Embedded newform 38.10.e.a.9.7

• Label: 38.10.e.a.9.7
• Level: $38$
• Weight: $10$
• Character: 38.9
• 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			9.7
Character	\(\chi\)	\(=\)	38.9
Dual form			38.10.e.a.17.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.77837 - 15.7569i) q^{2} +(199.355 - 167.278i) q^{3} +(-240.561 + 87.5572i) q^{4} +(1053.98 + 383.617i) q^{5} +(-3189.67 - 2676.45i) q^{6} +(-1724.05 + 2986.15i) q^{7} +(2048.00 + 3547.24i) q^{8} +(8342.29 - 47311.5i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 252 q^{3} + 4032 q^{6} + 14373 q^{7} + 86016 q^{8} + 12852 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.77837	−	15.7569i	−0.122788	−	0.696364i
							\(3\)	199.355	−	167.278i	1.42096	−	1.19232i	0.470130	−	0.882597i	\(-0.344207\pi\)
0.950826	−	0.309727i	\(-0.100238\pi\)
\(5\)	1053.98	+	383.617i	0.754167	+	0.274494i	0.690358	−	0.723468i	\(-0.257453\pi\)
							0.0638088	+	0.997962i	\(0.479675\pi\)
\(7\)	−1724.05	+	2986.15i	−0.271400	+	0.470078i	−0.969220	−	0.246194i	\(-0.920820\pi\)
							0.697821	+	0.716272i	\(0.254153\pi\)
\(11\)	−43255.3	−	74920.4i	−0.890784	−	1.54288i	−0.838938	−	0.544227i	\(-0.816823\pi\)
							−0.0518457	−	0.998655i	\(-0.516510\pi\)
\(13\)	8211.92	+	6890.62i	0.0797443	+	0.0669134i	0.681788	−	0.731549i	\(-0.261202\pi\)
							−0.602044	+	0.798463i	\(0.705647\pi\)
\(17\)	−74879.5	−	424663.i	−0.217442	−	1.23317i	−0.876619	−	0.481186i	\(-0.840206\pi\)
							0.659177	−	0.751988i	\(-0.270905\pi\)
\(19\)	496448.	+	276093.i	0.873941	+	0.486031i
							\(23\)	−103578.	+	37699.2i	−0.0771776	+	0.0280903i	−0.380320	−	0.924855i	\(-0.624186\pi\)
0.303143	+	0.952945i	\(0.401964\pi\)
\(29\)	−624950.	+	3.54427e6i	−0.164080	+	0.930542i	0.785929	+	0.618317i	\(0.212185\pi\)
							−0.950008	+	0.312225i	\(0.898926\pi\)
\(31\)	2.39656e6	−	4.15097e6i	0.466081	−	0.807275i	−0.533169	−	0.846009i	\(-0.678999\pi\)
							0.999250	+	0.0387335i	\(0.0123323\pi\)
\(37\)	1.01730e7			0.892363			0.446181	−	0.894943i	\(-0.352784\pi\)
							0.446181	+	0.894943i	\(0.352784\pi\)
\(41\)	−1.31108e7	+	1.10012e7i	−0.724604	+	0.608015i	−0.928655	−	0.370945i	\(-0.879034\pi\)
							0.204050	+	0.978960i	\(0.434589\pi\)
\(43\)	1.98208e7	+	7.21420e6i	0.884126	+	0.321796i	0.743874	−	0.668320i	\(-0.232986\pi\)
							0.140252	+	0.990116i	\(0.455209\pi\)
\(47\)	−9.59994e6	+	5.44440e7i	−0.286965	+	1.62746i	0.411218	+	0.911537i	\(0.365103\pi\)
							−0.698183	+	0.715920i	\(0.746008\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.e.a.9.7	✓	42
19.17	even	9	inner	38.10.e.a.17.7	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.e.a.9.7	✓	42	1.1	even	1	trivial
38.10.e.a.17.7	yes	42	19.17	even	9	inner