Embedded newform 576.2.bm.a.13.21

• Label: 576.2.bm.a.13.21
• Level: $576$
• Weight: $2$
• Character: 576.13
• Analytic conductor: $4.599$
• Analytic rank: $0$
• Dimension: $1504$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 576 = 2^{6} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	576.bm (of order \(48\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.59938315643\)
Analytic rank:	\(0\)
Dimension:	\(1504\)
Relative dimension:	\(94\) over \(\Q(\zeta_{48})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{48}]$

Embedding invariants

Embedding label			13.21
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.21

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15248 + 0.819623i) q^{2} +(1.33427 + 1.10441i) q^{3} +(0.656437 - 1.88920i) q^{4} +(0.128618 - 0.378896i) q^{5} +(-2.44292 - 0.179221i) q^{6} +(-1.38692 + 1.80746i) q^{7} +(0.791901 + 2.71531i) q^{8} +(0.560543 + 2.94717i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15248	+	0.819623i	−0.814929	+	0.579561i
							\(3\)	1.33427	+	1.10441i	0.770340	+	0.637633i
\(5\)	0.128618	−	0.378896i	0.0575196	−	0.169447i								−0.914373	−	0.404874i	\(-0.867315\pi\)
							0.971892	+	0.235427i	\(0.0756487\pi\)
\(7\)	−1.38692	+	1.80746i	−0.524205	+	0.683157i	−0.978157	−	0.207869i	\(-0.933347\pi\)
							0.453952	+	0.891026i	\(0.350014\pi\)
\(11\)	−5.67195	−	0.371759i	−1.71016	−	0.112090i	−0.821366	−	0.570401i	\(-0.806788\pi\)
							−0.888792	+	0.458312i	\(0.848454\pi\)
\(13\)	−1.49458	+	1.70424i	−0.414522	+	0.472672i	−0.920990	−	0.389587i	\(-0.872618\pi\)
							0.506467	+	0.862259i	\(0.330951\pi\)
\(17\)	0.935965	+	0.935965i	0.227005	+	0.227005i	0.811440	−	0.584435i	\(-0.198684\pi\)
							−0.584435	+	0.811440i	\(0.698684\pi\)
\(19\)	0.719896	−	3.61916i	0.165155	−	0.830292i	−0.806013	−	0.591898i	\(-0.798379\pi\)
							0.971168	−	0.238395i	\(-0.0766211\pi\)
\(23\)	−4.47337	+	3.43254i	−0.932763	+	0.715734i	−0.958846	−	0.283926i	\(-0.908363\pi\)
							0.0260834	+	0.999660i	\(0.491696\pi\)
\(29\)	−3.89440	+	7.89705i	−0.723171	+	1.46645i	0.154907	+	0.987929i	\(0.450492\pi\)
							−0.878078	+	0.478517i	\(0.841175\pi\)
\(31\)	3.85389	+	2.22504i	0.692179	+	0.399630i	0.804428	−	0.594050i	\(-0.202472\pi\)
							−0.112249	+	0.993680i	\(0.535805\pi\)
\(37\)	0.930003	+	4.67544i	0.152892	+	0.768638i	0.978798	+	0.204830i	\(0.0656641\pi\)
							−0.825906	+	0.563808i	\(0.809336\pi\)
\(41\)	−5.12418	+	3.93192i	−0.800262	+	0.614063i	−0.925795	−	0.378027i	\(-0.876603\pi\)
							0.125533	+	0.992089i	\(0.459936\pi\)
\(43\)	0.767985	−	11.7172i	0.117117	−	1.78685i	−0.385864	−	0.922556i	\(-0.626097\pi\)
							0.502981	−	0.864298i	\(-0.332237\pi\)
\(47\)	1.78780	−	6.67216i	0.260777	−	0.973234i	−0.704007	−	0.710193i	\(-0.748608\pi\)
							0.964785	−	0.263041i	\(-0.0847256\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.21	✓	1504
9.7	even	3	inner	576.2.bm.a.205.82	yes	1504
64.5	even	16	inner	576.2.bm.a.517.82	yes	1504
576.133	even	48	inner	576.2.bm.a.133.21	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.21	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.21	yes	1504	576.133	even	48	inner
576.2.bm.a.205.82	yes	1504	9.7	even	3	inner
576.2.bm.a.517.82	yes	1504	64.5	even	16	inner