Embedded newform 731.2.p.a.16.1

• Label: 731.2.p.a.16.1
• Level: $731$
• Weight: $2$
• Character: 731.16
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $384$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.p (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(384\)
Relative dimension:	\(64\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

Embedding invariants

Embedding label			16.1
Character	\(\chi\)	\(=\)	731.16
Dual form			731.2.p.a.594.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.608972 - 2.66808i) q^{2} +(-0.149961 - 0.0342275i) q^{3} +(-4.94587 + 2.38180i) q^{4} +(-1.75767 + 1.40169i) q^{5} +0.420950i q^{6} -1.31934i q^{7} +(5.95414 + 7.46625i) q^{8} +(-2.68159 - 1.29139i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 384 q - 12 q^{2} - 72 q^{4} - 8 q^{8} + 62 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{7}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.608972	−	2.66808i	−0.430608	−	1.88662i	−0.461582	−	0.887097i	\(-0.652718\pi\)
							0.0309740	−	0.999520i	\(-0.490139\pi\)
\(3\)	−0.149961	−	0.0342275i	−0.0865798	−	0.0197613i	0.179012	−	0.983847i	\(-0.442710\pi\)
							−0.265591	+	0.964086i	\(0.585567\pi\)
\(5\)	−1.75767	+	1.40169i	−0.786054	+	0.626857i	−0.932009	−	0.362434i	\(-0.881946\pi\)
							0.145956	+	0.989291i	\(0.453374\pi\)
\(7\)		−	1.31934i		−	0.498662i	−0.968418	−	0.249331i	\(-0.919789\pi\)
							0.968418	−	0.249331i	\(-0.0802107\pi\)
\(11\)	−2.12009	+	4.40240i	−0.639230	+	1.32737i	0.289699	+	0.957118i	\(0.406445\pi\)
							−0.928929	+	0.370257i	\(0.879270\pi\)
\(13\)	−2.22859	−	2.79457i	−0.618101	−	0.775074i	0.369975	−	0.929042i	\(-0.379366\pi\)
							−0.988076	+	0.153968i	\(0.950795\pi\)
\(17\)	2.79006	+	3.03571i	0.676690	+	0.736268i
							\(19\)	5.01431	−	2.41476i	1.15036	−	0.553985i	0.241220	−	0.970471i	\(-0.422453\pi\)
0.909142	+	0.416486i	\(0.136738\pi\)
\(23\)	3.11298	−	6.46417i	0.649102	−	1.34787i	−0.273408	−	0.961898i	\(-0.588151\pi\)
							0.922509	−	0.385975i	\(-0.126135\pi\)
\(29\)	8.59061	−	1.96075i	1.59524	−	0.364102i	0.669662	−	0.742666i	\(-0.266439\pi\)
							0.925575	+	0.378564i	\(0.123582\pi\)
\(31\)	0.464690	−	0.106062i	0.0834608	−	0.0190494i	−0.180587	−	0.983559i	\(-0.557800\pi\)
							0.264047	+	0.964510i	\(0.414942\pi\)
\(37\)			8.56761i			1.40851i	0.709949	+	0.704253i	\(0.248718\pi\)
							−0.709949	+	0.704253i	\(0.751282\pi\)
\(41\)	2.28890	−	0.522427i	0.357466	−	0.0815893i	−0.0400165	−	0.999199i	\(-0.512741\pi\)
							0.397483	+	0.917610i	\(0.369884\pi\)
\(43\)	−6.36919	+	1.55996i	−0.971292	+	0.237891i
							\(47\)	2.59962	−	1.25191i	0.379193	−	0.182610i	−0.234571	−	0.972099i	\(-0.575369\pi\)
0.613765	+	0.789489i	\(0.289654\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.p.a.16.1	✓	384
17.16	even	2	inner	731.2.p.a.16.2	yes	384
43.35	even	7	inner	731.2.p.a.594.2	yes	384
731.594	even	14	inner	731.2.p.a.594.1	yes	384

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.p.a.16.1	✓	384	1.1	even	1	trivial
731.2.p.a.16.2	yes	384	17.16	even	2	inner
731.2.p.a.594.1	yes	384	731.594	even	14	inner
731.2.p.a.594.2	yes	384	43.35	even	7	inner