Embedded newform 576.2.bl.a.347.11

• Label: 576.2.bl.a.347.11
• Level: $576$
• Weight: $2$
• Character: 576.347
• 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			347.11
Character	\(\chi\)	\(=\)	576.347
Dual form			576.2.bl.a.83.11

$q$-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 \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\)	\(65\)	\(127\)	\(325\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(-1\)	\(e\left(\frac{9}{16}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.bl.a.83.11	✓	1504	576.83	even	48	inner
576.2.bl.a.155.45	yes	1504	9.2	odd	6	inner
576.2.bl.a.275.45	yes	1504	64.19	odd	16	inner
576.2.bl.a.347.11	yes	1504	1.1	even	1	trivial