Embedded newform 576.2.bm.a.13.74

• Label: 576.2.bm.a.13.74
• Level: $576$
• Weight: $2$
• Character: 576.13
• 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			13.74
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.74

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.01475 + 0.985030i) q^{2} +(-1.67809 - 0.428964i) q^{3} +(0.0594319 + 1.99912i) q^{4} +(0.139774 - 0.411762i) q^{5} +(-1.28030 - 2.08826i) q^{6} +(0.527770 - 0.687803i) q^{7} +(-1.90888 + 2.08714i) q^{8} +(2.63198 + 1.43968i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)	\(1\)	\(e\left(\frac{15}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.01475	+	0.985030i	0.717536	+	0.696521i
							\(3\)	−1.67809	−	0.428964i	−0.968846	−	0.247662i
\(5\)	0.139774	−	0.411762i	0.0625089	−	0.184145i								−0.911189	−	0.411989i	\(-0.864834\pi\)
							0.973698	+	0.227844i	\(0.0731675\pi\)
\(7\)	0.527770	−	0.687803i	0.199478	−	0.259965i	−0.682894	−	0.730518i	\(-0.739279\pi\)
							0.882372	+	0.470553i	\(0.155945\pi\)
\(11\)	4.67545	+	0.306445i	1.40970	+	0.0923967i	0.751147	−	0.660136i	\(-0.229501\pi\)
							0.658555	+	0.752532i	\(0.271168\pi\)
\(13\)	−3.02137	+	3.44522i	−0.837979	+	0.955531i	−0.999509	−	0.0313363i	\(-0.990024\pi\)
							0.161530	+	0.986868i	\(0.448357\pi\)
\(17\)	2.54210	+	2.54210i	0.616549	+	0.616549i	0.944645	−	0.328095i	\(-0.106407\pi\)
							−0.328095	+	0.944645i	\(0.606407\pi\)
\(19\)	0.422578	−	2.12444i	0.0969461	−	0.487381i	−0.901556	−	0.432663i	\(-0.857574\pi\)
							0.998502	−	0.0547181i	\(-0.0174260\pi\)
\(23\)	−7.23315	+	5.55019i	−1.50822	+	1.15730i	−0.565012	+	0.825083i	\(0.691129\pi\)
							−0.943204	+	0.332213i	\(0.892205\pi\)
\(29\)	−1.64095	+	3.32753i	−0.304717	+	0.617906i	−0.994338	−	0.106265i	\(-0.966111\pi\)
							0.689620	+	0.724171i	\(0.257777\pi\)
\(31\)	7.63171	+	4.40617i	1.37070	+	0.791372i	0.991016	−	0.133746i	\(-0.0427006\pi\)
							0.379680	+	0.925118i	\(0.376034\pi\)
\(37\)	1.03248	+	5.19062i	0.169738	+	0.853332i	0.967986	+	0.251003i	\(0.0807605\pi\)
							−0.798248	+	0.602329i	\(0.794240\pi\)
\(41\)	−2.23083	+	1.71177i	−0.348397	+	0.267334i	−0.768104	−	0.640326i	\(-0.778799\pi\)
							0.419707	+	0.907660i	\(0.362133\pi\)
\(43\)	0.267329	−	4.07865i	0.0407673	−	0.621988i	−0.927775	−	0.373140i	\(-0.878281\pi\)
							0.968542	−	0.248849i	\(-0.0800522\pi\)
\(47\)	2.97061	−	11.0865i	0.433309	−	1.61713i	−0.311773	−	0.950157i	\(-0.600923\pi\)
							0.745082	−	0.666973i	\(-0.232410\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.74	✓	1504
9.7	even	3	inner	576.2.bm.a.205.55	yes	1504
64.5	even	16	inner	576.2.bm.a.517.55	yes	1504
576.133	even	48	inner	576.2.bm.a.133.74	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.74	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.74	yes	1504	576.133	even	48	inner
576.2.bm.a.205.55	yes	1504	9.7	even	3	inner
576.2.bm.a.517.55	yes	1504	64.5	even	16	inner