Embedded newform 38.10.e.b.9.5

• Label: 38.10.e.b.9.5
• 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			9.5
Character	\(\chi\)	\(=\)	38.9
Dual form			38.10.e.b.17.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(2.77837 + 15.7569i) q^{2} +(30.0038 - 25.1762i) q^{3} +(-240.561 + 87.5572i) q^{4} +(-2542.70 - 925.467i) q^{5} +(480.061 + 402.819i) q^{6} +(3969.67 - 6875.67i) q^{7} +(-2048.00 - 3547.24i) q^{8} +(-3151.53 + 17873.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 33 q^{3} + 528 q^{6} - 25809 q^{7} - 98304 q^{8} + 7953 q^{9}+O(q^{10}) \)

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
							\(3\)	30.0038	−	25.1762i	0.213861	−	0.179450i	−0.529564	−	0.848270i	\(-0.677645\pi\)
0.743425	+	0.668819i	\(0.233200\pi\)
\(5\)	−2542.70	−	925.467i	−1.81941	−	0.662210i	−0.995419	−	0.0956038i	\(-0.969522\pi\)
							−0.823988	−	0.566607i	\(-0.808256\pi\)
\(7\)	3969.67	−	6875.67i	0.624904	−	1.08237i	−0.363655	−	0.931534i	\(-0.618471\pi\)
							0.988559	−	0.150832i	\(-0.0481952\pi\)
\(11\)	33012.1	+	57178.6i	0.679838	+	1.17751i	0.975029	+	0.222076i	\(0.0712835\pi\)
							−0.295191	+	0.955438i	\(0.595383\pi\)
\(13\)	108084.	+	90693.4i	1.04958	+	0.880705i	0.993050	−	0.117696i	\(-0.0375508\pi\)
							0.0565336	+	0.998401i	\(0.481995\pi\)
\(17\)	−63200.9	−	358430.i	−0.183528	−	1.04084i	−0.927832	−	0.372999i	\(-0.878329\pi\)
							0.744303	−	0.667842i	\(-0.232782\pi\)
\(19\)	567999.	+	8049.39i	0.999900	+	0.0141701i
							\(23\)	−624830.	+	227420.i	−0.465572	+	0.169454i	−0.564145	−	0.825675i	\(-0.690794\pi\)
0.0985734	+	0.995130i	\(0.468572\pi\)
\(29\)	511837.	−	2.90277e6i	0.134382	−	0.762118i	−0.840906	−	0.541181i	\(-0.817977\pi\)
							0.975288	−	0.220937i	\(-0.0709115\pi\)
\(31\)	−2.09402e6	+	3.62695e6i	−0.407243	+	0.705366i	−0.994580	−	0.103977i	\(-0.966843\pi\)
							0.587337	+	0.809343i	\(0.300176\pi\)
\(37\)	2.56460e6			0.224963			0.112481	−	0.993654i	\(-0.464120\pi\)
							0.112481	+	0.993654i	\(0.464120\pi\)
\(41\)	1.46108e7	−	1.22599e7i	0.807506	−	0.677578i	−0.142505	−	0.989794i	\(-0.545516\pi\)
							0.950011	+	0.312216i	\(0.101071\pi\)
\(43\)	3.19768e7	+	1.16386e7i	1.42635	+	0.519149i	0.935883	−	0.352310i	\(-0.114604\pi\)
							0.490467	+	0.871460i	\(0.336826\pi\)
\(47\)	−1.75989e6	+	9.98086e6i	−0.0526073	+	0.298351i	−0.999748	−	0.0224689i	\(-0.992847\pi\)
							0.947140	+	0.320820i	\(0.103958\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.e.b.9.5	✓	48
19.17	even	9	inner	38.10.e.b.17.5	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.e.b.9.5	✓	48	1.1	even	1	trivial
38.10.e.b.17.5	yes	48	19.17	even	9	inner