Embedded newform 731.2.n.a.208.4

• Label: 731.2.n.a.208.4
• 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			208.4
Character	\(\chi\)	\(=\)	731.208
Dual form			731.2.n.a.608.61

$q$-expansion

\(f(q)\)	\(=\)	\(q-2.61030i q^{2} +(2.69586 + 0.722353i) q^{3} -4.81365 q^{4} +(1.96090 + 0.525421i) q^{5} +(1.88556 - 7.03699i) q^{6} +(-1.40725 + 0.377070i) q^{7} +7.34447i q^{8} +(4.14778 + 2.39472i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)		−	2.61030i		−	1.84576i	−0.385089	−	0.922879i	\(-0.625829\pi\)
							0.385089	−	0.922879i	\(-0.374171\pi\)
\(3\)	2.69586	+	0.722353i	1.55645	+	0.417051i	0.931539	−	0.363642i	\(-0.118467\pi\)
							0.624916	+	0.780692i	\(0.285133\pi\)
\(5\)	1.96090	+	0.525421i	0.876941	+	0.234976i	0.669087	−	0.743184i	\(-0.266685\pi\)
							0.207854	+	0.978160i	\(0.433352\pi\)
\(7\)	−1.40725	+	0.377070i	−0.531889	+	0.142519i	−0.514760	−	0.857334i	\(-0.672119\pi\)
							−0.0171285	+	0.999853i	\(0.505452\pi\)
\(11\)	2.06835	+	2.06835i	0.623630	+	0.623630i	0.946458	−	0.322827i	\(-0.104633\pi\)
							−0.322827	+	0.946458i	\(0.604633\pi\)
\(13\)	2.11987	−	3.67171i	0.587945	−	1.01835i	−0.406556	−	0.913626i	\(-0.633271\pi\)
							0.994501	−	0.104725i	\(-0.0333961\pi\)
\(17\)	−3.77722	−	1.65305i	−0.916111	−	0.400924i
							\(19\)	5.27367	−	3.04475i	1.20986	−	0.698515i	0.247133	−	0.968981i	\(-0.420511\pi\)
0.962729	+	0.270467i	\(0.0871781\pi\)
\(23\)	1.69586	−	6.32905i	0.353612	−	1.31970i	−0.528610	−	0.848865i	\(-0.677287\pi\)
							0.882222	−	0.470833i	\(-0.156047\pi\)
\(29\)	2.41210	+	9.00207i	0.447915	+	1.67164i	0.708123	+	0.706089i	\(0.249542\pi\)
							−0.260208	+	0.965553i	\(0.583791\pi\)
\(31\)	−0.304369	−	0.0815554i	−0.0546663	−	0.0146478i	0.231382	−	0.972863i	\(-0.425675\pi\)
							−0.286049	+	0.958215i	\(0.592342\pi\)
\(37\)	6.80362	+	1.82303i	1.11851	+	0.299704i	0.770280	−	0.637706i	\(-0.220116\pi\)
							0.348229	+	0.937409i	\(0.386783\pi\)
\(41\)	−1.58403	−	1.58403i	−0.247384	−	0.247384i	0.572512	−	0.819896i	\(-0.305969\pi\)
							−0.819896	+	0.572512i	\(0.805969\pi\)
\(43\)	−4.57115	+	4.70156i	−0.697093	+	0.716980i
							\(47\)	−2.60945			−0.380628			−0.190314	−	0.981723i	\(-0.560951\pi\)
−0.190314	+	0.981723i	\(0.560951\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.208.4	✓	256
17.13	even	4	inner	731.2.n.a.251.61	yes	256
43.6	even	3	inner	731.2.n.a.565.4	yes	256
731.608	even	12	inner	731.2.n.a.608.61	yes	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.4	✓	256	1.1	even	1	trivial
731.2.n.a.251.61	yes	256	17.13	even	4	inner
731.2.n.a.565.4	yes	256	43.6	even	3	inner
731.2.n.a.608.61	yes	256	731.608	even	12	inner