Embedded newform 576.2.bm.a.205.18

• Label: 576.2.bm.a.205.18
• Level: $576$
• Weight: $2$
• Character: 576.205
• 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			205.18
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.17666 - 0.784525i) q^{2} +(-1.47544 - 0.907238i) q^{3} +(0.769042 + 1.84623i) q^{4} +(-2.36898 - 2.70130i) q^{5} +(1.02433 + 2.22503i) q^{6} +(-4.85600 - 0.639304i) q^{7} +(0.543516 - 2.77571i) q^{8} +(1.35384 + 2.67715i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.17666	−	0.784525i	−0.832022	−	0.554743i
							\(3\)	−1.47544	−	0.907238i	−0.851845	−	0.523794i
\(5\)	−2.36898	−	2.70130i	−1.05944	−	1.20806i								−0.977393	−	0.211431i	\(-0.932188\pi\)
							−0.0820471	−	0.996628i	\(-0.526146\pi\)
\(7\)	−4.85600	−	0.639304i	−1.83540	−	0.241634i	−0.868298	−	0.496042i	\(-0.834786\pi\)
							−0.967097	+	0.254408i	\(0.918119\pi\)
\(11\)	−2.39238	−	4.85126i	−0.721329	−	1.46271i	−0.879798	−	0.475347i	\(-0.842323\pi\)
							0.158469	−	0.987364i	\(-0.449344\pi\)
\(13\)	−1.38803	−	4.08900i	−0.384970	−	1.13408i	−0.950865	−	0.309607i	\(-0.899802\pi\)
							0.565895	−	0.824478i	\(-0.308531\pi\)
\(17\)	2.53276	+	2.53276i	0.614283	+	0.614283i	0.944059	−	0.329776i	\(-0.106973\pi\)
							−0.329776	+	0.944059i	\(0.606973\pi\)
\(19\)	−0.255110	+	1.28253i	−0.0585263	+	0.294231i	−0.998952	−	0.0457674i	\(-0.985427\pi\)
							0.940426	+	0.339999i	\(0.110427\pi\)
\(23\)	−0.535956	−	4.07099i	−0.111754	−	0.848859i	−0.951881	−	0.306469i	\(-0.900853\pi\)
							0.840126	−	0.542391i	\(-0.182481\pi\)
\(29\)	−1.21586	+	0.0796918i	−0.225780	+	0.0147984i	−0.177873	−	0.984053i	\(-0.556922\pi\)
							−0.0479067	+	0.998852i	\(0.515255\pi\)
\(31\)	0.338001	−	0.195145i	0.0607067	−	0.0350490i	−0.469339	−	0.883018i	\(-0.655508\pi\)
							0.530046	+	0.847969i	\(0.322175\pi\)
\(37\)	−0.208376	−	1.04758i	−0.0342569	−	0.172221i	0.959869	−	0.280447i	\(-0.0904828\pi\)
							−0.994126	+	0.108226i	\(0.965483\pi\)
\(41\)	−0.197381	−	1.49926i	−0.0308257	−	0.234145i	0.969096	−	0.246685i	\(-0.0793414\pi\)
							−0.999921	+	0.0125407i	\(0.996008\pi\)
\(43\)	−1.18510	+	0.584427i	−0.180726	+	0.0891243i	−0.530401	−	0.847747i	\(-0.677959\pi\)
							0.349675	+	0.936871i	\(0.386292\pi\)
\(47\)	1.28250	−	0.343646i	0.187072	−	0.0501259i	−0.164067	−	0.986449i	\(-0.552461\pi\)
							0.351139	+	0.936323i	\(0.385795\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.18	yes	1504
9.4	even	3	inner	576.2.bm.a.13.82	✓	1504
64.5	even	16	inner	576.2.bm.a.133.82	yes	1504
576.517	even	48	inner	576.2.bm.a.517.18	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.82	✓	1504	9.4	even	3	inner
576.2.bm.a.133.82	yes	1504	64.5	even	16	inner
576.2.bm.a.205.18	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.18	yes	1504	576.517	even	48	inner