Embedded newform 384.3.u.a.235.41

• Label: 384.3.u.a.235.41
• 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			235.41
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.41

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.699284 - 1.87377i) q^{2} +(-1.09880 + 1.33889i) q^{3} +(-3.02200 - 2.62059i) q^{4} +(0.535294 + 1.76463i) q^{5} +(1.74040 + 2.99516i) q^{6} +(0.225582 - 1.13408i) q^{7} +(-7.02362 + 3.82999i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.699284	−	1.87377i	0.349642	−	0.936883i
							\(3\)	−1.09880	+	1.33889i	−0.366267	+	0.446298i
\(5\)	0.535294	+	1.76463i	0.107059	+	0.352925i								0.994059	−	0.108844i	\(-0.0347148\pi\)
							−0.887000	+	0.461769i	\(0.847215\pi\)
\(7\)	0.225582	−	1.13408i	0.0322261	−	0.162011i	−0.961321	−	0.275429i	\(-0.911180\pi\)
							0.993547	+	0.113418i	\(0.0361800\pi\)
\(11\)	17.6748	+	1.74082i	1.60680	+	0.158256i	0.861247	−	0.508187i	\(-0.169684\pi\)
							0.745555	+	0.666444i	\(0.232184\pi\)
\(13\)	4.38699	+	1.33078i	0.337461	+	0.102368i	0.454467	−	0.890764i	\(-0.349830\pi\)
							−0.117006	+	0.993131i	\(0.537330\pi\)
\(17\)	−5.46634	−	13.1969i	−0.321550	−	0.776289i	−0.999164	−	0.0408723i	\(-0.986986\pi\)
							0.677615	−	0.735417i	\(-0.263014\pi\)
\(19\)	14.4833	−	7.74151i	0.762281	−	0.407448i	−0.0439610	−	0.999033i	\(-0.513998\pi\)
							0.806242	+	0.591585i	\(0.201498\pi\)
\(23\)	−30.3660	−	20.2899i	−1.32026	−	0.882172i	−0.322342	−	0.946623i	\(-0.604470\pi\)
							−0.997921	+	0.0644517i	\(0.979470\pi\)
\(29\)	33.2829	−	3.27808i	1.14769	−	0.113037i	0.493763	−	0.869597i	\(-0.335621\pi\)
							0.653925	+	0.756559i	\(0.273121\pi\)
\(31\)	24.7519	−	24.7519i	0.798447	−	0.798447i	−0.184403	−	0.982851i	\(-0.559035\pi\)
							0.982851	+	0.184403i	\(0.0590353\pi\)
\(37\)	−19.4777	−	10.4110i	−0.526423	−	0.281379i	0.186701	−	0.982417i	\(-0.440220\pi\)
							−0.713124	+	0.701038i	\(0.752720\pi\)
\(41\)	46.7186	+	31.2164i	1.13948	+	0.761374i	0.974381	−	0.224903i	\(-0.0722065\pi\)
							0.165096	+	0.986277i	\(0.447207\pi\)
\(43\)	−37.3793	−	45.5469i	−0.869287	−	1.05923i	−0.997719	−	0.0675109i	\(-0.978494\pi\)
							0.128432	−	0.991718i	\(-0.459006\pi\)
\(47\)	3.34054	+	8.06478i	0.0710753	+	0.171591i	0.955425	−	0.295234i	\(-0.0953977\pi\)
							−0.884350	+	0.466825i	\(0.845398\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.41	yes	1024
128.67	odd	32	inner	384.3.u.a.67.41	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.41	✓	1024	128.67	odd	32	inner
384.3.u.a.235.41	yes	1024	1.1	even	1	trivial