Embedded newform 576.2.bm.a.13.7

• Label: 576.2.bm.a.13.7
• 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			13.7
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.35456 + 0.406401i) q^{2} +(0.757412 - 1.55767i) q^{3} +(1.66968 - 1.10099i) q^{4} +(-0.620708 + 1.82855i) q^{5} +(-0.392925 + 2.41777i) q^{6} +(-1.17718 + 1.53413i) q^{7} +(-1.81424 + 2.16992i) q^{8} +(-1.85265 - 2.35959i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.35456	+	0.406401i	−0.957820	+	0.287369i
							\(3\)	0.757412	−	1.55767i	0.437292	−	0.899319i
\(5\)	−0.620708	+	1.82855i	−0.277589	+	0.817751i								0.715312	+	0.698806i	\(0.246285\pi\)
							−0.992901	+	0.118946i	\(0.962049\pi\)
\(7\)	−1.17718	+	1.53413i	−0.444931	+	0.579845i	−0.960947	−	0.276734i	\(-0.910748\pi\)
							0.516016	+	0.856579i	\(0.327415\pi\)
\(11\)	−2.61156	−	0.171171i	−0.787416	−	0.0516100i	−0.333611	−	0.942711i	\(-0.608267\pi\)
							−0.453805	+	0.891101i	\(0.649934\pi\)
\(13\)	2.50352	−	2.85471i	0.694351	−	0.791755i	−0.291999	−	0.956419i	\(-0.594320\pi\)
							0.986350	+	0.164663i	\(0.0526538\pi\)
\(17\)	−3.52760	−	3.52760i	−0.855569	−	0.855569i	0.135244	−	0.990812i	\(-0.456818\pi\)
							−0.990812	+	0.135244i	\(0.956818\pi\)
\(19\)	1.42427	−	7.16030i	0.326750	−	1.64268i	−0.372663	−	0.927967i	\(-0.621555\pi\)
							0.699413	−	0.714718i	\(-0.253445\pi\)
\(23\)	7.28139	−	5.58721i	1.51828	−	1.16501i	0.582287	−	0.812983i	\(-0.302158\pi\)
							0.935988	−	0.352031i	\(-0.114509\pi\)
\(29\)	1.50594	−	3.05375i	0.279646	−	0.567067i	−0.711053	−	0.703139i	\(-0.751781\pi\)
							0.990699	+	0.136072i	\(0.0434479\pi\)
\(31\)	−5.33684	−	3.08122i	−0.958524	−	0.553404i	−0.0628057	−	0.998026i	\(-0.520005\pi\)
							−0.895719	+	0.444622i	\(0.853338\pi\)
\(37\)	0.105545	+	0.530609i	0.0173515	+	0.0872317i	0.988491	−	0.151281i	\(-0.0483400\pi\)
							−0.971139	+	0.238513i	\(0.923340\pi\)
\(41\)	−2.04249	+	1.56726i	−0.318983	+	0.244764i	−0.755825	−	0.654774i	\(-0.772764\pi\)
							0.436842	+	0.899538i	\(0.356097\pi\)
\(43\)	0.628799	−	9.59362i	0.0958909	−	1.46301i	−0.630745	−	0.775990i	\(-0.717251\pi\)
							0.726636	−	0.687023i	\(-0.241083\pi\)
\(47\)	0.188209	−	0.702405i	0.0274531	−	0.102456i	−0.950840	−	0.309683i	\(-0.899777\pi\)
							0.978293	+	0.207227i	\(0.0664438\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.7	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.7	yes	1504	576.133	even	48	inner
576.2.bm.a.205.70	yes	1504	9.7	even	3	inner
576.2.bm.a.517.70	yes	1504	64.5	even	16	inner