Embedded newform 38.5.f.a.15.3

• Label: 38.5.f.a.15.3
• Level: $38$
• Weight: $5$
• Character: 38.15
• Analytic conductor: $3.928$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.92805859719\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			15.3
Character	\(\chi\)	\(=\)	38.15
Dual form			38.5.f.a.33.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.967379 + 2.65785i) q^{2} +(10.2027 - 1.79900i) q^{3} +(-6.12836 - 5.14230i) q^{4} +(24.4832 - 20.5438i) q^{5} +(-5.08835 + 28.8575i) q^{6} +(6.11419 - 10.5901i) q^{7} +(19.5959 - 11.3137i) q^{8} +(24.7428 - 9.00564i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 12 q^{3} + 48 q^{6} + 90 q^{7} - 84 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{11}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.967379	+	2.65785i	−0.241845	+	0.664463i
							\(3\)	10.2027	−	1.79900i	1.13363	−	0.199889i	0.424812	−	0.905282i	\(-0.360340\pi\)
0.708817	+	0.705392i	\(0.249229\pi\)
\(5\)	24.4832	−	20.5438i	0.979326	−	0.821752i	−0.00466132	−	0.999989i	\(-0.501484\pi\)
							0.983988	+	0.178237i	\(0.0570393\pi\)
\(7\)	6.11419	−	10.5901i	0.124779	−	0.216124i	−0.796867	−	0.604154i	\(-0.793511\pi\)
							0.921647	+	0.388030i	\(0.126844\pi\)
\(11\)	88.0331	+	152.478i	0.727547	+	1.26015i	0.957917	+	0.287045i	\(0.0926729\pi\)
							−0.230370	+	0.973103i	\(0.573994\pi\)
\(13\)	−12.4512	−	2.19549i	−0.0736760	−	0.0129911i	0.136689	−	0.990614i	\(-0.456354\pi\)
							−0.210365	+	0.977623i	\(0.567465\pi\)
\(17\)	−37.6242	−	13.6941i	−0.130188	−	0.0473844i	0.276105	−	0.961128i	\(-0.410956\pi\)
							−0.406292	+	0.913743i	\(0.633179\pi\)
\(19\)	−357.921	−	47.0487i	−0.991471	−	0.130329i
							\(23\)	−554.047	−	464.900i	−1.04735	−	0.878829i	−0.0545352	−	0.998512i	\(-0.517368\pi\)
−0.992812	+	0.119683i	\(0.961812\pi\)
\(29\)	−182.242	−	500.706i	−0.216697	−	0.595370i	0.782946	−	0.622090i	\(-0.213716\pi\)
							−0.999643	+	0.0267195i	\(0.991494\pi\)
\(31\)	1467.99	+	847.542i	1.52756	+	0.881937i	0.999463	+	0.0327531i	\(0.0104275\pi\)
							0.528097	+	0.849184i	\(0.322906\pi\)
\(37\)			581.280i			0.424602i	0.977204	+	0.212301i	\(0.0680957\pi\)
							−0.977204	+	0.212301i	\(0.931904\pi\)
\(41\)	−2697.13	+	475.577i	−1.60448	+	0.282913i	−0.902957	−	0.429731i	\(-0.858608\pi\)
							−0.701525	+	0.712645i	\(0.747497\pi\)
\(43\)	896.694	−	752.416i	0.484962	−	0.406931i	−0.367255	−	0.930120i	\(-0.619702\pi\)
							0.852217	+	0.523189i	\(0.175258\pi\)
\(47\)	−402.184	+	146.383i	−0.182066	+	0.0662666i	−0.431445	−	0.902139i	\(-0.641996\pi\)
							0.249378	+	0.968406i	\(0.419774\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.5.f.a.15.3	✓	36
19.14	odd	18	inner	38.5.f.a.33.3	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.f.a.15.3	✓	36	1.1	even	1	trivial
38.5.f.a.33.3	yes	36	19.14	odd	18	inner