Embedded newform 576.2.bm.a.13.24

• Label: 576.2.bm.a.13.24
• 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			13.24
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.953638 - 1.04431i) q^{2} +(1.12996 - 1.31270i) q^{3} +(-0.181147 + 1.99178i) q^{4} +(1.16036 - 3.41831i) q^{5} +(-2.44844 + 0.0718207i) q^{6} +(-2.24164 + 2.92136i) q^{7} +(2.25278 - 1.71026i) q^{8} +(-0.446379 - 2.96661i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.953638	−	1.04431i	−0.674324	−	0.738435i
							\(3\)	1.12996	−	1.31270i	0.652383	−	0.757890i
\(5\)	1.16036	−	3.41831i	0.518928	−	1.52871i								−0.299495	−	0.954098i	\(-0.596818\pi\)
							0.818423	−	0.574616i	\(-0.194849\pi\)
\(7\)	−2.24164	+	2.92136i	−0.847260	+	1.10417i	0.146117	+	0.989267i	\(0.453322\pi\)
							−0.993377	+	0.114903i	\(0.963344\pi\)
\(11\)	4.11490	+	0.269705i	1.24069	+	0.0813191i	0.671573	−	0.740938i	\(-0.265619\pi\)
							0.569116	+	0.822257i	\(0.307286\pi\)
\(13\)	2.08461	−	2.37704i	0.578167	−	0.659273i	−0.386640	−	0.922231i	\(-0.626365\pi\)
							0.964807	+	0.262957i	\(0.0846979\pi\)
\(17\)	−4.41542	−	4.41542i	−1.07090	−	1.07090i	−0.997287	−	0.0736098i	\(-0.976548\pi\)
							−0.0736098	−	0.997287i	\(-0.523452\pi\)
\(19\)	0.330635	−	1.66221i	0.0758529	−	0.381338i	−0.924147	−	0.382037i	\(-0.875223\pi\)
							1.00000		0.000699247i	\(0.000222577\pi\)
\(23\)	−3.53021	+	2.70883i	−0.736100	+	0.564829i	−0.907339	−	0.420400i	\(-0.861890\pi\)
							0.171239	+	0.985230i	\(0.445223\pi\)
\(29\)	2.10605	−	4.27065i	0.391084	−	0.793039i	−0.608916	−	0.793234i	\(-0.708395\pi\)
							1.00000		0.000195289i	\(6.21625e-5\pi\)
\(31\)	4.26195	+	2.46064i	0.765469	+	0.441944i	0.831256	−	0.555890i	\(-0.187622\pi\)
							−0.0657867	+	0.997834i	\(0.520956\pi\)
\(37\)	1.25696	+	6.31914i	0.206642	+	1.03886i	0.935266	+	0.353946i	\(0.115160\pi\)
							−0.728624	+	0.684914i	\(0.759840\pi\)
\(41\)	−1.66673	+	1.27892i	−0.260299	+	0.199734i	−0.730663	−	0.682738i	\(-0.760789\pi\)
							0.470365	+	0.882472i	\(0.344122\pi\)
\(43\)	0.426056	−	6.50036i	0.0649730	−	0.991296i	−0.833571	−	0.552412i	\(-0.813707\pi\)
							0.898544	−	0.438884i	\(-0.144626\pi\)
\(47\)	−0.197201	+	0.735963i	−0.0287647	+	0.107351i	−0.978816	−	0.204743i	\(-0.934364\pi\)
							0.950051	+	0.312095i	\(0.101031\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.24	✓	1504
9.7	even	3	inner	576.2.bm.a.205.39	yes	1504
64.5	even	16	inner	576.2.bm.a.517.39	yes	1504
576.133	even	48	inner	576.2.bm.a.133.24	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.24	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.24	yes	1504	576.133	even	48	inner
576.2.bm.a.205.39	yes	1504	9.7	even	3	inner
576.2.bm.a.517.39	yes	1504	64.5	even	16	inner