0.707107 - 0.707107i) q^{64} +(-0.814313 + 0.624844i) q^{66} +(0.172572 + 0.867580i) q^{67} +(-1.49144 + 0.735499i) q^{68} +(-2.32599 - 1.78480i) q^{72} +(1.36603 - 0.366025i) q^{73} -1.98289i q^{75} +(-0.389345 - 0.0255190i) q^{76} +(-4.03906 - 2.33195i) q^{81} +(-1.85747 + 0.630526i) q^{82} +(-1.31587 - 1.50046i) q^{83} +(1.20711 - 0.158919i) q^{86} +(0.133975 - 0.500000i) q^{88} +(-0.607206 + 1.46593i) q^{89} +1.98289 q^{96} +(0.608761 + 0.793353i) q^{97} -1.00000 q^{98} +(0.198092 - 1.50465i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{3} - 8 q^{9}+O(q^{10}) $ 16 * q + 8 * q^3 - 8 * q^9	$ 16 q + 8 q^{3} - 8 q^{9} - 8 q^{12} - 16 q^{27} + 8 q^{36} + 8 q^{66} - 8 q^{68} + 8 q^{73} - 24 q^{81} + 8 q^{86} + 16 q^{88} - 16 q^{98}+O(q^{100}) $ 16 * q + 8 * q^3 - 8 * q^9 - 8 * q^12 - 16 * q^27 + 8 * q^36 + 8 * q^66 - 8 * q^68 + 8 * q^73 - 24 * q^81 + 8 * q^86 + 16 * q^88 - 16 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/776\mathbb{Z}\right)^\times$.

$n$	$389$	$393$	$583$
$\chi(n)$	$-1$	$e\left(\frac{29}{48}\right)$	$-1$

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 $

$2$	0.130526	−	0.991445i	0.130526	−	0.991445i
$2$	0.130526	−	0.991445i	0.130526	−	0.991445i
$3$	1.20711	+	1.57313i	1.20711	+	1.57313i	0.707107	+	0.707107i	$0.250000\pi$
$3$	1.20711	+	1.57313i	1.20711	+	1.57313i	0.500000	+	0.866025i	$0.333333\pi$
$4$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$5$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$5$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$6$	1.71723	−	0.991445i	1.71723	−	0.991445i
$7$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$7$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$8$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$9$	−0.758819	+	2.83195i	−0.758819	+	2.83195i
$10$		0			0
$11$	−0.513210	+	0.0675653i	−0.513210	+	0.0675653i	−0.382683	−	0.923880i	$-0.625000\pi$
$11$	−0.513210	+	0.0675653i	−0.513210	+	0.0675653i	−0.130526	+	0.991445i	$0.541667\pi$
$12$	−0.758819	−	1.83195i	−0.758819	−	1.83195i
$13$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$13$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$14$		0			0
$15$		0			0
$16$	0.866025	+	0.500000i	0.866025	+	0.500000i
$17$	1.25026	−	1.09645i	1.25026	−	1.09645i	0.258819	−	0.965926i	$-0.416667\pi$
$17$	1.25026	−	1.09645i	1.25026	−	1.09645i	0.991445	−	0.130526i	$-0.0416667\pi$
$18$	2.70868	+	1.12197i	2.70868	+	1.12197i
$19$	0.382683	−	0.0761205i	0.382683	−	0.0761205i		−	1.00000i	$-0.5\pi$
$19$	0.382683	−	0.0761205i	0.382683	−	0.0761205i	0.382683	+	0.923880i	$0.375000\pi$
$20$		0			0
$21$		0			0
$22$			0.517638i			0.517638i
$23$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$23$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$24$	−1.91532	+	0.513210i	−1.91532	+	0.513210i
$25$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$26$		0			0
$27$	−3.53906	+	1.46593i	−3.53906	+	1.46593i
$28$		0			0
$29$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$29$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$30$		0			0
$31$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$31$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$32$	0.608761	−	0.793353i	0.608761	−	0.793353i
$33$	−0.725788	−	0.725788i	−0.725788	−	0.725788i
$34$	−0.923880	−	1.38268i	−0.923880	−	1.38268i
$35$		0			0
$36$	1.46593	−	2.53906i	1.46593	−	2.53906i
$37$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$37$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$38$	−0.0255190	−	0.389345i	−0.0255190	−	0.389345i
$39$		0			0
$40$		0			0
$41$	−0.867580	−	1.75928i	−0.867580	−	1.75928i	−0.608761	−	0.793353i	$-0.708333\pi$
$41$	−0.867580	−	1.75928i	−0.867580	−	1.75928i	−0.258819	−	0.965926i	$-0.583333\pi$
$42$		0			0
$43$	0.315118	+	1.17604i	0.315118	+	1.17604i	0.923880	+	0.382683i	$0.125000\pi$
$43$	0.315118	+	1.17604i	0.315118	+	1.17604i	−0.608761	+	0.793353i	$0.708333\pi$
$44$	0.513210	+	0.0675653i	0.513210	+	0.0675653i
$45$		0			0
$46$		0			0
$47$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$47$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$48$	0.258819	+	1.96593i	0.258819	+	1.96593i
$49$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$50$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$51$	3.23407	+	0.643296i	3.23407	+	0.643296i
$52$		0			0
$53$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$53$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$54$	0.991445	+	3.70012i	0.991445	+	3.70012i
$55$		0			0
$56$		0			0
$57$	0.581687	+	0.510126i	0.581687	+	0.510126i
$58$		0			0
$59$	−0.0726721	−	1.10876i	−0.0726721	−	1.10876i	−0.866025	−	0.500000i	$-0.833333\pi$
$59$	−0.0726721	−	1.10876i	−0.0726721	−	1.10876i	0.793353	−	0.608761i	$-0.208333\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$65$		0			0
$66$	−0.814313	+	0.624844i	−0.814313	+	0.624844i
$67$	0.172572	+	0.867580i	0.172572	+	0.867580i	0.965926	+	0.258819i	$0.0833333\pi$
$67$	0.172572	+	0.867580i	0.172572	+	0.867580i	−0.793353	+	0.608761i	$0.791667\pi$
$68$	−1.49144	+	0.735499i	−1.49144	+	0.735499i
$69$		0			0
$70$		0			0
$71$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$71$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$72$	−2.32599	−	1.78480i	−2.32599	−	1.78480i
$73$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$73$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$74$		0			0
$75$		−	1.98289i		−	1.98289i
$76$	−0.389345	−	0.0255190i	−0.389345	−	0.0255190i
$77$		0			0
$78$		0			0
$79$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$79$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$80$		0			0
$81$	−4.03906	−	2.33195i	−4.03906	−	2.33195i
$82$	−1.85747	+	0.630526i	−1.85747	+	0.630526i
$83$	−1.31587	−	1.50046i	−1.31587	−	1.50046i	−0.707107	−	0.707107i	$-0.750000\pi$
$83$	−1.31587	−	1.50046i	−1.31587	−	1.50046i	−0.608761	−	0.793353i	$-0.708333\pi$
$84$		0			0
$85$		0			0
$86$	1.20711	−	0.158919i	1.20711	−	0.158919i
$87$		0			0
$88$	0.133975	−	0.500000i	0.133975	−	0.500000i
$89$	−0.607206	+	1.46593i	−0.607206	+	1.46593i	0.258819	+	0.965926i	$0.416667\pi$
$89$	−0.607206	+	1.46593i	−0.607206	+	1.46593i	−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	1.98289			1.98289
$97$	0.608761	+	0.793353i	0.608761	+	0.793353i
$97$	0.608761	+	0.793353i	0.608761	+	0.793353i
$98$	−1.00000			−1.00000
$99$	0.198092	−	1.50465i	0.198092	−	1.50465i
$100$	0.608761	+	0.793353i	0.608761	+	0.793353i
$101$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$101$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$102$	1.05992	−	3.12243i	1.05992	−	3.12243i
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.18270	+	0.583242i	1.18270	+	0.583242i	0.923880	−	0.382683i	$-0.125000\pi$
$107$	1.18270	+	0.583242i	1.18270	+	0.583242i	0.258819	+	0.965926i	$0.416667\pi$
$108$	3.79788	−	0.500000i	3.79788	−	0.500000i
$109$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$109$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.37413	−	0.793353i	−1.37413	−	0.793353i	−0.382683	−	0.923880i	$-0.625000\pi$
$113$	−1.37413	−	0.793353i	−1.37413	−	0.793353i	−0.991445	+	0.130526i	$0.958333\pi$
$114$	0.581687	−	0.510126i	0.581687	−	0.510126i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	−1.10876	−	0.0726721i	−1.10876	−	0.0726721i
$119$		0			0
$120$		0			0
$121$	−0.707107	+	0.189469i	−0.707107	+	0.189469i
$122$		0			0
$123$	1.72032	−	3.48846i	1.72032	−	3.48846i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$127$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$128$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$129$	−1.46968	+	1.91532i	−1.46968	+	1.91532i
$130$		0			0
$131$	0.732626	+	1.09645i	0.732626	+	1.09645i	0.991445	+	0.130526i	$0.0416667\pi$
$131$	0.732626	+	1.09645i	0.732626	+	1.09645i	−0.258819	+	0.965926i	$0.583333\pi$
$132$	0.513210	+	0.888905i	0.513210	+	0.888905i
$133$		0			0
$134$	0.882683	−	0.0578541i	0.882683	−	0.0578541i
$135$		0			0
$136$	0.534534	+	1.57469i	0.534534	+	1.57469i
$137$	−1.42388	−	1.24871i	−1.42388	−	1.24871i	−0.923880	−	0.382683i	$-0.875000\pi$
$137$	−1.42388	−	1.24871i	−1.42388	−	1.24871i	−0.500000	−	0.866025i	$-0.666667\pi$
$138$		0			0
$139$	−0.369474	+	1.85747i	−0.369474	+	1.85747i	0.130526	+	0.991445i	$0.458333\pi$
$139$	−0.369474	+	1.85747i	−0.369474	+	1.85747i	−0.500000	+	0.866025i	$0.666667\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−2.07313	+	2.07313i	−2.07313	+	2.07313i
$145$		0			0
$146$	−0.184592	−	1.40211i	−0.184592	−	1.40211i
$147$	1.40211	−	1.40211i	1.40211	−	1.40211i
$148$		0			0
$149$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$149$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$150$	−1.96593	−	0.258819i	−1.96593	−	0.258819i
$151$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$151$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$152$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$153$	2.15637	+	4.37270i	2.15637	+	4.37270i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$157$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−2.83920	+	3.70012i	−2.83920	+	3.70012i
$163$	−0.793353	+	0.608761i	−0.793353	+	0.608761i	−0.923880	−	0.382683i	$-0.875000\pi$
$163$	−0.793353	+	0.608761i	−0.793353	+	0.608761i	0.130526	+	0.991445i	$0.458333\pi$
$164$	0.382683	+	1.92388i	0.382683	+	1.92388i
$165$		0			0
$166$	−1.65938	+	1.10876i	−1.65938	+	1.10876i
$167$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$167$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$168$		0			0
$169$	0.793353	+	0.608761i	0.793353	+	0.608761i
$170$		0			0
$171$	−0.0748180	+	1.14150i	−0.0748180	+	1.14150i
$172$		−	1.21752i		−	1.21752i
$173$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$173$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$174$		0			0
$175$		0			0
$176$	−0.478235	−	0.198092i	−0.478235	−	0.198092i
$177$	1.65651	−	1.45272i	1.65651	−	1.45272i
$178$	1.37413	+	0.793353i	1.37413	+	0.793353i
$179$	1.05217	−	0.357164i	1.05217	−	0.357164i	0.258819	−	0.965926i	$-0.416667\pi$
$179$	1.05217	−	0.357164i	1.05217	−	0.357164i	0.793353	+	0.608761i	$0.208333\pi$
$180$		0			0
$181$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$181$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−0.567565	+	0.647184i	−0.567565	+	0.647184i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$191$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$192$	0.258819	−	1.96593i	0.258819	−	1.96593i
$193$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$193$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$194$	0.866025	−	0.500000i	0.866025	−	0.500000i
$195$		0			0
$196$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$197$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$197$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$198$	−1.46593	−	0.392794i	−1.46593	−	0.392794i
$199$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$199$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$200$	0.866025	−	0.500000i	0.866025	−	0.500000i
$201$	−1.15651	+	1.31874i	−1.15651	+	1.31874i
$202$		0			0
$203$		0			0
$204$	−2.95737	−	1.45841i	−2.95737	−	1.45841i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−0.191254	+	0.0649219i	−0.191254	+	0.0649219i
$210$		0			0
$211$	−1.34861	+	1.18270i	−1.34861	+	1.18270i	−0.382683	+	0.923880i	$0.625000\pi$
$211$	−1.34861	+	1.18270i	−1.34861	+	1.18270i	−0.965926	+	0.258819i	$0.916667\pi$
$212$		0			0
$213$		0			0
$214$	0.732626	−	1.09645i	0.732626	−	1.09645i
$215$		0			0
$216$		−	3.83065i		−	3.83065i
$217$		0			0
$218$		0			0
$219$	2.22474	+	1.70711i	2.22474	+	1.70711i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$223$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$224$		0			0
$225$	2.32599	−	1.78480i	2.32599	−	1.78480i
$226$	−0.965926	+	1.25882i	−0.965926	+	1.25882i
$227$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$227$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$228$	−0.429836	−	0.643296i	−0.429836	−	0.643296i
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.0420463	−	0.123864i	−0.0420463	−	0.123864i	0.923880	−	0.382683i	$-0.125000\pi$
$233$	−0.0420463	−	0.123864i	−0.0420463	−	0.123864i	−0.965926	+	0.258819i	$0.916667\pi$
$234$		0			0
$235$		0			0
$236$	−0.216773	+	1.08979i	−0.216773	+	1.08979i
$237$		0			0
$238$		0			0
$239$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$239$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$240$		0			0
$241$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$241$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$242$	0.0955518	+	0.725788i	0.0955518	+	0.725788i
$243$	−0.707107	−	5.37101i	−0.707107	−	5.37101i
$244$		0			0
$245$		0			0
$246$	−3.23407	−	2.16093i	−3.23407	−	2.16093i
$247$		0			0
$248$		0			0
$249$	0.772029	−	3.88125i	0.772029	−	3.88125i
$250$		0			0
$251$	1.47479	+	1.29335i	1.47479	+	1.29335i	0.866025	+	0.500000i	$0.166667\pi$
$251$	1.47479	+	1.29335i	1.47479	+	1.29335i	0.608761	+	0.793353i	$0.291667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.500000	+	0.866025i	0.500000	+	0.866025i
$257$	0.835400	+	1.25026i	0.835400	+	1.25026i	0.965926	+	0.258819i	$0.0833333\pi$
$257$	0.835400	+	1.25026i	0.835400	+	1.25026i	−0.130526	+	0.991445i	$0.541667\pi$
$258$	1.70711	+	1.70711i	1.70711	+	1.70711i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	1.18270	−	0.583242i	1.18270	−	0.583242i
$263$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$263$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$264$	0.948288	−	0.392794i	0.948288	−	0.392794i
$265$		0			0
$266$		0			0
$267$	−3.03906	+	0.814313i	−3.03906	+	0.814313i
$268$	0.0578541	−	0.882683i	0.0578541	−	0.882683i
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$271$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$272$	1.63099	−	0.324423i	1.63099	−	0.324423i
$273$		0			0
$274$	−1.42388	+	1.24871i	−1.42388	+	1.24871i
$275$	0.448288	+	0.258819i	0.448288	+	0.258819i
$276$		0			0
$277$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$277$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$278$	1.79335	+	0.608761i	1.79335	+	0.608761i
$279$		0			0
$280$		0			0
$281$	0.793353	+	0.391239i	0.793353	+	0.391239i	0.793353	−	0.608761i	$-0.208333\pi$
$281$	0.793353	+	0.391239i	0.793353	+	0.391239i			1.00000i	$0.5\pi$
$282$		0			0
$283$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$283$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.78480	+	2.32599i	1.78480	+	2.32599i
$289$	0.230427	−	1.75026i	0.230427	−	1.75026i
$290$		0			0
$291$	−0.513210	+	1.91532i	−0.513210	+	1.91532i
$292$	−1.41421			−1.41421
$293$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$293$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$294$	−1.20711	−	1.57313i	−1.20711	−	1.57313i
$295$		0			0
$296$		0			0
$297$	1.71723	−	0.991445i	1.71723	−	0.991445i
$298$		0			0
$299$		0			0
$300$	−0.513210	+	1.91532i	−0.513210	+	1.91532i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.369474	+	0.125419i	0.369474	+	0.125419i
$305$		0			0
$306$	4.61675	−	1.56718i	4.61675	−	1.56718i
$307$	0.226078	+	0.130526i	0.226078	+	0.130526i	0.608761	−	0.793353i	$-0.291667\pi$
$307$	0.226078	+	0.130526i	0.226078	+	0.130526i	−0.382683	+	0.923880i	$0.625000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$311$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$312$		0			0
$313$		−	1.21752i		−	1.21752i	−0.793353	−	0.608761i	$-0.791667\pi$
$313$		−	1.21752i		−	1.21752i	0.793353	−	0.608761i	$-0.208333\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$317$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	0.510126	+	2.56458i	0.510126	+	2.56458i
$322$		0			0
$323$	0.394993	−	0.514765i	0.394993	−	0.514765i
$324$	3.29788	+	3.29788i	3.29788	+	3.29788i
$325$		0			0
$326$	0.500000	+	0.866025i	0.500000	+	0.866025i
$327$		0			0
$328$	1.95737	−	0.128293i	1.95737	−	0.128293i
$329$		0			0
$330$		0			0
$331$	−0.483342	−	0.423880i	−0.483342	−	0.423880i	0.382683	−	0.923880i	$-0.375000\pi$
$331$	−0.483342	−	0.423880i	−0.483342	−	0.423880i	−0.866025	+	0.500000i	$0.833333\pi$
$332$	0.882683	+	1.78990i	0.882683	+	1.78990i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.128293	−	0.0255190i	−0.128293	−	0.0255190i	0.130526	−	0.991445i	$-0.458333\pi$
$337$	−0.128293	−	0.0255190i	−0.128293	−	0.0255190i	−0.258819	+	0.965926i	$0.583333\pi$
$338$	0.707107	−	0.707107i	0.707107	−	0.707107i
$339$	−0.410670	−	3.11935i	−0.410670	−	3.11935i
$340$		0			0
$341$		0			0
$342$	1.12197	+	0.223174i	1.12197	+	0.223174i
$343$		0			0
$344$	−1.20711	−	0.158919i	−1.20711	−	0.158919i
$345$		0			0
$346$		0			0
$347$	−0.0578541	−	0.117317i	−0.0578541	−	0.117317i	0.866025	−	0.500000i	$-0.166667\pi$
$347$	−0.0578541	−	0.117317i	−0.0578541	−	0.117317i	−0.923880	+	0.382683i	$0.875000\pi$
$348$		0			0
$349$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$349$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$350$		0			0
$351$		0			0
$352$	−0.258819	+	0.448288i	−0.258819	+	0.448288i
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$	−1.22407	−	1.83195i	−1.22407	−	1.83195i
$355$		0			0
$356$	0.965926	−	1.25882i	0.965926	−	1.25882i
$357$		0			0
$358$	−0.216773	−	1.08979i	−0.216773	−	1.08979i
$359$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$359$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$360$		0			0
$361$	−0.783227	+	0.324423i	−0.783227	+	0.324423i
$362$		0			0
$363$	−1.15161	−	0.883663i	−1.15161	−	0.883663i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$367$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$368$		0			0
$369$	5.64053	−	1.12197i	5.64053	−	1.12197i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$373$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$374$	0.567565	+	0.647184i	0.567565	+	0.647184i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.448288	−	1.67303i	0.448288	−	1.67303i	−0.258819	−	0.965926i	$-0.583333\pi$
$379$	0.448288	−	1.67303i	0.448288	−	1.67303i	0.707107	−	0.707107i	$-0.250000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$383$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$384$	−1.91532	−	0.513210i	−1.91532	−	0.513210i
$385$		0			0
$386$	0.130526	−	0.991445i	0.130526	−	0.991445i
$387$	−3.56960			−3.56960
$388$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$	0.965926	+	0.258819i	0.965926	+	0.258819i
$393$	−0.840506	+	2.47605i	−0.840506	+	2.47605i
$394$		0			0
$395$		0			0
$396$	−0.580775	+	1.40211i	−0.580775	+	1.40211i
$397$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$397$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$398$		0			0
$399$		0			0
$400$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$401$	−0.608761	−	0.206647i	−0.608761	−	0.206647i		−	1.00000i	$-0.5\pi$
$401$	−0.608761	−	0.206647i	−0.608761	−	0.206647i	−0.608761	+	0.793353i	$0.708333\pi$
$402$	1.15651	+	1.31874i	1.15651	+	1.31874i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−1.83195	+	2.74171i	−1.83195	+	2.74171i
$409$	1.99144	+	0.130526i	1.99144	+	0.130526i	1.00000			$0$
$409$	1.99144	+	0.130526i	1.99144	+	0.130526i	0.991445	+	0.130526i	$0.0416667\pi$
$410$		0			0
$411$	0.245609	−	3.74728i	0.245609	−	3.74728i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−3.36804	+	1.66093i	−3.36804	+	1.66093i
$418$	0.0394029	+	0.198092i	0.0394029	+	0.198092i
$419$	0.207107	−	0.158919i	0.207107	−	0.158919i	−0.500000	−	0.866025i	$-0.666667\pi$
$419$	0.207107	−	0.158919i	0.207107	−	0.158919i	0.707107	+	0.707107i	$0.250000\pi$
$420$		0			0
$421$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$421$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$422$	0.996552	+	1.49144i	0.996552	+	1.49144i
$423$		0			0
$424$		0			0
$425$	−1.65938	+	0.108761i	−1.65938	+	0.108761i
$426$		0			0
$427$		0			0
$428$	−0.991445	−	0.869474i	−0.991445	−	0.869474i
$429$		0			0
$430$		0			0
$431$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$431$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$432$	−3.79788	−	0.500000i	−3.79788	−	0.500000i
$433$	0.735499	+	0.491445i	0.735499	+	0.491445i	0.866025	−	0.500000i	$-0.166667\pi$
$433$	0.735499	+	0.491445i	0.735499	+	0.491445i	−0.130526	+	0.991445i	$0.541667\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	1.98289	−	1.98289i	1.98289	−	1.98289i
$439$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$439$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$440$		0			0
$441$	2.90677	+	0.382683i	2.90677	+	0.382683i
$442$		0			0
$443$	−0.125419	+	0.630526i	−0.125419	+	0.630526i	0.866025	+	0.500000i	$0.166667\pi$
$443$	−0.125419	+	0.630526i	−0.125419	+	0.630526i	−0.991445	+	0.130526i	$0.958333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.130526	+	0.226078i	−0.130526	+	0.226078i	−0.923880	−	0.382683i	$-0.875000\pi$
$449$	−0.130526	+	0.226078i	−0.130526	+	0.226078i	0.793353	+	0.608761i	$0.208333\pi$
$450$	−1.46593	−	2.53906i	−1.46593	−	2.53906i
$451$	0.564117	+	0.844261i	0.564117	+	0.844261i
$452$	1.12197	+	1.12197i	1.12197	+	1.12197i
$453$		0			0
$454$	−1.37413	+	1.05441i	−1.37413	+	1.05441i
$455$		0			0
$456$	−0.693897	+	0.342192i	−0.693897	+	0.342192i
$457$	−1.25026	+	0.835400i	−1.25026	+	0.835400i	−0.991445	−	0.130526i	$-0.958333\pi$
$457$	−1.25026	+	0.835400i	−1.25026	+	0.835400i	−0.258819	+	0.965926i	$0.583333\pi$
$458$		0			0
$459$	−2.81744	+	5.71320i	−2.81744	+	5.71320i
$460$		0			0
$461$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$461$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$	−0.128293	+	0.0255190i	−0.128293	+	0.0255190i
$467$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−0.707107	−	0.707107i	$-0.750000\pi$
$467$	−1.70711	−	0.707107i	−1.70711	−	0.707107i	−1.00000			$\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	1.05217	+	0.357164i	1.05217	+	0.357164i
$473$	−0.241181	−	0.582262i	−0.241181	−	0.582262i
$474$		0			0
$475$	−0.349942	−	0.172572i	−0.349942	−	0.172572i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.732051			0.732051
$485$		0			0
$486$	−5.41736			−5.41736
$487$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$487$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$488$		0			0
$489$	−1.91532	−	0.513210i	−1.91532	−	0.513210i
$490$		0			0
$491$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$491$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$492$	−2.56458	+	2.92434i	−2.56458	+	2.92434i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−3.74728	−	1.27203i	−3.74728	−	1.27203i
$499$	−0.257264	−	0.293353i	−0.257264	−	0.293353i	0.608761	−	0.793353i	$-0.291667\pi$
$499$	−0.257264	−	0.293353i	−0.257264	−	0.293353i	−0.866025	+	0.500000i	$0.833333\pi$
$500$		0			0
$501$		0			0
$502$	1.47479	−	1.29335i	1.47479	−	1.29335i
$503$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$503$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			1.98289i			1.98289i
$508$		0			0
$509$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$509$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$510$		0			0
$511$		0			0
$512$	0.923880	−	0.382683i	0.923880	−	0.382683i
$513$	−1.24275	+	0.830380i	−1.24275	+	0.830380i
$514$	1.34861	−	0.665060i	1.34861	−	0.665060i
$515$		0			0
$516$	1.91532	−	1.46968i	1.91532	−	1.46968i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$521$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$522$		0			0
$523$	−0.00855514	−	0.130526i	−0.00855514	−	0.130526i	0.991445	−	0.130526i	$-0.0416667\pi$
$523$	−0.00855514	−	0.130526i	−0.00855514	−	0.130526i	−1.00000			$\pi$
$524$	−0.423880	−	1.24871i	−0.423880	−	1.24871i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−0.265657	−	0.991445i	−0.265657	−	0.991445i
$529$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$530$		0			0
$531$	3.19510	+	0.635546i	3.19510	+	0.635546i
$532$		0			0
$533$		0			0
$534$	0.410670	+	3.11935i	0.410670	+	3.11935i
$535$		0			0
$536$	−0.867580	−	0.172572i	−0.867580	−	0.172572i
$537$	1.83195	+	1.22407i	1.83195	+	1.22407i
$538$		0			0
$539$	0.133975	+	0.500000i	0.133975	+	0.500000i
$540$		0			0
$541$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$541$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$542$		0			0
$543$		0			0
$544$	−0.108761	−	1.65938i	−0.108761	−	1.65938i
$545$		0			0
$546$		0			0
$547$	−0.608761	−	1.05441i	−0.608761	−	1.05441i	−0.991445	−	0.130526i	$-0.958333\pi$
$547$	−0.608761	−	1.05441i	−0.608761	−	1.05441i	0.382683	−	0.923880i	$-0.375000\pi$
$548$	1.05217	+	1.57469i	1.05217	+	1.57469i
$549$		0			0
$550$	0.315118	−	0.410670i	0.315118	−	0.410670i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.837633	−	1.69855i	0.837633	−	1.69855i
$557$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$557$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−1.70322	−	0.111635i	−1.70322	−	0.111635i
$562$	0.491445	−	0.735499i	0.491445	−	0.735499i
$563$	−1.63099	+	0.324423i	−1.63099	+	0.324423i	−0.923880	−	0.382683i	$-0.875000\pi$
$563$	−1.63099	+	0.324423i	−1.63099	+	0.324423i	−0.707107	+	0.707107i	$0.750000\pi$
$564$		0			0
$565$		0			0
$566$	1.22474	+	0.707107i	1.22474	+	0.707107i
$567$		0			0
$568$		0			0
$569$	1.69855	+	0.576581i	1.69855	+	0.576581i	0.991445	−	0.130526i	$-0.0416667\pi$
$569$	1.69855	+	0.576581i	1.69855	+	0.576581i	0.707107	+	0.707107i	$0.250000\pi$
$570$		0			0
$571$	0.258819	−	0.0340742i	0.258819	−	0.0340742i		−	1.00000i	$-0.5\pi$
$571$	0.258819	−	0.0340742i	0.258819	−	0.0340742i	0.258819	+	0.965926i	$0.416667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	2.53906	−	1.46593i	2.53906	−	1.46593i
$577$	0.576581	−	1.69855i	0.576581	−	1.69855i	−0.130526	−	0.991445i	$-0.541667\pi$
$577$	0.576581	−	1.69855i	0.576581	−	1.69855i	0.707107	−	0.707107i	$-0.250000\pi$
$578$	−1.70521	−	0.456911i	−1.70521	−	0.456911i
$579$	1.20711	+	1.57313i	1.20711	+	1.57313i
$580$		0			0
$581$		0			0
$582$	1.83195	+	0.758819i	1.83195	+	0.758819i
$583$		0			0
$584$	−0.184592	+	1.40211i	−0.184592	+	1.40211i
$585$		0			0
$586$		0			0
$587$	0.206647	−	0.608761i	0.206647	−	0.608761i	−0.793353	−	0.608761i	$-0.791667\pi$
$587$	0.206647	−	0.608761i	0.206647	−	0.608761i	1.00000			$0$
$588$	−1.71723	+	0.991445i	−1.71723	+	0.991445i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.991445	−	0.130526i	0.991445	−	0.130526i	0.382683	−	0.923880i	$-0.375000\pi$
$593$	0.991445	−	0.130526i	0.991445	−	0.130526i	0.608761	+	0.793353i	$0.291667\pi$
$594$	−0.758819	−	1.83195i	−0.758819	−	1.83195i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$599$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$600$	1.83195	+	0.758819i	1.83195	+	0.758819i
$601$	1.08979	−	0.216773i	1.08979	−	0.216773i	0.382683	−	0.923880i	$-0.375000\pi$
$601$	1.08979	−	0.216773i	1.08979	−	0.216773i	0.707107	+	0.707107i	$0.250000\pi$
$602$		0			0
$603$	−2.58790	−	0.169620i	−2.58790	−	0.169620i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$607$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$608$	0.172572	−	0.349942i	0.172572	−	0.349942i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.951161	−	4.78181i	−0.951161	−	4.78181i
$613$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$613$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$614$	0.158919	−	0.207107i	0.158919	−	0.207107i
$615$		0			0
$616$		0			0
$617$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	0.608761	−	0.793353i	$-0.291667\pi$
$617$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	−0.991445	+	0.130526i	$0.958333\pi$
$618$		0			0
$619$	−1.95737	+	0.128293i	−1.95737	+	0.128293i	−0.991445	−	0.130526i	$-0.958333\pi$
$619$	−1.95737	+	0.128293i	−1.95737	+	0.128293i	−0.965926	+	0.258819i	$0.916667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.258819	+	0.965926i	0.258819	+	0.965926i
$626$	−1.20711	−	0.158919i	−1.20711	−	0.158919i
$627$	−0.332994	−	0.222500i	−0.332994	−	0.222500i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$631$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$632$		0			0
$633$	−3.48846	−	0.693897i	−3.48846	−	0.693897i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	0.123864	+	1.88981i	0.123864	+	1.88981i	0.382683	+	0.923880i	$0.375000\pi$
$641$	0.123864	+	1.88981i	0.123864	+	1.88981i	−0.258819	+	0.965926i	$0.583333\pi$
$642$	2.60922	−	0.171017i	2.60922	−	0.171017i
$643$	0.382683	−	0.662827i	0.382683	−	0.662827i	−0.608761	−	0.793353i	$-0.708333\pi$
$643$	0.382683	−	0.662827i	0.382683	−	0.662827i	0.991445	+	0.130526i	$0.0416667\pi$
$644$		0			0
$645$		0			0
$646$	−0.458804	−	0.458804i	−0.458804	−	0.458804i
$647$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$647$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$648$	3.70012	−	2.83920i	3.70012	−	2.83920i
$649$	0.112210	+	0.564117i	0.112210	+	0.564117i
$650$		0			0
$651$		0			0
$652$	0.923880	−	0.382683i	0.923880	−	0.382683i
$653$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$653$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$654$		0			0
$655$		0			0
$656$	0.128293	−	1.95737i	0.128293	−	1.95737i
$657$			4.14626i			4.14626i
$658$		0			0
$659$	0.835400	−	1.25026i	0.835400	−	1.25026i	−0.130526	−	0.991445i	$-0.541667\pi$
$659$	0.835400	−	1.25026i	0.835400	−	1.25026i	0.965926	−	0.258819i	$-0.0833333\pi$
$660$		0			0
$661$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$661$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$662$	−0.483342	+	0.423880i	−0.483342	+	0.423880i
$663$		0			0
$664$	1.88981	−	0.641502i	1.88981	−	0.641502i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.226078	−	0.130526i	0.226078	−	0.130526i	−0.382683	−	0.923880i	$-0.625000\pi$
$673$	0.226078	−	0.130526i	0.226078	−	0.130526i	0.608761	+	0.793353i	$0.291667\pi$
$674$	−0.0420463	+	0.123864i	−0.0420463	+	0.123864i
$675$	3.70012	+	0.991445i	3.70012	+	0.991445i
$676$	−0.608761	−	0.793353i	−0.608761	−	0.793353i
$677$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$677$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$678$	−3.14626			−3.14626
$679$		0			0
$680$		0			0
$681$	0.448288	−	3.40508i	0.448288	−	3.40508i
$682$		0			0
$683$	0.252157	+	0.0675653i	0.252157	+	0.0675653i	0.382683	−	0.923880i	$-0.375000\pi$
$683$	0.252157	+	0.0675653i	0.252157	+	0.0675653i	−0.130526	+	0.991445i	$0.541667\pi$
$684$	0.367711	−	1.08324i	0.367711	−	1.08324i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−0.315118	+	1.17604i	−0.315118	+	1.17604i
$689$		0			0
$690$		0			0
$691$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$691$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$	−0.123864	+	0.0420463i	−0.123864	+	0.0420463i
$695$		0			0
$696$		0			0
$697$	−3.01367	−	1.24830i	−3.01367	−	1.24830i
$698$		0			0
$699$	0.144101	−	0.215662i	0.144101	−	0.215662i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	0.410670	+	0.315118i	0.410670	+	0.315118i
$705$		0			0
$706$		0			0
$707$		0			0
$708$	−1.97605	+	0.974481i	−1.97605	+	0.974481i
$709$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$709$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$710$		0			0
$711$		0			0
$712$	−1.12197	−	1.12197i	−1.12197	−	1.12197i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−1.10876	+	0.0726721i	−1.10876	+	0.0726721i
$717$		0			0
$718$		0			0
$719$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$719$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$720$		0			0
$721$		0			0
$722$	0.219416	+	0.818872i	0.219416	+	0.818872i
$723$		0			0
$724$		0			0
$725$		0			0
$726$	−1.02642	+	1.02642i	−1.02642	+	1.02642i
$727$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$727$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$728$		0			0
$729$	4.29788	−	4.29788i	4.29788	−	4.29788i
$730$		0			0
$731$	1.68345	+	1.12484i	1.68345	+	1.12484i
$732$		0			0
$733$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$733$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−0.147184	−	0.433591i	−0.147184	−	0.433591i
$738$	−0.376136	−	5.73872i	−0.376136	−	5.73872i
$739$	1.88981	−	0.123864i	1.88981	−	0.123864i	0.923880	−	0.382683i	$-0.125000\pi$
$739$	1.88981	−	0.123864i	1.88981	−	0.123864i	0.965926	+	0.258819i	$0.0833333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$743$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	5.24774	−	2.58790i	5.24774	−	2.58790i
$748$	0.715730	−	0.478235i	0.715730	−	0.478235i
$749$		0			0
$750$		0			0
$751$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$751$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$752$		0			0
$753$	−0.254391	+	3.88125i	−0.254391	+	3.88125i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$757$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$758$	−1.60021	−	0.662827i	−1.60021	−	0.662827i
$759$		0			0
$760$		0			0
$761$	−1.57469	+	0.534534i	−1.57469	+	0.534534i	−0.965926	−	0.258819i	$-0.916667\pi$
$761$	−1.57469	+	0.534534i	−1.57469	+	0.534534i	−0.608761	+	0.793353i	$0.708333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.758819	+	1.83195i	−0.758819	+	1.83195i
$769$	0.257264	−	0.293353i	0.257264	−	0.293353i	−0.608761	−	0.793353i	$-0.708333\pi$
$769$	0.257264	−	0.293353i	0.257264	−	0.293353i	0.866025	+	0.500000i	$0.166667\pi$
$770$		0			0
$771$	−0.958414	+	2.82340i	−0.958414	+	2.82340i
$772$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$773$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$773$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$774$	−0.465926	+	3.53906i	−0.465926	+	3.53906i
$775$		0			0
$776$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$777$		0			0
$778$		0			0
$779$	−0.465926	−	0.607206i	−0.465926	−	0.607206i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.382683	−	0.923880i	0.382683	−	0.923880i
$785$		0			0
$786$	2.34516	+	1.15651i	2.34516	+	1.15651i
$787$	0.758819	−	0.0999004i	0.758819	−	0.0999004i	0.258819	−	0.965926i	$-0.416667\pi$
$787$	0.758819	−	0.0999004i	0.758819	−	0.0999004i	0.500000	+	0.866025i	$0.333333\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	1.31431	+	0.758819i	1.31431	+	0.758819i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$797$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$798$		0			0
$799$		0			0
$800$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$801$	−3.69067	−	2.83195i	−3.69067	−	2.83195i
$802$	−0.284338	+	0.576581i	−0.284338	+	0.576581i
$803$	−0.676327	+	0.280144i	−0.676327	+	0.280144i
$804$	1.45841	−	0.974481i	1.45841	−	0.974481i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.923880	+	0.382683i	$0.125000\pi$
$809$	1.30656	+	1.30656i	1.30656	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$810$		0			0
$811$	−0.793353	−	1.37413i	−0.793353	−	1.37413i	−0.923880	−	0.382683i	$-0.875000\pi$
$811$	−0.793353	−	1.37413i	−0.793353	−	1.37413i	0.130526	−	0.991445i	$-0.458333\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	2.47914	+	2.17414i	2.47914	+	2.17414i
$817$	0.210111	+	0.426063i	0.210111	+	0.426063i
$818$	0.389345	−	1.95737i	0.389345	−	1.95737i
$819$		0			0
$820$		0			0
$821$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$821$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$822$	−3.68316	−	0.732626i	−3.68316	−	0.732626i
$823$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$823$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$824$		0			0
$825$	0.133975	+	1.01764i	0.133975	+	1.01764i
$826$		0			0
$827$	−1.75928	−	0.349942i	−1.75928	−	0.349942i	−0.793353	−	0.608761i	$-0.791667\pi$
$827$	−1.75928	−	0.349942i	−1.75928	−	0.349942i	−0.965926	+	0.258819i	$0.916667\pi$
$828$		0			0
$829$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$829$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.25026	−	1.09645i	−1.25026	−	1.09645i
$834$	1.20711	+	3.55602i	1.20711	+	3.55602i
$835$		0			0
$836$	0.201540	−	0.0132096i	0.201540	−	0.0132096i
$837$		0			0
$838$	−0.130526	−	0.226078i	−0.130526	−	0.226078i
$839$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$839$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$840$		0			0
$841$	0.608761	−	0.793353i	0.608761	−	0.793353i
$842$		0			0
$843$	0.342192	+	1.72032i	0.342192	+	1.72032i
$844$	1.60876	−	0.793353i	1.60876	−	0.793353i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−2.70868	+	0.725788i	−2.70868	+	0.725788i
$850$	−0.108761	+	1.65938i	−0.108761	+	1.65938i
$851$		0			0
$852$		0			0
$853$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$853$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$854$		0			0
$855$		0			0
$856$	−0.991445	+	0.869474i	−0.991445	+	0.869474i
$857$	−0.448288	−	0.258819i	−0.448288	−	0.258819i	0.258819	−	0.965926i	$-0.416667\pi$
$857$	−0.448288	−	0.258819i	−0.448288	−	0.258819i	−0.707107	+	0.707107i	$0.750000\pi$
$858$		0			0
$859$	−0.869474	−	0.991445i	−0.869474	−	0.991445i	0.130526	−	0.991445i	$-0.458333\pi$
$859$	−0.869474	−	0.991445i	−0.869474	−	0.991445i	−1.00000			$\pi$
$860$		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	$0.145833\pi$
$864$	−0.991445	+	3.70012i	−0.991445	+	3.70012i
$865$		0			0
$866$	0.583242	−	0.665060i	0.583242	−	0.665060i
$867$	3.03155	−	1.75026i	3.03155	−	1.75026i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−2.70868	+	1.12197i	−2.70868	+	1.12197i
$874$		0			0
$875$		0			0
$876$	−1.70711	−	2.22474i	−1.70711	−	2.22474i
$877$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$877$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.662827	+	1.60021i	−0.662827	+	1.60021i	0.130526	+	0.991445i	$0.458333\pi$
$881$	−0.662827	+	1.60021i	−0.662827	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$882$	0.758819	−	2.83195i	0.758819	−	2.83195i
$883$	−1.18270	−	0.583242i	−1.18270	−	0.583242i	−0.258819	−	0.965926i	$-0.583333\pi$
$883$	−1.18270	−	0.583242i	−1.18270	−	0.583242i	−0.923880	+	0.382683i	$0.875000\pi$
$884$		0			0
$885$		0			0
$886$	0.608761	+	0.206647i	0.608761	+	0.206647i
$887$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$887$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	2.23044	+	0.923880i	2.23044	+	0.923880i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.207107	+	0.158919i	0.207107	+	0.158919i
$899$		0			0
$900$	−2.70868	+	1.12197i	−2.70868	+	1.12197i
$901$		0			0
$902$	0.910670	−	0.449093i	0.910670	−	0.449093i
$903$		0			0
$904$	1.25882	−	0.965926i	1.25882	−	0.965926i
$905$		0			0
$906$		0			0
$907$	−1.10876	−	1.65938i	−1.10876	−	1.65938i	−0.608761	−	0.793353i	$-0.708333\pi$
$907$	−1.10876	−	1.65938i	−1.10876	−	1.65938i	−0.500000	−	0.866025i	$-0.666667\pi$
$908$	0.866025	+	1.50000i	0.866025	+	1.50000i
$909$		0			0
$910$		0			0
$911$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$911$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$912$	0.248693	+	0.732626i	0.248693	+	0.732626i
$913$	0.776695	+	0.681144i	0.776695	+	0.681144i
$914$	0.665060	+	1.34861i	0.665060	+	1.34861i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	5.29657	+	3.53906i	5.29657	+	3.53906i
$919$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$919$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$920$		0			0
$921$	0.0675653	+	0.513210i	0.0675653	+	0.513210i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.665060	+	1.34861i	0.665060	+	1.34861i	0.923880	+	0.382683i	$0.125000\pi$
$929$	0.665060	+	1.34861i	0.665060	+	1.34861i	−0.258819	+	0.965926i	$0.583333\pi$
$930$		0			0
$931$	−0.125419	−	0.369474i	−0.125419	−	0.369474i
$932$	0.00855514	+	0.130526i	0.00855514	+	0.130526i
$933$		0			0
$934$	−0.923880	+	1.60021i	−0.923880	+	1.60021i
$935$		0			0
$936$		0			0
$937$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$937$	−1.22474	−	1.22474i	−1.22474	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$938$		0			0
$939$	1.91532	−	1.46968i	1.91532	−	1.46968i
$940$		0			0
$941$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$941$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$942$		0			0
$943$		0			0
$944$	0.491445	−	0.996552i	0.491445	−	0.996552i
$945$		0			0
$946$	−0.608761	+	0.163117i	−0.608761	+	0.163117i
$947$	0.130526	−	1.99144i	0.130526	−	1.99144i		−	1.00000i	$-0.5\pi$
$947$	0.130526	−	1.99144i	0.130526	−	1.99144i	0.130526	−	0.991445i	$-0.458333\pi$
$948$		0			0
$949$		0			0
$950$	−0.216773	+	0.324423i	−0.216773	+	0.324423i
$951$		0			0
$952$		0			0
$953$	−0.835400	+	0.732626i	−0.835400	+	0.732626i	−0.965926	−	0.258819i	$-0.916667\pi$
$953$	−0.835400	+	0.732626i	−0.835400	+	0.732626i	0.130526	+	0.991445i	$0.458333\pi$
$954$		0			0
$955$		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			0
$963$	−2.54917	+	2.90677i	−2.54917	+	2.90677i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$967$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$968$	0.0955518	−	0.725788i	0.0955518	−	0.725788i
$969$	1.28659			1.28659
$970$		0			0
$971$	−0.765367			−0.765367			−0.382683	−	0.923880i	$-0.625000\pi$
$971$	−0.765367			−0.765367			−0.382683	+	0.923880i	$0.625000\pi$
$972$	−0.707107	+	5.37101i	−0.707107	+	5.37101i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.09645	−	1.25026i	1.09645	−	1.25026i	0.130526	−	0.991445i	$-0.458333\pi$
$977$	1.09645	−	1.25026i	1.09645	−	1.25026i	0.965926	−	0.258819i	$-0.0833333\pi$
$978$	−0.758819	+	1.83195i	−0.758819	+	1.83195i
$979$	0.212578	−	0.793353i	0.212578	−	0.793353i
$980$		0			0
$981$		0			0
$982$	0.382683	+	0.923880i	0.382683	+	0.923880i
$983$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$983$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$984$	2.56458	+	2.92434i	2.56458	+	2.92434i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$991$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$992$		0			0
$993$	0.0833732	−	1.27203i	0.0833732	−	1.27203i
$994$		0			0
$995$		0			0
$996$	−1.75026	+	3.54918i	−1.75026	+	3.54918i
$997$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$997$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$998$	−0.324423	+	0.216773i	−0.324423	+	0.216773i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	776.1.bp.a.723.1	yes	16
4.3	odd	2		3104.1.et.a.335.1		16
8.3	odd	2	CM	776.1.bp.a.723.1	yes	16
8.5	even	2		3104.1.et.a.335.1		16
97.86	even	48	inner	776.1.bp.a.571.1	✓	16
388.183	odd	48		3104.1.et.a.2511.1		16
776.571	odd	48	inner	776.1.bp.a.571.1	✓	16
776.765	even	48		3104.1.et.a.2511.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
776.1.bp.a.571.1	✓	16	97.86	even	48	inner
776.1.bp.a.571.1	✓	16	776.571	odd	48	inner
776.1.bp.a.723.1	yes	16	1.1	even	1	trivial
776.1.bp.a.723.1	yes	16	8.3	odd	2	CM
3104.1.et.a.335.1		16	4.3	odd	2
3104.1.et.a.335.1		16	8.5	even	2
3104.1.et.a.2511.1		16	388.183	odd	48
3104.1.et.a.2511.1		16	776.765	even	48

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:	\( N \)	\(=\)	\( 776 = 2^{3} \cdot 97 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	776.bp (of order \(48\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(0.130526 - 0.991445i) q^{2} +(1.20711 + 1.57313i) q^{3} +(-0.965926 - 0.258819i) q^{4} +(1.71723 - 0.991445i) q^{6} +(-0.382683 + 0.923880i) q^{8} +(-0.758819 + 2.83195i) q^{9} +O(q^{10})\)	\(q+(0.130526 - 0.991445i) q^{2} +(1.20711 + 1.57313i) q^{3} +(-0.965926 - 0.258819i) q^{4} +(1.71723 - 0.991445i) q^{6} +(-0.382683 + 0.923880i) q^{8} +(-0.758819 + 2.83195i) q^{9} +(-0.513210 + 0.0675653i) q^{11} +(-0.758819 - 1.83195i) q^{12} +(0.866025 + 0.500000i) q^{16} +(1.25026 - 1.09645i) q^{17} +(2.70868 + 1.12197i) q^{18} +(0.382683 - 0.0761205i) q^{19} +0.517638i q^{22} +(-1.91532 + 0.513210i) q^{24} +(-0.793353 - 0.608761i) q^{25} +(-3.53906 + 1.46593i) q^{27} +(0.608761 - 0.793353i) q^{32} +(-0.725788 - 0.725788i) q^{33} +(-0.923880 - 1.38268i) q^{34} +(1.46593 - 2.53906i) q^{36} +(-0.0255190 - 0.389345i) q^{38} +(-0.867580 - 1.75928i) q^{41} +(0.315118 + 1.17604i) q^{43} +(0.513210 + 0.0675653i) q^{44} +(0.258819 + 1.96593i) q^{48} +(-0.130526 - 0.991445i) q^{49} +(-0.707107 + 0.707107i) q^{50} +(3.23407 + 0.643296i) q^{51} +(0.991445 + 3.70012i) q^{54} +(0.581687 + 0.510126i) q^{57} +(-0.0726721 - 1.10876i) q^{59} +(-0.707107 - 0.707107i) q^{64} +(-0.814313 + 0.624844i) q^{66} +(0.172572 + 0.867580i) q^{67} +(-1.49144 + 0.735499i) q^{68} +(-2.32599 - 1.78480i) q^{72} +(1.36603 - 0.366025i) q^{73} -1.98289i q^{75} +(-0.389345 - 0.0255190i) q^{76} +(-4.03906 - 2.33195i) q^{81} +(-1.85747 + 0.630526i) q^{82} +(-1.31587 - 1.50046i) q^{83} +(1.20711 - 0.158919i) q^{86} +(0.133975 - 0.500000i) q^{88} +(-0.607206 + 1.46593i) q^{89} +1.98289 q^{96} +(0.608761 + 0.793353i) q^{97} -1.00000 q^{98} +(0.198092 - 1.50465i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{3} - 8 q^{9}+O(q^{10}) \) 16 * q + 8 * q^3 - 8 * q^9	\( 16 q + 8 q^{3} - 8 q^{9} - 8 q^{12} - 16 q^{27} + 8 q^{36} + 8 q^{66} - 8 q^{68} + 8 q^{73} - 24 q^{81} + 8 q^{86} + 16 q^{88} - 16 q^{98}+O(q^{100}) \) 16 * q + 8 * q^3 - 8 * q^9 - 8 * q^12 - 16 * q^27 + 8 * q^36 + 8 * q^66 - 8 * q^68 + 8 * q^73 - 24 * q^81 + 8 * q^86 + 16 * q^88 - 16 * q^98

Introduction

L-functions

Modular forms

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Database

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