Embedded newform 38.8.e.b.5.1

• Label: 38.8.e.b.5.1
• Level: $38$
• Weight: $8$
• Character: 38.5
• Analytic conductor: $11.871$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8706309684\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			5.1
Character	\(\chi\)	\(=\)	38.5
Dual form			38.8.e.b.23.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.51754 - 2.73616i) q^{2} +(-13.9835 + 79.3043i) q^{3} +(49.0268 - 41.1384i) q^{4} +(-397.617 - 333.640i) q^{5} +(111.868 + 634.435i) q^{6} +(170.178 + 294.757i) q^{7} +(256.000 - 443.405i) q^{8} +(-4038.53 - 1469.91i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 84 q^{3} + 672 q^{6} - 2988 q^{7} + 9216 q^{8} + 1428 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.51754	−	2.73616i	0.664463	−	0.241845i
							\(3\)	−13.9835	+	79.3043i	−0.299014	+	1.69579i	0.351411	+	0.936221i	\(0.385702\pi\)
−0.650425	+	0.759571i	\(0.725409\pi\)
\(5\)	−397.617	−	333.640i	−1.42256	−	1.19367i	−0.949953	−	0.312393i	\(-0.898869\pi\)
							−0.472605	−	0.881274i	\(-0.656686\pi\)
\(7\)	170.178	+	294.757i	0.187525	+	0.324804i	0.944425	−	0.328728i	\(-0.106620\pi\)
							−0.756899	+	0.653532i	\(0.773287\pi\)
\(11\)	1834.06	−	3176.69i	0.415470	−	0.719616i	−0.580007	−	0.814611i	\(-0.696950\pi\)
							0.995478	+	0.0949956i	\(0.0302837\pi\)
\(13\)	−1167.36	−	6620.41i	−0.147368	−	0.835763i	−0.965436	−	0.260641i	\(-0.916066\pi\)
							0.818068	−	0.575121i	\(-0.195045\pi\)
\(17\)	−13957.3	+	5080.06i	−0.689020	+	0.250783i	−0.662716	−	0.748871i	\(-0.730596\pi\)
							−0.0263044	+	0.999654i	\(0.508374\pi\)
\(19\)	−16141.0	−	25166.2i	−0.539875	−	0.841745i
							\(23\)	−58647.2	+	49210.8i	−1.00508	+	0.843360i	−0.987680	−	0.156488i	\(-0.949983\pi\)
−0.0173978	+	0.999849i	\(0.505538\pi\)
\(29\)	−161594.	−	58815.3i	−1.23036	−	0.447814i	−0.356638	−	0.934243i	\(-0.616077\pi\)
							−0.873720	+	0.486429i	\(0.838299\pi\)
\(31\)	−8167.49	−	14146.5i	−0.0492405	−	0.0852871i	0.840355	−	0.542037i	\(-0.182347\pi\)
							−0.889595	+	0.456750i	\(0.849013\pi\)
\(37\)	−402.433			−0.00130613			−0.000653067	−	1.00000i	\(-0.500208\pi\)
							−0.000653067		1.00000i	\(0.500208\pi\)
\(41\)	1578.35	−	8951.26i	0.00357651	−	0.0202834i	−0.982967	−	0.183781i	\(-0.941166\pi\)
							0.986544	+	0.163497i	\(0.0522775\pi\)
\(43\)	699861.	+	587253.i	1.34237	+	1.12638i	0.981011	+	0.193954i	\(0.0621314\pi\)
							0.361359	+	0.932427i	\(0.382313\pi\)
\(47\)	−1.25920e6	−	458313.i	−1.76911	−	0.643902i	−0.999989	−	0.00479493i	\(-0.998474\pi\)
							−0.769118	−	0.639107i	\(-0.779304\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.b.5.1	✓	36
19.4	even	9	inner	38.8.e.b.23.1	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.b.5.1	✓	36	1.1	even	1	trivial
38.8.e.b.23.1	yes	36	19.4	even	9	inner