Embedded newform 38.5.f.a.13.1

• Label: 38.5.f.a.13.1
• 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			13.1
Character	\(\chi\)	\(=\)	38.13
Dual form			38.5.f.a.3.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.81808 - 2.16670i) q^{2} +(-3.47598 + 9.55018i) q^{3} +(-1.38919 + 7.87846i) q^{4} +(-6.10286 - 34.6110i) q^{5} +(27.0120 - 9.83156i) q^{6} +(33.4829 - 57.9941i) q^{7} +(19.5959 - 11.3137i) q^{8} +(-17.0739 - 14.3267i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 12 q^{3} + 48 q^{6} + 90 q^{7} - 84 q^{9}+O(q^{10}) \)

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
							\(3\)	−3.47598	+	9.55018i	−0.386220	+	1.06113i	0.582469	+	0.812853i	\(0.302087\pi\)
−0.968689	+	0.248278i	\(0.920135\pi\)
\(5\)	−6.10286	−	34.6110i	−0.244114	−	1.38444i	−0.822541	−	0.568706i	\(-0.807444\pi\)
							0.578426	−	0.815735i	\(-0.303667\pi\)
\(7\)	33.4829	−	57.9941i	0.683325	−	1.18355i	−0.290636	−	0.956834i	\(-0.593867\pi\)
							0.973960	−	0.226719i	\(-0.0727999\pi\)
\(11\)	−92.1892	−	159.676i	−0.761894	−	1.31964i	−0.941873	−	0.335968i	\(-0.890936\pi\)
							0.179979	−	0.983670i	\(-0.442397\pi\)
\(13\)	−4.54627	−	12.4908i	−0.0269010	−	0.0739099i	0.925515	−	0.378711i	\(-0.123632\pi\)
							−0.952416	+	0.304801i	\(0.901410\pi\)
\(17\)	−183.691	+	154.135i	−0.635609	+	0.533340i	−0.902666	−	0.430341i	\(-0.858393\pi\)
							0.267057	+	0.963681i	\(0.413949\pi\)
\(19\)	356.862	+	54.5027i	0.988537	+	0.150977i
							\(23\)	152.780	−	866.457i	0.288808	−	1.63791i	−0.402547	−	0.915399i	\(-0.631875\pi\)
0.691355	−	0.722515i	\(-0.257014\pi\)
\(29\)	−361.177	+	430.434i	−0.429462	+	0.511813i	−0.936767	−	0.349954i	\(-0.886197\pi\)
							0.507305	+	0.861767i	\(0.330642\pi\)
\(31\)	−346.287	−	199.929i	−0.360340	−	0.208043i	0.308890	−	0.951098i	\(-0.400043\pi\)
							−0.669230	+	0.743055i	\(0.733376\pi\)
\(37\)			814.841i			0.595209i	0.954689	+	0.297605i	\(0.0961877\pi\)
							−0.954689	+	0.297605i	\(0.903812\pi\)
\(41\)	536.057	−	1472.81i	0.318892	−	0.876148i	−0.671886	−	0.740654i	\(-0.734516\pi\)
							0.990778	−	0.135494i	\(-0.0432621\pi\)
\(43\)	−23.4152	−	132.794i	−0.0126637	−	0.0718194i	0.977821	−	0.209442i	\(-0.0671649\pi\)
							−0.990485	+	0.137623i	\(0.956054\pi\)
\(47\)	3048.85	+	2558.29i	1.38019	+	1.15812i	0.969140	+	0.246510i	\(0.0792838\pi\)
							0.411054	+	0.911611i	\(0.365161\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.5.f.a.13.1	yes	36
19.3	odd	18	inner	38.5.f.a.3.1	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.f.a.3.1	✓	36	19.3	odd	18	inner
38.5.f.a.13.1	yes	36	1.1	even	1	trivial