Embedded newform 576.2.bm.a.205.17

• Label: 576.2.bm.a.205.17
• 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			205.17
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.19825 + 0.751129i) q^{2} +(0.809513 - 1.53124i) q^{3} +(0.871611 - 1.80008i) q^{4} +(2.13241 + 2.43155i) q^{5} +(0.180157 + 2.44286i) q^{6} +(-3.26846 - 0.430301i) q^{7} +(0.307686 + 2.81164i) q^{8} +(-1.68938 - 2.47911i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.19825	+	0.751129i	−0.847291	+	0.531128i
							\(3\)	0.809513	−	1.53124i	0.467373	−	0.884060i
\(5\)	2.13241	+	2.43155i	0.953642	+	1.08742i								0.996092	+	0.0883216i	\(0.0281503\pi\)
							−0.0424501	+	0.999099i	\(0.513516\pi\)
\(7\)	−3.26846	−	0.430301i	−1.23536	−	0.162638i	−0.515581	−	0.856841i	\(-0.672424\pi\)
							−0.719780	+	0.694202i	\(0.755757\pi\)
\(11\)	−2.09356	−	4.24532i	−0.631232	−	1.28001i	−0.943926	−	0.330158i	\(-0.892898\pi\)
							0.312693	−	0.949854i	\(-0.398769\pi\)
\(13\)	−1.82497	−	5.37620i	−0.506157	−	1.49109i	−0.836893	−	0.547367i	\(-0.815630\pi\)
							0.330736	−	0.943723i	\(-0.392703\pi\)
\(17\)	0.816625	+	0.816625i	0.198061	+	0.198061i	0.799168	−	0.601108i	\(-0.205274\pi\)
							−0.601108	+	0.799168i	\(0.705274\pi\)
\(19\)	−0.428672	+	2.15508i	−0.0983440	+	0.494409i	0.899949	+	0.435996i	\(0.143604\pi\)
							−0.998293	+	0.0584126i	\(0.981396\pi\)
\(23\)	−1.06407	−	8.08243i	−0.221874	−	1.68530i	−0.633336	−	0.773877i	\(-0.718315\pi\)
							0.411462	−	0.911427i	\(-0.365018\pi\)
\(29\)	5.53448	−	0.362749i	1.02773	−	0.0673608i	0.457800	−	0.889055i	\(-0.348638\pi\)
							0.569928	+	0.821695i	\(0.306971\pi\)
\(31\)	7.01027	−	4.04738i	1.25908	−	0.726932i	0.286186	−	0.958174i	\(-0.407612\pi\)
							0.972896	+	0.231243i	\(0.0742791\pi\)
\(37\)	0.116858	+	0.587485i	0.0192113	+	0.0965819i	0.989201	−	0.146563i	\(-0.0468210\pi\)
							−0.969990	+	0.243145i	\(0.921821\pi\)
\(41\)	0.609971	+	4.63319i	0.0952615	+	0.723583i	0.970707	+	0.240266i	\(0.0772347\pi\)
							−0.875446	+	0.483317i	\(0.839432\pi\)
\(43\)	−6.66329	+	3.28597i	−1.01614	+	0.501106i	−0.872547	−	0.488531i	\(-0.837533\pi\)
							−0.143595	+	0.989637i	\(0.545866\pi\)
\(47\)	5.32188	−	1.42599i	0.776276	−	0.208003i	0.151134	−	0.988513i	\(-0.451708\pi\)
							0.625142	+	0.780511i	\(0.285041\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.17	yes	1504
9.4	even	3	inner	576.2.bm.a.13.47	✓	1504
64.5	even	16	inner	576.2.bm.a.133.47	yes	1504
576.517	even	48	inner	576.2.bm.a.517.17	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.47	✓	1504	9.4	even	3	inner
576.2.bm.a.133.47	yes	1504	64.5	even	16	inner
576.2.bm.a.205.17	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.17	yes	1504	576.517	even	48	inner