Embedded newform 384.3.u.a.235.32

• Label: 384.3.u.a.235.32
• 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			235.32
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.32

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.0710332 + 1.99874i) q^{2} +(1.09880 - 1.33889i) q^{3} +(-3.98991 - 0.283953i) q^{4} +(0.819863 + 2.70273i) q^{5} +(2.59805 + 2.29132i) q^{6} +(0.909351 - 4.57162i) q^{7} +(0.850965 - 7.95461i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.0710332	+	1.99874i	−0.0355166	+	0.999369i
							\(3\)	1.09880	−	1.33889i	0.366267	−	0.446298i
\(5\)	0.819863	+	2.70273i	0.163973	+	0.540545i								0.999966	−	0.00819357i	\(-0.00260812\pi\)
							−0.835994	+	0.548739i	\(0.815108\pi\)
\(7\)	0.909351	−	4.57162i	0.129907	−	0.653088i	−0.859875	−	0.510504i	\(-0.829459\pi\)
							0.989783	−	0.142584i	\(-0.0455411\pi\)
\(11\)	−2.40205	−	0.236582i	−0.218369	−	0.0215074i	−0.0117589	−	0.999931i	\(-0.503743\pi\)
							−0.206610	+	0.978423i	\(0.566243\pi\)
\(13\)	13.9878	+	4.24315i	1.07598	+	0.326396i	0.778058	−	0.628193i	\(-0.216205\pi\)
							0.297926	+	0.954589i	\(0.403705\pi\)
\(17\)	8.19443	+	19.7831i	0.482025	+	1.16371i	0.958646	+	0.284603i	\(0.0918617\pi\)
							−0.476620	+	0.879109i	\(0.658138\pi\)
\(19\)	21.8331	−	11.6701i	1.14911	−	0.614214i	0.216925	−	0.976188i	\(-0.430397\pi\)
							0.932188	+	0.361975i	\(0.117897\pi\)
\(23\)	−0.873612	−	0.583729i	−0.0379831	−	0.0253795i	0.536434	−	0.843942i	\(-0.319771\pi\)
							−0.574417	+	0.818563i	\(0.694771\pi\)
\(29\)	6.19510	−	0.610165i	0.213624	−	0.0210402i	0.00936119	−	0.999956i	\(-0.497020\pi\)
							0.204263	+	0.978916i	\(0.434520\pi\)
\(31\)	−0.326429	+	0.326429i	−0.0105300	+	0.0105300i	−0.712352	−	0.701822i	\(-0.752370\pi\)
							0.701822	+	0.712352i	\(0.252370\pi\)
\(37\)	5.73358	+	3.06466i	0.154962	+	0.0828287i	0.547055	−	0.837097i	\(-0.315749\pi\)
							−0.392093	+	0.919926i	\(0.628249\pi\)
\(41\)	34.4785	+	23.0378i	0.840940	+	0.561898i	0.899767	−	0.436370i	\(-0.143736\pi\)
							−0.0588272	+	0.998268i	\(0.518736\pi\)
\(43\)	−11.6800	−	14.2321i	−0.271628	−	0.330980i	0.619274	−	0.785175i	\(-0.287427\pi\)
							−0.890902	+	0.454195i	\(0.849927\pi\)
\(47\)	15.3599	+	37.0821i	0.326807	+	0.788982i	0.998826	+	0.0484476i	\(0.0154274\pi\)
							−0.672019	+	0.740534i	\(0.734573\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.32	yes	1024
128.67	odd	32	inner	384.3.u.a.67.32	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.32	✓	1024	128.67	odd	32	inner
384.3.u.a.235.32	yes	1024	1.1	even	1	trivial