Embedded newform 576.2.bm.a.205.2

• Label: 576.2.bm.a.205.2
• 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			205.2
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41311 + 0.0559574i) q^{2} +(-1.05401 - 1.37443i) q^{3} +(1.99374 - 0.158147i) q^{4} +(1.11364 + 1.26986i) q^{5} +(1.56634 + 1.88324i) q^{6} +(-0.644387 - 0.0848352i) q^{7} +(-2.80851 + 0.335044i) q^{8} +(-0.778133 + 2.89733i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41311	+	0.0559574i	−0.999217	+	0.0395679i
							\(3\)	−1.05401	−	1.37443i	−0.608532	−	0.793529i
\(5\)	1.11364	+	1.26986i	0.498035	+	0.567900i								0.945175	−	0.326564i	\(-0.105891\pi\)
							−0.447140	+	0.894464i	\(0.647557\pi\)
\(7\)	−0.644387	−	0.0848352i	−0.243556	−	0.0320647i	0.00776008	−	0.999970i	\(-0.497530\pi\)
							−0.251316	+	0.967905i	\(0.580863\pi\)
\(11\)	−2.31665	−	4.69770i	−0.698497	−	1.41641i	−0.899544	−	0.436830i	\(-0.856101\pi\)
							0.201048	−	0.979581i	\(-0.435565\pi\)
\(13\)	1.12545	+	3.31547i	0.312143	+	0.919545i	0.983892	+	0.178765i	\(0.0572103\pi\)
							−0.671748	+	0.740779i	\(0.734456\pi\)
\(17\)	−2.81312	−	2.81312i	−0.682282	−	0.682282i	0.278232	−	0.960514i	\(-0.410252\pi\)
							−0.960514	+	0.278232i	\(0.910252\pi\)
\(19\)	0.851862	−	4.28260i	0.195431	−	0.982496i	−0.751175	−	0.660103i	\(-0.770513\pi\)
							0.946606	−	0.322393i	\(-0.104487\pi\)
\(23\)	0.0561046	+	0.426157i	0.0116986	+	0.0888598i	0.996298	−	0.0859652i	\(-0.0273974\pi\)
							−0.984600	+	0.174825i	\(0.944064\pi\)
\(29\)	−0.430131	+	0.0281922i	−0.0798732	+	0.00523517i	−0.105287	−	0.994442i	\(-0.533576\pi\)
							0.0254141	+	0.999677i	\(0.491910\pi\)
\(31\)	−1.34030	+	0.773823i	−0.240725	+	0.138983i	−0.615510	−	0.788129i	\(-0.711050\pi\)
							0.374785	+	0.927112i	\(0.377717\pi\)
\(37\)	−0.485089	−	2.43871i	−0.0797482	−	0.400921i	−0.999955	−	0.00948870i	\(-0.996980\pi\)
							0.920207	−	0.391433i	\(-0.128020\pi\)
\(41\)	−1.46420	−	11.1217i	−0.228669	−	1.73692i	−0.592607	−	0.805492i	\(-0.701901\pi\)
							0.363938	−	0.931423i	\(-0.381432\pi\)
\(43\)	3.21561	−	1.58576i	0.490376	−	0.241827i	−0.180252	−	0.983620i	\(-0.557691\pi\)
							0.670628	+	0.741794i	\(0.266025\pi\)
\(47\)	−11.7514	+	3.14878i	−1.71412	+	0.459296i	−0.976428	−	0.215844i	\(-0.930750\pi\)
							−0.737689	+	0.675141i	\(0.764083\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.2	yes	1504
9.4	even	3	inner	576.2.bm.a.13.63	✓	1504
64.5	even	16	inner	576.2.bm.a.133.63	yes	1504
576.517	even	48	inner	576.2.bm.a.517.2	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.63	✓	1504	9.4	even	3	inner
576.2.bm.a.133.63	yes	1504	64.5	even	16	inner
576.2.bm.a.205.2	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.2	yes	1504	576.517	even	48	inner