Embedded newform 38.5.f.a.13.3

• Label: 38.5.f.a.13.3
• Level: $38$
• Weight: $5$
• Character: 38.13
• Analytic conductor: $3.928$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.92805859719\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			13.3
Character	\(\chi\)	\(=\)	38.13
Dual form			38.5.f.a.3.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.81808 - 2.16670i) q^{2} +(0.470754 - 1.29339i) q^{3} +(-1.38919 + 7.87846i) q^{4} +(8.21061 + 46.5647i) q^{5} +(-3.65825 + 1.33149i) q^{6} +(33.5016 - 58.0265i) q^{7} +(19.5959 - 11.3137i) q^{8} +(60.5984 + 50.8481i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 12 q^{3} + 48 q^{6} + 90 q^{7} - 84 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.81808	−	2.16670i	−0.454519	−	0.541675i
							\(3\)	0.470754	−	1.29339i	0.0523060	−	0.143710i	−0.910788	−	0.412874i	\(-0.864525\pi\)
0.963094	+	0.269164i	\(0.0867474\pi\)
\(5\)	8.21061	+	46.5647i	0.328424	+	1.86259i	0.484428	+	0.874831i	\(0.339028\pi\)
							−0.156004	+	0.987756i	\(0.549861\pi\)
\(7\)	33.5016	−	58.0265i	0.683707	−	1.18421i	−0.290135	−	0.956986i	\(-0.593700\pi\)
							0.973841	−	0.227229i	\(-0.0729665\pi\)
\(11\)	3.72198	+	6.44665i	0.0307601	+	0.0532781i	0.880996	−	0.473124i	\(-0.156874\pi\)
							−0.850236	+	0.526402i	\(0.823541\pi\)
\(13\)	77.4823	+	212.881i	0.458475	+	1.25965i	0.926620	+	0.375998i	\(0.122700\pi\)
							−0.468145	+	0.883651i	\(0.655078\pi\)
\(17\)	12.5438	−	10.5255i	0.0434042	−	0.0364204i	−0.620828	−	0.783947i	\(-0.713203\pi\)
							0.664232	+	0.747527i	\(0.268759\pi\)
\(19\)	−106.087	−	345.060i	−0.293871	−	0.955845i
							\(23\)	−130.513	+	740.174i	−0.246716	+	1.39920i	0.569758	+	0.821813i	\(0.307037\pi\)
−0.816474	+	0.577382i	\(0.804074\pi\)
\(29\)	902.786	−	1075.90i	1.07347	−	1.27931i	0.115229	−	0.993339i	\(-0.463240\pi\)
							0.958239	−	0.285970i	\(-0.0923156\pi\)
\(31\)	−649.227	−	374.831i	−0.675574	−	0.390043i	0.122611	−	0.992455i	\(-0.460873\pi\)
							−0.798185	+	0.602412i	\(0.794206\pi\)
\(37\)		−	1153.46i		−	0.842558i	−0.906931	−	0.421279i	\(-0.861581\pi\)
							0.906931	−	0.421279i	\(-0.138419\pi\)
\(41\)	203.926	−	560.283i	0.121313	−	0.333303i	−0.864141	−	0.503250i	\(-0.832137\pi\)
							0.985453	+	0.169947i	\(0.0543596\pi\)
\(43\)	−213.032	−	1208.17i	−0.115215	−	0.653415i	−0.986644	−	0.162894i	\(-0.947917\pi\)
							0.871429	−	0.490522i	\(-0.163194\pi\)
\(47\)	−479.037	−	401.960i	−0.216857	−	0.181965i	0.527888	−	0.849314i	\(-0.322984\pi\)
							−0.744745	+	0.667350i	\(0.767429\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.5.f.a.13.3	yes	36
19.3	odd	18	inner	38.5.f.a.3.3	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.f.a.3.3	✓	36	19.3	odd	18	inner
38.5.f.a.13.3	yes	36	1.1	even	1	trivial