Embedded newform 731.2.n.a.608.48

• Label: 731.2.n.a.608.48
• Level: $731$
• Weight: $2$
• Character: 731.608
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $256$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.n (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(256\)
Relative dimension:	\(64\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			608.48
Character	\(\chi\)	\(=\)	731.608
Dual form			731.2.n.a.208.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+1.62839i q^{2} +(0.474953 - 0.127263i) q^{3} -0.651670 q^{4} +(2.37962 - 0.637618i) q^{5} +(0.207235 + 0.773411i) q^{6} +(4.72094 + 1.26497i) q^{7} +2.19561i q^{8} +(-2.38869 + 1.37911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			1.62839i			1.15145i	0.817644	+	0.575725i	\(0.195280\pi\)
							−0.817644	+	0.575725i	\(0.804720\pi\)
\(3\)	0.474953	−	0.127263i	0.274214	−	0.0734755i	−0.119091	−	0.992883i	\(-0.537998\pi\)
							0.393306	+	0.919408i	\(0.371331\pi\)
\(5\)	2.37962	−	0.637618i	1.06420	−	0.285152i	0.316092	−	0.948729i	\(-0.397629\pi\)
							0.748108	+	0.663577i	\(0.230962\pi\)
\(7\)	4.72094	+	1.26497i	1.78435	+	0.478114i	0.991365	−	0.131131i	\(-0.0418608\pi\)
							0.792982	+	0.609245i	\(0.208527\pi\)
\(11\)	0.797618	−	0.797618i	0.240491	−	0.240491i	−0.576562	−	0.817053i	\(-0.695606\pi\)
							0.817053	+	0.576562i	\(0.195606\pi\)
\(13\)	−2.17279	−	3.76339i	−0.602624	−	1.04378i	−0.992422	−	0.122875i	\(-0.960789\pi\)
							0.389798	−	0.920900i	\(-0.372545\pi\)
\(17\)	−2.22128	−	3.47360i	−0.538740	−	0.842472i
							\(19\)	7.38769	+	4.26529i	1.69485	+	0.978524i	0.950494	+	0.310744i	\(0.100578\pi\)
0.744359	+	0.667780i	\(0.232755\pi\)
\(23\)	−0.586854	−	2.19017i	−0.122368	−	0.456682i	0.877365	−	0.479824i	\(-0.159300\pi\)
							−0.999732	+	0.0231423i	\(0.992633\pi\)
\(29\)	2.33255	−	8.70518i	0.433143	−	1.61651i	−0.312329	−	0.949974i	\(-0.601109\pi\)
							0.745472	−	0.666537i	\(-0.232224\pi\)
\(31\)	−7.40168	+	1.98327i	−1.32938	+	0.356207i	−0.852484	−	0.522753i	\(-0.824905\pi\)
							−0.476897	+	0.878959i	\(0.658239\pi\)
\(37\)	−4.59597	+	1.23149i	−0.755572	+	0.202455i	−0.615988	−	0.787755i	\(-0.711243\pi\)
							−0.139584	+	0.990210i	\(0.544577\pi\)
\(41\)	0.206229	−	0.206229i	0.0322076	−	0.0322076i	−0.690820	−	0.723027i	\(-0.742750\pi\)
							0.723027	+	0.690820i	\(0.242750\pi\)
\(43\)	−6.29611	+	1.83277i	−0.960147	+	0.279494i
							\(47\)	11.0739			1.61529			0.807647	−	0.589666i	\(-0.200741\pi\)
0.807647	+	0.589666i	\(0.200741\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.608.48	yes	256
17.4	even	4	inner	731.2.n.a.565.17	yes	256
43.36	even	3	inner	731.2.n.a.251.48	yes	256
731.208	even	12	inner	731.2.n.a.208.17	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.17	✓	256	731.208	even	12	inner
731.2.n.a.251.48	yes	256	43.36	even	3	inner
731.2.n.a.565.17	yes	256	17.4	even	4	inner
731.2.n.a.608.48	yes	256	1.1	even	1	trivial