Embedded newform 38.5.f.a.15.1

• Label: 38.5.f.a.15.1
• Level: $38$
• Weight: $5$
• Character: 38.15
• 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			15.1
Character	\(\chi\)	\(=\)	38.15
Dual form			38.5.f.a.33.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.967379 + 2.65785i) q^{2} +(-10.7009 + 1.88685i) q^{3} +(-6.12836 - 5.14230i) q^{4} +(8.30053 - 6.96497i) q^{5} +(5.33682 - 30.2666i) q^{6} +(31.7080 - 54.9198i) q^{7} +(19.5959 - 11.3137i) q^{8} +(34.8331 - 12.6782i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 12 q^{3} + 48 q^{6} + 90 q^{7} - 84 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.967379	+	2.65785i	−0.241845	+	0.664463i
							\(3\)	−10.7009	+	1.88685i	−1.18898	+	0.209650i	−0.732931	−	0.680302i	\(-0.761848\pi\)
−0.456053	+	0.889953i	\(0.650737\pi\)
\(5\)	8.30053	−	6.96497i	0.332021	−	0.278599i	−0.461502	−	0.887139i	\(-0.652689\pi\)
							0.793523	+	0.608541i	\(0.208245\pi\)
\(7\)	31.7080	−	54.9198i	0.647101	−	1.12081i	−0.336711	−	0.941608i	\(-0.609314\pi\)
							0.983812	−	0.179204i	\(-0.0573522\pi\)
\(11\)	−76.3301	−	132.208i	−0.630827	−	1.09262i	−0.987383	−	0.158351i	\(-0.949382\pi\)
							0.356556	−	0.934274i	\(-0.383951\pi\)
\(13\)	25.4539	+	4.48821i	0.150615	+	0.0265575i	0.248447	−	0.968645i	\(-0.420080\pi\)
							−0.0978323	+	0.995203i	\(0.531191\pi\)
\(17\)	−328.706	−	119.639i	−1.13739	−	0.413976i	−0.296420	−	0.955058i	\(-0.595793\pi\)
							−0.840970	+	0.541081i	\(0.818015\pi\)
\(19\)	14.0048	−	360.728i	0.0387944	−	0.999247i
							\(23\)	−114.797	−	96.3260i	−0.217007	−	0.182091i	0.527803	−	0.849367i	\(-0.323016\pi\)
−0.744811	+	0.667276i	\(0.767460\pi\)
\(29\)	186.650	+	512.817i	0.221938	+	0.609771i	0.999826	−	0.0186320i	\(-0.00593111\pi\)
							−0.777888	+	0.628403i	\(0.783709\pi\)
\(31\)	−373.959	−	215.906i	−0.389136	−	0.224668i	0.292650	−	0.956220i	\(-0.405463\pi\)
							−0.681786	+	0.731552i	\(0.738796\pi\)
\(37\)		−	1759.28i		−	1.28508i	−0.766250	−	0.642542i	\(-0.777880\pi\)
							0.766250	−	0.642542i	\(-0.222120\pi\)
\(41\)	758.625	−	133.766i	0.451294	−	0.0795753i	0.0566185	−	0.998396i	\(-0.481968\pi\)
							0.394676	+	0.918821i	\(0.370857\pi\)
\(43\)	2061.95	−	1730.18i	1.11517	−	0.935741i	0.116822	−	0.993153i	\(-0.462729\pi\)
							0.998351	+	0.0574122i	\(0.0182849\pi\)
\(47\)	−3022.13	+	1099.96i	−1.36810	+	0.497947i	−0.918548	−	0.395309i	\(-0.870638\pi\)
							−0.449549	+	0.893256i	\(0.648415\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.5.f.a.15.1	✓	36
19.14	odd	18	inner	38.5.f.a.33.1	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.f.a.15.1	✓	36	1.1	even	1	trivial
38.5.f.a.33.1	yes	36	19.14	odd	18	inner