Embedded newform 731.2.be.a.59.4

• Label: 731.2.be.a.59.4
• 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			59.4
Character	\(\chi\)	\(=\)	731.59
Dual form			731.2.be.a.508.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.11046 - 1.32609i) q^{2} +(2.28054 + 1.61813i) q^{3} +(1.82776 + 3.79537i) q^{4} +(0.228454 + 0.792980i) q^{5} +(-2.66720 - 6.43919i) q^{6} +(0.870869 - 0.360726i) q^{7} +(0.617458 - 5.48010i) q^{8} +(1.59168 + 4.54877i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1536 q - 20 q^{2} - 20 q^{3} - 28 q^{5} - 48 q^{6} - 48 q^{7} - 36 q^{8} - 44 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.11046	−	1.32609i	−1.49232	−	0.937687i	−0.997644	−	0.0686090i	\(-0.978144\pi\)
							−0.494676	−	0.869077i	\(-0.664713\pi\)
\(3\)	2.28054	+	1.61813i	1.31667	+	0.934227i	0.999898	−	0.0142742i	\(-0.00454378\pi\)
							0.316772	+	0.948502i	\(0.397401\pi\)
\(5\)	0.228454	+	0.792980i	0.102168	+	0.354631i	0.995500	−	0.0947658i	\(-0.0302102\pi\)
							−0.893332	+	0.449397i	\(0.851639\pi\)
\(7\)	0.870869	−	0.360726i	0.329157	−	0.136341i	−0.211984	−	0.977273i	\(-0.567992\pi\)
							0.541141	+	0.840932i	\(0.317992\pi\)
\(11\)	0.256688	−	4.57075i	0.0773944	−	1.37813i	−0.683828	−	0.729643i	\(-0.739686\pi\)
							0.761222	−	0.648491i	\(-0.224600\pi\)
\(13\)	5.15897	−	4.11414i	1.43084	−	1.14106i	0.463952	−	0.885860i	\(-0.346431\pi\)
							0.966889	−	0.255198i	\(-0.0821406\pi\)
\(17\)	−1.52118	+	3.83223i	−0.368940	+	0.929453i
							\(19\)	2.58650	−	7.39180i	0.593384	−	1.69579i	−0.118020	−	0.993011i	\(-0.537655\pi\)
0.711404	−	0.702783i	\(-0.248060\pi\)
\(23\)	0.201912	−	3.59537i	0.0421015	−	0.749687i	−0.904340	−	0.426812i	\(-0.859637\pi\)
							0.946442	−	0.322875i	\(-0.104649\pi\)
\(29\)	−0.823030	+	4.84401i	−0.152833	+	0.899510i	0.800887	+	0.598815i	\(0.204361\pi\)
							−0.953720	+	0.300695i	\(0.902781\pi\)
\(31\)	−6.55311	−	1.11342i	−1.17697	−	0.199976i	−0.454097	−	0.890952i	\(-0.650038\pi\)
							−0.722877	+	0.690977i	\(0.757181\pi\)
\(37\)	1.32146	−	3.19028i	0.217246	−	0.524479i	−0.777257	−	0.629183i	\(-0.783390\pi\)
							0.994503	+	0.104704i	\(0.0333896\pi\)
\(41\)	5.12159	+	7.21821i	0.799858	+	1.12729i	0.989371	+	0.145416i	\(0.0464521\pi\)
							−0.189512	+	0.981878i	\(0.560691\pi\)
\(43\)	4.27073	+	4.97603i	0.651281	+	0.758837i
							\(47\)	1.24920	+	2.59400i	0.182215	+	0.378374i	0.971990	−	0.235024i	\(-0.0755167\pi\)
−0.789774	+	0.613397i	\(0.789802\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.be.a.59.4	✓	1536
17.15	even	8	inner	731.2.be.a.661.61	yes	1536
43.35	even	7	inner	731.2.be.a.637.61	yes	1536
731.508	even	56	inner	731.2.be.a.508.4	yes	1536

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.be.a.59.4	✓	1536	1.1	even	1	trivial
731.2.be.a.508.4	yes	1536	731.508	even	56	inner
731.2.be.a.637.61	yes	1536	43.35	even	7	inner
731.2.be.a.661.61	yes	1536	17.15	even	8	inner