Embedded newform 38.3.f.a.13.1

• Label: 38.3.f.a.13.1
• Level: $38$
• Weight: $3$
• Character: 38.13
• 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			13.1
Character	\(\chi\)	\(=\)	38.13
Dual form			38.3.f.a.3.1

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{5}{18}\right)\)

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

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.f.a.3.1	✓	24	19.3	odd	18	inner
38.3.f.a.13.1	yes	24	1.1	even	1	trivial
304.3.z.c.193.4		24	76.3	even	18
304.3.z.c.241.4		24	4.3	odd	2
342.3.z.b.127.3		24	3.2	odd	2
342.3.z.b.307.3		24	57.41	even	18
722.3.b.f.721.2		24	19.4	even	9
722.3.b.f.721.23		24	19.15	odd	18