Embedded newform 576.2.bm.a.13.22

• Label: 576.2.bm.a.13.22
• 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			13.22
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.22

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.08626 + 0.905557i) q^{2} +(-1.42087 + 0.990515i) q^{3} +(0.359934 - 1.96735i) q^{4} +(-0.994904 + 2.93089i) q^{5} +(0.646474 - 2.36264i) q^{6} +(2.14695 - 2.79796i) q^{7} +(1.39056 + 2.46299i) q^{8} +(1.03776 - 2.81479i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.08626	+	0.905557i	−0.768104	+	0.640325i
							\(3\)	−1.42087	+	0.990515i	−0.820341	+	0.571874i
\(5\)	−0.994904	+	2.93089i	−0.444935	+	1.31074i								0.461031	+	0.887384i	\(0.347480\pi\)
							−0.905966	+	0.423351i	\(0.860854\pi\)
\(7\)	2.14695	−	2.79796i	0.811471	−	1.05753i	−0.185718	−	0.982603i	\(-0.559461\pi\)
							0.997189	−	0.0749262i	\(-0.0238721\pi\)
\(11\)	−2.06614	−	0.135422i	−0.622965	−	0.0408313i	−0.249347	−	0.968414i	\(-0.580216\pi\)
							−0.373618	+	0.927583i	\(0.621883\pi\)
\(13\)	−1.77273	+	2.02141i	−0.491667	+	0.560638i	−0.943461	−	0.331484i	\(-0.892451\pi\)
							0.451794	+	0.892122i	\(0.350784\pi\)
\(17\)	−5.16851	−	5.16851i	−1.25355	−	1.25355i	−0.954119	−	0.299428i	\(-0.903204\pi\)
							−0.299428	−	0.954119i	\(-0.596796\pi\)
\(19\)	1.60715	−	8.07969i	0.368706	−	1.85361i	−0.136542	−	0.990634i	\(-0.543599\pi\)
							0.505248	−	0.862974i	\(-0.331401\pi\)
\(23\)	0.323603	−	0.248309i	0.0674759	−	0.0517761i	−0.574473	−	0.818524i	\(-0.694793\pi\)
							0.641949	+	0.766748i	\(0.278126\pi\)
\(29\)	1.36519	−	2.76832i	0.253509	−	0.514065i	−0.732498	−	0.680769i	\(-0.761646\pi\)
							0.986007	+	0.166704i	\(0.0533125\pi\)
\(31\)	7.84587	+	4.52982i	1.40916	+	0.813579i	0.995307	−	0.0967642i	\(-0.0308493\pi\)
							0.413853	+	0.910344i	\(0.364183\pi\)
\(37\)	−0.953475	−	4.79344i	−0.156750	−	0.788037i	−0.976534	−	0.215363i	\(-0.930906\pi\)
							0.819784	−	0.572673i	\(-0.194094\pi\)
\(41\)	3.09097	−	2.37178i	0.482728	−	0.370410i	−0.338541	−	0.940952i	\(-0.609933\pi\)
							0.821269	+	0.570541i	\(0.193267\pi\)
\(43\)	−0.702528	+	10.7185i	−0.107134	+	1.63456i	0.518952	+	0.854803i	\(0.326322\pi\)
							−0.626087	+	0.779753i	\(0.715344\pi\)
\(47\)	2.42064	−	9.03395i	0.353087	−	1.31774i	−0.529788	−	0.848130i	\(-0.677729\pi\)
							0.882875	−	0.469608i	\(-0.155605\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.22	✓	1504
9.7	even	3	inner	576.2.bm.a.205.85	yes	1504
64.5	even	16	inner	576.2.bm.a.517.85	yes	1504
576.133	even	48	inner	576.2.bm.a.133.22	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.22	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.22	yes	1504	576.133	even	48	inner
576.2.bm.a.205.85	yes	1504	9.7	even	3	inner
576.2.bm.a.517.85	yes	1504	64.5	even	16	inner