Embedded newform 38.8.e.b.5.2

• Label: 38.8.e.b.5.2
• Level: $38$
• Weight: $8$
• Character: 38.5
• 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			5.2
Character	\(\chi\)	\(=\)	38.5
Dual form			38.8.e.b.23.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.51754 - 2.73616i) q^{2} +(-11.7125 + 66.4251i) q^{3} +(49.0268 - 41.1384i) q^{4} +(272.185 + 228.391i) q^{5} +(93.7003 + 531.401i) q^{6} +(-194.092 - 336.178i) q^{7} +(256.000 - 443.405i) q^{8} +(-2220.00 - 808.013i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 84 q^{3} + 672 q^{6} - 2988 q^{7} + 9216 q^{8} + 1428 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.51754	−	2.73616i	0.664463	−	0.241845i
							\(3\)	−11.7125	+	66.4251i	−0.250453	+	1.42039i	0.557027	+	0.830495i	\(0.311942\pi\)
−0.807480	+	0.589895i	\(0.799169\pi\)
\(5\)	272.185	+	228.391i	0.973800	+	0.817115i	0.983142	−	0.182841i	\(-0.0585294\pi\)
							−0.00934243	+	0.999956i	\(0.502974\pi\)
\(7\)	−194.092	−	336.178i	−0.213877	−	0.370447i	0.739047	−	0.673654i	\(-0.235276\pi\)
							−0.952925	+	0.303207i	\(0.901943\pi\)
\(11\)	−3256.33	+	5640.14i	−0.737657	+	1.27766i	0.215891	+	0.976418i	\(0.430735\pi\)
							−0.953548	+	0.301242i	\(0.902599\pi\)
\(13\)	388.717	+	2204.53i	0.0490718	+	0.278300i	0.999463	−	0.0327559i	\(-0.0104284\pi\)
							−0.950392	+	0.311056i	\(0.899317\pi\)
\(17\)	−19633.9	+	7146.17i	−0.969251	+	0.352778i	−0.777652	−	0.628695i	\(-0.783589\pi\)
							−0.191599	+	0.981473i	\(0.561367\pi\)
\(19\)	29715.7	+	3293.58i	0.993914	+	0.110162i
							\(23\)	35240.1	−	29569.9i	0.603934	−	0.506761i	−0.288773	−	0.957397i	\(-0.593247\pi\)
0.892707	+	0.450637i	\(0.148803\pi\)
\(29\)	−22965.7	−	8358.82i	−0.174858	−	0.0636432i	0.253107	−	0.967438i	\(-0.418547\pi\)
							−0.427966	+	0.903795i	\(0.640770\pi\)
\(31\)	16288.4	+	28212.3i	0.0982000	+	0.170087i	0.910940	−	0.412540i	\(-0.135358\pi\)
							−0.812740	+	0.582627i	\(0.802025\pi\)
\(37\)	255897.			0.830538			0.415269	−	0.909699i	\(-0.363687\pi\)
							0.415269	+	0.909699i	\(0.363687\pi\)
\(41\)	104539.	−	592870.i	0.236883	−	1.34343i	−0.601728	−	0.798701i	\(-0.705521\pi\)
							0.838611	−	0.544731i	\(-0.183368\pi\)
\(43\)	668065.	+	560573.i	1.28138	+	1.07521i	0.993052	+	0.117677i	\(0.0375449\pi\)
							0.288331	+	0.957531i	\(0.406900\pi\)
\(47\)	−2629.38	−	957.015i	−0.00369412	−	0.00134455i	0.340172	−	0.940363i	\(-0.389515\pi\)
							−0.343867	+	0.939019i	\(0.611737\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.b.5.2	✓	36
19.4	even	9	inner	38.8.e.b.23.2	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.b.5.2	✓	36	1.1	even	1	trivial
38.8.e.b.23.2	yes	36	19.4	even	9	inner