Embedded newform 731.2.u.a.52.13

• Label: 731.2.u.a.52.13
• 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			52.13
Character	\(\chi\)	\(=\)	731.52
Dual form			731.2.u.a.239.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.140426 - 0.176089i) q^{2} +(1.06730 + 0.160869i) q^{3} +(0.433754 - 1.90040i) q^{4} +(3.39049 + 2.31160i) q^{5} +(-0.121549 - 0.210530i) q^{6} +(0.225543 - 0.390652i) q^{7} +(-0.801395 + 0.385931i) q^{8} +(-1.75348 - 0.540876i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 348 q - 2 q^{2} + 5 q^{3} - 54 q^{4} + 3 q^{5} + 8 q^{6} - 71 q^{7} - 26 q^{8} + 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.140426	−	0.176089i	−0.0992965	−	0.124514i	0.729701	−	0.683767i	\(-0.239659\pi\)
							−0.828997	+	0.559253i	\(0.811088\pi\)
\(3\)	1.06730	+	0.160869i	0.616204	+	0.0928778i	0.449725	−	0.893167i	\(-0.351522\pi\)
							0.166479	+	0.986045i	\(0.446760\pi\)
\(5\)	3.39049	+	2.31160i	1.51628	+	1.03378i	0.982090	+	0.188411i	\(0.0603338\pi\)
							0.534185	+	0.845368i	\(0.320619\pi\)
\(7\)	0.225543	−	0.390652i	0.0852472	−	0.147652i	−0.820249	−	0.572006i	\(-0.806165\pi\)
							0.905497	+	0.424354i	\(0.139499\pi\)
\(11\)	−0.992054	−	4.34647i	−0.299116	−	1.31051i	−0.871447	−	0.490489i	\(-0.836818\pi\)
							0.572332	−	0.820022i	\(-0.306039\pi\)
\(13\)	0.171918	−	2.29408i	0.0476814	−	0.636264i	−0.921256	−	0.388957i	\(-0.872835\pi\)
							0.968937	−	0.247307i	\(-0.0795456\pi\)
\(17\)	0.826239	−	0.563320i	0.200392	−	0.136625i
							\(19\)	8.11352	−	2.50269i	1.86137	−	0.574156i	0.863851	−	0.503747i	\(-0.168045\pi\)
0.997518	−	0.0704099i	\(-0.0224307\pi\)
\(23\)	−0.425760	+	0.395047i	−0.0887771	+	0.0823731i	−0.723313	−	0.690520i	\(-0.757382\pi\)
							0.634536	+	0.772893i	\(0.281191\pi\)
\(29\)	−3.31049	+	0.498977i	−0.614743	+	0.0926576i	−0.449031	−	0.893516i	\(-0.648231\pi\)
							−0.165713	+	0.986174i	\(0.552992\pi\)
\(31\)	−1.75105	+	4.46160i	−0.314498	+	0.801327i	0.682986	+	0.730431i	\(0.260681\pi\)
							−0.997484	+	0.0708955i	\(0.977414\pi\)
\(37\)	2.96836	+	5.14134i	0.487995	+	0.845232i	0.999905	−	0.0138074i	\(-0.00439519\pi\)
							−0.511910	+	0.859039i	\(0.671062\pi\)
\(41\)	4.90522	+	6.15095i	0.766066	+	0.960616i	0.999932	−	0.0116524i	\(-0.00370915\pi\)
							−0.233866	+	0.972269i	\(0.575138\pi\)
\(43\)	−0.281222	+	6.55141i	−0.0428860	+	0.999080i
							\(47\)	−0.134426	+	0.588958i	−0.0196080	+	0.0859084i	−0.983785	−	0.179351i	\(-0.942600\pi\)
0.964177	+	0.265259i	\(0.0854575\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.u.a.52.13	✓	348
43.24	even	21	inner	731.2.u.a.239.13	yes	348

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.u.a.52.13	✓	348	1.1	even	1	trivial
731.2.u.a.239.13	yes	348	43.24	even	21	inner