Embedded newform 576.2.bm.a.517.2

• Label: 576.2.bm.a.517.2
• Level: $576$
• Weight: $2$
• Character: 576.517
• 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			517.2
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.2

$q$-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 \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

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

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.63	✓	1504	64.13	even	16	inner
576.2.bm.a.133.63	yes	1504	9.7	even	3	inner
576.2.bm.a.205.2	yes	1504	576.205	even	48	inner
576.2.bm.a.517.2	yes	1504	1.1	even	1	trivial