Embedded newform 731.2.u.a.52.4

• Label: 731.2.u.a.52.4
• Level: $731$
• Weight: $2$
• Character: 731.52
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $348$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.u (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(348\)
Relative dimension:	\(29\) over \(\Q(\zeta_{21})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{21}]$

Embedding invariants

Embedding label			52.4
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.a.239.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.45147 - 1.82009i) q^{2} +(1.04795 + 0.157954i) q^{3} +(-0.760906 + 3.33375i) q^{4} +(0.648967 + 0.442458i) q^{5} +(-1.23359 - 2.13663i) q^{6} +(0.330221 - 0.571959i) q^{7} +(2.97727 - 1.43378i) q^{8} +(-1.79346 - 0.553209i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 2 q^{2} + 5 q^{3} - 54 q^{4} + 3 q^{5} + 8 q^{6} - 71 q^{7} - 26 q^{8} + 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{21}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.45147	−	1.82009i	−1.02634	−	1.28700i	−0.957212	−	0.289388i	\(-0.906548\pi\)
							−0.0691329	−	0.997607i	\(-0.522023\pi\)
\(3\)	1.04795	+	0.157954i	0.605037	+	0.0911946i	0.444414	−	0.895821i	\(-0.353412\pi\)
							0.160622	+	0.987016i	\(0.448650\pi\)
\(5\)	0.648967	+	0.442458i	0.290227	+	0.197873i	0.699677	−	0.714459i	\(-0.253327\pi\)
							−0.409450	+	0.912332i	\(0.634280\pi\)
\(7\)	0.330221	−	0.571959i	0.124812	−	0.216180i	−0.796848	−	0.604180i	\(-0.793501\pi\)
							0.921659	+	0.388000i	\(0.126834\pi\)
\(11\)	−0.424058	−	1.85792i	−0.127858	−	0.560184i	−0.997756	−	0.0669521i	\(-0.978673\pi\)
							0.869898	−	0.493232i	\(-0.164185\pi\)
\(13\)	0.110419	−	1.47344i	0.0306247	−	0.408659i	−0.960733	−	0.277474i	\(-0.910503\pi\)
							0.991358	−	0.131185i	\(-0.0418782\pi\)
\(17\)	0.826239	−	0.563320i	0.200392	−	0.136625i
							\(19\)	−1.17211	+	0.361549i	−0.268901	+	0.0829451i	−0.426274	−	0.904594i	\(-0.640174\pi\)
0.157372	+	0.987539i	\(0.449698\pi\)
\(23\)	2.99022	−	2.77452i	0.623504	−	0.578527i	−0.303847	−	0.952721i	\(-0.598271\pi\)
							0.927350	+	0.374194i	\(0.122081\pi\)
\(29\)	4.88281	−	0.735965i	0.906715	−	0.136665i	0.320903	−	0.947112i	\(-0.396014\pi\)
							0.585812	+	0.810447i	\(0.300776\pi\)
\(31\)	0.852335	−	2.17171i	0.153084	−	0.390051i	−0.833673	−	0.552259i	\(-0.813766\pi\)
							0.986757	+	0.162207i	\(0.0518613\pi\)
\(37\)	0.542002	+	0.938776i	0.0891047	+	0.154334i	0.907133	−	0.420844i	\(-0.138266\pi\)
							−0.818028	+	0.575178i	\(0.804933\pi\)
\(41\)	−5.02074	−	6.29581i	−0.784108	−	0.983241i	−0.999977	−	0.00681824i	\(-0.997830\pi\)
							0.215868	−	0.976422i	\(-0.430742\pi\)
\(43\)	−5.74852	−	3.15508i	−0.876641	−	0.481145i
							\(47\)	2.34546	−	10.2761i	0.342120	−	1.49893i	−0.452470	−	0.891779i	\(-0.649457\pi\)
0.794590	−	0.607146i	\(-0.207686\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.a.52.4	✓	348
43.24	even	21	inner	731.2.u.a.239.4	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.a.52.4	✓	348	1.1	even	1	trivial
731.2.u.a.239.4	yes	348	43.24	even	21	inner