Embedded newform 384.3.u.a.235.45

• Label: 384.3.u.a.235.45
• 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			235.45
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.45

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.20158 + 1.59881i) q^{2} +(1.09880 - 1.33889i) q^{3} +(-1.11240 + 3.84221i) q^{4} +(-0.743698 - 2.45164i) q^{5} +(3.46094 + 0.147988i) q^{6} +(2.32210 - 11.6740i) q^{7} +(-7.47961 + 2.83820i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.20158	+	1.59881i	0.600791	+	0.799406i
							\(3\)	1.09880	−	1.33889i	0.366267	−	0.446298i
\(5\)	−0.743698	−	2.45164i	−0.148740	−	0.490329i								0.850632	−	0.525761i	\(-0.176219\pi\)
							−0.999372	+	0.0354320i	\(0.988719\pi\)
\(7\)	2.32210	−	11.6740i	0.331729	−	1.66771i	−0.350492	−	0.936566i	\(-0.613986\pi\)
							0.682221	−	0.731146i	\(-0.261014\pi\)
\(11\)	13.4298	+	1.32272i	1.22089	+	0.120247i	0.687857	−	0.725846i	\(-0.258551\pi\)
							0.533035	+	0.846093i	\(0.321051\pi\)
\(13\)	−22.4757	−	6.81794i	−1.72890	−	0.524457i	−0.739946	−	0.672666i	\(-0.765149\pi\)
							−0.988956	+	0.148210i	\(0.952649\pi\)
\(17\)	−3.07528	−	7.42439i	−0.180899	−	0.436729i	0.807253	−	0.590205i	\(-0.200953\pi\)
							−0.988152	+	0.153476i	\(0.950953\pi\)
\(19\)	22.7800	−	12.1762i	1.19895	−	0.640851i	0.253725	−	0.967276i	\(-0.418344\pi\)
							0.945223	+	0.326425i	\(0.105844\pi\)
\(23\)	−9.90784	−	6.62021i	−0.430776	−	0.287835i	0.321218	−	0.947005i	\(-0.395908\pi\)
							−0.751994	+	0.659170i	\(0.770908\pi\)
\(29\)	23.7316	−	2.33736i	0.818332	−	0.0805986i	0.319816	−	0.947480i	\(-0.396379\pi\)
							0.498515	+	0.866881i	\(0.333879\pi\)
\(31\)	−25.1094	+	25.1094i	−0.809981	+	0.809981i	−0.984631	−	0.174650i	\(-0.944121\pi\)
							0.174650	+	0.984631i	\(0.444121\pi\)
\(37\)	21.5024	+	11.4933i	0.581145	+	0.310628i	0.735648	−	0.677364i	\(-0.236878\pi\)
							−0.154503	+	0.987992i	\(0.549378\pi\)
\(41\)	31.1412	+	20.8079i	0.759542	+	0.507510i	0.874004	−	0.485918i	\(-0.161515\pi\)
							−0.114463	+	0.993428i	\(0.536515\pi\)
\(43\)	39.2985	+	47.8854i	0.913919	+	1.11361i	0.993160	+	0.116758i	\(0.0372503\pi\)
							−0.0792411	+	0.996855i	\(0.525250\pi\)
\(47\)	2.72774	+	6.58534i	0.0580369	+	0.140114i	0.950238	−	0.311525i	\(-0.100840\pi\)
							−0.892201	+	0.451638i	\(0.850840\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.45	yes	1024
128.67	odd	32	inner	384.3.u.a.67.45	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.45	✓	1024	128.67	odd	32	inner
384.3.u.a.235.45	yes	1024	1.1	even	1	trivial