Embedded newform 38.6.e.b.5.1

• Label: 38.6.e.b.5.1
• 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			5.1
Character	\(\chi\)	\(=\)	38.5
Dual form			38.6.e.b.23.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.75877 + 1.36808i) q^{2} +(-5.14875 + 29.2000i) q^{3} +(12.2567 - 10.2846i) q^{4} +(68.5412 + 57.5129i) q^{5} +(-20.5950 - 116.800i) q^{6} +(68.7063 + 119.003i) q^{7} +(-32.0000 + 55.4256i) q^{8} +(-597.784 - 217.576i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 15 q^{3} + 60 q^{6} - 84 q^{7} - 960 q^{8} - 345 q^{9}+O(q^{10}) \)

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
							\(3\)	−5.14875	+	29.2000i	−0.330292	+	1.87318i	0.139231	+	0.990260i	\(0.455537\pi\)
−0.469523	+	0.882920i	\(0.655574\pi\)
\(5\)	68.5412	+	57.5129i	1.22610	+	1.02882i	0.998482	+	0.0550720i	\(0.0175388\pi\)
							0.227620	+	0.973750i	\(0.426906\pi\)
\(7\)	68.7063	+	119.003i	0.529970	+	0.917935i	0.999389	+	0.0349593i	\(0.0111302\pi\)
							−0.469419	+	0.882976i	\(0.655536\pi\)
\(11\)	152.529	−	264.188i	0.380077	−	0.658312i	−0.610996	−	0.791634i	\(-0.709231\pi\)
							0.991073	+	0.133321i	\(0.0425642\pi\)
\(13\)	15.9575	+	90.4993i	0.0261882	+	0.148521i	0.995098	−	0.0988935i	\(-0.0315303\pi\)
							−0.968910	+	0.247414i	\(0.920419\pi\)
\(17\)	54.3140	−	19.7687i	0.0455816	−	0.0165903i	−0.319129	−	0.947711i	\(-0.603390\pi\)
							0.364710	+	0.931121i	\(0.381168\pi\)
\(19\)	920.562	−	1276.19i	0.585018	−	0.811021i
							\(23\)	2223.62	−	1865.84i	0.876479	−	0.735453i	−0.0889731	−	0.996034i	\(-0.528359\pi\)
0.965452	+	0.260581i	\(0.0839141\pi\)
\(29\)	−5079.71	−	1848.86i	−1.12161	−	0.408234i	−0.286374	−	0.958118i	\(-0.592450\pi\)
							−0.835241	+	0.549884i	\(0.814672\pi\)
\(31\)	2969.66	+	5143.60i	0.555012	+	0.961310i	0.997903	+	0.0647339i	\(0.0206199\pi\)
							−0.442890	+	0.896576i	\(0.646047\pi\)
\(37\)	−287.050			−0.0344710			−0.0172355	−	0.999851i	\(-0.505486\pi\)
							−0.0172355	+	0.999851i	\(0.505486\pi\)
\(41\)	−240.363	+	1363.17i	−0.0223310	+	0.126645i	−0.993935	−	0.109966i	\(-0.964926\pi\)
							0.971604	+	0.236611i	\(0.0760368\pi\)
\(43\)	6184.72	+	5189.60i	0.510093	+	0.428018i	0.861162	−	0.508331i	\(-0.169737\pi\)
							−0.351069	+	0.936350i	\(0.614182\pi\)
\(47\)	−1477.74	−	537.854i	−0.0975784	−	0.0355156i	0.292770	−	0.956183i	\(-0.405423\pi\)
							−0.390348	+	0.920667i	\(0.627645\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.b.5.1	✓	30
19.2	odd	18		722.6.a.q.1.15		15
19.4	even	9	inner	38.6.e.b.23.1	yes	30
19.17	even	9		722.6.a.r.1.1		15

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.5.1	✓	30	1.1	even	1	trivial
38.6.e.b.23.1	yes	30	19.4	even	9	inner
722.6.a.q.1.15		15	19.2	odd	18
722.6.a.r.1.1		15	19.17	even	9