Embedded newform 38.3.f.a.3.1

• Label: 38.3.f.a.3.1
• Level: $38$
• Weight: $3$
• Character: 38.3
• Analytic conductor: $1.035$
• Analytic rank: $0$
• Dimension: $24$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.03542500457\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(4\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.1
Character	\(\chi\)	\(=\)	38.3
Dual form			38.3.f.a.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.909039 + 1.08335i) q^{2} +(-1.73025 - 4.75383i) q^{3} +(-0.347296 - 1.96962i) q^{4} +(0.678914 - 3.85031i) q^{5} +(6.72293 + 2.44695i) q^{6} +(1.77574 + 3.07567i) q^{7} +(2.44949 + 1.41421i) q^{8} +(-12.7107 + 10.6656i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{3} + 12 q^{6} - 18 q^{7} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.909039	+	1.08335i	−0.454519	+	0.541675i
							\(3\)	−1.73025	−	4.75383i	−0.576751	−	1.58461i	−0.793623	−	0.608410i	\(-0.791808\pi\)
0.216872	−	0.976200i	\(-0.430415\pi\)
\(5\)	0.678914	−	3.85031i	0.135783	−	0.770062i	−0.838529	−	0.544857i	\(-0.816584\pi\)
							0.974311	−	0.225205i	\(-0.0723051\pi\)
\(7\)	1.77574	+	3.07567i	0.253677	+	0.439381i	0.964535	−	0.263954i	\(-0.0850267\pi\)
							−0.710858	+	0.703335i	\(0.751693\pi\)
\(11\)	9.47515	−	16.4114i	0.861378	−	1.49195i	−0.00922204	−	0.999957i	\(-0.502936\pi\)
							0.870600	−	0.491992i	\(-0.163731\pi\)
\(13\)	−5.36377	+	14.7368i	−0.412598	+	1.13360i	0.543207	+	0.839599i	\(0.317210\pi\)
							−0.955804	+	0.294003i	\(0.905012\pi\)
\(17\)	15.2589	+	12.8037i	0.897580	+	0.753159i	0.969716	−	0.244236i	\(-0.0785370\pi\)
							−0.0721357	+	0.997395i	\(0.522981\pi\)
\(19\)	18.1819	−	5.51519i	0.956944	−	0.290273i
							\(23\)	−1.88549	−	10.6931i	−0.0819777	−	0.464919i	−0.997968	−	0.0637181i	\(-0.979704\pi\)
0.915990	−	0.401201i	\(-0.131407\pi\)
\(29\)	−14.3917	−	17.1514i	−0.496266	−	0.591427i	0.458533	−	0.888677i	\(-0.348375\pi\)
							−0.954800	+	0.297250i	\(0.903931\pi\)
\(31\)	−18.6178	+	10.7490i	−0.600574	+	0.346742i	−0.769267	−	0.638927i	\(-0.779379\pi\)
							0.168693	+	0.985669i	\(0.446045\pi\)
\(37\)			19.0629i			0.515213i	0.966250	+	0.257607i	\(0.0829338\pi\)
							−0.966250	+	0.257607i	\(0.917066\pi\)
\(41\)	−3.04521	−	8.36666i	−0.0742735	−	0.204065i	0.897000	−	0.442030i	\(-0.145742\pi\)
							−0.971274	+	0.237966i	\(0.923519\pi\)
\(43\)	−4.24035	+	24.0482i	−0.0986127	+	0.559261i	0.894968	+	0.446131i	\(0.147199\pi\)
							−0.993580	+	0.113129i	\(0.963913\pi\)
\(47\)	−59.2105	+	49.6835i	−1.25980	+	1.05710i	−0.264094	+	0.964497i	\(0.585073\pi\)
							−0.995704	+	0.0925983i	\(0.970483\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.3.f.a.3.1	✓	24
3.2	odd	2		342.3.z.b.307.3		24
4.3	odd	2		304.3.z.c.193.4		24
19.5	even	9		722.3.b.f.721.23		24
19.13	odd	18	inner	38.3.f.a.13.1	yes	24
19.14	odd	18		722.3.b.f.721.2		24
57.32	even	18		342.3.z.b.127.3		24
76.51	even	18		304.3.z.c.241.4		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.f.a.3.1	✓	24	1.1	even	1	trivial
38.3.f.a.13.1	yes	24	19.13	odd	18	inner
304.3.z.c.193.4		24	4.3	odd	2
304.3.z.c.241.4		24	76.51	even	18
342.3.z.b.127.3		24	57.32	even	18
342.3.z.b.307.3		24	3.2	odd	2
722.3.b.f.721.2		24	19.14	odd	18
722.3.b.f.721.23		24	19.5	even	9