Embedded newform 38.9.f.a.3.9

• Label: 38.9.f.a.3.9
• Level: $38$
• Weight: $9$
• Character: 38.3
• Analytic conductor: $15.480$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.4803871823\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(14\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.9
Character	\(\chi\)	\(=\)	38.3
Dual form			38.9.f.a.13.9

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.27231 - 8.66680i) q^{2} +(-36.7655 - 101.012i) q^{3} +(-22.2270 - 126.055i) q^{4} +(63.9854 - 362.879i) q^{5} +(-1142.82 - 415.954i) q^{6} +(-925.290 - 1602.65i) q^{7} +(-1254.14 - 724.077i) q^{8} +(-3825.78 + 3210.21i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 168 q^{3} - 2688 q^{6} - 6570 q^{7} - 2856 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.27231	−	8.66680i	0.454519	−	0.541675i
							\(3\)	−36.7655	−	101.012i	−0.453895	−	1.24707i	−0.929961	−	0.367657i	\(-0.880160\pi\)
0.476066	−	0.879409i	\(-0.342062\pi\)
\(5\)	63.9854	−	362.879i	0.102377	−	0.580607i	−0.889859	−	0.456235i	\(-0.849198\pi\)
							0.992236	−	0.124371i	\(-0.0396914\pi\)
\(7\)	−925.290	−	1602.65i	−0.385377	−	0.667492i	0.606445	−	0.795126i	\(-0.292595\pi\)
							−0.991821	+	0.127634i	\(0.959262\pi\)
\(11\)	9304.64	−	16116.1i	0.635520	−	1.10075i	−0.350885	−	0.936418i	\(-0.614119\pi\)
							0.986405	−	0.164334i	\(-0.0525474\pi\)
\(13\)	−10236.3	+	28124.0i	−0.358401	+	0.984700i	0.621183	+	0.783666i	\(0.286652\pi\)
							−0.979584	+	0.201034i	\(0.935570\pi\)
\(17\)	78091.3	+	65526.4i	0.934990	+	0.784550i	0.976706	−	0.214580i	\(-0.0688382\pi\)
							−0.0417166	+	0.999129i	\(0.513283\pi\)
\(19\)	−82787.4	+	100647.i	−0.635258	+	0.772300i
							\(23\)	−42660.6	−	241940.i	−0.152446	−	0.864563i	−0.961084	−	0.276257i	\(-0.910906\pi\)
0.808638	−	0.588306i	\(-0.200205\pi\)
\(29\)	−173310.	−	206543.i	−0.245037	−	0.292024i	0.629482	−	0.777015i	\(-0.283267\pi\)
							−0.874519	+	0.484991i	\(0.838823\pi\)
\(31\)	−240215.	+	138688.i	−0.260107	+	0.150173i	−0.624384	−	0.781118i	\(-0.714650\pi\)
							0.364276	+	0.931291i	\(0.381316\pi\)
\(37\)			1.17815e6i			0.628627i	0.949319	+	0.314313i	\(0.101774\pi\)
							−0.949319	+	0.314313i	\(0.898226\pi\)
\(41\)	−1.67955e6	−	4.61454e6i	−0.594372	−	1.63302i	−0.762299	−	0.647225i	\(-0.775929\pi\)
							0.167927	−	0.985799i	\(-0.446293\pi\)
\(43\)	499846.	−	2.83477e6i	0.146205	−	0.829171i	−0.820186	−	0.572096i	\(-0.806130\pi\)
							0.966392	−	0.257074i	\(-0.0827585\pi\)
\(47\)	1.32216e6	−	1.10942e6i	0.270952	−	0.227356i	−0.497180	−	0.867648i	\(-0.665631\pi\)
							0.768132	+	0.640292i	\(0.221187\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.f.a.3.9	✓	84
19.13	odd	18	inner	38.9.f.a.13.9	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.f.a.3.9	✓	84	1.1	even	1	trivial
38.9.f.a.13.9	yes	84	19.13	odd	18	inner