Embedded newform 38.6.e.a.9.2

• Label: 38.6.e.a.9.2
• Level: $38$
• Weight: $6$
• Character: 38.9
• Analytic conductor: $6.095$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.09458515289\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.2
Character	\(\chi\)	\(=\)	38.9
Dual form			38.6.e.a.17.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.694593 - 3.93923i) q^{2} +(-3.81408 + 3.20039i) q^{3} +(-15.0351 + 5.47232i) q^{4} +(-0.763769 - 0.277989i) q^{5} +(15.2563 + 12.8016i) q^{6} +(-52.7816 + 91.4204i) q^{7} +(32.0000 + 55.4256i) q^{8} +(-37.8918 + 214.895i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 18 q^{3} - 72 q^{6} + 438 q^{7} + 768 q^{8} + 378 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.694593	−	3.93923i	−0.122788	−	0.696364i
							\(3\)	−3.81408	+	3.20039i	−0.244673	+	0.205305i	−0.756875	−	0.653560i	\(-0.773275\pi\)
0.512201	+	0.858865i	\(0.328830\pi\)
\(5\)	−0.763769	−	0.277989i	−0.0136627	−	0.00497282i	0.335180	−	0.942154i	\(-0.391203\pi\)
							−0.348842	+	0.937181i	\(0.613425\pi\)
\(7\)	−52.7816	+	91.4204i	−0.407134	+	0.705177i	−0.994567	−	0.104096i	\(-0.966805\pi\)
							0.587433	+	0.809273i	\(0.300138\pi\)
\(11\)	145.339	+	251.735i	0.362161	+	0.627281i	0.988316	−	0.152417i	\(-0.0487058\pi\)
							−0.626155	+	0.779698i	\(0.715372\pi\)
\(13\)	251.789	+	211.276i	0.413216	+	0.346730i	0.825576	−	0.564292i	\(-0.190851\pi\)
							−0.412359	+	0.911021i	\(0.635295\pi\)
\(17\)	102.387	+	580.663i	0.0859252	+	0.487306i	0.997153	+	0.0754034i	\(0.0240245\pi\)
							−0.911228	+	0.411903i	\(0.864864\pi\)
\(19\)	−1528.90	+	372.236i	−0.971618	+	0.236556i
							\(23\)	−1809.60	+	658.642i	−0.713287	+	0.259615i	−0.673073	−	0.739576i	\(-0.735026\pi\)
−0.0402136	+	0.999191i	\(0.512804\pi\)
\(29\)	47.2279	−	267.843i	0.0104281	−	0.0591405i	−0.979150	−	0.203140i	\(-0.934885\pi\)
							0.989578	+	0.144000i	\(0.0459964\pi\)
\(31\)	324.254	−	561.625i	0.0606012	−	0.104964i	−0.834133	−	0.551563i	\(-0.814032\pi\)
							0.894734	+	0.446599i	\(0.147365\pi\)
\(37\)	5823.98			0.699384			0.349692	−	0.936865i	\(-0.386286\pi\)
							0.349692	+	0.936865i	\(0.386286\pi\)
\(41\)	3110.87	−	2610.33i	0.289016	−	0.242513i	−0.486739	−	0.873548i	\(-0.661814\pi\)
							0.775755	+	0.631034i	\(0.217369\pi\)
\(43\)	5142.59	+	1871.75i	0.424141	+	0.154375i	0.545267	−	0.838262i	\(-0.316428\pi\)
							−0.121126	+	0.992637i	\(0.538650\pi\)
\(47\)	−3538.09	+	20065.5i	−0.233628	+	1.32497i	0.611857	+	0.790969i	\(0.290423\pi\)
							−0.845484	+	0.534000i	\(0.820688\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.a.9.2	✓	24
19.6	even	9		722.6.a.o.1.5		12
19.13	odd	18		722.6.a.p.1.8		12
19.17	even	9	inner	38.6.e.a.17.2	yes	24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.a.9.2	✓	24	1.1	even	1	trivial
38.6.e.a.17.2	yes	24	19.17	even	9	inner
722.6.a.o.1.5		12	19.6	even	9
722.6.a.p.1.8		12	19.13	odd	18