Embedded newform 38.6.e.a.5.1

• Label: 38.6.e.a.5.1
• Level: $38$
• Weight: $6$
• Character: 38.5
• Analytic conductor: $6.095$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.09458515289\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			5.1
Character	\(\chi\)	\(=\)	38.5
Dual form			38.6.e.a.23.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.75877 - 1.36808i) q^{2} +(-2.69117 + 15.2624i) q^{3} +(12.2567 - 10.2846i) q^{4} +(-6.46132 - 5.42169i) q^{5} +(10.7647 + 61.0494i) q^{6} +(110.495 + 191.383i) q^{7} +(32.0000 - 55.4256i) q^{8} +(2.64806 + 0.963814i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 18 q^{3} - 72 q^{6} + 438 q^{7} + 768 q^{8} + 378 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.75877	−	1.36808i	0.664463	−	0.241845i
							\(3\)	−2.69117	+	15.2624i	−0.172638	+	0.979081i	0.768196	+	0.640215i	\(0.221155\pi\)
−0.940835	+	0.338866i	\(0.889957\pi\)
\(5\)	−6.46132	−	5.42169i	−0.115584	−	0.0969862i	0.583164	−	0.812354i	\(-0.301815\pi\)
							−0.698748	+	0.715368i	\(0.746259\pi\)
\(7\)	110.495	+	191.383i	0.852312	+	1.47625i	0.879117	+	0.476606i	\(0.158133\pi\)
							−0.0268053	+	0.999641i	\(0.508533\pi\)
\(11\)	−362.930	+	628.614i	−0.904360	+	1.56640i	−0.0825871	+	0.996584i	\(0.526318\pi\)
							−0.821773	+	0.569814i	\(0.807015\pi\)
\(13\)	−87.6873	−	497.299i	−0.143906	−	0.816130i	−0.968239	−	0.250026i	\(-0.919561\pi\)
							0.824333	−	0.566105i	\(-0.191550\pi\)
\(17\)	2027.50	−	737.951i	1.70153	−	0.619306i	0.705531	−	0.708680i	\(-0.250709\pi\)
							0.995998	+	0.0893737i	\(0.0284866\pi\)
\(19\)	−1016.37	−	1201.29i	−0.645905	−	0.763418i
							\(23\)	1475.28	−	1237.91i	0.581509	−	0.487944i	−0.303933	−	0.952693i	\(-0.598300\pi\)
0.885442	+	0.464749i	\(0.153856\pi\)
\(29\)	−2795.76	−	1017.57i	−0.617312	−	0.224683i	0.0143874	−	0.999896i	\(-0.495420\pi\)
							−0.631700	+	0.775213i	\(0.717642\pi\)
\(31\)	−636.652	−	1102.71i	−0.118987	−	0.206091i	0.800380	−	0.599493i	\(-0.204631\pi\)
							−0.919366	+	0.393402i	\(0.871298\pi\)
\(37\)	8224.53			0.987659			0.493829	−	0.869559i	\(-0.335597\pi\)
							0.493829	+	0.869559i	\(0.335597\pi\)
\(41\)	−373.661	+	2119.14i	−0.0347151	+	0.196879i	−0.997233	−	0.0743390i	\(-0.976315\pi\)
							0.962518	+	0.271218i	\(0.0874264\pi\)
\(43\)	755.936	+	634.305i	0.0623467	+	0.0523151i	0.673429	−	0.739252i	\(-0.264821\pi\)
							−0.611082	+	0.791567i	\(0.709265\pi\)
\(47\)	9607.06	+	3496.68i	0.634375	+	0.230894i	0.639134	−	0.769095i	\(-0.279293\pi\)
							−0.00475958	+	0.999989i	\(0.501515\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.a.5.1	✓	24
19.2	odd	18		722.6.a.p.1.10		12
19.4	even	9	inner	38.6.e.a.23.1	yes	24
19.17	even	9		722.6.a.o.1.3		12

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.a.5.1	✓	24	1.1	even	1	trivial
38.6.e.a.23.1	yes	24	19.4	even	9	inner
722.6.a.o.1.3		12	19.17	even	9
722.6.a.p.1.10		12	19.2	odd	18