Embedded newform 38.10.e.a.5.3

• Label: 38.10.e.a.5.3
• Level: $38$
• Weight: $10$
• Character: 38.5
• 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			5.3
Character	\(\chi\)	\(=\)	38.5
Dual form			38.10.e.a.23.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.0351 - 5.47232i) q^{2} +(-19.9921 + 113.381i) q^{3} +(196.107 - 164.554i) q^{4} +(-800.078 - 671.345i) q^{5} +(319.873 + 1814.09i) q^{6} +(928.665 + 1608.49i) q^{7} +(2048.00 - 3547.24i) q^{8} +(6040.49 + 2198.56i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 252 q^{3} + 4032 q^{6} + 14373 q^{7} + 86016 q^{8} + 12852 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.0351	−	5.47232i	0.664463	−	0.241845i
							\(3\)	−19.9921	+	113.381i	−0.142499	+	0.808152i	0.826842	+	0.562434i	\(0.190135\pi\)
−0.969341	+	0.245718i	\(0.920976\pi\)
\(5\)	−800.078	−	671.345i	−0.572489	−	0.480376i	0.309982	−	0.950743i	\(-0.399677\pi\)
							−0.882471	+	0.470367i	\(0.844122\pi\)
\(7\)	928.665	+	1608.49i	0.146190	+	0.253209i	0.929816	−	0.368024i	\(-0.119966\pi\)
							−0.783626	+	0.621233i	\(0.786632\pi\)
\(11\)	−24188.7	+	41896.0i	−0.498133	+	0.862791i	−0.999998	−	0.00215482i	\(-0.999314\pi\)
							0.501865	+	0.864946i	\(0.332647\pi\)
\(13\)	18578.1	+	105362.i	0.180408	+	1.02315i	0.931714	+	0.363192i	\(0.118313\pi\)
							−0.751306	+	0.659954i	\(0.770576\pi\)
\(17\)	−568675.	+	206981.i	−1.65137	+	0.601049i	−0.988972	−	0.148103i	\(-0.952683\pi\)
							−0.662397	+	0.749153i	\(0.730461\pi\)
\(19\)	−563667.	+	70474.7i	−0.992274	+	0.124063i
							\(23\)	−606992.	+	509327.i	−0.452281	+	0.379508i	−0.840281	−	0.542151i	\(-0.817610\pi\)
0.388001	+	0.921659i	\(0.373166\pi\)
\(29\)	3.58143e6	+	1.30353e6i	0.940297	+	0.342240i	0.766283	−	0.642503i	\(-0.222104\pi\)
							0.174014	+	0.984743i	\(0.444326\pi\)
\(31\)	2.15759e6	+	3.73706e6i	0.419606	+	0.726779i	0.995900	−	0.0904637i	\(-0.0288349\pi\)
							−0.576294	+	0.817243i	\(0.695502\pi\)
\(37\)	1.33008e7			1.16673			0.583364	−	0.812211i	\(-0.301736\pi\)
							0.583364	+	0.812211i	\(0.301736\pi\)
\(41\)	−5.01135e6	+	2.84208e7i	−0.276967	+	1.57076i	0.455678	+	0.890145i	\(0.349397\pi\)
							−0.732645	+	0.680611i	\(0.761714\pi\)
\(43\)	−1.92708e7	−	1.61701e7i	−0.859591	−	0.721282i	0.102289	−	0.994755i	\(-0.467383\pi\)
							−0.961880	+	0.273472i	\(0.911828\pi\)
\(47\)	1.01217e7	+	3.68399e6i	0.302560	+	0.110123i	0.488839	−	0.872374i	\(-0.337421\pi\)
							−0.186279	+	0.982497i	\(0.559643\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.e.a.5.3	✓	42
19.4	even	9	inner	38.10.e.a.23.3	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.e.a.5.3	✓	42	1.1	even	1	trivial
38.10.e.a.23.3	yes	42	19.4	even	9	inner