Embedded newform 38.10.e.a.9.4

• Label: 38.10.e.a.9.4
• Level: $38$
• Weight: $10$
• Character: 38.9
• 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			9.4
Character	\(\chi\)	\(=\)	38.9
Dual form			38.10.e.a.17.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.77837 - 15.7569i) q^{2} +(41.1535 - 34.5319i) q^{3} +(-240.561 + 87.5572i) q^{4} +(-479.910 - 174.673i) q^{5} +(-658.456 - 552.510i) q^{6} +(5188.46 - 8986.67i) q^{7} +(2048.00 + 3547.24i) q^{8} +(-2916.76 + 16541.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 252 q^{3} + 4032 q^{6} + 14373 q^{7} + 86016 q^{8} + 12852 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.77837	−	15.7569i	−0.122788	−	0.696364i
							\(3\)	41.1535	−	34.5319i	0.293333	−	0.246136i	−0.484230	−	0.874941i	\(-0.660900\pi\)
0.777563	+	0.628805i	\(0.216456\pi\)
\(5\)	−479.910	−	174.673i	−0.343396	−	0.124986i	0.164565	−	0.986366i	\(-0.447378\pi\)
							−0.507960	+	0.861381i	\(0.669600\pi\)
\(7\)	5188.46	−	8986.67i	0.816765	−	1.41468i	−0.0912886	−	0.995824i	\(-0.529099\pi\)
							0.908054	−	0.418854i	\(-0.137568\pi\)
\(11\)	−21056.4	−	36470.8i	−0.433628	−	0.751065i	0.563555	−	0.826079i	\(-0.309433\pi\)
							−0.997183	+	0.0750134i	\(0.976100\pi\)
\(13\)	−74943.4	−	62885.0i	−0.727760	−	0.610663i	0.201760	−	0.979435i	\(-0.435334\pi\)
							−0.929520	+	0.368772i	\(0.879778\pi\)
\(17\)	−13534.1	−	76755.8i	−0.0393015	−	0.222890i	0.958831	−	0.283978i	\(-0.0916542\pi\)
							−0.998132	+	0.0610876i	\(0.980543\pi\)
\(19\)	−38382.6	+	566758.i	−0.0675683	+	0.997715i
							\(23\)	−1.48851e6	+	541772.i	−1.10911	+	0.403684i	−0.830668	−	0.556769i	\(-0.812041\pi\)
−0.278444	+	0.960452i	\(0.589819\pi\)
\(29\)	102214.	−	579684.i	0.0268361	−	0.152195i	−0.968445	−	0.249227i	\(-0.919824\pi\)
							0.995281	+	0.0970314i	\(0.0309347\pi\)
\(31\)	−4.61577e6	+	7.99475e6i	−0.897670	+	1.55481i	−0.0672041	+	0.997739i	\(0.521408\pi\)
							−0.830465	+	0.557070i	\(0.811925\pi\)
\(37\)	−1.15071e7			−1.00939			−0.504694	−	0.863298i	\(-0.668395\pi\)
							−0.504694	+	0.863298i	\(0.668395\pi\)
\(41\)	1.71576e6	−	1.43969e6i	0.0948265	−	0.0795689i	−0.594142	−	0.804360i	\(-0.702508\pi\)
							0.688968	+	0.724791i	\(0.258064\pi\)
\(43\)	1.75381e7	+	6.38334e6i	0.782301	+	0.284734i	0.702132	−	0.712047i	\(-0.252232\pi\)
							0.0801692	+	0.996781i	\(0.474454\pi\)
\(47\)	8.76099e6	−	4.96861e7i	0.261886	−	1.48523i	−0.515871	−	0.856666i	\(-0.672531\pi\)
							0.777757	−	0.628565i	\(-0.216357\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.e.a.9.4	✓	42
19.17	even	9	inner	38.10.e.a.17.4	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.e.a.9.4	✓	42	1.1	even	1	trivial
38.10.e.a.17.4	yes	42	19.17	even	9	inner