Embedded newform 731.2.be.a.59.17

• Label: 731.2.be.a.59.17
• Level: $731$
• Weight: $2$
• Character: 731.59
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $1536$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.be (of order \(56\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(1536\)
Relative dimension:	\(64\) over \(\Q(\zeta_{56})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{56}]$

Embedding invariants

Embedding label			59.17
Character	\(\chi\)	\(=\)	731.59
Dual form			731.2.be.a.508.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.34805 - 0.847036i) q^{2} +(2.65624 + 1.88470i) q^{3} +(0.232002 + 0.481757i) q^{4} +(-0.0257895 - 0.0895172i) q^{5} +(-1.98433 - 4.79060i) q^{6} +(-1.60590 + 0.665187i) q^{7} +(-0.261197 + 2.31819i) q^{8} +(2.51266 + 7.18077i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 20 q^{2} - 20 q^{3} - 28 q^{5} - 48 q^{6} - 48 q^{7} - 36 q^{8} - 44 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{5}{8}\right)\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.34805	−	0.847036i	−0.953216	−	0.598945i	−0.0368643	−	0.999320i	\(-0.511737\pi\)
							−0.916351	+	0.400375i	\(0.868880\pi\)
\(3\)	2.65624	+	1.88470i	1.53358	+	1.08813i	0.963900	+	0.266265i	\(0.0857897\pi\)
							0.569679	+	0.821867i	\(0.307067\pi\)
\(5\)	−0.0257895	−	0.0895172i	−0.0115334	−	0.0400333i	0.954942	−	0.296792i	\(-0.0959168\pi\)
							−0.966476	+	0.256759i	\(0.917345\pi\)
\(7\)	−1.60590	+	0.665187i	−0.606975	+	0.251417i	−0.664934	−	0.746902i	\(-0.731541\pi\)
							0.0579597	+	0.998319i	\(0.481541\pi\)
\(11\)	0.140666	−	2.50479i	0.0424124	−	0.755223i	−0.903059	−	0.429517i	\(-0.858684\pi\)
							0.945471	−	0.325706i	\(-0.105602\pi\)
\(13\)	−2.54211	+	2.02727i	−0.705055	+	0.562263i	−0.909039	−	0.416712i	\(-0.863183\pi\)
							0.203984	+	0.978974i	\(0.434611\pi\)
\(17\)	−3.25774	+	2.52727i	−0.790119	+	0.612954i
							\(19\)	−2.09867	+	5.99765i	−0.481468	+	1.37596i	0.405487	+	0.914101i	\(0.367102\pi\)
−0.886955	+	0.461856i	\(0.847184\pi\)
\(23\)	−0.353882	+	6.30146i	−0.0737896	+	1.31395i	0.714932	+	0.699194i	\(0.246458\pi\)
							−0.788721	+	0.614751i	\(0.789257\pi\)
\(29\)	1.41065	−	8.30250i	0.261951	−	1.54174i	−0.483601	−	0.875288i	\(-0.660671\pi\)
							0.745553	−	0.666447i	\(-0.232186\pi\)
\(31\)	3.53640	+	0.600858i	0.635156	+	0.107917i	0.476155	−	0.879361i	\(-0.342030\pi\)
							0.159000	+	0.987279i	\(0.449173\pi\)
\(37\)	−3.07536	+	7.42458i	−0.505586	+	1.22059i	0.440815	+	0.897598i	\(0.354690\pi\)
							−0.946401	+	0.322995i	\(0.895310\pi\)
\(41\)	2.77323	+	3.90851i	0.433106	+	0.610406i	0.972854	−	0.231421i	\(-0.0743374\pi\)
							−0.539747	+	0.841827i	\(0.681480\pi\)
\(43\)	−0.407682	−	6.54475i	−0.0621710	−	0.998066i
							\(47\)	1.45931	+	3.03029i	0.212862	+	0.442013i	0.979873	−	0.199621i	\(-0.0639711\pi\)
−0.767011	+	0.641634i	\(0.778257\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.be.a.59.17	✓	1536
17.15	even	8	inner	731.2.be.a.661.48	yes	1536
43.35	even	7	inner	731.2.be.a.637.48	yes	1536
731.508	even	56	inner	731.2.be.a.508.17	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.be.a.59.17	✓	1536	1.1	even	1	trivial
731.2.be.a.508.17	yes	1536	731.508	even	56	inner
731.2.be.a.637.48	yes	1536	43.35	even	7	inner
731.2.be.a.661.48	yes	1536	17.15	even	8	inner