Embedded newform 384.3.u.a.235.3

• Label: 384.3.u.a.235.3
• Level: $384$
• Weight: $3$
• Character: 384.235
• Analytic conductor: $10.463$
• Analytic rank: $0$
• Dimension: $1024$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.u (of order \(32\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.4632421514\)
Analytic rank:	\(0\)
Dimension:	\(1024\)
Relative dimension:	\(64\) over \(\Q(\zeta_{32})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{32}]$

Embedding invariants

Embedding label			235.3
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.97899 - 0.289161i) q^{2} +(-1.09880 + 1.33889i) q^{3} +(3.83277 + 1.14449i) q^{4} +(2.38572 + 7.86467i) q^{5} +(2.56167 - 2.33192i) q^{6} +(2.36133 - 11.8712i) q^{7} +(-7.25406 - 3.37322i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(133\)	\(257\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{13}{32}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.97899	−	0.289161i	−0.989493	−	0.144581i
							\(3\)	−1.09880	+	1.33889i	−0.366267	+	0.446298i
\(5\)	2.38572	+	7.86467i	0.477145	+	1.57293i								0.783736	+	0.621094i	\(0.213311\pi\)
							−0.306592	+	0.951841i	\(0.599189\pi\)
\(7\)	2.36133	−	11.8712i	0.337333	−	1.69589i	−0.324214	−	0.945984i	\(-0.605100\pi\)
							0.661547	−	0.749904i	\(-0.269900\pi\)
\(11\)	0.949880	+	0.0935550i	0.0863527	+	0.00850500i	0.141101	−	0.989995i	\(-0.454936\pi\)
							−0.0547485	+	0.998500i	\(0.517436\pi\)
\(13\)	−23.6232	−	7.16602i	−1.81717	−	0.551232i	−0.817475	−	0.575964i	\(-0.804627\pi\)
							−0.999694	+	0.0247320i	\(0.992127\pi\)
\(17\)	−5.56476	−	13.4345i	−0.327339	−	0.790265i	−0.998788	−	0.0492151i	\(-0.984328\pi\)
							0.671450	−	0.741050i	\(-0.265672\pi\)
\(19\)	16.7745	−	8.96617i	0.882870	−	0.471904i	0.0333400	−	0.999444i	\(-0.489386\pi\)
							0.849530	+	0.527540i	\(0.176886\pi\)
\(23\)	−19.3840	−	12.9519i	−0.842781	−	0.563128i	0.0575458	−	0.998343i	\(-0.481672\pi\)
							−0.900327	+	0.435215i	\(0.856672\pi\)
\(29\)	43.5839	−	4.29264i	1.50289	−	0.148022i	0.687183	−	0.726484i	\(-0.258847\pi\)
							0.815709	+	0.578462i	\(0.196347\pi\)
\(31\)	−5.14591	+	5.14591i	−0.165997	+	0.165997i	−0.785217	−	0.619220i	\(-0.787449\pi\)
							0.619220	+	0.785217i	\(0.287449\pi\)
\(37\)	−39.8866	−	21.3199i	−1.07802	−	0.576212i	−0.165900	−	0.986143i	\(-0.553053\pi\)
							−0.912117	+	0.409930i	\(0.865553\pi\)
\(41\)	−25.7456	−	17.2026i	−0.627940	−	0.419576i	0.200469	−	0.979700i	\(-0.435754\pi\)
							−0.828409	+	0.560124i	\(0.810754\pi\)
\(43\)	−22.0447	−	26.8615i	−0.512667	−	0.624687i	0.450764	−	0.892643i	\(-0.351152\pi\)
							−0.963431	+	0.267957i	\(0.913652\pi\)
\(47\)	6.51499	+	15.7286i	0.138617	+	0.334650i	0.977909	−	0.209030i	\(-0.0670305\pi\)
							−0.839293	+	0.543680i	\(0.817030\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.3	yes	1024
128.67	odd	32	inner	384.3.u.a.67.3	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.3	✓	1024	128.67	odd	32	inner
384.3.u.a.235.3	yes	1024	1.1	even	1	trivial