Embedded newform 576.2.bm.a.13.8

• Label: 576.2.bm.a.13.8
• 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			13.8
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35449 - 0.406653i) q^{2} +(0.905946 + 1.47623i) q^{3} +(1.66927 + 1.10161i) q^{4} +(-1.35493 + 3.99150i) q^{5} +(-0.626776 - 2.36794i) q^{6} +(0.639597 - 0.833539i) q^{7} +(-1.81302 - 2.17093i) q^{8} +(-1.35852 + 2.67477i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35449	−	0.406653i	−0.957766	−	0.287547i
							\(3\)	0.905946	+	1.47623i	0.523048	+	0.852303i
\(5\)	−1.35493	+	3.99150i	−0.605943	+	1.78505i								0.0133735	+	0.999911i	\(0.495743\pi\)
							−0.619317	+	0.785141i	\(0.712590\pi\)
\(7\)	0.639597	−	0.833539i	0.241745	−	0.315048i	−0.656685	−	0.754165i	\(-0.728042\pi\)
							0.898430	+	0.439117i	\(0.144709\pi\)
\(11\)	−6.14620	−	0.402843i	−1.85315	−	0.121462i	−0.902298	−	0.431112i	\(-0.858121\pi\)
							−0.950850	+	0.309651i	\(0.899788\pi\)
\(13\)	−0.436988	+	0.498289i	−0.121199	+	0.138201i	−0.809257	−	0.587455i	\(-0.800130\pi\)
							0.688058	+	0.725655i	\(0.258463\pi\)
\(17\)	3.81355	+	3.81355i	0.924921	+	0.924921i	0.997372	−	0.0724509i	\(-0.0230820\pi\)
							−0.0724509	+	0.997372i	\(0.523082\pi\)
\(19\)	0.464562	−	2.33551i	0.106578	−	0.535803i	−0.890199	−	0.455573i	\(-0.849435\pi\)
							0.996776	−	0.0802300i	\(-0.0255655\pi\)
\(23\)	3.00197	−	2.30349i	0.625953	−	0.480311i	−0.246412	−	0.969165i	\(-0.579252\pi\)
							0.872366	+	0.488854i	\(0.162585\pi\)
\(29\)	2.11872	−	4.29633i	0.393436	−	0.797809i	−0.606560	−	0.795037i	\(-0.707451\pi\)
							0.999996	−	0.00277142i	\(-0.000882171\pi\)
\(31\)	−1.86974	−	1.07949i	−0.335815	−	0.193883i	0.322605	−	0.946534i	\(-0.395441\pi\)
							−0.658420	+	0.752651i	\(0.728775\pi\)
\(37\)	0.122483	+	0.615761i	0.0201360	+	0.101231i	0.989546	−	0.144215i	\(-0.0460657\pi\)
							−0.969410	+	0.245446i	\(0.921066\pi\)
\(41\)	1.80487	−	1.38493i	0.281874	−	0.216289i	−0.458142	−	0.888879i	\(-0.651485\pi\)
							0.740015	+	0.672590i	\(0.234818\pi\)
\(43\)	−0.583749	+	8.90629i	−0.0890209	+	1.35820i	0.686618	+	0.727019i	\(0.259095\pi\)
							−0.775638	+	0.631178i	\(0.782572\pi\)
\(47\)	1.75756	−	6.55930i	0.256366	−	0.956772i	−0.710959	−	0.703233i	\(-0.751739\pi\)
							0.967325	−	0.253539i	\(-0.0815945\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.8	✓	1504
9.7	even	3	inner	576.2.bm.a.205.54	yes	1504
64.5	even	16	inner	576.2.bm.a.517.54	yes	1504
576.133	even	48	inner	576.2.bm.a.133.8	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.8	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.8	yes	1504	576.133	even	48	inner
576.2.bm.a.205.54	yes	1504	9.7	even	3	inner
576.2.bm.a.517.54	yes	1504	64.5	even	16	inner