Embedded newform 38.6.e.b.17.5

• Label: 38.6.e.b.17.5
• Level: $38$
• Weight: $6$
• Character: 38.17
• 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			17.5
Character	\(\chi\)	\(=\)	38.17
Dual form			38.6.e.b.9.5

$q$-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 \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.9.5	✓	30	19.9	even	9	inner
38.6.e.b.17.5	yes	30	1.1	even	1	trivial
722.6.a.q.1.2		15	19.3	odd	18
722.6.a.r.1.14		15	19.16	even	9