Embedded newform 731.2.bd.a.7.10

• Label: 731.2.bd.a.7.10
• Level: $731$
• Weight: $2$
• Character: 731.7
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $1024$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.bd (of order \(48\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(1024\)
Relative dimension:	\(64\) over \(\Q(\zeta_{48})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{48}]$

Embedding invariants

Embedding label			7.10
Character	\(\chi\)	\(=\)	731.7
Dual form			731.2.bd.a.209.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.91170 - 0.791852i) q^{2} +(-0.155821 + 2.37736i) q^{3} +(1.61335 + 1.61335i) q^{4} +(0.654802 + 0.746659i) q^{5} +(2.18040 - 4.42142i) q^{6} +(2.00526 - 2.28656i) q^{7} +(-0.223006 - 0.538385i) q^{8} +(-2.65325 - 0.349307i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q - 24 q^{3} - 32 q^{4} - 24 q^{5} - 8 q^{6} - 24 q^{7} - 8 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{11}{16}\right)\)	\(e\left(\frac{5}{6}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.91170	−	0.791852i	−1.35178	−	0.559924i	−0.414990	−	0.909826i	\(-0.636215\pi\)
							−0.936786	+	0.349902i	\(0.886215\pi\)
\(3\)	−0.155821	+	2.37736i	−0.0899631	+	1.37257i	0.679404	+	0.733765i	\(0.262239\pi\)
							−0.769367	+	0.638807i	\(0.779428\pi\)
\(5\)	0.654802	+	0.746659i	0.292836	+	0.333916i	0.879486	−	0.475925i	\(-0.157887\pi\)
							−0.586649	+	0.809841i	\(0.699553\pi\)
\(7\)	2.00526	−	2.28656i	0.757916	−	0.864238i	−0.236295	−	0.971681i	\(-0.575933\pi\)
							0.994211	+	0.107443i	\(0.0342665\pi\)
\(11\)	−4.97783	+	3.32608i	−1.50087	+	1.00285i	−0.511207	+	0.859458i	\(0.670801\pi\)
							−0.989666	+	0.143393i	\(0.954199\pi\)
\(13\)	0.385204	+	1.43760i	0.106836	+	0.398719i	0.998547	−	0.0538871i	\(-0.0171611\pi\)
							−0.891711	+	0.452606i	\(0.850494\pi\)
\(17\)	−3.88137	+	1.39102i	−0.941372	+	0.337372i
							\(19\)	0.726022	+	5.51468i	0.166561	+	1.26515i	0.846700	+	0.532071i	\(0.178586\pi\)
−0.680139	+	0.733083i	\(0.738081\pi\)
\(23\)	2.66609	−	5.40630i	0.555919	−	1.12729i	−0.419668	−	0.907678i	\(-0.637853\pi\)
							0.975587	−	0.219614i	\(-0.0704799\pi\)
\(29\)	3.64733	+	1.23810i	0.677292	+	0.229910i	0.638827	−	0.769350i	\(-0.279420\pi\)
							0.0384646	+	0.999260i	\(0.487753\pi\)
\(31\)	−0.0171974	+	0.262382i	−0.00308875	+	0.0471252i	−0.999125	−	0.0418223i	\(-0.986684\pi\)
							0.996036	+	0.0889475i	\(0.0283503\pi\)
\(37\)	−0.452384	+	6.90205i	−0.0743715	+	1.13469i	0.783693	+	0.621148i	\(0.213334\pi\)
							−0.858065	+	0.513542i	\(0.828333\pi\)
\(41\)	0.0514023	−	0.258417i	0.00802768	−	0.0403579i	−0.976564	−	0.215228i	\(-0.930951\pi\)
							0.984592	+	0.174870i	\(0.0559505\pi\)
\(43\)	−5.52193	+	3.53671i	−0.842086	+	0.539343i
							\(47\)	−5.88224	+	5.88224i	−0.858013	+	0.858013i	−0.991104	−	0.133091i	\(-0.957510\pi\)
0.133091	+	0.991104i	\(0.457510\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bd.a.7.10	✓	1024
17.5	odd	16	inner	731.2.bd.a.566.55	yes	1024
43.37	odd	6	inner	731.2.bd.a.381.55	yes	1024
731.209	even	48	inner	731.2.bd.a.209.10	yes	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bd.a.7.10	✓	1024	1.1	even	1	trivial
731.2.bd.a.209.10	yes	1024	731.209	even	48	inner
731.2.bd.a.381.55	yes	1024	43.37	odd	6	inner
731.2.bd.a.566.55	yes	1024	17.5	odd	16	inner