Embedded newform 384.3.u.a.235.14

• Label: 384.3.u.a.235.14
• 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			235.14
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.65554 + 1.12213i) q^{2} +(-1.09880 + 1.33889i) q^{3} +(1.48163 - 3.71548i) q^{4} +(-2.43645 - 8.03191i) q^{5} +(0.316695 - 3.44959i) q^{6} +(2.01201 - 10.1151i) q^{7} +(1.71635 + 7.81371i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.65554	+	1.12213i	−0.827771	+	0.561067i
							\(3\)	−1.09880	+	1.33889i	−0.366267	+	0.446298i
\(5\)	−2.43645	−	8.03191i	−0.487291	−	1.60638i								−0.763856	−	0.645387i	\(-0.776696\pi\)
							0.276565	−	0.960995i	\(-0.410804\pi\)
\(7\)	2.01201	−	10.1151i	0.287431	−	1.44501i	−0.519554	−	0.854438i	\(-0.673902\pi\)
							0.806984	−	0.590573i	\(-0.201098\pi\)
\(11\)	18.7337	+	1.84511i	1.70307	+	0.167737i	0.902261	−	0.431189i	\(-0.141906\pi\)
							0.800804	+	0.598927i	\(0.204406\pi\)
\(13\)	−17.6268	−	5.34704i	−1.35591	−	0.411310i	−0.473086	−	0.881016i	\(-0.656860\pi\)
							−0.882823	+	0.469706i	\(0.844360\pi\)
\(17\)	−2.07942	−	5.02017i	−0.122319	−	0.295304i	0.850845	−	0.525417i	\(-0.176091\pi\)
							−0.973164	+	0.230113i	\(0.926091\pi\)
\(19\)	−27.5242	+	14.7120i	−1.44864	+	0.774316i	−0.993252	−	0.115975i	\(-0.963001\pi\)
							−0.455392	+	0.890291i	\(0.650501\pi\)
\(23\)	−5.94101	−	3.96966i	−0.258305	−	0.172594i	0.419675	−	0.907675i	\(-0.362144\pi\)
							−0.677979	+	0.735081i	\(0.737144\pi\)
\(29\)	9.90189	−	0.975251i	0.341445	−	0.0336293i	0.0741581	−	0.997246i	\(-0.476373\pi\)
							0.267286	+	0.963617i	\(0.413873\pi\)
\(31\)	12.7788	−	12.7788i	0.412221	−	0.412221i	−0.470291	−	0.882512i	\(-0.655851\pi\)
							0.882512	+	0.470291i	\(0.155851\pi\)
\(37\)	38.0375	+	20.3315i	1.02804	+	0.549499i	0.897070	−	0.441888i	\(-0.145691\pi\)
							0.130969	+	0.991386i	\(0.458191\pi\)
\(41\)	31.0906	+	20.7741i	0.758308	+	0.506685i	0.873599	−	0.486647i	\(-0.161780\pi\)
							−0.115291	+	0.993332i	\(0.536780\pi\)
\(43\)	−34.8504	−	42.4653i	−0.810474	−	0.987565i	−0.999984	−	0.00568659i	\(-0.998190\pi\)
							0.189510	−	0.981879i	\(-0.439310\pi\)
\(47\)	−1.92299	−	4.64251i	−0.0409147	−	0.0987769i	0.902101	−	0.431525i	\(-0.142024\pi\)
							−0.943016	+	0.332748i	\(0.892024\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.14	yes	1024
128.67	odd	32	inner	384.3.u.a.67.14	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.14	✓	1024	128.67	odd	32	inner
384.3.u.a.235.14	yes	1024	1.1	even	1	trivial