Embedded newform 38.8.e.b.9.6

• Label: 38.8.e.b.9.6
• Level: $38$
• Weight: $8$
• Character: 38.9
• Analytic conductor: $11.871$
• Analytic rank: $0$
• Dimension: $36$
• 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:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.6
Character	\(\chi\)	\(=\)	38.9
Dual form			38.8.e.b.17.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38919 - 7.87846i) q^{2} +(65.8294 - 55.2375i) q^{3} +(-60.1403 + 21.8893i) q^{4} +(133.061 + 48.4302i) q^{5} +(-526.636 - 441.900i) q^{6} +(558.915 - 968.069i) q^{7} +(256.000 + 443.405i) q^{8} +(902.570 - 5118.73i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 84 q^{3} + 672 q^{6} - 2988 q^{7} + 9216 q^{8} + 1428 q^{9}+O(q^{10}) \)

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
							\(3\)	65.8294	−	55.2375i	1.40765	−	1.18116i	0.450074	−	0.892991i	\(-0.351397\pi\)
0.957579	−	0.288170i	\(-0.0930470\pi\)
\(5\)	133.061	+	48.4302i	0.476053	+	0.173269i	0.568892	−	0.822412i	\(-0.307372\pi\)
							−0.0928396	+	0.995681i	\(0.529594\pi\)
\(7\)	558.915	−	968.069i	0.615889	−	1.06675i	−0.374339	−	0.927292i	\(-0.622130\pi\)
							0.990228	−	0.139459i	\(-0.0445364\pi\)
\(11\)	3439.29	+	5957.02i	0.779102	+	1.34944i	0.932460	+	0.361273i	\(0.117658\pi\)
							−0.153358	+	0.988171i	\(0.549009\pi\)
\(13\)	−6058.66	−	5083.82i	−0.764847	−	0.641783i	0.174537	−	0.984651i	\(-0.444157\pi\)
							−0.939384	+	0.342868i	\(0.888602\pi\)
\(17\)	3278.56	+	18593.6i	0.161850	+	0.917895i	0.952253	+	0.305309i	\(0.0987596\pi\)
							−0.790404	+	0.612586i	\(0.790129\pi\)
\(19\)	−29644.2	−	3884.87i	−0.991522	−	0.129939i
							\(23\)	22123.9	−	8052.45i	0.379153	−	0.138000i	−0.145411	−	0.989371i	\(-0.546450\pi\)
0.524564	+	0.851371i	\(0.324228\pi\)
\(29\)	−21462.7	+	121721.i	−0.163415	+	0.926773i	0.787268	+	0.616610i	\(0.211495\pi\)
							−0.950684	+	0.310163i	\(0.899617\pi\)
\(31\)	−116461.	+	201716.i	−0.702122	+	1.21611i	0.265598	+	0.964084i	\(0.414431\pi\)
							−0.967720	+	0.252028i	\(0.918903\pi\)
\(37\)	354734.			1.15132			0.575661	−	0.817688i	\(-0.304745\pi\)
							0.575661	+	0.817688i	\(0.304745\pi\)
\(41\)	375158.	−	314795.i	0.850102	−	0.713320i	−0.109710	−	0.993964i	\(-0.534992\pi\)
							0.959812	+	0.280643i	\(0.0905478\pi\)
\(43\)	620610.	+	225884.i	1.19036	+	0.433257i	0.859851	−	0.510545i	\(-0.170556\pi\)
							0.330512	+	0.943802i	\(0.392779\pi\)
\(47\)	19528.7	−	110753.i	0.0274366	−	0.155601i	−0.968012	−	0.250906i	\(-0.919272\pi\)
							0.995448	+	0.0953051i	\(0.0303827\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.b.9.6	✓	36
19.17	even	9	inner	38.8.e.b.17.6	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.b.9.6	✓	36	1.1	even	1	trivial
38.8.e.b.17.6	yes	36	19.17	even	9	inner