Embedded newform 731.2.s.b.214.13

• Label: 731.2.s.b.214.13
• Level: $731$
• Weight: $2$
• Character: 731.214
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.s (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(62\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			214.13
Character	\(\chi\)	\(=\)	731.214
Dual form			731.2.s.b.386.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.67323 - 0.693076i) q^{2} +(-1.35744 + 2.03155i) q^{3} +(0.905142 + 0.905142i) q^{4} +(0.494506 + 2.48605i) q^{5} +(3.67934 - 2.45845i) q^{6} +(0.453283 - 2.27881i) q^{7} +(0.498970 + 1.20462i) q^{8} +(-1.13651 - 2.74379i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{16}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.67323	−	0.693076i	−1.18315	−	0.490079i	−0.297635	−	0.954680i	\(-0.596198\pi\)
							−0.885520	+	0.464601i	\(0.846198\pi\)
\(3\)	−1.35744	+	2.03155i	−0.783719	+	1.17292i	0.197554	+	0.980292i	\(0.436700\pi\)
							−0.981272	+	0.192626i	\(0.938300\pi\)
\(5\)	0.494506	+	2.48605i	0.221150	+	1.11179i	0.918608	+	0.395169i	\(0.129314\pi\)
							−0.697459	+	0.716625i	\(0.745686\pi\)
\(7\)	0.453283	−	2.27881i	0.171325	−	0.861308i	−0.795517	−	0.605932i	\(-0.792800\pi\)
							0.966842	−	0.255377i	\(-0.0821995\pi\)
\(11\)	−1.66538	−	2.49242i	−0.502132	−	0.751493i	0.490659	−	0.871352i	\(-0.336756\pi\)
							−0.992791	+	0.119858i	\(0.961756\pi\)
\(13\)	−1.56255	+	1.56255i	−0.433372	+	0.433372i	−0.889774	−	0.456402i	\(-0.849138\pi\)
							0.456402	+	0.889774i	\(0.349138\pi\)
\(17\)	3.32990	−	2.43142i	0.807619	−	0.589705i
							\(19\)	−6.10359	−	2.52819i	−1.40026	−	0.580007i	−0.450441	−	0.892806i	\(-0.648733\pi\)
−0.949819	+	0.312799i	\(0.898733\pi\)
\(23\)	−0.517143	+	0.345544i	−0.107832	+	0.0720509i	−0.608317	−	0.793694i	\(-0.708155\pi\)
							0.500485	+	0.865745i	\(0.333155\pi\)
\(29\)	−1.46148	+	0.290706i	−0.271389	+	0.0539827i	−0.328909	−	0.944362i	\(-0.606681\pi\)
							0.0575195	+	0.998344i	\(0.481681\pi\)
\(31\)	−4.17085	+	6.24211i	−0.749106	+	1.12112i	0.239545	+	0.970885i	\(0.423002\pi\)
							−0.988651	+	0.150231i	\(0.951998\pi\)
\(37\)	6.27007	−	9.38382i	1.03079	−	1.54269i	0.204964	−	0.978769i	\(-0.434292\pi\)
							0.825829	−	0.563921i	\(-0.190708\pi\)
\(41\)	2.88097	+	0.573060i	0.449931	+	0.0894969i	0.414855	−	0.909888i	\(-0.363832\pi\)
							0.0350768	+	0.999385i	\(0.488832\pi\)
\(43\)	−3.14649	−	5.75323i	−0.479835	−	0.877359i
							\(47\)	4.78734	−	4.78734i	0.698305	−	0.698305i	−0.265740	−	0.964045i	\(-0.585616\pi\)
0.964045	+	0.265740i	\(0.0856162\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.214.13	✓	496
17.12	odd	16	inner	731.2.s.b.386.50	yes	496
43.42	odd	2	inner	731.2.s.b.214.50	yes	496
731.386	even	16	inner	731.2.s.b.386.13	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.13	✓	496	1.1	even	1	trivial
731.2.s.b.214.50	yes	496	43.42	odd	2	inner
731.2.s.b.386.13	yes	496	731.386	even	16	inner
731.2.s.b.386.50	yes	496	17.12	odd	16	inner