Embedded newform 38.10.e.a.17.1

• Label: 38.10.e.a.17.1
• Level: $38$
• Weight: $10$
• Character: 38.17
• Analytic conductor: $19.571$
• Analytic rank: $0$
• Dimension: $42$
• 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:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			17.1
Character	\(\chi\)	\(=\)	38.17
Dual form			38.10.e.a.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.77837 + 15.7569i) q^{2} +(-167.700 - 140.717i) q^{3} +(-240.561 - 87.5572i) q^{4} +(1637.73 - 596.085i) q^{5} +(2683.20 - 2251.47i) q^{6} +(1264.29 + 2189.82i) q^{7} +(2048.00 - 3547.24i) q^{8} +(4904.09 + 27812.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 252 q^{3} + 4032 q^{6} + 14373 q^{7} + 86016 q^{8} + 12852 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.77837	+	15.7569i	−0.122788	+	0.696364i
							\(3\)	−167.700	−	140.717i	−1.19533	−	1.00300i	−0.999751	−	0.0223113i	\(-0.992898\pi\)
−0.195577	−	0.980688i	\(-0.562658\pi\)
\(5\)	1637.73	−	596.085i	1.17186	−	0.426523i	0.318542	−	0.947909i	\(-0.396807\pi\)
							0.853321	+	0.521385i	\(0.174585\pi\)
\(7\)	1264.29	+	2189.82i	0.199025	+	0.344721i	0.948212	−	0.317637i	\(-0.102889\pi\)
							−0.749188	+	0.662358i	\(0.769556\pi\)
\(11\)	−9578.04	+	16589.7i	−0.197247	+	0.341641i	−0.947635	−	0.319356i	\(-0.896533\pi\)
							0.750388	+	0.660998i	\(0.229867\pi\)
\(13\)	85375.5	−	71638.6i	0.829065	−	0.695668i	−0.126011	−	0.992029i	\(-0.540218\pi\)
							0.955076	+	0.296361i	\(0.0957732\pi\)
\(17\)	31767.4	−	180162.i	0.0922491	−	0.523171i	−0.903307	−	0.428996i	\(-0.858868\pi\)
							0.995556	−	0.0941750i	\(-0.0300213\pi\)
\(19\)	−341929.	−	453621.i	−0.601929	−	0.798550i
							\(23\)	175354.	+	63823.5i	0.130659	+	0.0475560i	0.406522	−	0.913641i	\(-0.366741\pi\)
−0.275863	+	0.961197i	\(0.588964\pi\)
\(29\)	−667569.	−	3.78597e6i	−0.175269	−	0.994000i	−0.937833	−	0.347087i	\(-0.887171\pi\)
							0.762564	−	0.646913i	\(-0.223940\pi\)
\(31\)	−1.82078e6	−	3.15367e6i	−0.354102	−	0.613323i	0.632862	−	0.774265i	\(-0.281880\pi\)
							−0.986964	+	0.160942i	\(0.948547\pi\)
\(37\)	−1.95186e7			−1.71215			−0.856074	−	0.516853i	\(-0.827103\pi\)
							−0.856074	+	0.516853i	\(0.827103\pi\)
\(41\)	−5.59031e6	−	4.69083e6i	−0.308964	−	0.259252i	0.475099	−	0.879932i	\(-0.342412\pi\)
							−0.784064	+	0.620680i	\(0.786857\pi\)
\(43\)	−3.37097e7	+	1.22693e7i	−1.50365	+	0.547285i	−0.957003	−	0.290077i	\(-0.906319\pi\)
							−0.546649	+	0.837362i	\(0.684097\pi\)
\(47\)	−4.34780e6	−	2.46576e7i	−0.129966	−	0.737073i	−0.978234	−	0.207506i	\(-0.933465\pi\)
							0.848268	−	0.529567i	\(-0.177646\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.e.a.17.1	yes	42
19.9	even	9	inner	38.10.e.a.9.1	✓	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.e.a.9.1	✓	42	19.9	even	9	inner
38.10.e.a.17.1	yes	42	1.1	even	1	trivial