Embedded newform 38.8.e.a.9.4

• Label: 38.8.e.a.9.4
• Level: $38$
• Weight: $8$
• Character: 38.9
• Analytic conductor: $11.871$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8706309684\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.4
Character	\(\chi\)	\(=\)	38.9
Dual form			38.8.e.a.17.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38919 + 7.87846i) q^{2} +(28.3430 - 23.7826i) q^{3} +(-60.1403 + 21.8893i) q^{4} +(-54.5194 - 19.8434i) q^{5} +(226.744 + 190.261i) q^{6} +(-356.201 + 616.958i) q^{7} +(-256.000 - 443.405i) q^{8} +(-142.054 + 805.630i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 45 q^{3} + 360 q^{6} + 1806 q^{7} - 7680 q^{8} - 3105 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38919	+	7.87846i	0.122788	+	0.696364i
							\(3\)	28.3430	−	23.7826i	0.606069	−	0.508552i	−0.287321	−	0.957834i	\(-0.592765\pi\)
0.893390	+	0.449282i	\(0.148320\pi\)
\(5\)	−54.5194	−	19.8434i	−0.195055	−	0.0709941i	0.242646	−	0.970115i	\(-0.421985\pi\)
							−0.437700	+	0.899121i	\(0.644207\pi\)
\(7\)	−356.201	+	616.958i	−0.392511	+	0.679849i	−0.992780	−	0.119949i	\(-0.961727\pi\)
							0.600269	+	0.799798i	\(0.295060\pi\)
\(11\)	3641.45	+	6307.17i	0.824897	+	1.42876i	0.901998	+	0.431740i	\(0.142100\pi\)
							−0.0771018	+	0.997023i	\(0.524567\pi\)
\(13\)	3073.79	+	2579.22i	0.388037	+	0.325601i	0.815848	−	0.578267i	\(-0.196271\pi\)
							−0.427811	+	0.903868i	\(0.640715\pi\)
\(17\)	320.112	+	1815.45i	0.0158027	+	0.0896214i	0.991689	−	0.128658i	\(-0.0410668\pi\)
							−0.975886	+	0.218279i	\(0.929956\pi\)
\(19\)	−17260.6	+	24412.0i	−0.577321	+	0.816517i
							\(23\)	−43309.0	+	15763.2i	−0.742217	+	0.270145i	−0.685327	−	0.728235i	\(-0.740341\pi\)
−0.0568903	+	0.998380i	\(0.518119\pi\)
\(29\)	−14366.1	+	81474.4i	−0.109382	+	0.620338i	0.879997	+	0.474980i	\(0.157545\pi\)
							−0.989379	+	0.145358i	\(0.953567\pi\)
\(31\)	137821.	−	238713.i	0.830901	−	1.43916i	−0.0664230	−	0.997792i	\(-0.521159\pi\)
							0.897324	−	0.441372i	\(-0.145508\pi\)
\(37\)	169842.			0.551238			0.275619	−	0.961267i	\(-0.411117\pi\)
							0.275619	+	0.961267i	\(0.411117\pi\)
\(41\)	354627.	−	297567.i	0.803578	−	0.674282i	−0.145488	−	0.989360i	\(-0.546475\pi\)
							0.949066	+	0.315078i	\(0.102031\pi\)
\(43\)	−251443.	−	91517.7i	−0.482280	−	0.175536i	0.0894268	−	0.995993i	\(-0.471496\pi\)
							−0.571707	+	0.820458i	\(0.693719\pi\)
\(47\)	26312.5	−	149226.i	0.0369674	−	0.209653i	−0.960729	−	0.277488i	\(-0.910498\pi\)
							0.997697	+	0.0678353i	\(0.0216092\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.a.9.4	✓	30
19.17	even	9	inner	38.8.e.a.17.4	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.a.9.4	✓	30	1.1	even	1	trivial
38.8.e.a.17.4	yes	30	19.17	even	9	inner