Embedded newform 576.2.bm.a.205.49

• Label: 576.2.bm.a.205.49
• Level: $576$
• Weight: $2$
• Character: 576.205
• Analytic conductor: $4.599$
• Analytic rank: $0$
• Dimension: $1504$
• 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			205.49
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.49

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.102022 + 1.41053i) q^{2} +(-0.902375 + 1.47842i) q^{3} +(-1.97918 + 0.287809i) q^{4} +(2.43429 + 2.77578i) q^{5} +(-2.17741 - 1.12199i) q^{6} +(-3.87959 - 0.510758i) q^{7} +(-0.607882 - 2.76233i) q^{8} +(-1.37144 - 2.66817i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.102022	+	1.41053i	0.0721402	+	0.997395i
							\(3\)	−0.902375	+	1.47842i	−0.520986	+	0.853565i
\(5\)	2.43429	+	2.77578i	1.08865	+	1.24136i								0.968607	+	0.248596i	\(0.0799690\pi\)
							0.120040	+	0.992769i	\(0.461698\pi\)
\(7\)	−3.87959	−	0.510758i	−1.46635	−	0.193048i	−0.645309	−	0.763921i	\(-0.723272\pi\)
							−0.821038	+	0.570873i	\(0.806605\pi\)
\(11\)	−2.28024	−	4.62386i	−0.687517	−	1.39415i	−0.908090	−	0.418776i	\(-0.862459\pi\)
							0.220572	−	0.975371i	\(-0.429207\pi\)
\(13\)	−0.375306	−	1.10562i	−0.104091	−	0.306643i	0.882390	−	0.470519i	\(-0.155933\pi\)
							−0.986481	+	0.163877i	\(0.947600\pi\)
\(17\)	−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	\(0.858062\pi\)
\(23\)	0.483566	+	3.67305i	0.100830	+	0.765883i	0.964920	+	0.262545i	\(0.0845617\pi\)
							−0.864089	+	0.503338i	\(0.832105\pi\)
\(29\)	−4.89054	+	0.320543i	−0.908151	+	0.0595233i	−0.512315	−	0.858798i	\(-0.671212\pi\)
							−0.395836	+	0.918321i	\(0.629545\pi\)
\(31\)	−8.18857	+	4.72768i	−1.47071	+	0.849115i	−0.999459	−	0.0328831i	\(-0.989531\pi\)
							−0.471252	+	0.881999i	\(0.656198\pi\)
\(37\)	−1.51726	−	7.62778i	−0.249436	−	1.25400i	−0.878914	−	0.476981i	\(-0.841731\pi\)
							0.629478	−	0.777019i	\(-0.283269\pi\)
\(41\)	0.449417	+	3.41366i	0.0701872	+	0.533125i	0.990032	+	0.140842i	\(0.0449811\pi\)
							−0.919845	+	0.392282i	\(0.871686\pi\)
\(43\)	5.16975	−	2.54944i	0.788380	−	0.388786i	−0.00308916	−	0.999995i	\(-0.500983\pi\)
							0.791469	+	0.611209i	\(0.209317\pi\)
\(47\)	0.138241	−	0.0370415i	0.0201645	−	0.00540306i	−0.248723	−	0.968575i	\(-0.580011\pi\)
							0.268887	+	0.963172i	\(0.413344\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.15	✓	1504	9.4	even	3	inner
576.2.bm.a.133.15	yes	1504	64.5	even	16	inner
576.2.bm.a.205.49	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.49	yes	1504	576.517	even	48	inner