Embedded newform 384.3.u.a.235.11

• Label: 384.3.u.a.235.11
• 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			235.11
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68853 - 1.07186i) q^{2} +(1.09880 - 1.33889i) q^{3} +(1.70224 + 3.61972i) q^{4} +(1.00281 + 3.30581i) q^{5} +(-3.29046 + 1.08300i) q^{6} +(0.212740 - 1.06952i) q^{7} +(1.00554 - 7.93655i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68853	−	1.07186i	−0.844263	−	0.535929i
							\(3\)	1.09880	−	1.33889i	0.366267	−	0.446298i
\(5\)	1.00281	+	3.30581i	0.200562	+	0.661163i								0.998307	+	0.0581682i	\(0.0185260\pi\)
							−0.797745	+	0.602995i	\(0.793974\pi\)
\(7\)	0.212740	−	1.06952i	0.0303915	−	0.152788i	−0.962610	−	0.270892i	\(-0.912681\pi\)
							0.993001	+	0.118103i	\(0.0376814\pi\)
\(11\)	−9.91542	−	0.976583i	−0.901401	−	0.0887803i	−0.363311	−	0.931668i	\(-0.618354\pi\)
							−0.538090	+	0.842887i	\(0.680854\pi\)
\(13\)	3.75428	+	1.13885i	0.288790	+	0.0876036i	0.431356	−	0.902182i	\(-0.358035\pi\)
							−0.142566	+	0.989785i	\(0.545535\pi\)
\(17\)	−10.0859	−	24.3496i	−0.593289	−	1.43233i	−0.880308	−	0.474402i	\(-0.842664\pi\)
							0.287019	−	0.957925i	\(-0.407336\pi\)
\(19\)	24.5730	−	13.1346i	1.29332	−	0.691293i	0.326123	−	0.945327i	\(-0.394258\pi\)
							0.967195	+	0.254035i	\(0.0817578\pi\)
\(23\)	35.3828	+	23.6420i	1.53838	+	1.02791i	0.980156	+	0.198226i	\(0.0635181\pi\)
							0.558227	+	0.829688i	\(0.311482\pi\)
\(29\)	30.6523	−	3.01898i	1.05697	−	0.104103i	0.445433	−	0.895315i	\(-0.353050\pi\)
							0.611541	+	0.791212i	\(0.290550\pi\)
\(31\)	23.0334	−	23.0334i	0.743011	−	0.743011i	−0.230145	−	0.973156i	\(-0.573920\pi\)
							0.973156	+	0.230145i	\(0.0739200\pi\)
\(37\)	−10.2042	−	5.45425i	−0.275789	−	0.147412i	0.327704	−	0.944780i	\(-0.393725\pi\)
							−0.603493	+	0.797368i	\(0.706225\pi\)
\(41\)	−25.3320	−	16.9263i	−0.617853	−	0.412836i	0.206874	−	0.978368i	\(-0.433671\pi\)
							−0.824727	+	0.565532i	\(0.808671\pi\)
\(43\)	47.3162	+	57.6550i	1.10038	+	1.34081i	0.935026	+	0.354580i	\(0.115376\pi\)
							0.165352	+	0.986235i	\(0.447124\pi\)
\(47\)	−26.3138	−	63.5271i	−0.559868	−	1.35164i	−0.909871	−	0.414892i	\(-0.863819\pi\)
							0.350003	−	0.936749i	\(-0.386181\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.11	yes	1024
128.67	odd	32	inner	384.3.u.a.67.11	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.11	✓	1024	128.67	odd	32	inner
384.3.u.a.235.11	yes	1024	1.1	even	1	trivial