Embedded newform 576.2.bm.a.205.82

• Label: 576.2.bm.a.205.82
• Level: $576$
• Weight: $2$
• Character: 576.205
• 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			205.82
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.82

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.28606 + 0.588269i) q^{2} +(-0.509742 - 1.65534i) q^{3} +(1.30788 + 1.51309i) q^{4} +(0.263824 + 0.300834i) q^{5} +(0.318230 - 2.42873i) q^{6} +(2.25877 + 0.297372i) q^{7} +(0.791901 + 2.71531i) q^{8} +(-2.48033 + 1.68760i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.28606	+	0.588269i	0.909379	+	0.415969i
							\(3\)	−0.509742	−	1.65534i	−0.294300	−	0.955713i
\(5\)	0.263824	+	0.300834i	0.117986	+	0.134537i								0.807817	−	0.589433i	\(-0.200649\pi\)
							−0.689831	+	0.723970i	\(0.742315\pi\)
\(7\)	2.25877	+	0.297372i	0.853734	+	0.112396i	0.544676	−	0.838647i	\(-0.316653\pi\)
							0.309058	+	0.951043i	\(0.399986\pi\)
\(11\)	2.51402	+	5.09793i	0.758006	+	1.53708i	0.841305	+	0.540560i	\(0.181788\pi\)
							−0.0832989	+	0.996525i	\(0.526546\pi\)
\(13\)	−0.728628	−	2.14647i	−0.202085	−	0.595323i	0.797887	−	0.602807i	\(-0.205951\pi\)
							−0.999972	+	0.00748389i	\(0.997618\pi\)
\(17\)	0.935965	+	0.935965i	0.227005	+	0.227005i	0.811440	−	0.584435i	\(-0.198684\pi\)
							−0.584435	+	0.811440i	\(0.698684\pi\)
\(19\)	0.719896	−	3.61916i	0.165155	−	0.830292i	−0.806013	−	0.591898i	\(-0.798379\pi\)
							0.971168	−	0.238395i	\(-0.0766211\pi\)
\(23\)	−0.735980	−	5.59033i	−0.153463	−	1.16566i	−0.878772	−	0.477241i	\(-0.841637\pi\)
							0.725310	−	0.688422i	\(-0.241696\pi\)
\(29\)	8.78625	−	0.575881i	1.63156	−	0.106938i	0.778118	−	0.628118i	\(-0.216175\pi\)
							0.853447	+	0.521180i	\(0.174508\pi\)
\(31\)	−3.85389	+	2.22504i	−0.692179	+	0.399630i	−0.804428	−	0.594050i	\(-0.797528\pi\)
							0.112249	+	0.993680i	\(0.464195\pi\)
\(37\)	0.930003	+	4.67544i	0.152892	+	0.768638i	0.978798	+	0.204830i	\(0.0656641\pi\)
							−0.825906	+	0.563808i	\(0.809336\pi\)
\(41\)	−0.843054	−	6.40363i	−0.131663	−	1.00008i	−0.921941	−	0.387330i	\(-0.873397\pi\)
							0.790278	−	0.612748i	\(-0.209936\pi\)
\(43\)	−10.5314	+	5.19350i	−1.60602	+	0.792001i	−0.999994	−	0.00344539i	\(-0.998903\pi\)
							−0.606024	+	0.795446i	\(0.707237\pi\)
\(47\)	−6.67216	+	1.78780i	−0.973234	+	0.260777i	−0.710193	−	0.704007i	\(-0.751392\pi\)
							−0.263041	+	0.964785i	\(0.584726\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.82	yes	1504
9.4	even	3	inner	576.2.bm.a.13.21	✓	1504
64.5	even	16	inner	576.2.bm.a.133.21	yes	1504
576.517	even	48	inner	576.2.bm.a.517.82	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.21	✓	1504	9.4	even	3	inner
576.2.bm.a.133.21	yes	1504	64.5	even	16	inner
576.2.bm.a.205.82	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.82	yes	1504	576.517	even	48	inner