Embedded newform 38.10.e.b.9.8

• Label: 38.10.e.b.9.8
• Level: $38$
• Weight: $10$
• Character: 38.9
• Analytic conductor: $19.571$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.5713617742\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(8\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.8
Character	\(\chi\)	\(=\)	38.9
Dual form			38.10.e.b.17.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.77837 + 15.7569i) q^{2} +(194.492 - 163.198i) q^{3} +(-240.561 + 87.5572i) q^{4} +(-1439.52 - 523.944i) q^{5} +(3111.88 + 2611.17i) q^{6} +(-2289.54 + 3965.60i) q^{7} +(-2048.00 - 3547.24i) q^{8} +(7775.61 - 44097.7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 33 q^{3} + 528 q^{6} - 25809 q^{7} - 98304 q^{8} + 7953 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	2.77837	+	15.7569i	0.122788	+	0.696364i
							\(3\)	194.492	−	163.198i	1.38630	−	1.16324i	0.419484	−	0.907763i	\(-0.362211\pi\)
0.966814	−	0.255480i	\(-0.0822335\pi\)
\(5\)	−1439.52	−	523.944i	−1.03004	−	0.374904i	−0.228943	−	0.973440i	\(-0.573527\pi\)
							−0.801096	+	0.598536i	\(0.795749\pi\)
\(7\)	−2289.54	+	3965.60i	−0.360419	+	0.624264i	−0.988030	−	0.154263i	\(-0.950700\pi\)
							0.627611	+	0.778527i	\(0.284033\pi\)
\(11\)	−3174.03	−	5497.57i	−0.0653647	−	0.113215i	0.831491	−	0.555538i	\(-0.187488\pi\)
							−0.896856	+	0.442323i	\(0.854154\pi\)
\(13\)	−84703.4	−	71074.6i	−0.822538	−	0.690191i	0.131027	−	0.991379i	\(-0.458172\pi\)
							−0.953565	+	0.301188i	\(0.902617\pi\)
\(17\)	27982.6	+	158697.i	0.0812583	+	0.460839i	0.998101	+	0.0615912i	\(0.0196175\pi\)
							−0.916843	+	0.399248i	\(0.869271\pi\)
\(19\)	−542551.	−	168302.i	−0.955102	−	0.296278i
							\(23\)	−2.13185e6	+	775929.i	−1.58848	+	0.578159i	−0.977025	−	0.213123i	\(-0.931637\pi\)
−0.611452	+	0.791281i	\(0.709414\pi\)
\(29\)	940211.	−	5.33220e6i	0.246851	−	1.39996i	−0.569305	−	0.822127i	\(-0.692788\pi\)
							0.816155	−	0.577833i	\(-0.196101\pi\)
\(31\)	1.02507e6	−	1.77547e6i	0.199354	−	0.345292i	−0.748965	−	0.662610i	\(-0.769449\pi\)
							0.948319	+	0.317318i	\(0.102782\pi\)
\(37\)	4.21679e6			0.369891			0.184946	−	0.982749i	\(-0.440789\pi\)
							0.184946	+	0.982749i	\(0.440789\pi\)
\(41\)	−2.95132e6	+	2.47645e6i	−0.163113	+	0.136868i	−0.720691	−	0.693257i	\(-0.756175\pi\)
							0.557578	+	0.830125i	\(0.311731\pi\)
\(43\)	3.45625e7	+	1.25797e7i	1.54169	+	0.561130i	0.966450	−	0.256853i	\(-0.0826858\pi\)
							0.575242	+	0.817983i	\(0.304908\pi\)
\(47\)	2.12930e6	−	1.20758e7i	0.0636496	−	0.360975i	−0.936303	−	0.351194i	\(-0.885776\pi\)
							0.999952	−	0.00978039i	\(-0.00311324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.e.b.9.8	✓	48
19.17	even	9	inner	38.10.e.b.17.8	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.e.b.9.8	✓	48	1.1	even	1	trivial
38.10.e.b.17.8	yes	48	19.17	even	9	inner