Embedded newform 731.2.u.a.52.11

• Label: 731.2.u.a.52.11
• Level: $731$
• Weight: $2$
• Character: 731.52
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $348$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.u (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(348\)
Relative dimension:	\(29\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			52.11
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.a.239.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.592982 - 0.743576i) q^{2} +(-3.02986 - 0.456678i) q^{3} +(0.243764 - 1.06800i) q^{4} +(2.13428 + 1.45513i) q^{5} +(1.45708 + 2.52373i) q^{6} +(-2.14683 + 3.71842i) q^{7} +(-2.65246 + 1.27736i) q^{8} +(6.10478 + 1.88307i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 2 q^{2} + 5 q^{3} - 54 q^{4} + 3 q^{5} + 8 q^{6} - 71 q^{7} - 26 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.592982	−	0.743576i	−0.419302	−	0.525788i	0.526656	−	0.850079i	\(-0.323446\pi\)
							−0.945957	+	0.324291i	\(0.894874\pi\)
\(3\)	−3.02986	−	0.456678i	−1.74929	−	0.263663i	−0.805118	−	0.593115i	\(-0.797898\pi\)
							−0.944173	+	0.329451i	\(0.893136\pi\)
\(5\)	2.13428	+	1.45513i	0.954478	+	0.650752i	0.937087	−	0.349097i	\(-0.113512\pi\)
							0.0173910	+	0.999849i	\(0.494464\pi\)
\(7\)	−2.14683	+	3.71842i	−0.811426	+	1.40543i	0.100440	+	0.994943i	\(0.467975\pi\)
							−0.911866	+	0.410488i	\(0.865358\pi\)
\(11\)	−1.27472	−	5.58492i	−0.384343	−	1.68392i	−0.683690	−	0.729773i	\(-0.739626\pi\)
							0.299347	−	0.954144i	\(-0.403231\pi\)
\(13\)	−0.296907	+	3.96194i	−0.0823471	+	1.09885i	0.791022	+	0.611787i	\(0.209549\pi\)
							−0.873370	+	0.487058i	\(0.838070\pi\)
\(17\)	0.826239	−	0.563320i	0.200392	−	0.136625i
							\(19\)	0.842305	−	0.259817i	0.193238	−	0.0596060i	−0.196626	−	0.980479i	\(-0.562998\pi\)
0.389864	+	0.920873i	\(0.372522\pi\)
\(23\)	5.90025	−	5.47463i	1.23029	−	1.14154i	0.245202	−	0.969472i	\(-0.421146\pi\)
							0.985085	−	0.172068i	\(-0.0550448\pi\)
\(29\)	−1.34974	+	0.203441i	−0.250640	+	0.0377780i	−0.273161	−	0.961968i	\(-0.588069\pi\)
							0.0225203	+	0.999746i	\(0.492831\pi\)
\(31\)	0.353295	−	0.900181i	0.0634537	−	0.161677i	−0.895630	−	0.444799i	\(-0.853275\pi\)
							0.959084	+	0.283122i	\(0.0913702\pi\)
\(37\)	−5.01764	−	8.69080i	−0.824894	−	1.42876i	−0.902000	−	0.431736i	\(-0.857901\pi\)
							0.0771055	−	0.997023i	\(-0.475432\pi\)
\(41\)	−2.43898	−	3.05839i	−0.380905	−	0.477640i	0.554010	−	0.832510i	\(-0.313097\pi\)
							−0.934916	+	0.354869i	\(0.884525\pi\)
\(43\)	−6.34130	−	1.66970i	−0.967039	−	0.254627i
							\(47\)	0.768088	−	3.36521i	0.112037	−	0.490867i	−0.887510	−	0.460788i	\(-0.847567\pi\)
0.999548	−	0.0300792i	\(-0.00957594\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.a.52.11	✓	348
43.24	even	21	inner	731.2.u.a.239.11	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.a.52.11	✓	348	1.1	even	1	trivial
731.2.u.a.239.11	yes	348	43.24	even	21	inner