Embedded newform 731.2.u.a.52.2

• Label: 731.2.u.a.52.2
• 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			52.2
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.a.239.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62919 - 2.04294i) q^{2} +(-1.11228 - 0.167649i) q^{3} +(-1.07429 + 4.70679i) q^{4} +(3.54342 + 2.41587i) q^{5} +(1.46961 + 2.54545i) q^{6} +(1.14433 - 1.98204i) q^{7} +(6.65741 - 3.20604i) q^{8} +(-1.65766 - 0.511320i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 2 q^{2} + 5 q^{3} - 54 q^{4} + 3 q^{5} + 8 q^{6} - 71 q^{7} - 26 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62919	−	2.04294i	−1.15201	−	1.44457i	−0.875274	−	0.483627i	\(-0.839319\pi\)
							−0.276735	−	0.960946i	\(-0.589252\pi\)
\(3\)	−1.11228	−	0.167649i	−0.642175	−	0.0967923i	−0.180121	−	0.983644i	\(-0.557649\pi\)
							−0.462053	+	0.886852i	\(0.652887\pi\)
\(5\)	3.54342	+	2.41587i	1.58467	+	1.08041i	0.952049	+	0.305944i	\(0.0989721\pi\)
							0.632618	+	0.774464i	\(0.281980\pi\)
\(7\)	1.14433	−	1.98204i	0.432517	−	0.749141i	−0.564572	−	0.825384i	\(-0.690959\pi\)
							0.997089	+	0.0762422i	\(0.0242922\pi\)
\(11\)	−0.186428	−	0.816795i	−0.0562102	−	0.246273i	0.939015	−	0.343875i	\(-0.111740\pi\)
							−0.995225	+	0.0976025i	\(0.968883\pi\)
\(13\)	−0.369869	+	4.93556i	−0.102583	+	1.36888i	0.673939	+	0.738787i	\(0.264601\pi\)
							−0.776522	+	0.630090i	\(0.783018\pi\)
\(17\)	0.826239	−	0.563320i	0.200392	−	0.136625i
							\(19\)	−0.872745	+	0.269206i	−0.200221	+	0.0617601i	−0.393244	−	0.919434i	\(-0.628647\pi\)
0.193022	+	0.981194i	\(0.438171\pi\)
\(23\)	4.36607	−	4.05112i	0.910388	−	0.844716i	−0.0779558	−	0.996957i	\(-0.524839\pi\)
							0.988343	+	0.152240i	\(0.0486488\pi\)
\(29\)	−3.50772	+	0.528704i	−0.651368	+	0.0981779i	−0.466410	−	0.884569i	\(-0.654453\pi\)
							−0.184958	+	0.982746i	\(0.559215\pi\)
\(31\)	0.653233	−	1.66441i	0.117324	−	0.298937i	−0.860133	−	0.510069i	\(-0.829620\pi\)
							0.977458	+	0.211132i	\(0.0677151\pi\)
\(37\)	−0.880423	−	1.52494i	−0.144741	−	0.250698i	0.784536	−	0.620084i	\(-0.212901\pi\)
							−0.929276	+	0.369386i	\(0.879568\pi\)
\(41\)	7.33432	+	9.19695i	1.14543	+	1.43632i	0.881754	+	0.471710i	\(0.156363\pi\)
							0.263675	+	0.964612i	\(0.415065\pi\)
\(43\)	0.527203	+	6.53621i	0.0803976	+	0.996763i
							\(47\)	−1.70075	+	7.45149i	−0.248081	+	1.08691i	0.685367	+	0.728198i	\(0.259642\pi\)
−0.933447	+	0.358714i	\(0.883215\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.a.52.2	✓	348
43.24	even	21	inner	731.2.u.a.239.2	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.a.52.2	✓	348	1.1	even	1	trivial
731.2.u.a.239.2	yes	348	43.24	even	21	inner