Embedded newform 38.9.d.a.27.11

• Label: 38.9.d.a.27.11
• Level: $38$
• Weight: $9$
• Character: 38.27
• Analytic conductor: $15.480$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	38.d (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.4803871823\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			27.11
Character	\(\chi\)	\(=\)	38.27
Dual form			38.9.d.a.31.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(9.79796 + 5.65685i) q^{2} +(61.4665 + 35.4877i) q^{3} +(64.0000 + 110.851i) q^{4} +(219.968 - 380.996i) q^{5} +(401.497 + 695.414i) q^{6} +2502.16 q^{7} +1448.15i q^{8} +(-761.746 - 1319.38i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 168 q^{3} + 1536 q^{4} - 558 q^{5} + 896 q^{6} + 11992 q^{7} + 18448 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	9.79796	+	5.65685i	0.612372	+	0.353553i
							\(3\)	61.4665	+	35.4877i	0.758846	+	0.438120i	0.828881	−	0.559425i	\(-0.188978\pi\)
−0.0700355	+	0.997545i	\(0.522311\pi\)
\(5\)	219.968	−	380.996i	0.351949	−	0.609594i	−0.634642	−	0.772807i	\(-0.718852\pi\)
							0.986591	+	0.163212i	\(0.0521856\pi\)
\(7\)	2502.16			1.04213			0.521066	−	0.853517i	\(-0.325535\pi\)
							0.521066	+	0.853517i	\(0.325535\pi\)
\(11\)	23414.2			1.59922			0.799610	−	0.600519i	\(-0.205039\pi\)
							0.799610	+	0.600519i	\(0.205039\pi\)
\(13\)	−15118.4	+	8728.61i	−0.529337	+	0.305613i	−0.740747	−	0.671785i	\(-0.765528\pi\)
							0.211409	+	0.977398i	\(0.432195\pi\)
\(17\)	−60654.4	+	105057.i	−0.726217	+	1.25785i	0.232254	+	0.972655i	\(0.425390\pi\)
							−0.958471	+	0.285190i	\(0.907943\pi\)
\(19\)	−10436.4	−	129902.i	−0.0800821	−	0.996788i
							\(23\)	82002.8	+	142033.i	0.293033	+	0.507549i	0.974526	−	0.224277i	\(-0.0720019\pi\)
−0.681492	+	0.731826i	\(0.738669\pi\)
\(29\)	−44629.3	+	25766.7i	−0.0630998	+	0.0364307i	−0.531218	−	0.847235i	\(-0.678265\pi\)
							0.468118	+	0.883666i	\(0.344932\pi\)
\(31\)			1.20623e6i			1.30612i	0.757305	+	0.653061i	\(0.226516\pi\)
							−0.757305	+	0.653061i	\(0.773484\pi\)
\(37\)		−	1.62790e6i		−	0.868601i	−0.900768	−	0.434301i	\(-0.856996\pi\)
							0.900768	−	0.434301i	\(-0.143004\pi\)
\(41\)	−2.26211e6	−	1.30603e6i	−0.800533	−	0.462188i	0.0431246	−	0.999070i	\(-0.486269\pi\)
							−0.843657	+	0.536882i	\(0.819602\pi\)
\(43\)	1.26121e6	−	2.18449e6i	0.368906	−	0.638963i	−0.620489	−	0.784215i	\(-0.713066\pi\)
							0.989395	+	0.145252i	\(0.0463993\pi\)
\(47\)	−4.49308e6	−	7.78224e6i	−0.920772	−	1.59482i	−0.798223	−	0.602362i	\(-0.794226\pi\)
							−0.122550	−	0.992462i	\(-0.539107\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.d.a.27.11	✓	24
19.12	odd	6	inner	38.9.d.a.31.11	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.d.a.27.11	✓	24	1.1	even	1	trivial
38.9.d.a.31.11	yes	24	19.12	odd	6	inner