Embedded newform 38.3.f.a.13.2

• Label: 38.3.f.a.13.2
• 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			13.2
Character	\(\chi\)	\(=\)	38.13
Dual form			38.3.f.a.3.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.909039 - 1.08335i) q^{2} +(0.836494 - 2.29825i) q^{3} +(-0.347296 + 1.96962i) q^{4} +(-0.852562 - 4.83512i) q^{5} +(-3.25021 + 1.18298i) q^{6} +(0.583107 - 1.00997i) q^{7} +(2.44949 - 1.41421i) q^{8} +(2.31218 + 1.94015i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{3} + 12 q^{6} - 18 q^{7} + 6 q^{9}+O(q^{10}) \)

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
							\(3\)	0.836494	−	2.29825i	0.278831	−	0.766083i	−0.718665	−	0.695357i	\(-0.755246\pi\)
0.997496	−	0.0707256i	\(-0.0225315\pi\)
\(5\)	−0.852562	−	4.83512i	−0.170512	−	0.967024i	−0.943197	−	0.332234i	\(-0.892198\pi\)
							0.772685	−	0.634790i	\(-0.218913\pi\)
\(7\)	0.583107	−	1.00997i	0.0833010	−	0.144282i	−0.821365	−	0.570403i	\(-0.806787\pi\)
							0.904666	+	0.426121i	\(0.140120\pi\)
\(11\)	1.75917	+	3.04697i	0.159924	+	0.276997i	0.934841	−	0.355066i	\(-0.115542\pi\)
							−0.774917	+	0.632063i	\(0.782208\pi\)
\(13\)	7.84716	+	21.5599i	0.603627	+	1.65845i	0.743861	+	0.668334i	\(0.232992\pi\)
							−0.140234	+	0.990118i	\(0.544785\pi\)
\(17\)	7.26534	−	6.09634i	0.427373	−	0.358609i	−0.403586	−	0.914942i	\(-0.632237\pi\)
							0.830959	+	0.556333i	\(0.187792\pi\)
\(19\)	−18.9314	+	1.61270i	−0.996391	+	0.0848790i
							\(23\)	3.50841	−	19.8972i	0.152540	−	0.865095i	−0.808461	−	0.588550i	\(-0.799699\pi\)
0.961001	−	0.276546i	\(-0.0891897\pi\)
\(29\)	−25.7239	+	30.6565i	−0.887030	+	1.05712i	0.110965	+	0.993824i	\(0.464606\pi\)
							−0.997995	+	0.0632969i	\(0.979839\pi\)
\(31\)	−43.3050	−	25.0022i	−1.39694	−	0.806521i	−0.402866	−	0.915259i	\(-0.631986\pi\)
							−0.994070	+	0.108738i	\(0.965319\pi\)
\(37\)			47.7325i			1.29007i	0.764154	+	0.645034i	\(0.223157\pi\)
							−0.764154	+	0.645034i	\(0.776843\pi\)
\(41\)	7.79790	−	21.4246i	0.190193	−	0.522550i	−0.807543	−	0.589809i	\(-0.799203\pi\)
							0.997736	+	0.0672587i	\(0.0214253\pi\)
\(43\)	5.98145	+	33.9225i	0.139103	+	0.788895i	0.971914	+	0.235337i	\(0.0756193\pi\)
							−0.832810	+	0.553558i	\(0.813270\pi\)
\(47\)	−15.8422	−	13.2932i	−0.337068	−	0.282834i	0.458504	−	0.888692i	\(-0.348385\pi\)
							−0.795573	+	0.605858i	\(0.792830\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.3.f.a.13.2	yes	24
3.2	odd	2		342.3.z.b.127.4		24
4.3	odd	2		304.3.z.c.241.2		24
19.3	odd	18	inner	38.3.f.a.3.2	✓	24
19.4	even	9		722.3.b.f.721.9		24
19.15	odd	18		722.3.b.f.721.16		24
57.41	even	18		342.3.z.b.307.4		24
76.3	even	18		304.3.z.c.193.2		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.f.a.3.2	✓	24	19.3	odd	18	inner
38.3.f.a.13.2	yes	24	1.1	even	1	trivial
304.3.z.c.193.2		24	76.3	even	18
304.3.z.c.241.2		24	4.3	odd	2
342.3.z.b.127.4		24	3.2	odd	2
342.3.z.b.307.4		24	57.41	even	18
722.3.b.f.721.9		24	19.4	even	9
722.3.b.f.721.16		24	19.15	odd	18