Embedded newform 384.3.u.a.235.29

• Label: 384.3.u.a.235.29
• 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			235.29
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.29

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.363461 - 1.96670i) q^{2} +(-1.09880 + 1.33889i) q^{3} +(-3.73579 + 1.42964i) q^{4} +(2.57886 + 8.50138i) q^{5} +(3.03257 + 1.67437i) q^{6} +(-0.889106 + 4.46984i) q^{7} +(4.16947 + 6.82755i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.363461	−	1.96670i	−0.181731	−	0.983348i
							\(3\)	−1.09880	+	1.33889i	−0.366267	+	0.446298i
\(5\)	2.57886	+	8.50138i	0.515773	+	1.70028i								0.697205	+	0.716872i	\(0.254427\pi\)
							−0.181432	+	0.983403i	\(0.558073\pi\)
\(7\)	−0.889106	+	4.46984i	−0.127015	+	0.638548i	0.863856	+	0.503738i	\(0.168042\pi\)
							−0.990871	+	0.134810i	\(0.956958\pi\)
\(11\)	6.31958	+	0.622425i	0.574508	+	0.0565841i	0.381099	−	0.924534i	\(-0.375546\pi\)
							0.193409	+	0.981118i	\(0.438046\pi\)
\(13\)	−16.3435	−	4.95775i	−1.25719	−	0.381365i	−0.409723	−	0.912210i	\(-0.634375\pi\)
							−0.847469	+	0.530845i	\(0.821875\pi\)
\(17\)	−5.53108	−	13.3532i	−0.325357	−	0.785482i	−0.998925	−	0.0463567i	\(-0.985239\pi\)
							0.673567	−	0.739126i	\(-0.264761\pi\)
\(19\)	−24.4488	+	13.0681i	−1.28678	+	0.687797i	−0.965800	−	0.259289i	\(-0.916512\pi\)
							−0.320978	+	0.947087i	\(0.604012\pi\)
\(23\)	8.87615	+	5.93086i	0.385920	+	0.257863i	0.733362	−	0.679838i	\(-0.237950\pi\)
							−0.347443	+	0.937701i	\(0.612950\pi\)
\(29\)	−28.6943	+	2.82614i	−0.989458	+	0.0974531i	−0.579777	−	0.814775i	\(-0.696860\pi\)
							−0.409682	+	0.912229i	\(0.634360\pi\)
\(31\)	41.1292	−	41.1292i	1.32675	−	1.32675i	0.418558	−	0.908190i	\(-0.362536\pi\)
							0.908190	−	0.418558i	\(-0.137464\pi\)
\(37\)	52.0839	+	27.8394i	1.40767	+	0.752417i	0.987882	−	0.155205i	\(-0.0496037\pi\)
							0.419790	+	0.907621i	\(0.362104\pi\)
\(41\)	−59.4143	−	39.6994i	−1.44913	−	0.968277i	−0.997086	−	0.0762815i	\(-0.975695\pi\)
							−0.452043	−	0.891996i	\(-0.649305\pi\)
\(43\)	5.44118	+	6.63009i	0.126539	+	0.154188i	0.832426	−	0.554136i	\(-0.186951\pi\)
							−0.705887	+	0.708324i	\(0.749451\pi\)
\(47\)	2.74631	+	6.63018i	0.0584322	+	0.141068i	0.950399	−	0.311033i	\(-0.100675\pi\)
							−0.891967	+	0.452100i	\(0.850675\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.29	yes	1024
128.67	odd	32	inner	384.3.u.a.67.29	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.29	✓	1024	128.67	odd	32	inner
384.3.u.a.235.29	yes	1024	1.1	even	1	trivial