Embedded newform 576.2.bm.a.133.35

• Label: 576.2.bm.a.133.35
• 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			133.35
Character	\(\chi\)	\(=\)	576.133
Dual form			576.2.bm.a.13.35

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.597638 + 1.28173i) q^{2} +(1.25745 + 1.19114i) q^{3} +(-1.28566 - 1.53202i) q^{4} +(-0.632104 - 1.86212i) q^{5} +(-2.27822 + 0.899843i) q^{6} +(-1.03864 - 1.35358i) q^{7} +(2.73199 - 0.732272i) q^{8} +(0.162374 + 2.99560i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.597638	+	1.28173i	−0.422594	+	0.906319i
							\(3\)	1.25745	+	1.19114i	0.725991	+	0.687705i
\(5\)	−0.632104	−	1.86212i	−0.282685	−	0.832764i								−0.991815	−	0.127686i	\(-0.959245\pi\)
							0.709129	−	0.705079i	\(-0.249088\pi\)
\(7\)	−1.03864	−	1.35358i	−0.392567	−	0.511604i	0.554699	−	0.832051i	\(-0.312833\pi\)
							−0.947266	+	0.320447i	\(0.896167\pi\)
\(11\)	−5.45054	+	0.357247i	−1.64340	+	0.107714i	−0.858807	−	0.512299i	\(-0.828794\pi\)
							−0.784591	+	0.620013i	\(0.787127\pi\)
\(13\)	−4.17789	−	4.76397i	−1.15874	−	1.32129i	−0.938274	−	0.345893i	\(-0.887576\pi\)
							−0.220464	−	0.975395i	\(-0.570757\pi\)
\(17\)	−2.66682	+	2.66682i	−0.646799	+	0.646799i	−0.952218	−	0.305419i	\(-0.901203\pi\)
							0.305419	+	0.952218i	\(0.401203\pi\)
\(19\)	−0.0844392	−	0.424505i	−0.0193717	−	0.0973880i	0.969890	−	0.243544i	\(-0.0783099\pi\)
							−0.989262	+	0.146156i	\(0.953310\pi\)
\(23\)	−1.98705	−	1.52472i	−0.414329	−	0.317926i	0.380476	−	0.924791i	\(-0.375760\pi\)
							−0.794805	+	0.606865i	\(0.792427\pi\)
\(29\)	−3.02838	−	6.14095i	−0.562357	−	1.14035i	−0.973481	−	0.228767i	\(-0.926530\pi\)
							0.411124	−	0.911579i	\(-0.365136\pi\)
\(31\)	−4.60137	+	2.65660i	−0.826430	+	0.477140i	−0.852629	−	0.522517i	\(-0.824993\pi\)
							0.0261987	+	0.999657i	\(0.491660\pi\)
\(37\)	0.0591265	−	0.297249i	0.00972033	−	0.0488674i	−0.975621	−	0.219463i	\(-0.929569\pi\)
							0.985341	+	0.170596i	\(0.0545693\pi\)
\(41\)	6.21223	+	4.76681i	0.970187	+	0.744451i	0.966798	−	0.255543i	\(-0.0822544\pi\)
							0.00338960	+	0.999994i	\(0.498921\pi\)
\(43\)	−0.447490	−	6.82737i	−0.0682415	−	1.04116i	−0.885424	−	0.464785i	\(-0.846132\pi\)
							0.817182	−	0.576380i	\(-0.195535\pi\)
\(47\)	2.77339	+	10.3504i	0.404541	+	1.50977i	0.804901	+	0.593409i	\(0.202218\pi\)
							−0.400361	+	0.916358i	\(0.631115\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.133.35	yes	1504
9.4	even	3	inner	576.2.bm.a.517.30	yes	1504
64.13	even	16	inner	576.2.bm.a.205.30	yes	1504
576.13	even	48	inner	576.2.bm.a.13.35	✓	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.35	✓	1504	576.13	even	48	inner
576.2.bm.a.133.35	yes	1504	1.1	even	1	trivial
576.2.bm.a.205.30	yes	1504	64.13	even	16	inner
576.2.bm.a.517.30	yes	1504	9.4	even	3	inner