Embedded newform 38.9.f.a.3.4

• Label: 38.9.f.a.3.4
• 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			3.4
Character	\(\chi\)	\(=\)	38.3
Dual form			38.9.f.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.27231 + 8.66680i) q^{2} +(6.44442 + 17.7059i) q^{3} +(-22.2270 - 126.055i) q^{4} +(-26.7224 + 151.550i) q^{5} +(-200.319 - 72.9103i) q^{6} +(-536.717 - 929.622i) q^{7} +(1254.14 + 724.077i) q^{8} +(4754.05 - 3989.12i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 168 q^{3} - 2688 q^{6} - 6570 q^{7} - 2856 q^{9}+O(q^{10}) \)

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
							\(3\)	6.44442	+	17.7059i	0.0795608	+	0.218591i	0.973095	−	0.230403i	\(-0.0740045\pi\)
−0.893534	+	0.448995i	\(0.851782\pi\)
\(5\)	−26.7224	+	151.550i	−0.0427558	+	0.242480i	−0.998694	−	0.0510871i	\(-0.983731\pi\)
							0.955938	+	0.293567i	\(0.0948425\pi\)
\(7\)	−536.717	−	929.622i	−0.223539	−	0.387181i	0.732341	−	0.680938i	\(-0.238428\pi\)
							−0.955880	+	0.293757i	\(0.905094\pi\)
\(11\)	−7296.55	+	12638.0i	−0.498364	+	0.863192i	−0.999998	−	0.00188799i	\(-0.999399\pi\)
							0.501634	+	0.865080i	\(0.332732\pi\)
\(13\)	17724.3	−	48697.1i	0.620577	−	1.70502i	−0.0849940	−	0.996381i	\(-0.527087\pi\)
							0.705571	−	0.708639i	\(-0.250691\pi\)
\(17\)	50693.2	+	42536.7i	0.606952	+	0.509293i	0.893672	−	0.448721i	\(-0.148120\pi\)
							−0.286720	+	0.958014i	\(0.592565\pi\)
\(19\)	80704.1	+	102325.i	0.619272	+	0.785177i
							\(23\)	61365.7	+	348022.i	0.219288	+	1.24364i	0.873309	+	0.487167i	\(0.161970\pi\)
−0.654021	+	0.756476i	\(0.726919\pi\)
\(29\)	160818.	+	191656.i	0.227375	+	0.270975i	0.867655	−	0.497166i	\(-0.165626\pi\)
							−0.640280	+	0.768142i	\(0.721182\pi\)
\(31\)	1.14088e6	−	658689.i	1.23536	−	0.713236i	0.267219	−	0.963636i	\(-0.413895\pi\)
							0.968143	+	0.250400i	\(0.0805620\pi\)
\(37\)			2.93733e6i			1.56728i	0.621215	+	0.783640i	\(0.286639\pi\)
							−0.621215	+	0.783640i	\(0.713361\pi\)
\(41\)	−1.60095e6	−	4.39856e6i	−0.566554	−	1.55659i	−0.809847	−	0.586641i	\(-0.800450\pi\)
							0.243293	−	0.969953i	\(-0.421772\pi\)
\(43\)	958056.	−	5.43341e6i	0.280232	−	1.58927i	−0.441605	−	0.897209i	\(-0.645591\pi\)
							0.721837	−	0.692063i	\(-0.243298\pi\)
\(47\)	−3.31038e6	+	2.77774e6i	−0.678401	+	0.569246i	−0.915539	−	0.402230i	\(-0.868235\pi\)
							0.237138	+	0.971476i	\(0.423791\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.f.a.3.4	✓	84
19.13	odd	18	inner	38.9.f.a.13.4	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.f.a.3.4	✓	84	1.1	even	1	trivial
38.9.f.a.13.4	yes	84	19.13	odd	18	inner