Embedded newform 38.6.e.a.9.4

• Label: 38.6.e.a.9.4
• Level: $38$
• Weight: $6$
• Character: 38.9
• 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			9.4
Character	\(\chi\)	\(=\)	38.9
Dual form			38.6.e.a.17.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.694593 - 3.93923i) q^{2} +(15.2107 - 12.7633i) q^{3} +(-15.0351 + 5.47232i) q^{4} +(-77.4900 - 28.2040i) q^{5} +(-60.8427 - 51.0531i) q^{6} +(31.0980 - 53.8633i) q^{7} +(32.0000 + 55.4256i) q^{8} +(26.2669 - 148.967i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 18 q^{3} - 72 q^{6} + 438 q^{7} + 768 q^{8} + 378 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.694593	−	3.93923i	−0.122788	−	0.696364i
							\(3\)	15.2107	−	12.7633i	0.975765	−	0.818764i	−0.00768044	−	0.999971i	\(-0.502445\pi\)
0.983445	+	0.181207i	\(0.0580003\pi\)
\(5\)	−77.4900	−	28.2040i	−1.38618	−	0.504529i	−0.462136	−	0.886809i	\(-0.652917\pi\)
							−0.924047	+	0.382280i	\(0.875139\pi\)
\(7\)	31.0980	−	53.8633i	0.239876	−	0.415478i	−0.720802	−	0.693141i	\(-0.756226\pi\)
							0.960679	+	0.277663i	\(0.0895598\pi\)
\(11\)	23.0319	+	39.8924i	0.0573915	+	0.0994050i	0.893294	−	0.449473i	\(-0.148388\pi\)
							−0.835902	+	0.548878i	\(0.815055\pi\)
\(13\)	−470.595	−	394.876i	−0.772305	−	0.648041i	0.168993	−	0.985617i	\(-0.445948\pi\)
							−0.941298	+	0.337576i	\(0.890393\pi\)
\(17\)	−265.830	−	1507.60i	−0.223091	−	1.26521i	−0.866301	−	0.499522i	\(-0.833509\pi\)
							0.643210	−	0.765690i	\(-0.277602\pi\)
\(19\)	1059.07	+	1163.81i	0.673042	+	0.739604i
							\(23\)	1547.62	−	563.288i	0.610021	−	0.222030i	−0.0184913	−	0.999829i	\(-0.505886\pi\)
0.628513	+	0.777799i	\(0.283664\pi\)
\(29\)	1410.66	−	8000.26i	0.311478	−	1.76648i	−0.279842	−	0.960046i	\(-0.590282\pi\)
							0.591321	−	0.806436i	\(-0.298607\pi\)
\(31\)	3137.20	−	5433.79i	0.586324	−	1.01554i	−0.408384	−	0.912810i	\(-0.633908\pi\)
							0.994709	−	0.102734i	\(-0.0327589\pi\)
\(37\)	7660.34			0.919907			0.459953	−	0.887943i	\(-0.347866\pi\)
							0.459953	+	0.887943i	\(0.347866\pi\)
\(41\)	−3980.33	+	3339.89i	−0.369793	+	0.310293i	−0.808680	−	0.588249i	\(-0.799817\pi\)
							0.438886	+	0.898543i	\(0.355373\pi\)
\(43\)	7859.62	+	2860.67i	0.648232	+	0.235937i	0.645147	−	0.764058i	\(-0.276796\pi\)
							0.00308448	+	0.999995i	\(0.499018\pi\)
\(47\)	−1318.75	+	7478.98i	−0.0870797	+	0.493853i	0.909809	+	0.415028i	\(0.136228\pi\)
							−0.996888	+	0.0788256i	\(0.974883\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.a.9.4	✓	24
19.6	even	9		722.6.a.o.1.10		12
19.13	odd	18		722.6.a.p.1.3		12
19.17	even	9	inner	38.6.e.a.17.4	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.a.9.4	✓	24	1.1	even	1	trivial
38.6.e.a.17.4	yes	24	19.17	even	9	inner
722.6.a.o.1.10		12	19.6	even	9
722.6.a.p.1.3		12	19.13	odd	18