Embedded newform 38.8.e.a.9.1

• Label: 38.8.e.a.9.1
• Level: $38$
• Weight: $8$
• Character: 38.9
• Analytic conductor: $11.871$
• Analytic rank: $0$
• Dimension: $30$
• 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:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.1
Character	\(\chi\)	\(=\)	38.9
Dual form			38.8.e.a.17.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.38919 + 7.87846i) q^{2} +(-46.9776 + 39.4189i) q^{3} +(-60.1403 + 21.8893i) q^{4} +(-248.833 - 90.5677i) q^{5} +(-375.821 - 315.351i) q^{6} +(81.2847 - 140.789i) q^{7} +(-256.000 - 443.405i) q^{8} +(273.278 - 1549.84i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 45 q^{3} + 360 q^{6} + 1806 q^{7} - 7680 q^{8} - 3105 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.38919	+	7.87846i	0.122788	+	0.696364i
							\(3\)	−46.9776	+	39.4189i	−1.00454	+	0.842908i	−0.987607	−	0.156949i	\(-0.949834\pi\)
−0.0169317	+	0.999857i	\(0.505390\pi\)
\(5\)	−248.833	−	90.5677i	−0.890251	−	0.324025i	−0.143912	−	0.989591i	\(-0.545968\pi\)
							−0.746339	+	0.665566i	\(0.768190\pi\)
\(7\)	81.2847	−	140.789i	0.0895706	−	0.155141i	−0.817759	−	0.575561i	\(-0.804784\pi\)
							0.907330	+	0.420420i	\(0.138117\pi\)
\(11\)	926.087	+	1604.03i	0.209786	+	0.363361i	0.951647	−	0.307193i	\(-0.0993898\pi\)
							−0.741861	+	0.670554i	\(0.766056\pi\)
\(13\)	3434.27	+	2881.70i	0.433544	+	0.363787i	0.833287	−	0.552841i	\(-0.186456\pi\)
							−0.399743	+	0.916627i	\(0.630901\pi\)
\(17\)	2.63223	+	14.9281i	0.000129943	+	0.000736943i	0.984873	−	0.173280i	\(-0.0554365\pi\)
							−0.984743	+	0.174017i	\(0.944325\pi\)
\(19\)	2655.91	−	29779.5i	0.0888333	−	0.996047i
							\(23\)	18055.4	−	6571.63i	0.309428	−	0.112623i	−0.182638	−	0.983180i	\(-0.558464\pi\)
0.492067	+	0.870557i	\(0.336242\pi\)
\(29\)	22722.9	−	128868.i	0.173010	−	0.981186i	−0.767407	−	0.641161i	\(-0.778453\pi\)
							0.940416	−	0.340025i	\(-0.110436\pi\)
\(31\)	74880.8	−	129697.i	0.451445	−	0.781925i	−0.547031	−	0.837112i	\(-0.684242\pi\)
							0.998476	+	0.0551868i	\(0.0175754\pi\)
\(37\)	−167136.			−0.542454			−0.271227	−	0.962515i	\(-0.587429\pi\)
							−0.271227	+	0.962515i	\(0.587429\pi\)
\(41\)	−321108.	+	269442.i	−0.727626	+	0.610551i	−0.929483	−	0.368864i	\(-0.879747\pi\)
							0.201857	+	0.979415i	\(0.435302\pi\)
\(43\)	−732234.	−	266511.i	−1.40446	−	0.511183i	−0.474963	−	0.880006i	\(-0.657539\pi\)
							−0.929500	+	0.368823i	\(0.879761\pi\)
\(47\)	−8858.33	+	50238.1i	−0.0124454	+	0.0705815i	−0.990398	−	0.138246i	\(-0.955854\pi\)
							0.977953	+	0.208827i	\(0.0669647\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.a.9.1	✓	30
19.17	even	9	inner	38.8.e.a.17.1	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.a.9.1	✓	30	1.1	even	1	trivial
38.8.e.a.17.1	yes	30	19.17	even	9	inner