Embedded newform 731.2.n.a.608.64

• Label: 731.2.n.a.608.64
• 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			608.64
Character	\(\chi\)	\(=\)	731.608
Dual form			731.2.n.a.208.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+2.76205i q^{2} +(1.78973 - 0.479556i) q^{3} -5.62893 q^{4} +(-4.22937 + 1.13326i) q^{5} +(1.32456 + 4.94332i) q^{6} +(-0.560575 - 0.150206i) q^{7} -10.0233i q^{8} +(0.375075 - 0.216550i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 256 q - 6 q^{3} - 264 q^{4} + 2 q^{5} - 2 q^{6}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)			2.76205i			1.95307i	0.215369	+	0.976533i	\(0.430904\pi\)
							−0.215369	+	0.976533i	\(0.569096\pi\)
\(3\)	1.78973	−	0.479556i	1.03330	−	0.276872i	0.297966	−	0.954577i	\(-0.403692\pi\)
							0.735334	+	0.677705i	\(0.237025\pi\)
\(5\)	−4.22937	+	1.13326i	−1.89143	+	0.506808i	−0.893049	+	0.449959i	\(0.851438\pi\)
							−0.998383	+	0.0568483i	\(0.981895\pi\)
\(7\)	−0.560575	−	0.150206i	−0.211878	−	0.0567724i	0.151319	−	0.988485i	\(-0.451648\pi\)
							−0.363196	+	0.931713i	\(0.618315\pi\)
\(11\)	0.496365	−	0.496365i	0.149660	−	0.149660i	−0.628306	−	0.777966i	\(-0.716252\pi\)
							0.777966	+	0.628306i	\(0.216252\pi\)
\(13\)	1.00946	+	1.74844i	0.279975	+	0.484931i	0.971378	−	0.237538i	\(-0.0763405\pi\)
							−0.691403	+	0.722469i	\(0.743007\pi\)
\(17\)	4.04096	−	0.818933i	0.980076	−	0.198620i
							\(19\)	−2.90471	−	1.67703i	−0.666385	−	0.384738i	0.128320	−	0.991733i	\(-0.459041\pi\)
−0.794706	+	0.606995i	\(0.792375\pi\)
\(23\)	−2.15954	−	8.05951i	−0.450295	−	1.68052i	−0.701564	−	0.712607i	\(-0.747514\pi\)
							0.251269	−	0.967917i	\(-0.419152\pi\)
\(29\)	1.14362	−	4.26803i	0.212364	−	0.792554i	−0.774714	−	0.632312i	\(-0.782106\pi\)
							0.987078	−	0.160242i	\(-0.0512274\pi\)
\(31\)	−6.88080	+	1.84370i	−1.23583	+	0.331139i	−0.816846	−	0.576856i	\(-0.804279\pi\)
							−0.418982	+	0.907995i	\(0.637613\pi\)
\(37\)	−1.64735	+	0.441407i	−0.270823	+	0.0725669i	−0.391675	−	0.920104i	\(-0.628104\pi\)
							0.120852	+	0.992671i	\(0.461438\pi\)
\(41\)	−0.356059	+	0.356059i	−0.0556071	+	0.0556071i	−0.734364	−	0.678756i	\(-0.762519\pi\)
							0.678756	+	0.734364i	\(0.262519\pi\)
\(43\)	−6.31083	+	1.78140i	−0.962393	+	0.271662i
							\(47\)	−4.18728			−0.610778			−0.305389	−	0.952228i	\(-0.598786\pi\)
−0.305389	+	0.952228i	\(0.598786\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.n.a.608.64	yes	256
17.4	even	4	inner	731.2.n.a.565.1	yes	256
43.36	even	3	inner	731.2.n.a.251.64	yes	256
731.208	even	12	inner	731.2.n.a.208.1	✓	256

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.n.a.208.1	✓	256	731.208	even	12	inner
731.2.n.a.251.64	yes	256	43.36	even	3	inner
731.2.n.a.565.1	yes	256	17.4	even	4	inner
731.2.n.a.608.64	yes	256	1.1	even	1	trivial