Embedded newform 38.9.d.a.31.5

• Label: 38.9.d.a.31.5
• Level: $38$
• Weight: $9$
• Character: 38.31
• 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			31.5
Character	\(\chi\)	\(=\)	38.31
Dual form			38.9.d.a.27.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.79796 + 5.65685i) q^{2} +(65.0070 - 37.5318i) q^{3} +(64.0000 - 110.851i) q^{4} +(393.618 + 681.767i) q^{5} +(-424.624 + 735.470i) q^{6} -1847.62 q^{7} +1448.15i q^{8} +(-463.230 + 802.337i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 168 q^{3} + 1536 q^{4} - 558 q^{5} + 896 q^{6} + 11992 q^{7} + 18448 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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
							\(3\)	65.0070	−	37.5318i	0.802555	−	0.463355i	−0.0418087	−	0.999126i	\(-0.513312\pi\)
0.844364	+	0.535770i	\(0.179979\pi\)
\(5\)	393.618	+	681.767i	0.629789	+	1.09083i	0.987594	+	0.157031i	\(0.0501922\pi\)
							−0.357804	+	0.933797i	\(0.616475\pi\)
\(7\)	−1847.62			−0.769520			−0.384760	−	0.923017i	\(-0.625716\pi\)
							−0.384760	+	0.923017i	\(0.625716\pi\)
\(11\)	12831.0			0.876372			0.438186	−	0.898884i	\(-0.355621\pi\)
							0.438186	+	0.898884i	\(0.355621\pi\)
\(13\)	−2950.60	−	1703.53i	−0.103309	−	0.0596453i	0.447455	−	0.894306i	\(-0.352330\pi\)
							−0.550764	+	0.834661i	\(0.685664\pi\)
\(17\)	30078.6	+	52097.6i	0.360132	+	0.623767i	0.987982	−	0.154567i	\(-0.0493984\pi\)
							−0.627850	+	0.778334i	\(0.716065\pi\)
\(19\)	−103601.	+	79059.4i	−0.794968	+	0.606651i
							\(23\)	−140500.	+	243353.i	−0.502070	+	0.869611i	0.497927	+	0.867219i	\(0.334095\pi\)
−0.999997	+	0.00239207i	\(0.999239\pi\)
\(29\)	923354.	+	533099.i	1.30550	+	0.753730i	0.981341	−	0.192274i	\(-0.0615861\pi\)
							0.324157	+	0.946003i	\(0.394919\pi\)
\(31\)			1.27364e6i			1.37912i	0.724231	+	0.689558i	\(0.242195\pi\)
							−0.724231	+	0.689558i	\(0.757805\pi\)
\(37\)		−	2.07897e6i		−	1.10928i	−0.832090	−	0.554641i	\(-0.812856\pi\)
							0.832090	−	0.554641i	\(-0.187144\pi\)
\(41\)	1.35413e6	−	781809.i	0.479210	−	0.276672i	−0.240877	−	0.970556i	\(-0.577435\pi\)
							0.720087	+	0.693884i	\(0.244102\pi\)
\(43\)	1.53682e6	+	2.66185e6i	0.449520	+	0.778592i	0.998355	−	0.0573392i	\(-0.0182617\pi\)
							−0.548835	+	0.835931i	\(0.684928\pi\)
\(47\)	548871.	−	950673.i	0.112481	−	0.194823i	−0.804289	−	0.594238i	\(-0.797454\pi\)
							0.916770	+	0.399416i	\(0.130787\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.d.a.31.5	yes	24
19.8	odd	6	inner	38.9.d.a.27.5	✓	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.d.a.27.5	✓	24	19.8	odd	6	inner
38.9.d.a.31.5	yes	24	1.1	even	1	trivial