Embedded newform 38.5.f.a.3.5

• Label: 38.5.f.a.3.5
• Level: $38$
• Weight: $5$
• Character: 38.3
• 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			3.5
Character	\(\chi\)	\(=\)	38.3
Dual form			38.5.f.a.13.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.81808 - 2.16670i) q^{2} +(-0.486419 - 1.33642i) q^{3} +(-1.38919 - 7.87846i) q^{4} +(3.87248 - 21.9619i) q^{5} +(-3.77998 - 1.37580i) q^{6} +(-12.4808 - 21.6173i) q^{7} +(-19.5959 - 11.3137i) q^{8} +(60.5002 - 50.7657i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 12 q^{3} + 48 q^{6} + 90 q^{7} - 84 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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
							\(3\)	−0.486419	−	1.33642i	−0.0540465	−	0.148492i	0.909732	−	0.415197i	\(-0.136287\pi\)
−0.963778	+	0.266705i	\(0.914065\pi\)
\(5\)	3.87248	−	21.9619i	0.154899	−	0.878477i	−0.803979	−	0.594658i	\(-0.797287\pi\)
							0.958878	−	0.283819i	\(-0.0916014\pi\)
\(7\)	−12.4808	−	21.6173i	−0.254710	−	0.441170i	0.710107	−	0.704094i	\(-0.248647\pi\)
							−0.964817	+	0.262924i	\(0.915313\pi\)
\(11\)	−23.8209	+	41.2590i	−0.196867	+	0.340984i	−0.947511	−	0.319723i	\(-0.896410\pi\)
							0.750644	+	0.660707i	\(0.229743\pi\)
\(13\)	−16.3525	+	44.9280i	−0.0967601	+	0.265846i	−0.978624	−	0.205658i	\(-0.934067\pi\)
							0.881864	+	0.471504i	\(0.156289\pi\)
\(17\)	200.647	+	168.363i	0.694282	+	0.582572i	0.920140	−	0.391589i	\(-0.128074\pi\)
							−0.225859	+	0.974160i	\(0.572519\pi\)
\(19\)	357.128	+	52.7313i	0.989274	+	0.146070i
							\(23\)	108.960	+	617.944i	0.205974	+	1.16814i	0.895900	+	0.444256i	\(0.146532\pi\)
−0.689926	+	0.723880i	\(0.742357\pi\)
\(29\)	−115.667	−	137.846i	−0.137535	−	0.163907i	0.692881	−	0.721052i	\(-0.256341\pi\)
							−0.830415	+	0.557145i	\(0.811897\pi\)
\(31\)	−307.796	+	177.706i	−0.320287	+	0.184918i	−0.651521	−	0.758631i	\(-0.725869\pi\)
							0.331233	+	0.943549i	\(0.392535\pi\)
\(37\)		−	596.017i		−	0.435367i	−0.976019	−	0.217683i	\(-0.930150\pi\)
							0.976019	−	0.217683i	\(-0.0698501\pi\)
\(41\)	−862.537	−	2369.80i	−0.513109	−	1.40976i	−0.877980	−	0.478698i	\(-0.841109\pi\)
							0.364870	−	0.931058i	\(-0.381113\pi\)
\(43\)	103.512	−	587.044i	0.0559825	−	0.317493i	−0.943938	−	0.330124i	\(-0.892910\pi\)
							0.999920	+	0.0126310i	\(0.00402067\pi\)
\(47\)	−627.241	+	526.318i	−0.283948	+	0.238261i	−0.773626	−	0.633643i	\(-0.781559\pi\)
							0.489678	+	0.871904i	\(0.337114\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.5.f.a.3.5	✓	36
19.13	odd	18	inner	38.5.f.a.13.5	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.f.a.3.5	✓	36	1.1	even	1	trivial
38.5.f.a.13.5	yes	36	19.13	odd	18	inner