Embedded newform 576.2.bm.a.13.53

• Label: 576.2.bm.a.13.53
• 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			13.53
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.53

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.252933 + 1.39141i) q^{2} +(-0.850205 + 1.50902i) q^{3} +(-1.87205 + 0.703868i) q^{4} +(0.645564 - 1.90177i) q^{5} +(-2.31472 - 0.801304i) q^{6} +(1.89668 - 2.47180i) q^{7} +(-1.45287 - 2.42676i) q^{8} +(-1.55430 - 2.56596i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.252933	+	1.39141i	0.178851	+	0.983876i
							\(3\)	−0.850205	+	1.50902i	−0.490866	+	0.871235i
\(5\)	0.645564	−	1.90177i	0.288705	−	0.850498i								−0.701721	−	0.712452i	\(-0.747585\pi\)
							0.990426	−	0.138046i	\(-0.0440821\pi\)
\(7\)	1.89668	−	2.47180i	0.716876	−	0.934251i	−0.282813	−	0.959175i	\(-0.591268\pi\)
							0.999689	+	0.0249241i	\(0.00793441\pi\)
\(11\)	4.63108	+	0.303537i	1.39632	+	0.0915199i	0.744891	−	0.667187i	\(-0.232502\pi\)
							0.651433	+	0.758707i	\(0.274168\pi\)
\(13\)	0.0402318	−	0.0458756i	0.0111583	−	0.0127236i	−0.746234	−	0.665684i	\(-0.768140\pi\)
							0.757392	+	0.652960i	\(0.226473\pi\)
\(17\)	−2.39965	−	2.39965i	−0.582002	−	0.582002i	0.353451	−	0.935453i	\(-0.385008\pi\)
							−0.935453	+	0.353451i	\(0.885008\pi\)
\(19\)	0.881841	−	4.43331i	0.202308	−	1.01707i	−0.737494	−	0.675353i	\(-0.763991\pi\)
							0.939802	−	0.341718i	\(-0.111009\pi\)
\(23\)	−0.0971987	+	0.0745832i	−0.0202673	+	0.0155517i	−0.618845	−	0.785513i	\(-0.712399\pi\)
							0.598578	+	0.801064i	\(0.295733\pi\)
\(29\)	0.875850	−	1.77605i	0.162641	−	0.329804i	−0.800262	−	0.599650i	\(-0.795306\pi\)
							0.962903	+	0.269847i	\(0.0869730\pi\)
\(31\)	−1.81908	−	1.05024i	−0.326716	−	0.188630i	0.327666	−	0.944794i	\(-0.393738\pi\)
							−0.654382	+	0.756164i	\(0.727071\pi\)
\(37\)	0.634863	+	3.19167i	0.104371	+	0.524708i	0.997231	+	0.0743726i	\(0.0236954\pi\)
							−0.892860	+	0.450335i	\(0.851305\pi\)
\(41\)	−4.57307	+	3.50904i	−0.714193	+	0.548020i	−0.900704	−	0.434433i	\(-0.856949\pi\)
							0.186511	+	0.982453i	\(0.440282\pi\)
\(43\)	0.494051	−	7.53777i	0.0753421	−	1.14950i	−0.778041	−	0.628214i	\(-0.783786\pi\)
							0.853383	−	0.521285i	\(-0.174547\pi\)
\(47\)	−2.20262	+	8.22030i	−0.321285	+	1.19905i	0.596708	+	0.802458i	\(0.296475\pi\)
							−0.917994	+	0.396595i	\(0.870192\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.53	✓	1504
9.7	even	3	inner	576.2.bm.a.205.74	yes	1504
64.5	even	16	inner	576.2.bm.a.517.74	yes	1504
576.133	even	48	inner	576.2.bm.a.133.53	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.53	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.53	yes	1504	576.133	even	48	inner
576.2.bm.a.205.74	yes	1504	9.7	even	3	inner
576.2.bm.a.517.74	yes	1504	64.5	even	16	inner