Embedded newform 384.3.u.a.235.58

• Label: 384.3.u.a.235.58
• 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			235.58
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.58

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.89182 - 0.648860i) q^{2} +(-1.09880 + 1.33889i) q^{3} +(3.15796 - 2.45505i) q^{4} +(0.348095 + 1.14752i) q^{5} +(-1.20998 + 3.24591i) q^{6} +(2.24192 - 11.2709i) q^{7} +(4.38130 - 6.69359i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.89182	−	0.648860i	0.945910	−	0.324430i
							\(3\)	−1.09880	+	1.33889i	−0.366267	+	0.446298i
\(5\)	0.348095	+	1.14752i	0.0696190	+	0.229503i								0.984845	−	0.173438i	\(-0.0554878\pi\)
							−0.915226	+	0.402942i	\(0.867988\pi\)
\(7\)	2.24192	−	11.2709i	0.320275	−	1.61013i	−0.400051	−	0.916493i	\(-0.631008\pi\)
							0.720325	−	0.693636i	\(-0.243992\pi\)
\(11\)	−2.51088	−	0.247300i	−0.228262	−	0.0224818i	−0.0167605	−	0.999860i	\(-0.505335\pi\)
							−0.211501	+	0.977378i	\(0.567835\pi\)
\(13\)	−6.19014	−	1.87776i	−0.476164	−	0.144443i	0.0430748	−	0.999072i	\(-0.486285\pi\)
							−0.519239	+	0.854629i	\(0.673785\pi\)
\(17\)	0.544847	+	1.31538i	0.0320499	+	0.0773752i	0.939094	−	0.343660i	\(-0.111667\pi\)
							−0.907044	+	0.421035i	\(0.861667\pi\)
\(19\)	7.16265	−	3.82851i	0.376981	−	0.201501i	−0.272014	−	0.962293i	\(-0.587690\pi\)
							0.648995	+	0.760793i	\(0.275190\pi\)
\(23\)	1.79855	+	1.20175i	0.0781976	+	0.0522500i	0.594054	−	0.804425i	\(-0.297526\pi\)
							−0.515857	+	0.856675i	\(0.672526\pi\)
\(29\)	−15.8902	+	1.56505i	−0.547939	+	0.0539673i	−0.368199	−	0.929747i	\(-0.620026\pi\)
							−0.179740	+	0.983714i	\(0.557526\pi\)
\(31\)	30.6766	−	30.6766i	0.989569	−	0.989569i	−0.0103770	−	0.999946i	\(-0.503303\pi\)
							0.999946	+	0.0103770i	\(0.00330318\pi\)
\(37\)	−10.1075	−	5.40260i	−0.273177	−	0.146016i	0.329120	−	0.944288i	\(-0.393248\pi\)
							−0.602297	+	0.798272i	\(0.705748\pi\)
\(41\)	53.5561	+	35.7850i	1.30625	+	0.872805i	0.996941	−	0.0781581i	\(-0.0249039\pi\)
							0.309304	+	0.950963i	\(0.399904\pi\)
\(43\)	30.4410	+	37.0924i	0.707929	+	0.862614i	0.995462	−	0.0951563i	\(-0.0303351\pi\)
							−0.287533	+	0.957771i	\(0.592835\pi\)
\(47\)	−9.37587	−	22.6353i	−0.199487	−	0.481603i	0.792203	−	0.610258i	\(-0.208934\pi\)
							−0.991689	+	0.128655i	\(0.958934\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.58	yes	1024
128.67	odd	32	inner	384.3.u.a.67.58	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.58	✓	1024	128.67	odd	32	inner
384.3.u.a.235.58	yes	1024	1.1	even	1	trivial