Embedded newform 731.2.u.a.52.6

• Label: 731.2.u.a.52.6
• 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			52.6
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.a.239.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.951366 - 1.19298i) q^{2} +(0.415637 + 0.0626473i) q^{3} +(-0.0730511 + 0.320058i) q^{4} +(2.78277 + 1.89726i) q^{5} +(-0.320687 - 0.555446i) q^{6} +(-1.33438 + 2.31122i) q^{7} +(-2.29821 + 1.10676i) q^{8} +(-2.69789 - 0.832188i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 2 q^{2} + 5 q^{3} - 54 q^{4} + 3 q^{5} + 8 q^{6} - 71 q^{7} - 26 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.951366	−	1.19298i	−0.672718	−	0.843561i	0.321943	−	0.946759i	\(-0.395664\pi\)
							−0.994661	+	0.103198i	\(0.967093\pi\)
\(3\)	0.415637	+	0.0626473i	0.239968	+	0.0361694i	0.267925	−	0.963440i	\(-0.413662\pi\)
							−0.0279568	+	0.999609i	\(0.508900\pi\)
\(5\)	2.78277	+	1.89726i	1.24449	+	0.848479i	0.992853	−	0.119342i	\(-0.0380785\pi\)
							0.251638	+	0.967822i	\(0.419031\pi\)
\(7\)	−1.33438	+	2.31122i	−0.504350	+	0.873559i	0.495638	+	0.868529i	\(0.334934\pi\)
							−0.999987	+	0.00502996i	\(0.998399\pi\)
\(11\)	0.569649	+	2.49579i	0.171756	+	0.752510i	0.985276	+	0.170973i	\(0.0546912\pi\)
							−0.813520	+	0.581537i	\(0.802452\pi\)
\(13\)	−0.0339696	+	0.453293i	−0.00942149	+	0.125721i	−0.999931	−	0.0117210i	\(-0.996269\pi\)
							0.990510	+	0.137442i	\(0.0438881\pi\)
\(17\)	0.826239	−	0.563320i	0.200392	−	0.136625i
							\(19\)	−6.76091	+	2.08546i	−1.55106	+	0.478438i	−0.947735	−	0.319058i	\(-0.896633\pi\)
−0.603323	+	0.797497i	\(0.706157\pi\)
\(23\)	−4.09873	+	3.80307i	−0.854644	+	0.792994i	−0.979975	−	0.199121i	\(-0.936191\pi\)
							0.125331	+	0.992115i	\(0.460001\pi\)
\(29\)	4.53622	−	0.683726i	0.842356	−	0.126965i	0.286331	−	0.958131i	\(-0.407564\pi\)
							0.556024	+	0.831166i	\(0.312326\pi\)
\(31\)	−3.60801	+	9.19307i	−0.648018	+	1.65112i	0.106076	+	0.994358i	\(0.466171\pi\)
							−0.754094	+	0.656766i	\(0.771924\pi\)
\(37\)	−4.89075	−	8.47103i	−0.804035	−	1.39263i	−0.916941	−	0.399023i	\(-0.869349\pi\)
							0.112906	−	0.993606i	\(-0.463984\pi\)
\(41\)	−4.80940	−	6.03079i	−0.751101	−	0.941851i	0.248540	−	0.968622i	\(-0.420049\pi\)
							−0.999642	+	0.0267702i	\(0.991478\pi\)
\(43\)	6.42217	−	1.32504i	0.979372	−	0.202067i
							\(47\)	−0.782205	+	3.42706i	−0.114096	+	0.499889i	0.885296	+	0.465028i	\(0.153956\pi\)
−0.999392	+	0.0348604i	\(0.988901\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.a.52.6	✓	348
43.24	even	21	inner	731.2.u.a.239.6	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.a.52.6	✓	348	1.1	even	1	trivial
731.2.u.a.239.6	yes	348	43.24	even	21	inner