Embedded newform 576.2.bl.a.83.11

• Label: 576.2.bl.a.83.11
• Level: $576$
• Weight: $2$
• Character: 576.83
• 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			83.11
Character	\(\chi\)	\(=\)	576.83
Dual form			576.2.bl.a.347.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.33110 + 0.477678i) q^{2} +(0.934604 + 1.45826i) q^{3} +(1.54365 - 1.27167i) q^{4} +(1.57302 + 1.79369i) q^{5} +(-1.94063 - 1.49464i) q^{6} +(-0.379595 + 2.88331i) q^{7} +(-1.44729 + 2.43009i) q^{8} +(-1.25303 + 2.72579i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1504 q - 24 q^{2} - 16 q^{3} - 8 q^{4} - 24 q^{5} - 16 q^{6} - 8 q^{7} - 16 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(-1\)	\(e\left(\frac{7}{16}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.33110	+	0.477678i	−0.941229	+	0.337770i
							\(3\)	0.934604	+	1.45826i	0.539594	+	0.841925i
\(5\)	1.57302	+	1.79369i	0.703478	+	0.802163i								0.987661	−	0.156607i	\(-0.0500557\pi\)
							−0.284183	+	0.958770i	\(0.591722\pi\)
\(7\)	−0.379595	+	2.88331i	−0.143473	+	1.08979i	0.756547	+	0.653940i	\(0.226885\pi\)
							−0.900020	+	0.435849i	\(0.856448\pi\)
\(11\)	4.15967	−	2.05132i	1.25419	−	0.618497i	0.310891	−	0.950446i	\(-0.399373\pi\)
							0.943298	+	0.331948i	\(0.107706\pi\)
\(13\)	−4.81605	+	1.63483i	−1.33573	+	0.453420i	−0.895738	−	0.444582i	\(-0.853352\pi\)
							−0.439993	+	0.898001i	\(0.645019\pi\)
\(17\)	0.291199	−	0.291199i	0.0706261	−	0.0706261i	−0.670911	−	0.741538i	\(-0.734097\pi\)
							0.741538	+	0.670911i	\(0.234097\pi\)
\(19\)	−1.13368	+	5.69939i	−0.260084	+	1.30753i	0.601075	+	0.799193i	\(0.294739\pi\)
							−0.861158	+	0.508337i	\(0.830261\pi\)
\(23\)	−1.09286	−	8.30109i	−0.227877	−	1.73090i	−0.597635	−	0.801768i	\(-0.703893\pi\)
							0.369758	−	0.929128i	\(-0.379440\pi\)
\(29\)	5.91340	−	0.387584i	1.09809	−	0.0719726i	0.494432	−	0.869216i	\(-0.335376\pi\)
							0.603658	+	0.797244i	\(0.293709\pi\)
\(31\)	−1.09637	−	1.89897i	−0.196914	−	0.341064i	0.750613	−	0.660743i	\(-0.229758\pi\)
							−0.947526	+	0.319678i	\(0.896425\pi\)
\(37\)	7.87307	−	1.56605i	1.29432	−	0.257457i	0.500570	−	0.865696i	\(-0.333124\pi\)
							0.793754	+	0.608239i	\(0.208124\pi\)
\(41\)	8.57820	−	1.12934i	1.33969	−	0.176374i	0.573615	−	0.819125i	\(-0.305541\pi\)
							0.766075	+	0.642752i	\(0.222207\pi\)
\(43\)	0.566293	−	0.279265i	0.0863588	−	0.0425874i	−0.398596	−	0.917126i	\(-0.630503\pi\)
							0.484955	+	0.874539i	\(0.338836\pi\)
\(47\)	−12.3323	+	3.30442i	−1.79885	+	0.482000i	−0.993797	−	0.111208i	\(-0.964528\pi\)
							−0.805050	+	0.593207i	\(0.797861\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	576.2.bl.a.83.11	✓	1504
9.5	odd	6	inner	576.2.bl.a.275.45	yes	1504
64.27	odd	16	inner	576.2.bl.a.155.45	yes	1504
576.347	even	48	inner	576.2.bl.a.347.11	yes	1504

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.83.11	✓	1504	1.1	even	1	trivial
576.2.bl.a.155.45	yes	1504	64.27	odd	16	inner
576.2.bl.a.275.45	yes	1504	9.5	odd	6	inner
576.2.bl.a.347.11	yes	1504	576.347	even	48	inner