Embedded newform 38.5.f.a.13.6

• Label: 38.5.f.a.13.6
• 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			13.6
Character	\(\chi\)	\(=\)	38.13
Dual form			38.5.f.a.3.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.81808 + 2.16670i) q^{2} +(4.86583 - 13.3688i) q^{3} +(-1.38919 + 7.87846i) q^{4} +(1.39867 + 7.93225i) q^{5} +(37.8126 - 13.7627i) q^{6} +(43.6122 - 75.5386i) q^{7} +(-19.5959 + 11.3137i) q^{8} +(-92.9980 - 78.0346i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 12 q^{3} + 48 q^{6} + 90 q^{7} - 84 q^{9}+O(q^{10}) \)

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
							\(3\)	4.86583	−	13.3688i	0.540648	−	1.48542i	−0.305354	−	0.952239i	\(-0.598775\pi\)
0.846002	−	0.533180i	\(-0.179003\pi\)
\(5\)	1.39867	+	7.93225i	0.0559468	+	0.317290i	0.999919	−	0.0127352i	\(-0.00405385\pi\)
							−0.943972	+	0.330025i	\(0.892943\pi\)
\(7\)	43.6122	−	75.5386i	0.890045	−	1.54160i	0.0502251	−	0.998738i	\(-0.484006\pi\)
							0.839820	−	0.542865i	\(-0.182661\pi\)
\(11\)	76.3319	+	132.211i	0.630842	+	1.09265i	0.987380	+	0.158370i	\(0.0506237\pi\)
							−0.356538	+	0.934281i	\(0.616043\pi\)
\(13\)	8.37534	+	23.0111i	0.0495582	+	0.136160i	0.962002	−	0.273041i	\(-0.0880296\pi\)
							−0.912444	+	0.409201i	\(0.865807\pi\)
\(17\)	−315.298	+	264.566i	−1.09100	+	0.915455i	−0.996787	−	0.0800974i	\(-0.974477\pi\)
							−0.0942097	+	0.995552i	\(0.530032\pi\)
\(19\)	−187.458	+	308.514i	−0.519273	+	0.854608i
							\(23\)	30.2035	−	171.293i	0.0570955	−	0.323805i	−0.942860	−	0.333188i	\(-0.891876\pi\)
0.999956	+	0.00938320i	\(0.00298681\pi\)
\(29\)	−639.626	+	762.276i	−0.760554	+	0.906393i	−0.997883	−	0.0650366i	\(-0.979284\pi\)
							0.237329	+	0.971429i	\(0.423728\pi\)
\(31\)	−91.0276	−	52.5548i	−0.0947217	−	0.0546876i	0.451891	−	0.892073i	\(-0.350750\pi\)
							−0.546613	+	0.837386i	\(0.684083\pi\)
\(37\)			1350.03i			0.986142i	0.869989	+	0.493071i	\(0.164126\pi\)
							−0.869989	+	0.493071i	\(0.835874\pi\)
\(41\)	346.562	−	952.170i	0.206164	−	0.566431i	−0.792915	−	0.609332i	\(-0.791438\pi\)
							0.999079	+	0.0429010i	\(0.0136600\pi\)
\(43\)	−311.940	−	1769.10i	−0.168707	−	0.956787i	−0.945159	−	0.326610i	\(-0.894093\pi\)
							0.776452	−	0.630177i	\(-0.217018\pi\)
\(47\)	1684.53	+	1413.49i	0.762575	+	0.639876i	0.938796	−	0.344474i	\(-0.111943\pi\)
							−0.176221	+	0.984351i	\(0.556387\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.5.f.a.13.6	yes	36
19.3	odd	18	inner	38.5.f.a.3.6	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.f.a.3.6	✓	36	19.3	odd	18	inner
38.5.f.a.13.6	yes	36	1.1	even	1	trivial