Embedded newform 38.6.e.b.5.5

• Label: 38.6.e.b.5.5
• Level: $38$
• Weight: $6$
• Character: 38.5
• Analytic conductor: $6.095$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.09458515289\)
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			5.5
Character	\(\chi\)	\(=\)	38.5
Dual form			38.6.e.b.23.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.75877 + 1.36808i) q^{2} +(4.41021 - 25.0115i) q^{3} +(12.2567 - 10.2846i) q^{4} +(-71.5678 - 60.0525i) q^{5} +(17.6408 + 100.046i) q^{6} +(91.0882 + 157.769i) q^{7} +(-32.0000 + 55.4256i) q^{8} +(-377.781 - 137.501i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 15 q^{3} + 60 q^{6} - 84 q^{7} - 960 q^{8} - 345 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.75877	+	1.36808i	−0.664463	+	0.241845i
							\(3\)	4.41021	−	25.0115i	0.282915	−	1.60449i	−0.429725	−	0.902960i	\(-0.641389\pi\)
0.712640	−	0.701530i	\(-0.247499\pi\)
\(5\)	−71.5678	−	60.0525i	−1.28024	−	1.07425i	−0.993210	−	0.116331i	\(-0.962887\pi\)
							−0.287033	−	0.957921i	\(-0.592669\pi\)
\(7\)	91.0882	+	157.769i	0.702615	+	1.21696i	0.967545	+	0.252697i	\(0.0813176\pi\)
							−0.264931	+	0.964267i	\(0.585349\pi\)
\(11\)	−198.935	+	344.565i	−0.495711	+	0.858597i	−0.999988	−	0.00494549i	\(-0.998426\pi\)
							0.504277	+	0.863542i	\(0.331759\pi\)
\(13\)	−119.944	−	680.234i	−0.196842	−	1.11635i	−0.909771	−	0.415111i	\(-0.863743\pi\)
							0.712928	−	0.701237i	\(-0.247368\pi\)
\(17\)	−1158.23	+	421.562i	−0.972015	+	0.353785i	−0.778731	−	0.627358i	\(-0.784136\pi\)
							−0.193284	+	0.981143i	\(0.561914\pi\)
\(19\)	785.549	−	1363.46i	0.499217	−	0.866477i
							\(23\)	216.010	−	181.254i	0.0851441	−	0.0714444i	−0.599222	−	0.800583i	\(-0.704523\pi\)
0.684366	+	0.729138i	\(0.260079\pi\)
\(29\)	−4078.07	−	1484.30i	−0.900451	−	0.327737i	−0.150017	−	0.988683i	\(-0.547933\pi\)
							−0.750433	+	0.660946i	\(0.770155\pi\)
\(31\)	−1343.51	−	2327.02i	−0.251093	−	0.434906i	0.712734	−	0.701435i	\(-0.247457\pi\)
							−0.963827	+	0.266528i	\(0.914123\pi\)
\(37\)	−3855.72			−0.463021			−0.231511	−	0.972832i	\(-0.574367\pi\)
							−0.231511	+	0.972832i	\(0.574367\pi\)
\(41\)	3396.47	−	19262.3i	0.315550	−	1.78957i	−0.253568	−	0.967318i	\(-0.581604\pi\)
							0.569118	−	0.822256i	\(-0.307285\pi\)
\(43\)	−284.262	−	238.524i	−0.0234449	−	0.0196726i	0.630990	−	0.775791i	\(-0.282649\pi\)
							−0.654435	+	0.756118i	\(0.727093\pi\)
\(47\)	−12719.0	−	4629.35i	−0.839865	−	0.305686i	−0.113964	−	0.993485i	\(-0.536355\pi\)
							−0.725901	+	0.687799i	\(0.758577\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.b.5.5	✓	30
19.2	odd	18		722.6.a.q.1.3		15
19.4	even	9	inner	38.6.e.b.23.5	yes	30
19.17	even	9		722.6.a.r.1.13		15

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.5.5	✓	30	1.1	even	1	trivial
38.6.e.b.23.5	yes	30	19.4	even	9	inner
722.6.a.q.1.3		15	19.2	odd	18
722.6.a.r.1.13		15	19.17	even	9