Embedded newform 731.2.y.a.4.14

• Label: 731.2.y.a.4.14
• Level: $731$
• Weight: $2$
• Character: 731.4
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $768$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.y (of order \(28\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(64\) over \(\Q(\zeta_{28})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{28}]$

Embedding invariants

Embedding label			4.14
Character	\(\chi\)	\(=\)	731.4
Dual form			731.2.y.a.183.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.48420 - 1.18361i) q^{2} +(0.454616 + 0.0512230i) q^{3} +(0.356876 + 1.56358i) q^{4} +(-2.90786 + 1.01751i) q^{5} +(-0.614114 - 0.614114i) q^{6} +(-3.31960 - 3.31960i) q^{7} +(-0.326349 + 0.677670i) q^{8} +(-2.72073 - 0.620989i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 768 q - 14 q^{3} + 104 q^{4} - 6 q^{5} - 36 q^{6} - 28 q^{7}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{2}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.48420	−	1.18361i	−1.04949	−	0.836939i	−0.0625509	−	0.998042i	\(-0.519924\pi\)
							−0.986938	+	0.161103i	\(0.948495\pi\)
\(3\)	0.454616	+	0.0512230i	0.262473	+	0.0295736i	0.242220	−	0.970221i	\(-0.422124\pi\)
							0.0202527	+	0.999795i	\(0.493553\pi\)
\(5\)	−2.90786	+	1.01751i	−1.30044	+	0.455042i	−0.889631	−	0.456681i	\(-0.849038\pi\)
							−0.410806	+	0.911723i	\(0.634752\pi\)
\(7\)	−3.31960	−	3.31960i	−1.25469	−	1.25469i	−0.953593	−	0.301098i	\(-0.902647\pi\)
							−0.301098	−	0.953593i	\(-0.597353\pi\)
\(11\)	−4.14287	+	2.60314i	−1.24912	+	0.784876i	−0.983798	−	0.179283i	\(-0.942622\pi\)
							−0.265325	+	0.964159i	\(0.585479\pi\)
\(13\)	5.50008	+	2.64870i	1.52545	+	0.734617i	0.993678	−	0.112271i	\(-0.0358123\pi\)
							0.531771	+	0.846888i	\(0.321527\pi\)
\(17\)	−1.70523	−	3.75396i	−0.413580	−	0.910468i
							\(19\)	2.70823	−	0.618135i	0.621310	−	0.141810i	0.0997332	−	0.995014i	\(-0.468201\pi\)
0.521577	+	0.853204i	\(0.325344\pi\)
\(23\)	4.42218	−	2.77864i	0.922088	−	0.579386i	0.0147973	−	0.999891i	\(-0.495290\pi\)
							0.907291	+	0.420504i	\(0.138147\pi\)
\(29\)	0.180546	+	1.60239i	0.0335266	+	0.297557i	0.999275	+	0.0380759i	\(0.0121229\pi\)
							−0.965748	+	0.259481i	\(0.916449\pi\)
\(31\)	−0.0846010	−	0.750855i	−0.0151948	−	0.134857i	0.983824	−	0.179135i	\(-0.0573299\pi\)
							−0.999019	+	0.0442777i	\(0.985901\pi\)
\(37\)	0.385583	−	0.385583i	0.0633895	−	0.0633895i	−0.674701	−	0.738091i	\(-0.735728\pi\)
							0.738091	+	0.674701i	\(0.235728\pi\)
\(41\)	3.78531	−	0.426501i	0.591165	−	0.0666083i	0.188686	−	0.982037i	\(-0.439577\pi\)
							0.402479	+	0.915429i	\(0.368149\pi\)
\(43\)	5.14800	+	4.06179i	0.785062	+	0.619417i
							\(47\)	2.66094	+	11.6584i	0.388139	+	1.70055i	0.671060	+	0.741403i	\(0.265839\pi\)
−0.282921	+	0.959143i	\(0.591304\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.y.a.4.14	✓	768
17.13	even	4	inner	731.2.y.a.47.51	yes	768
43.11	even	7	inner	731.2.y.a.140.51	yes	768
731.183	even	28	inner	731.2.y.a.183.14	yes	768

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.y.a.4.14	✓	768	1.1	even	1	trivial
731.2.y.a.47.51	yes	768	17.13	even	4	inner
731.2.y.a.140.51	yes	768	43.11	even	7	inner
731.2.y.a.183.14	yes	768	731.183	even	28	inner