Embedded newform 576.2.bm.a.13.13

• Label: 576.2.bm.a.13.13
• 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			13.13
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28635 + 0.587628i) q^{2} +(-0.713799 + 1.57813i) q^{3} +(1.30939 - 1.51179i) q^{4} +(1.42869 - 4.20879i) q^{5} +(-0.00915756 - 2.44947i) q^{6} +(-1.84067 + 2.39881i) q^{7} +(-0.795959 + 2.71412i) q^{8} +(-1.98098 - 2.25293i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28635	+	0.587628i	−0.909586	+	0.415516i
							\(3\)	−0.713799	+	1.57813i	−0.412112	+	0.911133i
\(5\)	1.42869	−	4.20879i	0.638931	−	1.88223i								0.214680	−	0.976684i	\(-0.431129\pi\)
							0.424251	−	0.905545i	\(-0.360538\pi\)
\(7\)	−1.84067	+	2.39881i	−0.695709	+	0.906666i	−0.998930	−	0.0462546i	\(-0.985271\pi\)
							0.303221	+	0.952920i	\(0.401938\pi\)
\(11\)	2.46757	+	0.161733i	0.744001	+	0.0487644i	0.432689	−	0.901543i	\(-0.357565\pi\)
							0.311312	+	0.950308i	\(0.399232\pi\)
\(13\)	−1.59009	+	1.81315i	−0.441011	+	0.502877i	−0.929055	−	0.369943i	\(-0.879377\pi\)
							0.488044	+	0.872819i	\(0.337711\pi\)
\(17\)	0.475619	+	0.475619i	0.115355	+	0.115355i	0.762428	−	0.647073i	\(-0.224007\pi\)
							−0.647073	+	0.762428i	\(0.724007\pi\)
\(19\)	1.38102	−	6.94286i	0.316828	−	1.59280i	−0.414024	−	0.910266i	\(-0.635877\pi\)
							0.730852	−	0.682536i	\(-0.239123\pi\)
\(23\)	0.293604	−	0.225290i	0.0612207	−	0.0469763i	−0.577698	−	0.816251i	\(-0.696049\pi\)
							0.638918	+	0.769274i	\(0.279382\pi\)
\(29\)	1.73739	−	3.52308i	0.322626	−	0.654220i	−0.673780	−	0.738932i	\(-0.735330\pi\)
							0.996405	+	0.0847120i	\(0.0269970\pi\)
\(31\)	−2.80303	−	1.61833i	−0.503439	−	0.290661i	0.226694	−	0.973966i	\(-0.427208\pi\)
							−0.730133	+	0.683305i	\(0.760542\pi\)
\(37\)	−1.42013	−	7.13945i	−0.233467	−	1.17372i	−0.902568	−	0.430547i	\(-0.858321\pi\)
							0.669101	−	0.743172i	\(-0.266679\pi\)
\(41\)	4.29572	−	3.29622i	0.670879	−	0.514784i	−0.216231	−	0.976342i	\(-0.569376\pi\)
							0.887110	+	0.461559i	\(0.152710\pi\)
\(43\)	−0.101073	+	1.54207i	−0.0154134	+	0.235163i	0.983200	+	0.182529i	\(0.0584284\pi\)
							−0.998614	+	0.0526342i	\(0.983238\pi\)
\(47\)	3.10834	−	11.6005i	0.453399	−	1.69211i	−0.239355	−	0.970932i	\(-0.576936\pi\)
							0.692753	−	0.721175i	\(-0.256397\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.13	✓	1504
9.7	even	3	inner	576.2.bm.a.205.78	yes	1504
64.5	even	16	inner	576.2.bm.a.517.78	yes	1504
576.133	even	48	inner	576.2.bm.a.133.13	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.13	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.13	yes	1504	576.133	even	48	inner
576.2.bm.a.205.78	yes	1504	9.7	even	3	inner
576.2.bm.a.517.78	yes	1504	64.5	even	16	inner