Embedded newform 38.8.e.b.5.6

• Label: 38.8.e.b.5.6
• 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			5.6
Character	\(\chi\)	\(=\)	38.5
Dual form			38.8.e.b.23.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.51754 - 2.73616i) q^{2} +(11.6928 - 66.3132i) q^{3} +(49.0268 - 41.1384i) q^{4} +(-157.410 - 132.083i) q^{5} +(-93.5425 - 530.506i) q^{6} +(-83.9701 - 145.440i) q^{7} +(256.000 - 443.405i) q^{8} +(-2205.61 - 802.777i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 84 q^{3} + 672 q^{6} - 2988 q^{7} + 9216 q^{8} + 1428 q^{9}+O(q^{10}) \)

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
							\(3\)	11.6928	−	66.3132i	0.250031	−	1.41800i	−0.558480	−	0.829518i	\(-0.688615\pi\)
0.808512	−	0.588480i	\(-0.200274\pi\)
\(5\)	−157.410	−	132.083i	−0.563168	−	0.472554i	0.316203	−	0.948692i	\(-0.397592\pi\)
							−0.879371	+	0.476138i	\(0.842036\pi\)
\(7\)	−83.9701	−	145.440i	−0.0925298	−	0.160266i	0.816045	−	0.577988i	\(-0.196162\pi\)
							−0.908575	+	0.417722i	\(0.862829\pi\)
\(11\)	−154.304	+	267.262i	−0.0349544	+	0.0605428i	−0.882973	−	0.469423i	\(-0.844462\pi\)
							0.848019	+	0.529966i	\(0.177795\pi\)
\(13\)	960.146	+	5445.26i	0.121209	+	0.687411i	0.983487	+	0.180976i	\(0.0579258\pi\)
							−0.862278	+	0.506435i	\(0.830963\pi\)
\(17\)	−13585.1	+	4944.58i	−0.670645	+	0.244095i	−0.654825	−	0.755780i	\(-0.727258\pi\)
							−0.0158193	+	0.999875i	\(0.505036\pi\)
\(19\)	187.469	−	29897.1i	0.00627036	−	0.999980i
							\(23\)	52368.5	−	43942.4i	0.897475	−	0.753071i	−0.0722200	−	0.997389i	\(-0.523008\pi\)
0.969695	+	0.244318i	\(0.0785639\pi\)
\(29\)	−103380.	−	37627.1i	−0.787122	−	0.286489i	−0.0829831	−	0.996551i	\(-0.526445\pi\)
							−0.704139	+	0.710062i	\(0.748667\pi\)
\(31\)	32959.1	+	57086.9i	0.198705	+	0.344168i	0.948109	−	0.317946i	\(-0.102993\pi\)
							−0.749403	+	0.662114i	\(0.769660\pi\)
\(37\)	396889.			1.28814			0.644071	−	0.764966i	\(-0.277244\pi\)
							0.644071	+	0.764966i	\(0.277244\pi\)
\(41\)	39105.5	−	221778.i	0.0886123	−	0.502546i	−0.907906	−	0.419173i	\(-0.862320\pi\)
							0.996518	−	0.0833721i	\(-0.0265690\pi\)
\(43\)	407301.	+	341766.i	0.781224	+	0.655525i	0.943557	−	0.331211i	\(-0.107457\pi\)
							−0.162332	+	0.986736i	\(0.551902\pi\)
\(47\)	902527.	+	328493.i	1.26800	+	0.461513i	0.886444	−	0.462835i	\(-0.153168\pi\)
							0.381551	+	0.924348i	\(0.375390\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.b.5.6	✓	36
19.4	even	9	inner	38.8.e.b.23.6	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.b.5.6	✓	36	1.1	even	1	trivial
38.8.e.b.23.6	yes	36	19.4	even	9	inner