Embedded newform 38.8.e.b.17.2

• Label: 38.8.e.b.17.2
• Level: $38$
• Weight: $8$
• Character: 38.17
• 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			17.2
Character	\(\chi\)	\(=\)	38.17
Dual form			38.8.e.b.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38919 + 7.87846i) q^{2} +(-33.6098 - 28.2020i) q^{3} +(-60.1403 - 21.8893i) q^{4} +(202.712 - 73.7810i) q^{5} +(268.878 - 225.616i) q^{6} +(234.915 + 406.885i) q^{7} +(256.000 - 443.405i) q^{8} +(-45.5015 - 258.052i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 84 q^{3} + 672 q^{6} - 2988 q^{7} + 9216 q^{8} + 1428 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

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
							\(3\)	−33.6098	−	28.2020i	−0.718689	−	0.603052i	0.208333	−	0.978058i	\(-0.433196\pi\)
−0.927022	+	0.375006i	\(0.877641\pi\)
\(5\)	202.712	−	73.7810i	0.725244	−	0.263967i	0.0470934	−	0.998890i	\(-0.485004\pi\)
							0.678150	+	0.734923i	\(0.262782\pi\)
\(7\)	234.915	+	406.885i	0.258862	+	0.448362i	0.965937	−	0.258776i	\(-0.0833192\pi\)
							−0.707075	+	0.707138i	\(0.749986\pi\)
\(11\)	−1940.70	+	3361.39i	−0.439627	+	0.761456i	−0.997661	−	0.0683623i	\(-0.978223\pi\)
							0.558034	+	0.829818i	\(0.311556\pi\)
\(13\)	−7912.23	+	6639.15i	−0.998842	+	0.838128i	−0.986824	−	0.161799i	\(-0.948270\pi\)
							−0.0120188	+	0.999928i	\(0.503826\pi\)
\(17\)	−425.431	+	2412.74i	−0.0210019	+	0.119108i	−0.993506	−	0.113777i	\(-0.963705\pi\)
							0.972504	+	0.232884i	\(0.0748163\pi\)
\(19\)	−25701.4	+	15274.4i	−0.859647	+	0.510889i
							\(23\)	−11617.2	−	4228.30i	−0.199092	−	0.0724634i	0.240550	−	0.970637i	\(-0.422672\pi\)
−0.439641	+	0.898173i	\(0.644894\pi\)
\(29\)	9651.48	+	54736.2i	0.0734854	+	0.416756i	0.999252	+	0.0386617i	\(0.0123095\pi\)
							−0.925767	+	0.378095i	\(0.876579\pi\)
\(31\)	131255.	+	227340.i	0.791316	+	1.37060i	0.925152	+	0.379597i	\(0.123937\pi\)
							−0.133836	+	0.991004i	\(0.542729\pi\)
\(37\)	33134.1			0.107540			0.0537700	−	0.998553i	\(-0.482876\pi\)
							0.0537700	+	0.998553i	\(0.482876\pi\)
\(41\)	−237287.	−	199107.i	−0.537687	−	0.451173i	0.333059	−	0.942906i	\(-0.391919\pi\)
							−0.870746	+	0.491733i	\(0.836364\pi\)
\(43\)	180794.	−	65803.8i	0.346773	−	0.126215i	−0.162761	−	0.986666i	\(-0.552040\pi\)
							0.509534	+	0.860450i	\(0.329818\pi\)
\(47\)	17993.8	+	102048.i	0.0252802	+	0.143371i	0.994835	−	0.101502i	\(-0.0323648\pi\)
							−0.969555	+	0.244873i	\(0.921254\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.b.17.2	yes	36
19.9	even	9	inner	38.8.e.b.9.2	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.b.9.2	✓	36	19.9	even	9	inner
38.8.e.b.17.2	yes	36	1.1	even	1	trivial