Embedded newform 576.2.bl.a.11.18

• Label: 576.2.bl.a.11.18
• Level: $576$
• Weight: $2$
• Character: 576.11
• 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.bl (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			11.18
Character	\(\chi\)	\(=\)	576.11
Dual form			576.2.bl.a.419.18

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.18516 - 0.771613i) q^{2} +(-0.450297 - 1.67249i) q^{3} +(0.809226 + 1.82898i) q^{4} +(-3.24873 - 1.60210i) q^{5} +(-0.756842 + 2.32963i) q^{6} +(1.96855 + 2.56546i) q^{7} +(0.452197 - 2.79205i) q^{8} +(-2.59447 + 1.50624i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{5}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.18516	−	0.771613i	−0.838037	−	0.545613i
							\(3\)	−0.450297	−	1.67249i	−0.259979	−	0.965614i
\(5\)	−3.24873	−	1.60210i	−1.45288	−	0.716480i								−0.467048	−	0.884232i	\(-0.654683\pi\)
							−0.985829	+	0.167752i	\(0.946349\pi\)
\(7\)	1.96855	+	2.56546i	0.744042	+	0.969654i	0.999996	+	0.00296521i	\(0.000943855\pi\)
							−0.255954	+	0.966689i	\(0.582389\pi\)
\(11\)	4.16177	+	3.64978i	1.25482	+	1.10045i	0.990715	+	0.135956i	\(0.0434106\pi\)
							0.264107	+	0.964493i	\(0.414923\pi\)
\(13\)	4.89891	+	0.321091i	1.35871	+	0.0890547i	0.727261	−	0.686361i	\(-0.240793\pi\)
							0.631451	+	0.775416i	\(0.282460\pi\)
\(17\)	−1.39317	−	1.39317i	−0.337893	−	0.337893i	0.517681	−	0.855574i	\(-0.326795\pi\)
							−0.855574	+	0.517681i	\(0.826795\pi\)
\(19\)	2.51433	−	3.76296i	0.576826	−	0.863282i	−0.422240	−	0.906484i	\(-0.638756\pi\)
							0.999066	+	0.0432024i	\(0.0137560\pi\)
\(23\)	−0.984046	+	1.28243i	−0.205188	+	0.267406i	−0.884600	−	0.466351i	\(-0.845568\pi\)
							0.679412	+	0.733757i	\(0.262235\pi\)
\(29\)	0.345628	−	1.01819i	0.0641816	−	0.189073i	−0.910109	−	0.414370i	\(-0.864002\pi\)
							0.974290	+	0.225297i	\(0.0723353\pi\)
\(31\)	−4.41373	−	7.64480i	−0.792729	−	1.37305i	−0.924271	−	0.381736i	\(-0.875326\pi\)
							0.131543	−	0.991311i	\(-0.458007\pi\)
\(37\)	−3.61737	+	2.41705i	−0.594692	+	0.397361i	−0.816176	−	0.577803i	\(-0.803910\pi\)
							0.221484	+	0.975164i	\(0.428910\pi\)
\(41\)	3.15074	+	2.41765i	0.492063	+	0.377573i	0.824780	−	0.565453i	\(-0.191299\pi\)
							−0.332717	+	0.943027i	\(0.607965\pi\)
\(43\)	1.41288	+	1.23906i	0.215462	+	0.188956i	0.760273	−	0.649604i	\(-0.225065\pi\)
							−0.544811	+	0.838559i	\(0.683398\pi\)
\(47\)	2.50570	+	9.35139i	0.365494	+	1.36404i	0.866751	+	0.498742i	\(0.166204\pi\)
							−0.501257	+	0.865298i	\(0.667129\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.11.18	✓	1504
9.5	odd	6	inner	576.2.bl.a.203.15	yes	1504
64.35	odd	16	inner	576.2.bl.a.227.15	yes	1504
576.419	even	48	inner	576.2.bl.a.419.18	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.11.18	✓	1504	1.1	even	1	trivial
576.2.bl.a.203.15	yes	1504	9.5	odd	6	inner
576.2.bl.a.227.15	yes	1504	64.35	odd	16	inner
576.2.bl.a.419.18	yes	1504	576.419	even	48	inner