Embedded newform 731.2.v.a.638.9

• Label: 731.2.v.a.638.9
• Level: $731$
• Weight: $2$
• Character: 731.638
• 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			638.9
Character	\(\chi\)	\(=\)	731.638
Dual form			731.2.v.a.393.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.57826 + 1.57826i) q^{2} +(-0.397900 - 3.02235i) q^{3} -2.98178i q^{4} +(-3.75316 + 0.494113i) q^{5} +(5.39803 + 4.14206i) q^{6} +(-4.91759 - 0.647412i) q^{7} +(1.54951 + 1.54951i) q^{8} +(-6.07851 + 1.62873i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 512 q - 16 q^{2} - 4 q^{3} - 12 q^{5} - 4 q^{6} - 4 q^{7} + 16 q^{8} - 28 q^{9}+O(q^{10}) \)

Character values

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

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

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	−0.123673	+	0.992323i	\(0.539467\pi\)
							−0.992323	+	0.123673i	\(0.960533\pi\)
\(3\)	−0.397900	−	3.02235i	−0.229728	−	1.74496i	−0.585763	−	0.810483i	\(-0.699205\pi\)
							0.356035	−	0.934473i	\(-0.384128\pi\)
\(5\)	−3.75316	+	0.494113i	−1.67846	+	0.220974i	−0.908728	−	0.417389i	\(-0.862945\pi\)
							−0.769736	+	0.638363i	\(0.779612\pi\)
\(7\)	−4.91759	−	0.647412i	−1.85867	−	0.244699i	−0.883864	−	0.467743i	\(-0.845067\pi\)
							−0.974808	+	0.223045i	\(0.928400\pi\)
\(11\)	−1.17529	+	0.486822i	−0.354364	+	0.146782i	−0.552762	−	0.833339i	\(-0.686426\pi\)
							0.198398	+	0.980121i	\(0.436426\pi\)
\(13\)	−0.0344993	−	0.0199182i	−0.00956839	−	0.00552431i	0.495208	−	0.868774i	\(-0.335092\pi\)
							−0.504777	+	0.863250i	\(0.668425\pi\)
\(17\)	−4.06891	−	0.666290i	−0.986856	−	0.161599i
							\(19\)	−3.21094	−	0.860369i	−0.736640	−	0.197382i	−0.129056	−	0.991637i	\(-0.541195\pi\)
−0.607584	+	0.794255i	\(0.707861\pi\)
\(23\)	−3.10670	−	2.38385i	−0.647792	−	0.497068i	0.231821	−	0.972758i	\(-0.425532\pi\)
							−0.879613	+	0.475690i	\(0.842198\pi\)
\(29\)	−4.32460	−	5.63593i	−0.803058	−	1.04657i	−0.997838	−	0.0657190i	\(-0.979066\pi\)
							0.194780	−	0.980847i	\(-0.437601\pi\)
\(31\)	−1.08756	−	8.26083i	−0.195331	−	1.48369i	−0.755412	−	0.655250i	\(-0.772563\pi\)
							0.560081	−	0.828438i	\(-0.310770\pi\)
\(37\)	−0.0781442	−	0.593564i	−0.0128468	−	0.0975814i	0.983820	−	0.179159i	\(-0.0573376\pi\)
							−0.996667	+	0.0815774i	\(0.974004\pi\)
\(41\)	3.30350	+	7.97535i	0.515920	+	1.24554i	0.940390	+	0.340099i	\(0.110461\pi\)
							−0.424470	+	0.905442i	\(0.639539\pi\)
\(43\)	−0.218814	−	6.55379i	−0.0333688	−	0.999443i
							\(47\)		−	0.806211i		−	0.117598i	−0.998270	−	0.0587990i	\(-0.981273\pi\)
0.998270	−	0.0587990i	\(-0.0187271\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.v.a.36.9	✓	512	17.2	even	8	inner
731.2.v.a.264.9	yes	512	43.6	even	3	inner
731.2.v.a.393.9	yes	512	731.393	even	24	inner
731.2.v.a.638.9	yes	512	1.1	even	1	trivial