Embedded newform 38.10.e.b.9.6

• Label: 38.10.e.b.9.6
• 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			9.6
Character	\(\chi\)	\(=\)	38.9
Dual form			38.10.e.b.17.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.77837 + 15.7569i) q^{2} +(99.1452 - 83.1927i) q^{3} +(-240.561 + 87.5572i) q^{4} +(750.555 + 273.180i) q^{5} +(1586.32 + 1331.08i) q^{6} +(-2034.77 + 3524.32i) q^{7} +(-2048.00 - 3547.24i) q^{8} +(-509.171 + 2887.65i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 33 q^{3} + 528 q^{6} - 25809 q^{7} - 98304 q^{8} + 7953 q^{9}+O(q^{10}) \)

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
							\(3\)	99.1452	−	83.1927i	0.706685	−	0.592979i	−0.216982	−	0.976176i	\(-0.569621\pi\)
0.923667	+	0.383196i	\(0.125177\pi\)
\(5\)	750.555	+	273.180i	0.537053	+	0.195471i	0.596285	−	0.802773i	\(-0.296643\pi\)
							−0.0592316	+	0.998244i	\(0.518865\pi\)
\(7\)	−2034.77	+	3524.32i	−0.320312	+	0.554797i	−0.980552	−	0.196258i	\(-0.937121\pi\)
							0.660240	+	0.751054i	\(0.270454\pi\)
\(11\)	−19162.4	−	33190.2i	−0.394623	−	0.683508i	0.598430	−	0.801175i	\(-0.295792\pi\)
							−0.993053	+	0.117668i	\(0.962458\pi\)
\(13\)	142893.	+	119901.i	1.38760	+	1.16434i	0.966301	+	0.257414i	\(0.0828705\pi\)
							0.421300	+	0.906921i	\(0.361574\pi\)
\(17\)	87387.1	+	495597.i	0.253762	+	1.43916i	0.799230	+	0.601026i	\(0.205241\pi\)
							−0.545467	+	0.838132i	\(0.683648\pi\)
\(19\)	506805.	−	256586.i	0.892175	−	0.451691i
							\(23\)	94432.5	−	34370.6i	0.0703633	−	0.0256101i	−0.306599	−	0.951839i	\(-0.599191\pi\)
0.376962	+	0.926229i	\(0.376969\pi\)
\(29\)	−1.06460e6	+	6.03764e6i	−0.279509	+	1.58517i	0.444757	+	0.895651i	\(0.353290\pi\)
							−0.724266	+	0.689521i	\(0.757821\pi\)
\(31\)	−3.23708e6	+	5.60678e6i	−0.629543	+	1.09040i	0.358100	+	0.933683i	\(0.383425\pi\)
							−0.987643	+	0.156718i	\(0.949909\pi\)
\(37\)	7.51618e6			0.659309			0.329655	−	0.944102i	\(-0.393068\pi\)
							0.329655	+	0.944102i	\(0.393068\pi\)
\(41\)	1.43862e7	−	1.20715e7i	0.795097	−	0.667166i	−0.151904	−	0.988395i	\(-0.548541\pi\)
							0.947001	+	0.321229i	\(0.104096\pi\)
\(43\)	−2.19797e7	−	7.99994e6i	−0.980422	−	0.356844i	−0.198418	−	0.980118i	\(-0.563580\pi\)
							−0.782004	+	0.623273i	\(0.785802\pi\)
\(47\)	5.93532e6	−	3.36609e7i	0.177421	−	1.00620i	−0.757892	−	0.652380i	\(-0.773771\pi\)
							0.935313	−	0.353822i	\(-0.115118\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.e.b.9.6	✓	48
19.17	even	9	inner	38.10.e.b.17.6	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.e.b.9.6	✓	48	1.1	even	1	trivial
38.10.e.b.17.6	yes	48	19.17	even	9	inner