Embedded newform 38.8.e.a.5.5

• Label: 38.8.e.a.5.5
• 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			5.5
Character	\(\chi\)	\(=\)	38.5
Dual form			38.8.e.a.23.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.51754 + 2.73616i) q^{2} +(11.5048 - 65.2472i) q^{3} +(49.0268 - 41.1384i) q^{4} +(158.287 + 132.818i) q^{5} +(92.0387 + 521.978i) q^{6} +(668.912 + 1158.59i) q^{7} +(-256.000 + 443.405i) q^{8} +(-2069.73 - 753.319i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 45 q^{3} + 360 q^{6} + 1806 q^{7} - 7680 q^{8} - 3105 q^{9}+O(q^{10}) \)

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
							\(3\)	11.5048	−	65.2472i	0.246012	−	1.39520i	−0.572119	−	0.820171i	\(-0.693878\pi\)
0.818131	−	0.575032i	\(-0.195011\pi\)
\(5\)	158.287	+	132.818i	0.566304	+	0.475185i	0.880417	−	0.474200i	\(-0.157263\pi\)
							−0.314113	+	0.949385i	\(0.601707\pi\)
\(7\)	668.912	+	1158.59i	0.737099	+	1.27669i	0.953796	+	0.300454i	\(0.0971383\pi\)
							−0.216697	+	0.976239i	\(0.569528\pi\)
\(11\)	3797.30	−	6577.12i	0.860203	−	1.48991i	−0.0115306	−	0.999934i	\(-0.503670\pi\)
							0.871733	−	0.489981i	\(-0.162996\pi\)
\(13\)	1974.16	+	11196.0i	0.249218	+	1.41339i	0.810489	+	0.585754i	\(0.199201\pi\)
							−0.561271	+	0.827632i	\(0.689687\pi\)
\(17\)	1875.53	−	682.637i	0.0925876	−	0.0336991i	−0.295311	−	0.955401i	\(-0.595423\pi\)
							0.387899	+	0.921702i	\(0.373201\pi\)
\(19\)	23749.0	−	18161.9i	0.794343	−	0.607469i
							\(23\)	−5803.54	+	4869.75i	−0.0994594	+	0.0834563i	−0.691161	−	0.722700i	\(-0.742901\pi\)
0.591702	+	0.806157i	\(0.298456\pi\)
\(29\)	185378.	+	67472.0i	1.41145	+	0.513725i	0.931555	−	0.363602i	\(-0.118453\pi\)
							0.479894	+	0.877327i	\(0.340675\pi\)
\(31\)	−122615.	−	212375.i	−0.739224	−	1.28037i	−0.952845	−	0.303457i	\(-0.901859\pi\)
							0.213621	−	0.976917i	\(-0.431474\pi\)
\(37\)	−431556.			−1.40065			−0.700327	−	0.713822i	\(-0.746963\pi\)
							−0.700327	+	0.713822i	\(0.746963\pi\)
\(41\)	−34114.5	+	193473.i	−0.0773029	+	0.438406i	0.921451	+	0.388495i	\(0.127005\pi\)
							−0.998754	+	0.0499112i	\(0.984106\pi\)
\(43\)	265254.	+	222574.i	0.508771	+	0.426909i	0.860696	−	0.509119i	\(-0.170029\pi\)
							−0.351926	+	0.936028i	\(0.614473\pi\)
\(47\)	783011.	+	284993.i	1.10008	+	0.400397i	0.827347	−	0.561692i	\(-0.189849\pi\)
							0.272736	+	0.962089i	\(0.412071\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.a.5.5	✓	30
19.4	even	9	inner	38.8.e.a.23.5	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.a.5.5	✓	30	1.1	even	1	trivial
38.8.e.a.23.5	yes	30	19.4	even	9	inner