Embedded newform 731.2.u.a.52.19

• Label: 731.2.u.a.52.19
• 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			52.19
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.a.239.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.607502 + 0.761783i) q^{2} +(2.59131 + 0.390577i) q^{3} +(0.233787 - 1.02429i) q^{4} +(-0.368169 - 0.251013i) q^{5} +(1.27669 + 2.21129i) q^{6} +(0.0206336 - 0.0357384i) q^{7} +(2.67804 - 1.28968i) q^{8} +(3.69561 + 1.13994i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 2 q^{2} + 5 q^{3} - 54 q^{4} + 3 q^{5} + 8 q^{6} - 71 q^{7} - 26 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.607502	+	0.761783i	0.429569	+	0.538662i	0.948761	−	0.315995i	\(-0.102338\pi\)
							−0.519192	+	0.854658i	\(0.673767\pi\)
\(3\)	2.59131	+	0.390577i	1.49609	+	0.225500i	0.845520	−	0.533944i	\(-0.179291\pi\)
							0.650573	+	0.759444i	\(0.274529\pi\)
\(5\)	−0.368169	−	0.251013i	−0.164650	−	0.112257i	0.478192	−	0.878255i	\(-0.341292\pi\)
							−0.642842	+	0.765999i	\(0.722245\pi\)
\(7\)	0.0206336	−	0.0357384i	0.00779877	−	0.0135079i	−0.862100	−	0.506739i	\(-0.830851\pi\)
							0.869898	+	0.493231i	\(0.164184\pi\)
\(11\)	0.772175	+	3.38312i	0.232820	+	1.02005i	0.947288	+	0.320382i	\(0.103811\pi\)
							−0.714469	+	0.699667i	\(0.753332\pi\)
\(13\)	0.178381	−	2.38032i	0.0494739	−	0.660183i	−0.916249	−	0.400609i	\(-0.868799\pi\)
							0.965723	−	0.259574i	\(-0.0835823\pi\)
\(17\)	0.826239	−	0.563320i	0.200392	−	0.136625i
							\(19\)	−3.68541	+	1.13680i	−0.845490	+	0.260799i	−0.687087	−	0.726575i	\(-0.741111\pi\)
−0.158403	+	0.987374i	\(0.550635\pi\)
\(23\)	1.17311	−	1.08848i	0.244609	−	0.226964i	−0.548334	−	0.836259i	\(-0.684738\pi\)
							0.792944	+	0.609295i	\(0.208547\pi\)
\(29\)	1.20731	−	0.181973i	0.224192	−	0.0337915i	−0.0359857	−	0.999352i	\(-0.511457\pi\)
							0.260177	+	0.965561i	\(0.416219\pi\)
\(31\)	−3.68466	+	9.38837i	−0.661785	+	1.68620i	0.0637055	+	0.997969i	\(0.479708\pi\)
							−0.725491	+	0.688232i	\(0.758387\pi\)
\(37\)	−0.788613	−	1.36592i	−0.129647	−	0.224556i	0.793893	−	0.608058i	\(-0.208051\pi\)
							−0.923540	+	0.383502i	\(0.874718\pi\)
\(41\)	3.74763	+	4.69938i	0.585281	+	0.733920i	0.983004	−	0.183586i	\(-0.0587706\pi\)
							−0.397722	+	0.917506i	\(0.630199\pi\)
\(43\)	5.55552	+	3.48370i	0.847209	+	0.531260i
							\(47\)	−1.65399	+	7.24661i	−0.241259	+	1.05703i	0.698613	+	0.715500i	\(0.253801\pi\)
−0.939872	+	0.341526i	\(0.889056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.a.52.19	✓	348
43.24	even	21	inner	731.2.u.a.239.19	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.a.52.19	✓	348	1.1	even	1	trivial
731.2.u.a.239.19	yes	348	43.24	even	21	inner