Embedded newform 384.3.u.a.235.15

• Label: 384.3.u.a.235.15
• 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			235.15
Character	\(\chi\)	\(=\)	384.235
Dual form			384.3.u.a.67.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.62747 + 1.16248i) q^{2} +(-1.09880 + 1.33889i) q^{3} +(1.29730 - 3.78378i) q^{4} +(1.48706 + 4.90218i) q^{5} +(0.231831 - 3.45634i) q^{6} +(0.591864 - 2.97550i) q^{7} +(2.28725 + 7.66606i) q^{8} +(-0.585271 - 2.94236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 1024 q+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.62747	+	1.16248i	−0.813733	+	0.581238i
							\(3\)	−1.09880	+	1.33889i	−0.366267	+	0.446298i
\(5\)	1.48706	+	4.90218i	0.297412	+	0.980435i								0.970546	+	0.240915i	\(0.0774476\pi\)
							−0.673134	+	0.739520i	\(0.735052\pi\)
\(7\)	0.591864	−	2.97550i	0.0845520	−	0.425072i	−0.915204	−	0.402990i	\(-0.867971\pi\)
							0.999756	−	0.0220811i	\(-0.00702921\pi\)
\(11\)	−9.37494	−	0.923351i	−0.852267	−	0.0839410i	−0.337557	−	0.941305i	\(-0.609601\pi\)
							−0.514710	+	0.857364i	\(0.672101\pi\)
\(13\)	−0.511084	−	0.155036i	−0.0393142	−	0.0119258i	0.270566	−	0.962701i	\(-0.412789\pi\)
							−0.309881	+	0.950776i	\(0.600289\pi\)
\(17\)	9.20123	+	22.2137i	0.541249	+	1.30669i	0.923842	+	0.382774i	\(0.125031\pi\)
							−0.382593	+	0.923917i	\(0.624969\pi\)
\(19\)	−16.9165	+	9.04203i	−0.890340	+	0.475896i	−0.852104	−	0.523372i	\(-0.824674\pi\)
							−0.0382355	+	0.999269i	\(0.512174\pi\)
\(23\)	0.551998	+	0.368833i	0.0239999	+	0.0160362i	0.567512	−	0.823365i	\(-0.307906\pi\)
							−0.543512	+	0.839401i	\(0.682906\pi\)
\(29\)	−41.4835	+	4.08577i	−1.43046	+	0.140888i	−0.783386	−	0.621535i	\(-0.786509\pi\)
							−0.647078	+	0.762424i	\(0.724009\pi\)
\(31\)	−9.99437	+	9.99437i	−0.322399	+	0.322399i	−0.849687	−	0.527288i	\(-0.823209\pi\)
							0.527288	+	0.849687i	\(0.323209\pi\)
\(37\)	−37.1097	−	19.8356i	−1.00297	−	0.536096i	−0.113713	−	0.993514i	\(-0.536275\pi\)
							−0.889252	+	0.457417i	\(0.848775\pi\)
\(41\)	−11.2652	−	7.52714i	−0.274760	−	0.183589i	0.410554	−	0.911836i	\(-0.365335\pi\)
							−0.685314	+	0.728247i	\(0.740335\pi\)
\(43\)	−32.6914	−	39.8346i	−0.760265	−	0.926386i	0.238735	−	0.971085i	\(-0.423267\pi\)
							−0.999001	+	0.0446986i	\(0.985767\pi\)
\(47\)	−15.3139	−	36.9710i	−0.325828	−	0.786617i	−0.998893	−	0.0470347i	\(-0.985023\pi\)
							0.673066	−	0.739583i	\(-0.264977\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.u.a.235.15	yes	1024
128.67	odd	32	inner	384.3.u.a.67.15	✓	1024

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.u.a.67.15	✓	1024	128.67	odd	32	inner
384.3.u.a.235.15	yes	1024	1.1	even	1	trivial