Embedded newform 576.2.bm.a.133.1

• Label: 576.2.bm.a.133.1
• Level: $576$
• Weight: $2$
• Character: 576.133
• 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			133.1
Character	\(\chi\)	\(=\)	576.133
Dual form			576.2.bm.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41382 + 0.0331870i) q^{2} +(-0.374386 - 1.69110i) q^{3} +(1.99780 - 0.0938413i) q^{4} +(-0.000471804 - 0.00138989i) q^{5} +(0.585439 + 2.37850i) q^{6} +(1.11009 + 1.44669i) q^{7} +(-2.82142 + 0.198976i) q^{8} +(-2.71967 + 1.26625i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{1}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41382	+	0.0331870i	−0.999725	+	0.0234668i
							\(3\)	−0.374386	−	1.69110i	−0.216152	−	0.976360i
\(5\)	−0.000471804	−	0.00138989i	−0.000210997	−	0.000621577i								0.946825	−	0.321750i	\(-0.104271\pi\)
							−0.947036	+	0.321129i	\(0.895938\pi\)
\(7\)	1.11009	+	1.44669i	0.419574	+	0.546799i	0.954543	−	0.298072i	\(-0.0963436\pi\)
							−0.534970	+	0.844871i	\(0.679677\pi\)
\(11\)	−0.469317	+	0.0307607i	−0.141504	+	0.00927469i	−0.135991	−	0.990710i	\(-0.543422\pi\)
							−0.00551367	+	0.999985i	\(0.501755\pi\)
\(13\)	−3.13750	−	3.57763i	−0.870185	−	0.992256i	−1.00000		0.000717789i	\(-0.999772\pi\)
							0.129815	−	0.991538i	\(-0.458562\pi\)
\(17\)	−4.32364	+	4.32364i	−1.04864	+	1.04864i	−0.0498810	+	0.998755i	\(0.515884\pi\)
							−0.998755	+	0.0498810i	\(0.984116\pi\)
\(19\)	−1.25108	−	6.28962i	−0.287018	−	1.44294i	−0.807906	−	0.589312i	\(-0.799399\pi\)
							0.520888	−	0.853625i	\(-0.325601\pi\)
\(23\)	−3.02225	−	2.31906i	−0.630183	−	0.483557i	0.243598	−	0.969876i	\(-0.421672\pi\)
							−0.873781	+	0.486320i	\(0.838339\pi\)
\(29\)	−3.60750	−	7.31528i	−0.669896	−	1.35841i	−0.920660	−	0.390366i	\(-0.872349\pi\)
							0.250764	−	0.968048i	\(-0.419318\pi\)
\(31\)	−3.92540	+	2.26633i	−0.705022	+	0.407045i	−0.809215	−	0.587512i	\(-0.800107\pi\)
							0.104193	+	0.994557i	\(0.466774\pi\)
\(37\)	−0.838744	+	4.21665i	−0.137889	+	0.693213i	0.848554	+	0.529108i	\(0.177473\pi\)
							−0.986443	+	0.164105i	\(0.947527\pi\)
\(41\)	−5.59749	−	4.29510i	−0.874181	−	0.670782i	0.0709483	−	0.997480i	\(-0.477397\pi\)
							−0.945129	+	0.326698i	\(0.894064\pi\)
\(43\)	0.577512	+	8.81113i	0.0880697	+	1.34368i	0.781836	+	0.623484i	\(0.214283\pi\)
							−0.693766	+	0.720200i	\(0.744050\pi\)
\(47\)	1.77421	+	6.62143i	0.258795	+	0.965835i	0.965940	+	0.258767i	\(0.0833161\pi\)
							−0.707145	+	0.707068i	\(0.750017\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.133.1	yes	1504
9.4	even	3	inner	576.2.bm.a.517.64	yes	1504
64.13	even	16	inner	576.2.bm.a.205.64	yes	1504
576.13	even	48	inner	576.2.bm.a.13.1	✓	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.1	✓	1504	576.13	even	48	inner
576.2.bm.a.133.1	yes	1504	1.1	even	1	trivial
576.2.bm.a.205.64	yes	1504	64.13	even	16	inner
576.2.bm.a.517.64	yes	1504	9.4	even	3	inner