Embedded newform 731.2.v.a.36.9

• Label: 731.2.v.a.36.9
• Level: $731$
• Weight: $2$
• Character: 731.36
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $512$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.v (of order \(24\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(512\)
Relative dimension:	\(64\) over \(\Q(\zeta_{24})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{24}]$

Embedding invariants

Embedding label			36.9
Character	\(\chi\)	\(=\)	731.36
Dual form			731.2.v.a.264.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.57826 - 1.57826i) q^{2} +(-2.41848 - 1.85577i) q^{3} +2.98178i q^{4} +(2.30449 - 3.00327i) q^{5} +(0.888110 + 6.74586i) q^{6} +(3.01947 + 3.93505i) q^{7} +(1.54951 - 1.54951i) q^{8} +(1.62873 + 6.07851i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q - 16 q^{2} - 4 q^{3} - 12 q^{5} - 4 q^{6} - 4 q^{7} + 16 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{7}{8}\right)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.57826	−	1.57826i	−1.11600	−	1.11600i	−0.992323	−	0.123673i	\(-0.960533\pi\)
							−0.123673	−	0.992323i	\(-0.539467\pi\)
\(3\)	−2.41848	−	1.85577i	−1.39631	−	1.07143i	−0.987295	−	0.158901i	\(-0.949205\pi\)
							−0.409017	−	0.912527i	\(-0.634128\pi\)
\(5\)	2.30449	−	3.00327i	1.03060	−	1.34311i	0.0928946	−	0.995676i	\(-0.470388\pi\)
							0.937706	−	0.347429i	\(-0.112945\pi\)
\(7\)	3.01947	+	3.93505i	1.14125	+	1.48731i	0.847010	+	0.531577i	\(0.178400\pi\)
							0.294242	+	0.955731i	\(0.404933\pi\)
\(11\)	−1.17529	−	0.486822i	−0.354364	−	0.146782i	0.198398	−	0.980121i	\(-0.436426\pi\)
							−0.552762	+	0.833339i	\(0.686426\pi\)
\(13\)	0.0344993	+	0.0199182i	0.00956839	+	0.00552431i	0.504777	−	0.863250i	\(-0.331575\pi\)
							−0.495208	+	0.868774i	\(0.664908\pi\)
\(17\)	2.61148	+	3.19064i	0.633377	+	0.773843i
							\(19\)	0.860369	−	3.21094i	0.197382	−	0.736640i	−0.794255	−	0.607584i	\(-0.792139\pi\)
0.991637	−	0.129056i	\(-0.0411947\pi\)
\(23\)	−0.511129	−	3.88241i	−0.106578	−	0.809538i	−0.958344	−	0.285616i	\(-0.907802\pi\)
							0.851766	−	0.523922i	\(-0.175532\pi\)
\(29\)	7.04316	+	0.927250i	1.30788	+	0.172186i	0.752048	−	0.659108i	\(-0.229066\pi\)
							0.555833	+	0.831294i	\(0.312399\pi\)
\(31\)	−6.61031	−	5.07227i	−1.18725	−	0.911006i	−0.189757	−	0.981831i	\(-0.560770\pi\)
							−0.997489	+	0.0708251i	\(0.977437\pi\)
\(37\)	−0.474970	−	0.364457i	−0.0780845	−	0.0599164i	0.568982	−	0.822350i	\(-0.307338\pi\)
							−0.647066	+	0.762434i	\(0.724004\pi\)
\(41\)	3.30350	−	7.97535i	0.515920	−	1.24554i	−0.424470	−	0.905442i	\(-0.639539\pi\)
							0.940390	−	0.340099i	\(-0.110461\pi\)
\(43\)	−6.55379	+	0.218814i	−0.999443	+	0.0333688i
							\(47\)			0.806211i			0.117598i	0.998270	+	0.0587990i	\(0.0187271\pi\)
−0.998270	+	0.0587990i	\(0.981273\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.v.a.36.9	✓	512
17.9	even	8	inner	731.2.v.a.638.9	yes	512
43.6	even	3	inner	731.2.v.a.393.9	yes	512
731.264	even	24	inner	731.2.v.a.264.9	yes	512

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.v.a.36.9	✓	512	1.1	even	1	trivial
731.2.v.a.264.9	yes	512	731.264	even	24	inner
731.2.v.a.393.9	yes	512	43.6	even	3	inner
731.2.v.a.638.9	yes	512	17.9	even	8	inner