Embedded newform 576.2.bm.a.517.28

• Label: 576.2.bm.a.517.28
• Level: $576$
• Weight: $2$
• Character: 576.517
• 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			517.28
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.28

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.836570 - 1.14024i) q^{2} +(-1.64723 + 0.535390i) q^{3} +(-0.600300 + 1.90778i) q^{4} +(-2.22047 + 2.53196i) q^{5} +(1.98850 + 1.43034i) q^{6} +(3.36080 - 0.442458i) q^{7} +(2.67753 - 0.911510i) q^{8} +(2.42672 - 1.76382i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.836570	−	1.14024i	−0.591545	−	0.806272i
							\(3\)	−1.64723	+	0.535390i	−0.951027	+	0.309107i
\(5\)	−2.22047	+	2.53196i	−0.993023	+	1.13233i								−0.00181713	+	0.999998i	\(0.500578\pi\)
							−0.991206	+	0.132328i	\(0.957755\pi\)
\(7\)	3.36080	−	0.442458i	1.27026	−	0.167233i	0.534905	−	0.844912i	\(-0.320347\pi\)
							0.735358	+	0.677679i	\(0.237014\pi\)
\(11\)	−0.977694	+	1.98257i	−0.294786	+	0.597766i	−0.992999	−	0.118124i	\(-0.962312\pi\)
							0.698213	+	0.715890i	\(0.253979\pi\)
\(13\)	0.602631	−	1.77529i	0.167140	−	0.492378i	−0.830797	−	0.556575i	\(-0.812115\pi\)
							0.997937	+	0.0641969i	\(0.0204486\pi\)
\(17\)	−1.12378	+	1.12378i	−0.272556	+	0.272556i	−0.830128	−	0.557572i	\(-0.811733\pi\)
							0.557572	+	0.830128i	\(0.311733\pi\)
\(19\)	0.386267	+	1.94189i	0.0886157	+	0.445501i	0.999463	+	0.0327584i	\(0.0104292\pi\)
							−0.910848	+	0.412743i	\(0.864571\pi\)
\(23\)	−0.921094	+	6.99640i	−0.192061	+	1.45885i	0.575461	+	0.817829i	\(0.304822\pi\)
							−0.767523	+	0.641022i	\(0.778511\pi\)
\(29\)	3.27466	+	0.214632i	0.608088	+	0.0398562i	0.366337	−	0.930482i	\(-0.380612\pi\)
							0.241751	+	0.970338i	\(0.422278\pi\)
\(31\)	−6.32219	−	3.65012i	−1.13550	−	0.655581i	−0.190187	−	0.981748i	\(-0.560909\pi\)
							−0.945312	+	0.326167i	\(0.894243\pi\)
\(37\)	−1.66107	+	8.35075i	−0.273078	+	1.37286i	0.564001	+	0.825774i	\(0.309261\pi\)
							−0.837079	+	0.547082i	\(0.815739\pi\)
\(41\)	0.295357	−	2.24346i	0.0461270	−	0.350369i	−0.952781	−	0.303657i	\(-0.901792\pi\)
							0.998908	−	0.0467123i	\(-0.0148744\pi\)
\(43\)	−3.30740	−	1.63103i	−0.504374	−	0.248730i	0.172257	−	0.985052i	\(-0.444894\pi\)
							−0.676631	+	0.736322i	\(0.736561\pi\)
\(47\)	−5.09425	−	1.36500i	−0.743072	−	0.199106i	−0.132629	−	0.991166i	\(-0.542342\pi\)
							−0.610443	+	0.792060i	\(0.709009\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.28	yes	1504
9.7	even	3	inner	576.2.bm.a.133.36	yes	1504
64.13	even	16	inner	576.2.bm.a.13.36	✓	1504
576.205	even	48	inner	576.2.bm.a.205.28	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.36	✓	1504	64.13	even	16	inner
576.2.bm.a.133.36	yes	1504	9.7	even	3	inner
576.2.bm.a.205.28	yes	1504	576.205	even	48	inner
576.2.bm.a.517.28	yes	1504	1.1	even	1	trivial