Embedded newform 38.10.e.b.17.4

• Label: 38.10.e.b.17.4
• Level: $38$
• Weight: $10$
• Character: 38.17
• 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			17.4
Character	\(\chi\)	\(=\)	38.17
Dual form			38.10.e.b.9.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.77837 - 15.7569i) q^{2} +(15.1406 + 12.7045i) q^{3} +(-240.561 - 87.5572i) q^{4} +(-8.62619 + 3.13968i) q^{5} +(242.250 - 203.272i) q^{6} +(80.6742 + 139.732i) q^{7} +(-2048.00 + 3547.24i) q^{8} +(-3350.08 - 18999.3i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 33 q^{3} + 528 q^{6} - 25809 q^{7} - 98304 q^{8} + 7953 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

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
							\(3\)	15.1406	+	12.7045i	0.107919	+	0.0905549i	0.695151	−	0.718864i	\(-0.255338\pi\)
−0.587231	+	0.809419i	\(0.699782\pi\)
\(5\)	−8.62619	+	3.13968i	−0.00617240	+	0.00224657i	−0.345104	−	0.938564i	\(-0.612157\pi\)
							0.338932	+	0.940811i	\(0.389934\pi\)
\(7\)	80.6742	+	139.732i	0.0126997	+	0.0219965i	0.872305	−	0.488961i	\(-0.162624\pi\)
							−0.859606	+	0.510958i	\(0.829291\pi\)
\(11\)	−29207.8	+	50589.3i	−0.601494	+	1.04182i	0.391101	+	0.920348i	\(0.372094\pi\)
							−0.992595	+	0.121470i	\(0.961239\pi\)
\(13\)	−24467.7	+	20530.9i	−0.237601	+	0.199371i	−0.753811	−	0.657091i	\(-0.771787\pi\)
							0.516210	+	0.856462i	\(0.327342\pi\)
\(17\)	−82728.8	+	469178.i	−0.240235	+	1.36244i	0.591068	+	0.806622i	\(0.298706\pi\)
							−0.831303	+	0.555819i	\(0.812405\pi\)
\(19\)	−496652.	−	275725.i	−0.874302	−	0.485383i
							\(23\)	1.26754e6	+	461346.i	0.944463	+	0.343757i	0.767927	−	0.640537i	\(-0.221288\pi\)
0.176537	+	0.984294i	\(0.443511\pi\)
\(29\)	1.14324e6	+	6.48361e6i	0.300154	+	1.70226i	0.645480	+	0.763777i	\(0.276657\pi\)
							−0.345325	+	0.938483i	\(0.612231\pi\)
\(31\)	2.08519e6	+	3.61165e6i	0.405525	+	0.702389i	0.994382	−	0.105847i	\(-0.0337555\pi\)
							−0.588858	+	0.808237i	\(0.700422\pi\)
\(37\)	409106.			0.0358862			0.0179431	−	0.999839i	\(-0.494288\pi\)
							0.0179431	+	0.999839i	\(0.494288\pi\)
\(41\)	−8.99505e6	−	7.54774e6i	−0.497137	−	0.417147i	0.359439	−	0.933169i	\(-0.382968\pi\)
							−0.856576	+	0.516021i	\(0.827413\pi\)
\(43\)	−3.68190e7	+	1.34010e7i	−1.64234	+	0.597764i	−0.987447	−	0.157953i	\(-0.949510\pi\)
							−0.654898	+	0.755718i	\(0.727288\pi\)
\(47\)	1.34042e6	+	7.60188e6i	0.0400682	+	0.227238i	0.998266	−	0.0588697i	\(-0.0187497\pi\)
							−0.958198	+	0.286108i	\(0.907639\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.e.b.17.4	yes	48
19.9	even	9	inner	38.10.e.b.9.4	✓	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.e.b.9.4	✓	48	19.9	even	9	inner
38.10.e.b.17.4	yes	48	1.1	even	1	trivial