Embedded newform 38.9.f.a.3.14

• Label: 38.9.f.a.3.14
• 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			3.14
Character	\(\chi\)	\(=\)	38.3
Dual form			38.9.f.a.13.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.27231 - 8.66680i) q^{2} +(49.7737 + 136.752i) q^{3} +(-22.2270 - 126.055i) q^{4} +(202.576 - 1148.87i) q^{5} +(1547.17 + 563.125i) q^{6} +(-1213.32 - 2101.54i) q^{7} +(-1254.14 - 724.077i) q^{8} +(-11197.7 + 9395.96i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 168 q^{3} - 2688 q^{6} - 6570 q^{7} - 2856 q^{9}+O(q^{10}) \)

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
							\(3\)	49.7737	+	136.752i	0.614490	+	1.68830i	0.720084	+	0.693887i	\(0.244103\pi\)
−0.105594	+	0.994409i	\(0.533675\pi\)
\(5\)	202.576	−	1148.87i	0.324122	−	1.83819i	−0.191655	−	0.981462i	\(-0.561385\pi\)
							0.515777	−	0.856723i	\(-0.327503\pi\)
\(7\)	−1213.32	−	2101.54i	−0.505341	−	0.875277i	−0.999981	−	0.00617882i	\(-0.998033\pi\)
							0.494639	−	0.869098i	\(-0.335300\pi\)
\(11\)	9874.14	−	17102.5i	0.674417	−	1.16812i	−0.302222	−	0.953237i	\(-0.597728\pi\)
							0.976639	−	0.214887i	\(-0.0689382\pi\)
\(13\)	1510.70	−	4150.62i	0.0528939	−	0.145325i	−0.910432	−	0.413658i	\(-0.864251\pi\)
							0.963326	+	0.268334i	\(0.0864730\pi\)
\(17\)	−17132.1	−	14375.6i	−0.205124	−	0.172119i	0.534439	−	0.845207i	\(-0.320523\pi\)
							−0.739562	+	0.673088i	\(0.764968\pi\)
\(19\)	126633.	−	30782.0i	0.971704	−	0.236201i
							\(23\)	24717.4	+	140179.i	0.0883265	+	0.500924i	0.996589	+	0.0825239i	\(0.0262981\pi\)
−0.908263	+	0.418401i	\(0.862591\pi\)
\(29\)	533240.	+	635491.i	0.753930	+	0.898498i	0.997448	−	0.0713979i	\(-0.0227460\pi\)
							−0.243518	+	0.969896i	\(0.578302\pi\)
\(31\)	624528.	−	360572.i	0.676247	−	0.390431i	−0.122192	−	0.992506i	\(-0.538993\pi\)
							0.798440	+	0.602075i	\(0.205659\pi\)
\(37\)			1.56299e6i			0.833967i	0.908914	+	0.416984i	\(0.136913\pi\)
							−0.908914	+	0.416984i	\(0.863087\pi\)
\(41\)	−856355.	−	2.35282e6i	−0.303053	−	0.832631i	−0.993966	−	0.109693i	\(-0.965013\pi\)
							0.690913	−	0.722938i	\(-0.257209\pi\)
\(43\)	732776.	−	4.15578e6i	0.214337	−	1.21557i	−0.667716	−	0.744416i	\(-0.732728\pi\)
							0.882053	−	0.471150i	\(-0.156161\pi\)
\(47\)	−3.07931e6	+	2.58385e6i	−0.631047	+	0.529511i	−0.901254	−	0.433291i	\(-0.857352\pi\)
							0.270207	+	0.962802i	\(0.412908\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.f.a.3.14	✓	84
19.13	odd	18	inner	38.9.f.a.13.14	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.f.a.3.14	✓	84	1.1	even	1	trivial
38.9.f.a.13.14	yes	84	19.13	odd	18	inner