Embedded newform 576.2.bm.a.517.70

• Label: 576.2.bm.a.517.70
• Level: $576$
• Weight: $2$
• Character: 576.517
• 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			517.70
Character	\(\chi\)	\(=\)	576.517
Dual form			576.2.bm.a.205.70

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.02923 - 0.969885i) q^{2} +(1.72895 + 0.103664i) q^{3} +(0.118648 - 1.99648i) q^{4} +(-1.27321 + 1.45182i) q^{5} +(1.88003 - 1.57018i) q^{6} +(1.91718 - 0.252402i) q^{7} +(-1.81424 - 2.16992i) q^{8} +(2.97851 + 0.358461i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.02923	−	0.969885i	0.727779	−	0.685812i
							\(3\)	1.72895	+	0.103664i	0.998207	+	0.0598507i
\(5\)	−1.27321	+	1.45182i	−0.569399	+	0.649275i								−0.962839	−	0.270075i	\(-0.912951\pi\)
							0.393440	+	0.919350i	\(0.371285\pi\)
\(7\)	1.91718	−	0.252402i	0.724626	−	0.0953988i	0.240815	−	0.970571i	\(-0.422585\pi\)
							0.483811	+	0.875172i	\(0.339252\pi\)
\(11\)	1.15754	−	2.34727i	0.349012	−	0.707727i	−0.649606	−	0.760271i	\(-0.725066\pi\)
							0.998619	+	0.0525437i	\(0.0167329\pi\)
\(13\)	1.22050	−	3.59547i	0.338505	−	0.997203i	−0.635778	−	0.771872i	\(-0.719320\pi\)
							0.974282	−	0.225331i	\(-0.0723462\pi\)
\(17\)	−3.52760	+	3.52760i	−0.855569	+	0.855569i	−0.990812	−	0.135244i	\(-0.956818\pi\)
							0.135244	+	0.990812i	\(0.456818\pi\)
\(19\)	1.42427	+	7.16030i	0.326750	+	1.64268i	0.699413	+	0.714718i	\(0.253445\pi\)
							−0.372663	+	0.927967i	\(0.621555\pi\)
\(23\)	1.19797	−	9.09948i	0.249794	−	1.89737i	−0.163127	−	0.986605i	\(-0.552158\pi\)
							0.412920	−	0.910767i	\(-0.364509\pi\)
\(29\)	−3.39759	−	0.222690i	−0.630917	−	0.0413525i	−0.253410	−	0.967359i	\(-0.581552\pi\)
							−0.377507	+	0.926007i	\(0.623219\pi\)
\(31\)	5.33684	+	3.08122i	0.958524	+	0.553404i	0.895719	−	0.444622i	\(-0.146662\pi\)
							0.0628057	+	0.998026i	\(0.479995\pi\)
\(37\)	0.105545	−	0.530609i	0.0173515	−	0.0872317i	−0.971139	−	0.238513i	\(-0.923340\pi\)
							0.988491	+	0.151281i	\(0.0483400\pi\)
\(41\)	−0.336040	+	2.55247i	−0.0524806	+	0.398630i	0.944963	+	0.327176i	\(0.106097\pi\)
							−0.997444	+	0.0714534i	\(0.977236\pi\)
\(43\)	−8.62272	−	4.25225i	−1.31495	−	0.648463i	−0.356655	−	0.934236i	\(-0.616083\pi\)
							−0.958297	+	0.285774i	\(0.907749\pi\)
\(47\)	−0.702405	−	0.188209i	−0.102456	−	0.0274531i	0.207227	−	0.978293i	\(-0.433556\pi\)
							−0.309683	+	0.950840i	\(0.600223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.517.70	yes	1504
9.7	even	3	inner	576.2.bm.a.133.7	yes	1504
64.13	even	16	inner	576.2.bm.a.13.7	✓	1504
576.205	even	48	inner	576.2.bm.a.205.70	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.7	✓	1504	64.13	even	16	inner
576.2.bm.a.133.7	yes	1504	9.7	even	3	inner
576.2.bm.a.205.70	yes	1504	576.205	even	48	inner
576.2.bm.a.517.70	yes	1504	1.1	even	1	trivial