Embedded newform 731.2.n.a.208.17

• Label: 731.2.n.a.208.17
• Level: $731$
• Weight: $2$
• Character: 731.208
• 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			208.17
Character	\(\chi\)	\(=\)	731.208
Dual form			731.2.n.a.608.48

$q$-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 \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	1.62839i		−	1.15145i	−0.817644	−	0.575725i	\(-0.804720\pi\)
							0.817644	−	0.575725i	\(-0.195280\pi\)
\(3\)	0.474953	+	0.127263i	0.274214	+	0.0734755i	0.393306	−	0.919408i	\(-0.371331\pi\)
							−0.119091	+	0.992883i	\(0.537998\pi\)
\(5\)	2.37962	+	0.637618i	1.06420	+	0.285152i	0.748108	−	0.663577i	\(-0.230962\pi\)
							0.316092	+	0.948729i	\(0.397629\pi\)
\(7\)	4.72094	−	1.26497i	1.78435	−	0.478114i	0.792982	−	0.609245i	\(-0.208527\pi\)
							0.991365	+	0.131131i	\(0.0418608\pi\)
\(11\)	0.797618	+	0.797618i	0.240491	+	0.240491i	0.817053	−	0.576562i	\(-0.195606\pi\)
							−0.576562	+	0.817053i	\(0.695606\pi\)
\(13\)	−2.17279	+	3.76339i	−0.602624	+	1.04378i	0.389798	+	0.920900i	\(0.372545\pi\)
							−0.992422	+	0.122875i	\(0.960789\pi\)
\(17\)	−2.22128	+	3.47360i	−0.538740	+	0.842472i
							\(19\)	7.38769	−	4.26529i	1.69485	−	0.978524i	0.744359	−	0.667780i	\(-0.232755\pi\)
0.950494	−	0.310744i	\(-0.100578\pi\)
\(23\)	−0.586854	+	2.19017i	−0.122368	+	0.456682i	−0.999732	−	0.0231423i	\(-0.992633\pi\)
							0.877365	+	0.479824i	\(0.159300\pi\)
\(29\)	2.33255	+	8.70518i	0.433143	+	1.61651i	0.745472	+	0.666537i	\(0.232224\pi\)
							−0.312329	+	0.949974i	\(0.601109\pi\)
\(31\)	−7.40168	−	1.98327i	−1.32938	−	0.356207i	−0.476897	−	0.878959i	\(-0.658239\pi\)
							−0.852484	+	0.522753i	\(0.824905\pi\)
\(37\)	−4.59597	−	1.23149i	−0.755572	−	0.202455i	−0.139584	−	0.990210i	\(-0.544577\pi\)
							−0.615988	+	0.787755i	\(0.711243\pi\)
\(41\)	0.206229	+	0.206229i	0.0322076	+	0.0322076i	0.723027	−	0.690820i	\(-0.242750\pi\)
							−0.690820	+	0.723027i	\(0.742750\pi\)
\(43\)	−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.208.17	✓	256
17.13	even	4	inner	731.2.n.a.251.48	yes	256
43.6	even	3	inner	731.2.n.a.565.17	yes	256
731.608	even	12	inner	731.2.n.a.608.48	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.17	✓	256	1.1	even	1	trivial
731.2.n.a.251.48	yes	256	17.13	even	4	inner
731.2.n.a.565.17	yes	256	43.6	even	3	inner
731.2.n.a.608.48	yes	256	731.608	even	12	inner