Embedded newform 38.8.e.b.9.3

• Label: 38.8.e.b.9.3
• Level: $38$
• Weight: $8$
• Character: 38.9
• Analytic conductor: $11.871$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8706309684\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.3
Character	\(\chi\)	\(=\)	38.9
Dual form			38.8.e.b.17.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38919 - 7.87846i) q^{2} +(-12.0854 + 10.1408i) q^{3} +(-60.1403 + 21.8893i) q^{4} +(-307.410 - 111.888i) q^{5} +(96.6830 + 81.1267i) q^{6} +(444.097 - 769.199i) q^{7} +(256.000 + 443.405i) q^{8} +(-336.549 + 1908.66i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 84 q^{3} + 672 q^{6} - 2988 q^{7} + 9216 q^{8} + 1428 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38919	−	7.87846i	−0.122788	−	0.696364i
							\(3\)	−12.0854	+	10.1408i	−0.258426	+	0.216845i	−0.762790	−	0.646646i	\(-0.776171\pi\)
0.504365	+	0.863491i	\(0.331727\pi\)
\(5\)	−307.410	−	111.888i	−1.09982	−	0.400303i	−0.272572	−	0.962135i	\(-0.587874\pi\)
							−0.827251	+	0.561832i	\(0.810097\pi\)
\(7\)	444.097	−	769.199i	0.489367	−	0.847609i	−0.510558	−	0.859843i	\(-0.670561\pi\)
							0.999925	+	0.0122347i	\(0.00389453\pi\)
\(11\)	1665.01	+	2883.88i	0.377175	+	0.653286i	0.990650	−	0.136428i	\(-0.0435623\pi\)
							−0.613475	+	0.789714i	\(0.710229\pi\)
\(13\)	2839.61	+	2382.72i	0.358474	+	0.300795i	0.804182	−	0.594383i	\(-0.202604\pi\)
							−0.445708	+	0.895178i	\(0.647048\pi\)
\(17\)	3206.13	+	18182.8i	0.158274	+	0.897616i	0.955731	+	0.294241i	\(0.0950667\pi\)
							−0.797457	+	0.603375i	\(0.793822\pi\)
\(19\)	26846.2	+	13158.8i	0.897935	+	0.440129i
							\(23\)	39510.6	−	14380.7i	0.677120	−	0.246452i	0.0195096	−	0.999810i	\(-0.493790\pi\)
0.657610	+	0.753358i	\(0.271567\pi\)
\(29\)	−31376.8	+	177946.i	−0.238899	+	1.35487i	0.595345	+	0.803470i	\(0.297015\pi\)
							−0.834244	+	0.551395i	\(0.814096\pi\)
\(31\)	−8989.89	+	15571.0i	−0.0541987	+	0.0938749i	−0.891852	−	0.452328i	\(-0.850594\pi\)
							0.837653	+	0.546203i	\(0.183927\pi\)
\(37\)	−21044.8			−0.0683028			−0.0341514	−	0.999417i	\(-0.510873\pi\)
							−0.0341514	+	0.999417i	\(0.510873\pi\)
\(41\)	−360569.	+	302553.i	−0.817043	+	0.685581i	−0.952278	−	0.305233i	\(-0.901266\pi\)
							0.135235	+	0.990814i	\(0.456821\pi\)
\(43\)	−653299.	−	237782.i	−1.25306	−	0.456077i	−0.371627	−	0.928382i	\(-0.621200\pi\)
							−0.881435	+	0.472305i	\(0.843422\pi\)
\(47\)	−95540.2	+	541835.i	−0.134228	+	0.761246i	0.841166	+	0.540777i	\(0.181870\pi\)
							−0.975394	+	0.220469i	\(0.929241\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.b.9.3	✓	36
19.17	even	9	inner	38.8.e.b.17.3	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.b.9.3	✓	36	1.1	even	1	trivial
38.8.e.b.17.3	yes	36	19.17	even	9	inner