Embedded newform 38.10.e.a.5.6

• Label: 38.10.e.a.5.6
• Level: $38$
• Weight: $10$
• Character: 38.5
• Analytic conductor: $19.571$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.5713617742\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			5.6
Character	\(\chi\)	\(=\)	38.5
Dual form			38.10.e.a.23.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.0351 - 5.47232i) q^{2} +(26.9513 - 152.849i) q^{3} +(196.107 - 164.554i) q^{4} +(-1711.99 - 1436.53i) q^{5} +(-431.221 - 2445.58i) q^{6} +(-910.825 - 1577.60i) q^{7} +(2048.00 - 3547.24i) q^{8} +(-4140.35 - 1506.96i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 252 q^{3} + 4032 q^{6} + 14373 q^{7} + 86016 q^{8} + 12852 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.0351	−	5.47232i	0.664463	−	0.241845i
							\(3\)	26.9513	−	152.849i	0.192103	−	1.08947i	−0.724381	−	0.689400i	\(-0.757874\pi\)
0.916484	−	0.400071i	\(-0.131015\pi\)
\(5\)	−1711.99	−	1436.53i	−1.22500	−	1.02790i	−0.998548	−	0.0538748i	\(-0.982843\pi\)
							−0.226452	−	0.974022i	\(-0.572713\pi\)
\(7\)	−910.825	−	1577.60i	−0.143382	−	0.248344i	0.785386	−	0.619006i	\(-0.212464\pi\)
							−0.928768	+	0.370662i	\(0.879131\pi\)
\(11\)	−40102.0	+	69458.7i	−0.825846	+	1.43041i	0.0754254	+	0.997151i	\(0.475969\pi\)
							−0.901271	+	0.433255i	\(0.857365\pi\)
\(13\)	−6693.16	−	37958.8i	−0.0649960	−	0.368610i	−0.999906	−	0.0137251i	\(-0.995631\pi\)
							0.934910	−	0.354885i	\(-0.115480\pi\)
\(17\)	626504.	−	228029.i	1.81930	−	0.662171i	0.823857	−	0.566797i	\(-0.191818\pi\)
							0.995442	−	0.0953731i	\(-0.0304044\pi\)
\(19\)	−444386.	+	353849.i	−0.782292	+	0.622912i
							\(23\)	−289176.	+	242648.i	−0.215470	+	0.180801i	−0.744134	−	0.668030i	\(-0.767138\pi\)
0.528664	+	0.848831i	\(0.322693\pi\)
\(29\)	−6.88491e6	−	2.50590e6i	−1.80762	−	0.657920i	−0.997422	−	0.0717640i	\(-0.977137\pi\)
							−0.810198	−	0.586156i	\(-0.800641\pi\)
\(31\)	−3.25923e6	−	5.64515e6i	−0.633852	−	1.09786i	−0.986757	−	0.162204i	\(-0.948140\pi\)
							0.352906	−	0.935659i	\(-0.385194\pi\)
\(37\)	−1.53347e7			−1.34514			−0.672569	−	0.740034i	\(-0.734809\pi\)
							−0.672569	+	0.740034i	\(0.734809\pi\)
\(41\)	−173650.	+	984816.i	−0.00959724	+	0.0544287i	−0.989230	−	0.146368i	\(-0.953242\pi\)
							0.979633	+	0.200796i	\(0.0643529\pi\)
\(43\)	9.41908e6	+	7.90355e6i	0.420146	+	0.352544i	0.828219	−	0.560405i	\(-0.189355\pi\)
							−0.408073	+	0.912949i	\(0.633799\pi\)
\(47\)	−1.97178e7	−	7.17671e6i	−0.589412	−	0.214528i	0.0300587	−	0.999548i	\(-0.490431\pi\)
							−0.619471	+	0.785020i	\(0.712653\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.e.a.5.6	✓	42
19.4	even	9	inner	38.10.e.a.23.6	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.e.a.5.6	✓	42	1.1	even	1	trivial
38.10.e.a.23.6	yes	42	19.4	even	9	inner