Embedded newform 384.3.u.a.235.43

• Label: 384.3.u.a.235.43
• 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			235.43
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.43

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.961577 + 1.75367i) q^{2} +(1.09880 - 1.33889i) q^{3} +(-2.15074 + 3.37258i) q^{4} +(-1.93451 - 6.37721i) q^{5} +(3.40456 + 0.639489i) q^{6} +(-0.200299 + 1.00697i) q^{7} +(-7.98251 - 0.528691i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.961577	+	1.75367i	0.480789	+	0.876837i
							\(3\)	1.09880	−	1.33889i	0.366267	−	0.446298i
\(5\)	−1.93451	−	6.37721i	−0.386901	−	1.27544i								−0.906925	−	0.421291i	\(-0.861577\pi\)
							0.520024	−	0.854152i	\(-0.325923\pi\)
\(7\)	−0.200299	+	1.00697i	−0.0286141	+	0.143853i	−0.992451	−	0.122639i	\(-0.960864\pi\)
							0.963837	+	0.266491i	\(0.0858644\pi\)
\(11\)	−18.2109	−	1.79362i	−1.65554	−	0.163056i	−0.773359	−	0.633968i	\(-0.781425\pi\)
							−0.882180	+	0.470912i	\(0.843925\pi\)
\(13\)	11.6950	+	3.54765i	0.899619	+	0.272896i	0.706026	−	0.708186i	\(-0.250486\pi\)
							0.193592	+	0.981082i	\(0.437986\pi\)
\(17\)	−9.47053	−	22.8639i	−0.557090	−	1.34493i	−0.912060	−	0.410057i	\(-0.865509\pi\)
							0.354970	−	0.934878i	\(-0.384491\pi\)
\(19\)	−2.08580	+	1.11488i	−0.109779	+	0.0586780i	−0.525372	−	0.850873i	\(-0.676074\pi\)
							0.415593	+	0.909550i	\(0.363574\pi\)
\(23\)	−23.0293	−	15.3877i	−1.00127	−	0.669030i	−0.0570627	−	0.998371i	\(-0.518173\pi\)
							−0.944211	+	0.329341i	\(0.893173\pi\)
\(29\)	−6.59134	+	0.649190i	−0.227287	+	0.0223859i	−0.211020	−	0.977482i	\(-0.567678\pi\)
							−0.0162678	+	0.999868i	\(0.505178\pi\)
\(31\)	17.1000	−	17.1000i	0.551612	−	0.551612i	−0.375294	−	0.926906i	\(-0.622458\pi\)
							0.926906	+	0.375294i	\(0.122458\pi\)
\(37\)	−50.1728	−	26.8179i	−1.35602	−	0.724809i	−0.376551	−	0.926396i	\(-0.622890\pi\)
							−0.979471	+	0.201587i	\(0.935390\pi\)
\(41\)	59.8257	+	39.9743i	1.45916	+	0.974982i	0.996060	+	0.0886783i	\(0.0282643\pi\)
							0.463104	+	0.886304i	\(0.346736\pi\)
\(43\)	−30.7831	−	37.5093i	−0.715885	−	0.872309i	0.280269	−	0.959921i	\(-0.409576\pi\)
							−0.996155	+	0.0876127i	\(0.972076\pi\)
\(47\)	20.8846	+	50.4200i	0.444354	+	1.07277i	0.974405	+	0.224800i	\(0.0721728\pi\)
							−0.530051	+	0.847966i	\(0.677827\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.43	yes	1024
128.67	odd	32	inner	384.3.u.a.67.43	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.43	✓	1024	128.67	odd	32	inner
384.3.u.a.235.43	yes	1024	1.1	even	1	trivial