Embedded newform 576.2.bm.a.13.25

• Label: 576.2.bm.a.13.25
• 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			13.25
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.950049 + 1.04757i) q^{2} +(1.70894 - 0.282012i) q^{3} +(-0.194815 - 1.99049i) q^{4} +(0.762090 - 2.24504i) q^{5} +(-1.32815 + 2.05816i) q^{6} +(0.172281 - 0.224521i) q^{7} +(2.27026 + 1.68698i) q^{8} +(2.84094 - 0.963882i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.950049	+	1.04757i	−0.671786	+	0.740745i
							\(3\)	1.70894	−	0.282012i	0.986656	−	0.162820i
\(5\)	0.762090	−	2.24504i	0.340817	−	1.00401i								−0.632502	−	0.774559i	\(-0.717972\pi\)
							0.973319	−	0.229456i	\(-0.0736946\pi\)
\(7\)	0.172281	−	0.224521i	0.0651161	−	0.0848610i	−0.759660	−	0.650321i	\(-0.774635\pi\)
							0.824776	+	0.565460i	\(0.191301\pi\)
\(11\)	0.890807	+	0.0583866i	0.268589	+	0.0176042i	0.199102	−	0.979979i	\(-0.436197\pi\)
							0.0694861	+	0.997583i	\(0.477864\pi\)
\(13\)	−0.972436	+	1.10885i	−0.269705	+	0.307540i	−0.870800	−	0.491638i	\(-0.836398\pi\)
							0.601094	+	0.799178i	\(0.294732\pi\)
\(17\)	−3.08704	−	3.08704i	−0.748717	−	0.748717i	0.225522	−	0.974238i	\(-0.427591\pi\)
							−0.974238	+	0.225522i	\(0.927591\pi\)
\(19\)	−0.658744	+	3.31173i	−0.151126	+	0.759763i	0.828666	+	0.559744i	\(0.189100\pi\)
							−0.979792	+	0.200019i	\(0.935900\pi\)
\(23\)	6.76461	−	5.19067i	1.41052	−	1.08233i	0.426455	−	0.904509i	\(-0.359762\pi\)
							0.984063	−	0.177820i	\(-0.0569045\pi\)
\(29\)	2.30142	−	4.66682i	0.427363	−	0.866606i	−0.571576	−	0.820549i	\(-0.693668\pi\)
							0.998939	−	0.0460569i	\(-0.0146656\pi\)
\(31\)	−2.39713	−	1.38398i	−0.430536	−	0.248570i	0.269039	−	0.963129i	\(-0.413294\pi\)
							−0.699575	+	0.714559i	\(0.746627\pi\)
\(37\)	−1.25131	−	6.29074i	−0.205713	−	1.03419i	−0.936254	−	0.351323i	\(-0.885732\pi\)
							0.730541	−	0.682869i	\(-0.239268\pi\)
\(41\)	−4.82178	+	3.69988i	−0.753036	+	0.577825i	−0.912354	−	0.409403i	\(-0.865737\pi\)
							0.159318	+	0.987227i	\(0.449070\pi\)
\(43\)	−0.0912444	+	1.39212i	−0.0139146	+	0.212296i	0.985240	+	0.171180i	\(0.0547580\pi\)
							−0.999154	+	0.0411162i	\(0.986909\pi\)
\(47\)	−1.23844	+	4.62193i	−0.180646	+	0.674178i	0.814875	+	0.579636i	\(0.196805\pi\)
							−0.995521	+	0.0945421i	\(0.969861\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.25	✓	1504
9.7	even	3	inner	576.2.bm.a.205.88	yes	1504
64.5	even	16	inner	576.2.bm.a.517.88	yes	1504
576.133	even	48	inner	576.2.bm.a.133.25	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.25	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.25	yes	1504	576.133	even	48	inner
576.2.bm.a.205.88	yes	1504	9.7	even	3	inner
576.2.bm.a.517.88	yes	1504	64.5	even	16	inner