Embedded newform 38.8.e.a.5.4

• Label: 38.8.e.a.5.4
• Level: $38$
• Weight: $8$
• Character: 38.5
• Analytic conductor: $11.871$
• Analytic rank: $0$
• Dimension: $30$
• 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:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			5.4
Character	\(\chi\)	\(=\)	38.5
Dual form			38.8.e.a.23.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.51754 + 2.73616i) q^{2} +(7.10410 - 40.2894i) q^{3} +(49.0268 - 41.1384i) q^{4} +(-241.265 - 202.445i) q^{5} +(56.8328 + 322.315i) q^{6} +(-369.505 - 640.002i) q^{7} +(-256.000 + 443.405i) q^{8} +(482.343 + 175.558i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 45 q^{3} + 360 q^{6} + 1806 q^{7} - 7680 q^{8} - 3105 q^{9}+O(q^{10}) \)

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
							\(3\)	7.10410	−	40.2894i	0.151909	−	0.861521i	−0.809648	−	0.586916i	\(-0.800342\pi\)
0.961557	−	0.274605i	\(-0.0885471\pi\)
\(5\)	−241.265	−	202.445i	−0.863175	−	0.724290i	0.0994749	−	0.995040i	\(-0.468284\pi\)
							−0.962650	+	0.270751i	\(0.912728\pi\)
\(7\)	−369.505	−	640.002i	−0.407171	−	0.705242i	0.587400	−	0.809297i	\(-0.300152\pi\)
							−0.994572	+	0.104055i	\(0.966818\pi\)
\(11\)	195.375	−	338.399i	0.0442582	−	0.0766574i	−0.843048	−	0.537839i	\(-0.819241\pi\)
							0.887306	+	0.461181i	\(0.152574\pi\)
\(13\)	838.196	+	4753.65i	0.105814	+	0.600102i	0.990892	+	0.134659i	\(0.0429939\pi\)
							−0.885078	+	0.465443i	\(0.845895\pi\)
\(17\)	−14799.2	+	5386.48i	−0.730580	+	0.265909i	−0.680410	−	0.732831i	\(-0.738198\pi\)
							−0.0501695	+	0.998741i	\(0.515976\pi\)
\(19\)	−25682.4	+	15306.5i	−0.859008	+	0.511961i
							\(23\)	−68573.6	+	57540.1i	−1.17519	+	0.986104i	−0.175195	+	0.984534i	\(0.556055\pi\)
−0.999999	+	0.00157043i	\(0.999500\pi\)
\(29\)	−28375.0	−	10327.6i	−0.216044	−	0.0786336i	0.231731	−	0.972780i	\(-0.425561\pi\)
							−0.447775	+	0.894146i	\(0.647783\pi\)
\(31\)	103301.	+	178923.i	0.622786	+	1.07870i	0.988965	+	0.148152i	\(0.0473326\pi\)
							−0.366178	+	0.930545i	\(0.619334\pi\)
\(37\)	209313.			0.679344			0.339672	−	0.940544i	\(-0.389684\pi\)
							0.339672	+	0.940544i	\(0.389684\pi\)
\(41\)	31720.8	−	179898.i	0.0718788	−	0.407645i	−0.927545	−	0.373711i	\(-0.878085\pi\)
							0.999424	−	0.0339340i	\(-0.0108036\pi\)
\(43\)	−359022.	−	301255.i	−0.688623	−	0.577823i	0.229889	−	0.973217i	\(-0.426164\pi\)
							−0.918512	+	0.395394i	\(0.870608\pi\)
\(47\)	−333660.	−	121442.i	−0.468772	−	0.170619i	0.0968238	−	0.995302i	\(-0.469132\pi\)
							−0.565596	+	0.824682i	\(0.691354\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.a.5.4	✓	30
19.4	even	9	inner	38.8.e.a.23.4	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.a.5.4	✓	30	1.1	even	1	trivial
38.8.e.a.23.4	yes	30	19.4	even	9	inner