Embedded newform 576.2.bm.a.13.15

• Label: 576.2.bm.a.13.15
• 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			13.15
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.27256 - 0.616911i) q^{2} +(-1.71120 + 0.267919i) q^{3} +(1.23884 + 1.57012i) q^{4} +(1.18675 - 3.49605i) q^{5} +(2.34290 + 0.714716i) q^{6} +(2.38213 - 3.10445i) q^{7} +(-0.607882 - 2.76233i) q^{8} +(2.85644 - 0.916929i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.27256	−	0.616911i	−0.899839	−	0.436222i
							\(3\)	−1.71120	+	0.267919i	−0.987964	+	0.154683i
\(5\)	1.18675	−	3.49605i	0.530730	−	1.56348i								−0.269013	−	0.963136i	\(-0.586698\pi\)
							0.799743	−	0.600342i	\(-0.204969\pi\)
\(7\)	2.38213	−	3.10445i	0.900359	−	1.17337i	−0.0838712	−	0.996477i	\(-0.526728\pi\)
							0.984230	−	0.176894i	\(-0.0566049\pi\)
\(11\)	5.14450	+	0.337188i	1.55112	+	0.101666i	0.816715	−	0.577041i	\(-0.195793\pi\)
							0.734409	+	0.678707i	\(0.237459\pi\)
\(13\)	−0.769839	+	0.877833i	−0.213515	+	0.243467i	−0.848677	−	0.528912i	\(-0.822600\pi\)
							0.635162	+	0.772379i	\(0.280933\pi\)
\(17\)	−0.383134	−	0.383134i	−0.0929236	−	0.0929236i	0.659117	−	0.752041i	\(-0.270930\pi\)
							−0.752041	+	0.659117i	\(0.770930\pi\)
\(19\)	−1.01877	+	5.12169i	−0.233721	+	1.17500i	0.668497	+	0.743715i	\(0.266938\pi\)
							−0.902218	+	0.431281i	\(0.858062\pi\)
\(23\)	2.93917	−	2.25530i	0.612859	−	0.470263i	−0.255090	−	0.966917i	\(-0.582105\pi\)
							0.867949	+	0.496654i	\(0.165438\pi\)
\(29\)	2.16767	−	4.39560i	0.402527	−	0.816243i	−0.597371	−	0.801965i	\(-0.703788\pi\)
							0.999898	−	0.0142785i	\(-0.00454514\pi\)
\(31\)	8.18857	+	4.72768i	1.47071	+	0.849115i	0.999459	−	0.0328831i	\(-0.0104689\pi\)
							0.471252	+	0.881999i	\(0.343802\pi\)
\(37\)	−1.51726	−	7.62778i	−0.249436	−	1.25400i	−0.878914	−	0.476981i	\(-0.841731\pi\)
							0.629478	−	0.777019i	\(-0.283269\pi\)
\(41\)	2.73161	−	2.09604i	0.426606	−	0.327346i	−0.373043	−	0.927814i	\(-0.621686\pi\)
							0.799649	+	0.600468i	\(0.205019\pi\)
\(43\)	−0.376997	+	5.75186i	−0.0574914	+	0.877150i	0.867566	+	0.497322i	\(0.165683\pi\)
							−0.925057	+	0.379828i	\(0.875983\pi\)
\(47\)	−0.0370415	+	0.138241i	−0.00540306	+	0.0201645i	−0.968575	−	0.248723i	\(-0.919989\pi\)
							0.963172	+	0.268887i	\(0.0866559\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.15	✓	1504
9.7	even	3	inner	576.2.bm.a.205.49	yes	1504
64.5	even	16	inner	576.2.bm.a.517.49	yes	1504
576.133	even	48	inner	576.2.bm.a.133.15	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.15	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.15	yes	1504	576.133	even	48	inner
576.2.bm.a.205.49	yes	1504	9.7	even	3	inner
576.2.bm.a.517.49	yes	1504	64.5	even	16	inner