Embedded newform 576.2.bm.a.205.93

• Label: 576.2.bm.a.205.93
• 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			205.93
Character	\(\chi\)	\(=\)	576.205
Dual form			576.2.bm.a.517.93

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.41235 - 0.0725673i) q^{2} +(-1.51508 - 0.839360i) q^{3} +(1.98947 - 0.204981i) q^{4} +(-1.38021 - 1.57383i) q^{5} +(-2.20074 - 1.07552i) q^{6} +(2.49216 + 0.328099i) q^{7} +(2.79495 - 0.433875i) q^{8} +(1.59095 + 2.54340i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.41235	−	0.0725673i	0.998683	−	0.0513128i
							\(3\)	−1.51508	−	0.839360i	−0.874733	−	0.484604i
\(5\)	−1.38021	−	1.57383i	−0.617250	−	0.703839i								0.355766	−	0.934575i	\(-0.384220\pi\)
							−0.973016	+	0.230736i	\(0.925887\pi\)
\(7\)	2.49216	+	0.328099i	0.941947	+	0.124010i	0.585832	−	0.810433i	\(-0.300768\pi\)
							0.356115	+	0.934442i	\(0.384101\pi\)
\(11\)	−1.07736	−	2.18466i	−0.324835	−	0.658700i	0.671794	−	0.740738i	\(-0.265524\pi\)
							−0.996629	+	0.0820375i	\(0.973857\pi\)
\(13\)	−1.11630	−	3.28851i	−0.309606	−	0.912069i	−0.984688	−	0.174326i	\(-0.944225\pi\)
							0.675082	−	0.737742i	\(-0.264108\pi\)
\(17\)	−1.22795	−	1.22795i	−0.297823	−	0.297823i	0.542338	−	0.840160i	\(-0.317539\pi\)
							−0.840160	+	0.542338i	\(0.817539\pi\)
\(19\)	0.214484	−	1.07828i	0.0492059	−	0.247375i	−0.948352	−	0.317220i	\(-0.897251\pi\)
							0.997558	+	0.0698453i	\(0.0222506\pi\)
\(23\)	−0.00305513	−	0.0232060i	−0.000637039	−	0.00483880i	0.991123	−	0.132945i	\(-0.0424434\pi\)
							−0.991760	+	0.128106i	\(0.959110\pi\)
\(29\)	−7.22264	+	0.473397i	−1.34121	+	0.0879076i	−0.719037	−	0.694972i	\(-0.755417\pi\)
							−0.622174	+	0.782879i	\(0.713750\pi\)
\(31\)	7.56639	−	4.36846i	1.35896	−	0.784599i	0.369480	−	0.929239i	\(-0.379536\pi\)
							0.989484	+	0.144640i	\(0.0462024\pi\)
\(37\)	1.63587	+	8.22409i	0.268936	+	1.35203i	0.845061	+	0.534670i	\(0.179564\pi\)
							−0.576125	+	0.817361i	\(0.695436\pi\)
\(41\)	0.138446	+	1.05160i	0.0216217	+	0.164233i	0.998849	−	0.0479613i	\(-0.0152724\pi\)
							−0.977228	+	0.212194i	\(0.931939\pi\)
\(43\)	1.20936	−	0.596388i	0.184425	−	0.0909484i	−0.347732	−	0.937594i	\(-0.613048\pi\)
							0.532157	+	0.846646i	\(0.321382\pi\)
\(47\)	1.62162	−	0.434511i	0.236537	−	0.0633800i	−0.138603	−	0.990348i	\(-0.544261\pi\)
							0.375140	+	0.926968i	\(0.377595\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.205.93	yes	1504
9.4	even	3	inner	576.2.bm.a.13.34	✓	1504
64.5	even	16	inner	576.2.bm.a.133.34	yes	1504
576.517	even	48	inner	576.2.bm.a.517.93	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.34	✓	1504	9.4	even	3	inner
576.2.bm.a.133.34	yes	1504	64.5	even	16	inner
576.2.bm.a.205.93	yes	1504	1.1	even	1	trivial
576.2.bm.a.517.93	yes	1504	576.517	even	48	inner