Embedded newform 731.2.u.a.52.12

• Label: 731.2.u.a.52.12
• 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			52.12
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.a.239.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.250802 - 0.314495i) q^{2} +(-2.19641 - 0.331055i) q^{3} +(0.409036 - 1.79210i) q^{4} +(-1.91395 - 1.30491i) q^{5} +(0.446747 + 0.773789i) q^{6} +(-0.720517 + 1.24797i) q^{7} +(-1.39103 + 0.669886i) q^{8} +(1.84788 + 0.569996i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 2 q^{2} + 5 q^{3} - 54 q^{4} + 3 q^{5} + 8 q^{6} - 71 q^{7} - 26 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.250802	−	0.314495i	−0.177344	−	0.222382i	0.685213	−	0.728343i	\(-0.259709\pi\)
							−0.862556	+	0.505961i	\(0.831138\pi\)
\(3\)	−2.19641	−	0.331055i	−1.26810	−	0.191135i	−0.519678	−	0.854362i	\(-0.673948\pi\)
							−0.748418	+	0.663228i	\(0.769186\pi\)
\(5\)	−1.91395	−	1.30491i	−0.855943	−	0.583572i	0.0538899	−	0.998547i	\(-0.482838\pi\)
							−0.909833	+	0.414975i	\(0.863790\pi\)
\(7\)	−0.720517	+	1.24797i	−0.272330	+	0.471689i	−0.969458	−	0.245257i	\(-0.921128\pi\)
							0.697128	+	0.716947i	\(0.254461\pi\)
\(11\)	−0.588643	−	2.57901i	−0.177482	−	0.777602i	−0.982787	−	0.184741i	\(-0.940855\pi\)
							0.805305	−	0.592861i	\(-0.202002\pi\)
\(13\)	0.342305	−	4.56774i	0.0949383	−	1.26686i	−0.722713	−	0.691149i	\(-0.757105\pi\)
							0.817651	−	0.575714i	\(-0.195276\pi\)
\(17\)	0.826239	−	0.563320i	0.200392	−	0.136625i
							\(19\)	−7.46738	+	2.30338i	−1.71314	+	0.528432i	−0.987222	−	0.159354i	\(-0.949059\pi\)
−0.725914	+	0.687786i	\(0.758583\pi\)
\(23\)	−0.537497	+	0.498725i	−0.112076	+	0.103991i	−0.734220	−	0.678911i	\(-0.762452\pi\)
							0.622144	+	0.782902i	\(0.286262\pi\)
\(29\)	5.90432	−	0.889934i	1.09641	−	0.165257i	0.424162	−	0.905586i	\(-0.360569\pi\)
							0.672244	+	0.740330i	\(0.265331\pi\)
\(31\)	−2.36672	+	6.03030i	−0.425075	+	1.08307i	0.544604	+	0.838693i	\(0.316680\pi\)
							−0.969679	+	0.244381i	\(0.921415\pi\)
\(37\)	5.49593	+	9.51923i	0.903525	+	1.56495i	0.822885	+	0.568208i	\(0.192363\pi\)
							0.0806396	+	0.996743i	\(0.474304\pi\)
\(41\)	−1.34811	−	1.69048i	−0.210540	−	0.264008i	0.665337	−	0.746543i	\(-0.268288\pi\)
							−0.875877	+	0.482535i	\(0.839716\pi\)
\(43\)	5.81073	−	3.03898i	0.886128	−	0.463440i
							\(47\)	1.27965	−	5.60653i	0.186657	−	0.817796i	−0.791707	−	0.610902i	\(-0.790807\pi\)
0.978363	−	0.206895i	\(-0.0663358\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.a.52.12	✓	348
43.24	even	21	inner	731.2.u.a.239.12	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.a.52.12	✓	348	1.1	even	1	trivial
731.2.u.a.239.12	yes	348	43.24	even	21	inner