Embedded newform 384.3.u.a.235.56

• Label: 384.3.u.a.235.56
• 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			235.56
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.56

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.74060 - 0.985045i) q^{2} +(-1.09880 + 1.33889i) q^{3} +(2.05937 - 3.42914i) q^{4} +(0.214012 + 0.705502i) q^{5} +(-0.593704 + 3.41285i) q^{6} +(-1.42721 + 7.17506i) q^{7} +(0.206693 - 7.99733i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.74060	−	0.985045i	0.870300	−	0.492522i
							\(3\)	−1.09880	+	1.33889i	−0.366267	+	0.446298i
\(5\)	0.214012	+	0.705502i	0.0428023	+	0.141100i								0.975737	−	0.218944i	\(-0.0702614\pi\)
							−0.932935	+	0.360045i	\(0.882761\pi\)
\(7\)	−1.42721	+	7.17506i	−0.203887	+	1.02501i	0.734285	+	0.678841i	\(0.237517\pi\)
							−0.938172	+	0.346168i	\(0.887483\pi\)
\(11\)	12.8796	+	1.26853i	1.17087	+	0.115321i	0.664699	−	0.747111i	\(-0.268560\pi\)
							0.506173	+	0.862432i	\(0.331060\pi\)
\(13\)	12.7072	+	3.85470i	0.977480	+	0.296515i	0.738309	−	0.674462i	\(-0.235624\pi\)
							0.239171	+	0.970978i	\(0.423124\pi\)
\(17\)	2.65819	+	6.41744i	0.156364	+	0.377497i	0.982576	−	0.185864i	\(-0.0595083\pi\)
							−0.826211	+	0.563360i	\(0.809508\pi\)
\(19\)	−6.28392	+	3.35883i	−0.330733	+	0.176780i	−0.628402	−	0.777889i	\(-0.716291\pi\)
							0.297669	+	0.954669i	\(0.403791\pi\)
\(23\)	30.2056	+	20.1827i	1.31329	+	0.877510i	0.997455	−	0.0713057i	\(-0.0227166\pi\)
							0.315831	+	0.948815i	\(0.397717\pi\)
\(29\)	12.7073	−	1.25156i	0.438182	−	0.0431572i	0.123479	−	0.992347i	\(-0.460595\pi\)
							0.314704	+	0.949190i	\(0.398095\pi\)
\(31\)	−4.12360	+	4.12360i	−0.133019	+	0.133019i	−0.770482	−	0.637462i	\(-0.779984\pi\)
							0.637462	+	0.770482i	\(0.279984\pi\)
\(37\)	−10.4910	−	5.60754i	−0.283539	−	0.151555i	0.323496	−	0.946229i	\(-0.395142\pi\)
							−0.607036	+	0.794674i	\(0.707642\pi\)
\(41\)	−46.3898	−	30.9967i	−1.13146	−	0.756017i	−0.158584	−	0.987346i	\(-0.550693\pi\)
							−0.972876	+	0.231329i	\(0.925693\pi\)
\(43\)	−2.33295	−	2.84271i	−0.0542547	−	0.0661095i	0.745176	−	0.666868i	\(-0.232365\pi\)
							−0.799431	+	0.600758i	\(0.794865\pi\)
\(47\)	11.9497	+	28.8491i	0.254248	+	0.613810i	0.998538	−	0.0540467i	\(-0.0172120\pi\)
							−0.744290	+	0.667857i	\(0.767212\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.56	yes	1024
128.67	odd	32	inner	384.3.u.a.67.56	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.56	✓	1024	128.67	odd	32	inner
384.3.u.a.235.56	yes	1024	1.1	even	1	trivial