Embedded newform 38.6.e.b.9.5

• Label: 38.6.e.b.9.5
• Level: $38$
• Weight: $6$
• Character: 38.9
• Analytic conductor: $6.095$
• Analytic rank: $0$
• Dimension: $30$
• 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:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			9.5
Character	\(\chi\)	\(=\)	38.9
Dual form			38.6.e.b.17.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.694593 + 3.93923i) q^{2} +(19.8464 - 16.6531i) q^{3} +(-15.0351 + 5.47232i) q^{4} +(17.3464 + 6.31358i) q^{5} +(79.3855 + 66.6124i) q^{6} +(61.0467 - 105.736i) q^{7} +(-32.0000 - 55.4256i) q^{8} +(74.3569 - 421.699i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 15 q^{3} + 60 q^{6} - 84 q^{7} - 960 q^{8} - 345 q^{9}+O(q^{10}) \)

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
							\(3\)	19.8464	−	16.6531i	1.27315	−	1.06830i	0.278996	−	0.960292i	\(-0.409998\pi\)
0.994150	−	0.108004i	\(-0.0344460\pi\)
\(5\)	17.3464	+	6.31358i	0.310302	+	0.112941i	0.492477	−	0.870325i	\(-0.336091\pi\)
							−0.182175	+	0.983266i	\(0.558314\pi\)
\(7\)	61.0467	−	105.736i	0.470887	−	0.815601i	−0.528558	−	0.848897i	\(-0.677267\pi\)
							0.999446	+	0.0332962i	\(0.0106005\pi\)
\(11\)	35.5887	+	61.6414i	0.0886809	+	0.153600i	0.906954	−	0.421230i	\(-0.138402\pi\)
							−0.818273	+	0.574830i	\(0.805068\pi\)
\(13\)	413.918	+	347.319i	0.679291	+	0.569993i	0.915799	−	0.401637i	\(-0.131558\pi\)
							−0.236508	+	0.971630i	\(0.576003\pi\)
\(17\)	38.4150	+	217.862i	0.0322388	+	0.182835i	0.996675	−	0.0814808i	\(-0.0259649\pi\)
							−0.964436	+	0.264316i	\(0.914854\pi\)
\(19\)	511.729	+	1488.03i	0.325204	+	0.945644i
							\(23\)	−3869.48	+	1408.38i	−1.52522	+	0.555136i	−0.962446	−	0.271474i	\(-0.912489\pi\)
−0.562776	+	0.826609i	\(0.690267\pi\)
\(29\)	964.187	−	5468.17i	0.212895	−	1.20739i	−0.671626	−	0.740890i	\(-0.734404\pi\)
							0.884522	−	0.466499i	\(-0.154485\pi\)
\(31\)	−3775.51	+	6539.37i	−0.705620	+	1.22217i	0.260847	+	0.965380i	\(0.415998\pi\)
							−0.966467	+	0.256790i	\(0.917335\pi\)
\(37\)	6113.82			0.734190			0.367095	−	0.930184i	\(-0.380352\pi\)
							0.367095	+	0.930184i	\(0.380352\pi\)
\(41\)	−15601.8	+	13091.5i	−1.44949	+	1.21627i	−0.516545	+	0.856260i	\(0.672782\pi\)
							−0.932949	+	0.360009i	\(0.882774\pi\)
\(43\)	−4795.60	−	1745.46i	−0.395523	−	0.143959i	0.136599	−	0.990626i	\(-0.456383\pi\)
							−0.532122	+	0.846668i	\(0.678605\pi\)
\(47\)	912.747	−	5176.44i	0.0602706	−	0.341812i	−0.939729	−	0.341919i	\(-0.888923\pi\)
							1.00000		0.000107525i	\(3.42263e-5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.b.9.5	✓	30
19.6	even	9		722.6.a.r.1.14		15
19.13	odd	18		722.6.a.q.1.2		15
19.17	even	9	inner	38.6.e.b.17.5	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.9.5	✓	30	1.1	even	1	trivial
38.6.e.b.17.5	yes	30	19.17	even	9	inner
722.6.a.q.1.2		15	19.13	odd	18
722.6.a.r.1.14		15	19.6	even	9