Embedded newform 576.2.bm.a.133.34

• Label: 576.2.bm.a.133.34
• Level: $576$
• Weight: $2$
• Character: 576.133
• 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			133.34
Character	\(\chi\)	\(=\)	576.133
Dual form			576.2.bm.a.13.34

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.643330 - 1.25942i) q^{2} +(0.195670 - 1.72096i) q^{3} +(-1.17225 + 1.62044i) q^{4} +(-0.672871 - 1.98222i) q^{5} +(-2.29329 + 0.860717i) q^{6} +(-1.53022 - 1.99422i) q^{7} +(2.79495 + 0.433875i) q^{8} +(-2.92343 - 0.673482i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.643330	−	1.25942i	−0.454903	−	0.890541i
							\(3\)	0.195670	−	1.72096i	0.112970	−	0.993598i
\(5\)	−0.672871	−	1.98222i	−0.300917	−	0.886474i								−0.987249	−	0.159185i	\(-0.949113\pi\)
							0.686332	−	0.727289i	\(-0.259220\pi\)
\(7\)	−1.53022	−	1.99422i	−0.578369	−	0.753745i	0.408939	−	0.912562i	\(-0.365899\pi\)
							−0.987308	+	0.158817i	\(0.949232\pi\)
\(11\)	2.43065	−	0.159313i	0.732869	−	0.0480348i	0.305601	−	0.952160i	\(-0.401143\pi\)
							0.427268	+	0.904125i	\(0.359476\pi\)
\(13\)	−2.28978	−	2.61100i	−0.635072	−	0.724161i	0.341373	−	0.939928i	\(-0.389108\pi\)
							−0.976445	+	0.215767i	\(0.930775\pi\)
\(17\)	−1.22795	+	1.22795i	−0.297823	+	0.297823i	−0.840160	−	0.542338i	\(-0.817539\pi\)
							0.542338	+	0.840160i	\(0.317539\pi\)
\(19\)	0.214484	+	1.07828i	0.0492059	+	0.247375i	0.997558	−	0.0698453i	\(-0.0222506\pi\)
							−0.948352	+	0.317220i	\(0.897251\pi\)
\(23\)	−0.0185695	−	0.0142488i	−0.00387200	−	0.00297109i	0.606824	−	0.794837i	\(-0.292443\pi\)
							−0.610696	+	0.791865i	\(0.709110\pi\)
\(29\)	3.20135	+	6.49169i	0.594475	+	1.20548i	0.961382	+	0.275219i	\(0.0887502\pi\)
							−0.366906	+	0.930258i	\(0.619583\pi\)
\(31\)	−7.56639	+	4.36846i	−1.35896	+	0.784599i	−0.989484	−	0.144640i	\(-0.953798\pi\)
							−0.369480	+	0.929239i	\(0.620464\pi\)
\(37\)	1.63587	−	8.22409i	0.268936	−	1.35203i	−0.576125	−	0.817361i	\(-0.695436\pi\)
							0.845061	−	0.534670i	\(-0.179564\pi\)
\(41\)	0.841492	+	0.645699i	0.131419	+	0.100841i	0.672379	−	0.740207i	\(-0.265272\pi\)
							−0.540960	+	0.841048i	\(0.681939\pi\)
\(43\)	−0.0881905	−	1.34553i	−0.0134489	−	0.205191i	−0.999295	−	0.0375389i	\(-0.988048\pi\)
							0.985846	−	0.167652i	\(-0.0536185\pi\)
\(47\)	−0.434511	−	1.62162i	−0.0633800	−	0.236537i	0.926968	−	0.375140i	\(-0.122405\pi\)
							−0.990348	+	0.138603i	\(0.955739\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.34	✓	1504	576.13	even	48	inner
576.2.bm.a.133.34	yes	1504	1.1	even	1	trivial
576.2.bm.a.205.93	yes	1504	64.13	even	16	inner
576.2.bm.a.517.93	yes	1504	9.4	even	3	inner