Embedded newform 384.3.u.a.235.25

• Label: 384.3.u.a.235.25
• 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			235.25
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.25

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.714150 + 1.86815i) q^{2} +(1.09880 - 1.33889i) q^{3} +(-2.97998 - 2.66828i) q^{4} +(0.536964 + 1.77013i) q^{5} +(1.71655 + 3.00890i) q^{6} +(0.988537 - 4.96971i) q^{7} +(7.11291 - 3.66150i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.714150	+	1.86815i	−0.357075	+	0.934076i
							\(3\)	1.09880	−	1.33889i	0.366267	−	0.446298i
\(5\)	0.536964	+	1.77013i	0.107393	+	0.354026i								0.994122	−	0.108262i	\(-0.0345285\pi\)
							−0.886730	+	0.462288i	\(0.847029\pi\)
\(7\)	0.988537	−	4.96971i	0.141220	−	0.709959i	−0.843682	−	0.536843i	\(-0.819617\pi\)
							0.984902	−	0.173115i	\(-0.0553833\pi\)
\(11\)	0.536126	+	0.0528038i	0.0487387	+	0.00480035i	0.122357	−	0.992486i	\(-0.460955\pi\)
							−0.0736184	+	0.997286i	\(0.523455\pi\)
\(13\)	−13.7404	−	4.16811i	−1.05696	−	0.320624i	−0.286475	−	0.958088i	\(-0.592484\pi\)
							−0.770480	+	0.637464i	\(0.779984\pi\)
\(17\)	−7.75526	−	18.7229i	−0.456192	−	1.10134i	−0.969927	−	0.243396i	\(-0.921739\pi\)
							0.513735	−	0.857949i	\(-0.328261\pi\)
\(19\)	−28.7684	+	15.3770i	−1.51413	+	0.809318i	−0.998861	−	0.0477121i	\(-0.984807\pi\)
							−0.515266	+	0.857030i	\(0.672307\pi\)
\(23\)	−16.9267	−	11.3100i	−0.735941	−	0.491740i	0.130232	−	0.991484i	\(-0.458428\pi\)
							−0.866174	+	0.499743i	\(0.833428\pi\)
\(29\)	38.6008	−	3.80185i	1.33106	−	0.131098i	0.592657	−	0.805455i	\(-0.298079\pi\)
							0.738406	+	0.674356i	\(0.235579\pi\)
\(31\)	8.41783	−	8.41783i	0.271543	−	0.271543i	−0.558178	−	0.829721i	\(-0.688499\pi\)
							0.829721	+	0.558178i	\(0.188499\pi\)
\(37\)	−20.9620	−	11.2044i	−0.566541	−	0.302823i	0.163148	−	0.986602i	\(-0.447835\pi\)
							−0.729689	+	0.683779i	\(0.760335\pi\)
\(41\)	−29.5962	−	19.7755i	−0.721858	−	0.482330i	0.139568	−	0.990212i	\(-0.455429\pi\)
							−0.861427	+	0.507882i	\(0.830429\pi\)
\(43\)	−47.5348	−	57.9214i	−1.10546	−	1.34701i	−0.932121	−	0.362146i	\(-0.882044\pi\)
							−0.173340	−	0.984862i	\(-0.555456\pi\)
\(47\)	−19.7636	−	47.7135i	−0.420502	−	1.01518i	−0.982200	−	0.187839i	\(-0.939852\pi\)
							0.561698	−	0.827342i	\(-0.310148\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.25	yes	1024
128.67	odd	32	inner	384.3.u.a.67.25	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.25	✓	1024	128.67	odd	32	inner
384.3.u.a.235.25	yes	1024	1.1	even	1	trivial