Embedded newform 731.2.bd.a.7.11

• Label: 731.2.bd.a.7.11
• 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			7.11
Character	\(\chi\)	\(=\)	731.7
Dual form			731.2.bd.a.209.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.86110 - 0.770892i) q^{2} +(0.138846 - 2.11838i) q^{3} +(1.45520 + 1.45520i) q^{4} +(2.61856 + 2.98590i) q^{5} +(-1.89145 + 3.83547i) q^{6} +(-0.831524 + 0.948172i) q^{7} +(-0.0446788 - 0.107864i) q^{8} +(-1.49391 - 0.196677i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q - 24 q^{3} - 32 q^{4} - 24 q^{5} - 8 q^{6} - 24 q^{7} - 8 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.86110	−	0.770892i	−1.31599	−	0.545103i	−0.389367	−	0.921083i	\(-0.627306\pi\)
							−0.926628	+	0.375980i	\(0.877306\pi\)
\(3\)	0.138846	−	2.11838i	0.0801627	−	1.22305i	−0.748437	−	0.663205i	\(-0.769196\pi\)
							0.828600	−	0.559841i	\(-0.189138\pi\)
\(5\)	2.61856	+	2.98590i	1.17106	+	1.33534i	0.931192	+	0.364529i	\(0.118770\pi\)
							0.239866	+	0.970806i	\(0.422897\pi\)
\(7\)	−0.831524	+	0.948172i	−0.314287	+	0.358375i	−0.887318	−	0.461158i	\(-0.847434\pi\)
							0.573031	+	0.819534i	\(0.305767\pi\)
\(11\)	−1.52602	+	1.01965i	−0.460113	+	0.307437i	−0.763937	−	0.645291i	\(-0.776736\pi\)
							0.303825	+	0.952728i	\(0.401736\pi\)
\(13\)	−1.03067	−	3.84652i	−0.285857	−	1.06683i	−0.948211	−	0.317642i	\(-0.897109\pi\)
							0.662354	−	0.749191i	\(-0.269558\pi\)
\(17\)	−2.81582	+	3.01184i	−0.682937	+	0.730477i
							\(19\)	0.408852	+	3.10554i	0.0937971	+	0.712460i	0.972135	+	0.234420i	\(0.0753191\pi\)
−0.878338	+	0.478040i	\(0.841348\pi\)
\(23\)	−2.82630	+	5.73116i	−0.589324	+	1.19503i	0.374185	+	0.927354i	\(0.377922\pi\)
							−0.963509	+	0.267676i	\(0.913744\pi\)
\(29\)	8.56007	+	2.90575i	1.58957	+	0.539585i	0.969527	−	0.244986i	\(-0.0787833\pi\)
							0.620039	+	0.784571i	\(0.287117\pi\)
\(31\)	−0.377202	+	5.75500i	−0.0677476	+	1.03363i	0.819722	+	0.572762i	\(0.194128\pi\)
							−0.887470	+	0.460866i	\(0.847539\pi\)
\(37\)	−0.369974	+	5.64471i	−0.0608233	+	0.927985i	0.853010	+	0.521895i	\(0.174775\pi\)
							−0.913833	+	0.406090i	\(0.866892\pi\)
\(41\)	1.47383	−	7.40944i	0.230173	−	1.15716i	−0.676863	−	0.736109i	\(-0.736661\pi\)
							0.907037	−	0.421051i	\(-0.138339\pi\)
\(43\)	4.25317	−	4.99105i	0.648602	−	0.761128i
							\(47\)	−7.09391	+	7.09391i	−1.03475	+	1.03475i	−0.0353785	+	0.999374i	\(0.511264\pi\)
−0.999374	+	0.0353785i	\(0.988736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.bd.a.7.11	✓	1024
17.5	odd	16	inner	731.2.bd.a.566.54	yes	1024
43.37	odd	6	inner	731.2.bd.a.381.54	yes	1024
731.209	even	48	inner	731.2.bd.a.209.11	yes	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.bd.a.7.11	✓	1024	1.1	even	1	trivial
731.2.bd.a.209.11	yes	1024	731.209	even	48	inner
731.2.bd.a.381.54	yes	1024	43.37	odd	6	inner
731.2.bd.a.566.54	yes	1024	17.5	odd	16	inner