Embedded newform 576.2.bl.a.11.2

• Label: 576.2.bl.a.11.2
• 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			11.2
Character	\(\chi\)	\(=\)	576.11
Dual form			576.2.bl.a.419.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41165 + 0.0851593i) q^{2} +(-1.66601 - 0.473724i) q^{3} +(1.98550 - 0.240430i) q^{4} +(-1.72799 - 0.852148i) q^{5} +(2.39216 + 0.526855i) q^{6} +(0.749706 + 0.977036i) q^{7} +(-2.78234 + 0.508485i) q^{8} +(2.55117 + 1.57846i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41165	+	0.0851593i	−0.998185	+	0.0602167i
							\(3\)	−1.66601	−	0.473724i	−0.961871	−	0.273505i
\(5\)	−1.72799	−	0.852148i	−0.772778	−	0.381092i								0.0127492	−	0.999919i	\(-0.495942\pi\)
							−0.785528	+	0.618827i	\(0.787608\pi\)
\(7\)	0.749706	+	0.977036i	0.283362	+	0.369285i	0.913249	−	0.407402i	\(-0.133565\pi\)
							−0.629887	+	0.776687i	\(0.716899\pi\)
\(11\)	−1.03280	−	0.905738i	−0.311400	−	0.273090i	0.489352	−	0.872086i	\(-0.337233\pi\)
							−0.800752	+	0.598996i	\(0.795567\pi\)
\(13\)	1.67090	+	0.109517i	0.463425	+	0.0303745i	0.295328	−	0.955396i	\(-0.404571\pi\)
							0.168097	+	0.985770i	\(0.446238\pi\)
\(17\)	2.19144	+	2.19144i	0.531502	+	0.531502i	0.921019	−	0.389517i	\(-0.127358\pi\)
							−0.389517	+	0.921019i	\(0.627358\pi\)
\(19\)	−0.262896	+	0.393452i	−0.0603126	+	0.0902642i	−0.860400	−	0.509619i	\(-0.829786\pi\)
							0.800088	+	0.599883i	\(0.204786\pi\)
\(23\)	−1.13581	+	1.48022i	−0.236833	+	0.308646i	−0.896615	−	0.442810i	\(-0.853982\pi\)
							0.659783	+	0.751456i	\(0.270648\pi\)
\(29\)	2.37989	−	7.01094i	0.441935	−	1.30190i	−0.466740	−	0.884395i	\(-0.654571\pi\)
							0.908675	−	0.417505i	\(-0.137095\pi\)
\(31\)	0.773053	+	1.33897i	0.138844	+	0.240486i	0.927059	−	0.374914i	\(-0.122328\pi\)
							−0.788215	+	0.615400i	\(0.788995\pi\)
\(37\)	1.46296	−	0.977517i	0.240509	−	0.160703i	−0.429475	−	0.903079i	\(-0.641301\pi\)
							0.669983	+	0.742376i	\(0.266301\pi\)
\(41\)	−1.02065	−	0.783171i	−0.159398	−	0.122311i	0.525983	−	0.850495i	\(-0.323698\pi\)
							−0.685381	+	0.728185i	\(0.740364\pi\)
\(43\)	−6.95908	−	6.10295i	−1.06125	−	0.930691i	−0.0635157	−	0.997981i	\(-0.520231\pi\)
							−0.997733	+	0.0672903i	\(0.978565\pi\)
\(47\)	−2.32556	−	8.67912i	−0.339218	−	1.26598i	−0.899223	−	0.437490i	\(-0.855867\pi\)
							0.560005	−	0.828489i	\(-0.310799\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.11.2	✓	1504
9.5	odd	6	inner	576.2.bl.a.203.36	yes	1504
64.35	odd	16	inner	576.2.bl.a.227.36	yes	1504
576.419	even	48	inner	576.2.bl.a.419.2	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.11.2	✓	1504	1.1	even	1	trivial
576.2.bl.a.203.36	yes	1504	9.5	odd	6	inner
576.2.bl.a.227.36	yes	1504	64.35	odd	16	inner
576.2.bl.a.419.2	yes	1504	576.419	even	48	inner