Embedded newform 576.2.bm.a.13.36

• Label: 576.2.bm.a.13.36
• 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			13.36
Character	\(\chi\)	\(=\)	576.13
Dual form			576.2.bm.a.133.36

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.569193 - 1.29461i) q^{2} +(-0.135731 + 1.72672i) q^{3} +(-1.35204 + 1.47377i) q^{4} +(-1.08251 + 3.18896i) q^{5} +(2.31269 - 0.807120i) q^{6} +(-2.06358 + 2.68931i) q^{7} +(2.67753 + 0.911510i) q^{8} +(-2.96315 - 0.468740i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 8 q^{2} - 16 q^{3} - 8 q^{4} - 8 q^{5} - 16 q^{6} - 8 q^{7} - 32 q^{8} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.569193	−	1.29461i	−0.402480	−	0.915429i
							\(3\)	−0.135731	+	1.72672i	−0.0783642	+	0.996925i
\(5\)	−1.08251	+	3.18896i	−0.484111	+	1.42615i								0.381004	+	0.924573i	\(0.375578\pi\)
							−0.865115	+	0.501573i	\(0.832755\pi\)
\(7\)	−2.06358	+	2.68931i	−0.779960	+	1.01646i	0.219208	+	0.975678i	\(0.429653\pi\)
							−0.999168	+	0.0407854i	\(0.987014\pi\)
\(11\)	2.20580	+	0.144576i	0.665074	+	0.0435912i	0.394201	−	0.919024i	\(-0.371021\pi\)
							0.270872	+	0.962615i	\(0.412688\pi\)
\(13\)	1.23613	−	1.40954i	0.342842	−	0.390936i	−0.554565	−	0.832141i	\(-0.687115\pi\)
							0.897407	+	0.441204i	\(0.145449\pi\)
\(17\)	−1.12378	−	1.12378i	−0.272556	−	0.272556i	0.557572	−	0.830128i	\(-0.311733\pi\)
							−0.830128	+	0.557572i	\(0.811733\pi\)
\(19\)	0.386267	−	1.94189i	0.0886157	−	0.445501i	−0.910848	−	0.412743i	\(-0.864571\pi\)
							0.999463	−	0.0327584i	\(-0.0104292\pi\)
\(23\)	−5.59851	+	4.29589i	−1.16737	+	0.895755i	−0.995991	−	0.0894498i	\(-0.971489\pi\)
							−0.171380	+	0.985205i	\(0.554822\pi\)
\(29\)	−1.45145	+	2.94325i	−0.269528	+	0.546548i	−0.988990	−	0.147984i	\(-0.952722\pi\)
							0.719462	+	0.694532i	\(0.244388\pi\)
\(31\)	6.32219	+	3.65012i	1.13550	+	0.655581i	0.945312	−	0.326167i	\(-0.105757\pi\)
							0.190187	+	0.981748i	\(0.439091\pi\)
\(37\)	−1.66107	−	8.35075i	−0.273078	−	1.37286i	−0.837079	−	0.547082i	\(-0.815739\pi\)
							0.564001	−	0.825774i	\(-0.309261\pi\)
\(41\)	1.79521	−	1.37752i	0.280365	−	0.215132i	−0.459000	−	0.888436i	\(-0.651792\pi\)
							0.739365	+	0.673304i	\(0.235126\pi\)
\(43\)	0.241188	−	3.67981i	0.0367808	−	0.561166i	−0.939208	−	0.343347i	\(-0.888439\pi\)
							0.975989	−	0.217819i	\(-0.0698941\pi\)
\(47\)	1.36500	−	5.09425i	0.199106	−	0.743072i	−0.792060	−	0.610443i	\(-0.790991\pi\)
							0.991166	−	0.132629i	\(-0.0423419\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bm.a.13.36	✓	1504
9.7	even	3	inner	576.2.bm.a.205.28	yes	1504
64.5	even	16	inner	576.2.bm.a.517.28	yes	1504
576.133	even	48	inner	576.2.bm.a.133.36	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bm.a.13.36	✓	1504	1.1	even	1	trivial
576.2.bm.a.133.36	yes	1504	576.133	even	48	inner
576.2.bm.a.205.28	yes	1504	9.7	even	3	inner
576.2.bm.a.517.28	yes	1504	64.5	even	16	inner