Embedded newform 731.2.n.a.208.2

• Label: 731.2.n.a.208.2
• 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			208.2
Character	\(\chi\)	\(=\)	731.208
Dual form			731.2.n.a.608.63

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.67881i q^{2} +(-0.393175 - 0.105351i) q^{3} -5.17600 q^{4} +(3.35902 + 0.900047i) q^{5} +(-0.282214 + 1.05324i) q^{6} +(-1.82716 + 0.489587i) q^{7} +8.50788i q^{8} +(-2.45459 - 1.41716i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	2.67881i		−	1.89420i	−0.320937	−	0.947101i	\(-0.603998\pi\)
							0.320937	−	0.947101i	\(-0.396002\pi\)
\(3\)	−0.393175	−	0.105351i	−0.226999	−	0.0608243i	0.143526	−	0.989646i	\(-0.454156\pi\)
							−0.370526	+	0.928822i	\(0.620822\pi\)
\(5\)	3.35902	+	0.900047i	1.50220	+	0.402513i	0.913836	−	0.406084i	\(-0.133106\pi\)
							0.588363	+	0.808597i	\(0.299773\pi\)
\(7\)	−1.82716	+	0.489587i	−0.690602	+	0.185046i	−0.587017	−	0.809574i	\(-0.699698\pi\)
							−0.103585	+	0.994621i	\(0.533031\pi\)
\(11\)	−2.86515	−	2.86515i	−0.863875	−	0.863875i	0.127911	−	0.991786i	\(-0.459173\pi\)
							−0.991786	+	0.127911i	\(0.959173\pi\)
\(13\)	−2.42672	+	4.20320i	−0.673050	+	1.16576i	0.303985	+	0.952677i	\(0.401683\pi\)
							−0.977035	+	0.213080i	\(0.931651\pi\)
\(17\)	−0.418714	−	4.10179i	−0.101553	−	0.994830i
							\(19\)	−5.21121	+	3.00869i	−1.19553	+	0.690241i	−0.959556	−	0.281518i	\(-0.909162\pi\)
−0.235977	+	0.971759i	\(0.575829\pi\)
\(23\)	−0.191642	+	0.715219i	−0.0399602	+	0.149133i	−0.983023	−	0.183480i	\(-0.941264\pi\)
							0.943063	+	0.332613i	\(0.107930\pi\)
\(29\)	2.53130	+	9.44693i	0.470050	+	1.75425i	0.639578	+	0.768726i	\(0.279109\pi\)
							−0.169528	+	0.985525i	\(0.554224\pi\)
\(31\)	−6.00571	−	1.60922i	−1.07866	−	0.289025i	−0.324611	−	0.945848i	\(-0.605233\pi\)
							−0.754045	+	0.656822i	\(0.771900\pi\)
\(37\)	−9.45652	−	2.53387i	−1.55464	−	0.416565i	−0.623679	−	0.781680i	\(-0.714363\pi\)
							−0.930962	+	0.365115i	\(0.881030\pi\)
\(41\)	4.08647	+	4.08647i	0.638199	+	0.638199i	0.950111	−	0.311912i	\(-0.100969\pi\)
							−0.311912	+	0.950111i	\(0.600969\pi\)
\(43\)	0.144398	−	6.55585i	0.0220205	−	0.999758i
							\(47\)	1.64427			0.239841			0.119921	−	0.992783i	\(-0.461736\pi\)
0.119921	+	0.992783i	\(0.461736\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.208.2	✓	256
17.13	even	4	inner	731.2.n.a.251.63	yes	256
43.6	even	3	inner	731.2.n.a.565.2	yes	256
731.608	even	12	inner	731.2.n.a.608.63	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.2	✓	256	1.1	even	1	trivial
731.2.n.a.251.63	yes	256	17.13	even	4	inner
731.2.n.a.565.2	yes	256	43.6	even	3	inner
731.2.n.a.608.63	yes	256	731.608	even	12	inner