Embedded newform 38.6.e.a.17.3

• Label: 38.6.e.a.17.3
• Level: $38$
• Weight: $6$
• Character: 38.17
• Analytic conductor: $6.095$
• Analytic rank: $0$
• Dimension: $24$
• 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:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			17.3
Character	\(\chi\)	\(=\)	38.17
Dual form			38.6.e.a.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.694593 + 3.93923i) q^{2} +(9.08943 + 7.62694i) q^{3} +(-15.0351 - 5.47232i) q^{4} +(86.6641 - 31.5432i) q^{5} +(-36.3577 + 30.5078i) q^{6} +(77.2983 + 133.885i) q^{7} +(32.0000 - 55.4256i) q^{8} +(-17.7489 - 100.659i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 18 q^{3} - 72 q^{6} + 438 q^{7} + 768 q^{8} + 378 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.694593	+	3.93923i	−0.122788	+	0.696364i
							\(3\)	9.08943	+	7.62694i	0.583087	+	0.489268i	0.885959	−	0.463763i	\(-0.153501\pi\)
−0.302872	+	0.953031i	\(0.597945\pi\)
\(5\)	86.6641	−	31.5432i	1.55030	−	0.564261i	0.581808	−	0.813326i	\(-0.302345\pi\)
							0.968487	+	0.249065i	\(0.0801231\pi\)
\(7\)	77.2983	+	133.885i	0.596245	+	1.03273i	0.993370	+	0.114962i	\(0.0366747\pi\)
							−0.397125	+	0.917765i	\(0.629992\pi\)
\(11\)	−295.747	+	512.249i	−0.736951	+	1.27644i	0.216911	+	0.976191i	\(0.430402\pi\)
							−0.953862	+	0.300245i	\(0.902932\pi\)
\(13\)	−59.9700	+	50.3208i	−0.0984183	+	0.0825828i	−0.690669	−	0.723171i	\(-0.742684\pi\)
							0.592251	+	0.805754i	\(0.298239\pi\)
\(17\)	107.127	−	607.546i	0.0899034	−	0.509867i	−0.906287	−	0.422663i	\(-0.861096\pi\)
							0.996190	−	0.0872046i	\(-0.0277934\pi\)
\(19\)	1154.29	+	1069.45i	0.733551	+	0.679634i
							\(23\)	−1564.98	−	569.607i	−0.616865	−	0.224520i	0.0146395	−	0.999893i	\(-0.495340\pi\)
−0.631504	+	0.775372i	\(0.717562\pi\)
\(29\)	−962.144	−	5456.59i	−0.212444	−	1.20483i	−0.885287	−	0.465046i	\(-0.846038\pi\)
							0.672843	−	0.739786i	\(-0.265073\pi\)
\(31\)	−3104.33	−	5376.85i	−0.580181	−	1.00490i	−0.995457	−	0.0952076i	\(-0.969649\pi\)
							0.415277	−	0.909695i	\(-0.363685\pi\)
\(37\)	−532.286			−0.0639206			−0.0319603	−	0.999489i	\(-0.510175\pi\)
							−0.0319603	+	0.999489i	\(0.510175\pi\)
\(41\)	13614.5	+	11423.9i	1.26486	+	1.06134i	0.995147	+	0.0984030i	\(0.0313734\pi\)
							0.269713	+	0.962941i	\(0.413071\pi\)
\(43\)	−18265.2	+	6648.00i	−1.50645	+	0.548302i	−0.957721	−	0.287699i	\(-0.907110\pi\)
							−0.548727	+	0.836001i	\(0.684887\pi\)
\(47\)	−2337.26	−	13255.3i	−0.154334	−	0.875273i	−0.959392	−	0.282076i	\(-0.908977\pi\)
							0.805058	−	0.593197i	\(-0.202134\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.a.17.3	yes	24
19.3	odd	18		722.6.a.p.1.4		12
19.9	even	9	inner	38.6.e.a.9.3	✓	24
19.16	even	9		722.6.a.o.1.9		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.a.9.3	✓	24	19.9	even	9	inner
38.6.e.a.17.3	yes	24	1.1	even	1	trivial
722.6.a.o.1.9		12	19.16	even	9
722.6.a.p.1.4		12	19.3	odd	18