Embedded newform 38.3.f.a.29.2

• Label: 38.3.f.a.29.2
• Level: $38$
• Weight: $3$
• Character: 38.29
• 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			29.2
Character	\(\chi\)	\(=\)	38.29
Dual form			38.3.f.a.21.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39273 + 0.245576i) q^{2} +(0.272417 + 0.324654i) q^{3} +(1.87939 - 0.684040i) q^{4} +(7.98876 + 2.90767i) q^{5} +(-0.459130 - 0.385256i) q^{6} +(-2.58545 + 4.47813i) q^{7} +(-2.44949 + 1.41421i) q^{8} +(1.53164 - 8.68639i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{3} + 12 q^{6} - 18 q^{7} + 6 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{17}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39273	+	0.245576i	−0.696364	+	0.122788i
							\(3\)	0.272417	+	0.324654i	0.0908057	+	0.108218i	0.809534	−	0.587073i	\(-0.199720\pi\)
−0.718728	+	0.695291i	\(0.755275\pi\)
\(5\)	7.98876	+	2.90767i	1.59775	+	0.581534i	0.978966	−	0.204025i	\(-0.0654023\pi\)
							0.618787	+	0.785559i	\(0.287625\pi\)
\(7\)	−2.58545	+	4.47813i	−0.369350	+	0.639733i	−0.989464	−	0.144779i	\(-0.953753\pi\)
							0.620114	+	0.784512i	\(0.287086\pi\)
\(11\)	−4.81869	−	8.34622i	−0.438063	−	0.758747i	0.559477	−	0.828846i	\(-0.311002\pi\)
							−0.997540	+	0.0700987i	\(0.977669\pi\)
\(13\)	−8.78112	+	10.4649i	−0.675471	+	0.804995i	−0.989518	−	0.144413i	\(-0.953871\pi\)
							0.314047	+	0.949407i	\(0.398315\pi\)
\(17\)	−3.08077	−	17.4719i	−0.181222	−	1.02776i	−0.930715	−	0.365747i	\(-0.880814\pi\)
							0.749493	−	0.662012i	\(-0.230297\pi\)
\(19\)	−17.6183	+	7.11298i	−0.927281	+	0.374367i
							\(23\)	14.1083	−	5.13501i	0.613406	−	0.223261i	−0.0165870	−	0.999862i	\(-0.505280\pi\)
0.629993	+	0.776601i	\(0.283058\pi\)
\(29\)	11.5878	+	2.04324i	0.399579	+	0.0704566i	0.369827	−	0.929101i	\(-0.379417\pi\)
							0.0297524	+	0.999557i	\(0.490528\pi\)
\(31\)	−44.1349	−	25.4813i	−1.42371	−	0.821977i	−0.427092	−	0.904208i	\(-0.640462\pi\)
							−0.996613	+	0.0822315i	\(0.973795\pi\)
\(37\)		−	17.8330i		−	0.481972i	−0.970529	−	0.240986i	\(-0.922529\pi\)
							0.970529	−	0.240986i	\(-0.0774708\pi\)
\(41\)	−15.8487	−	18.8877i	−0.386553	−	0.460676i	0.537318	−	0.843380i	\(-0.319437\pi\)
							−0.923871	+	0.382704i	\(0.874993\pi\)
\(43\)	14.2358	+	5.18140i	0.331065	+	0.120498i	0.502204	−	0.864749i	\(-0.332523\pi\)
							−0.171139	+	0.985247i	\(0.554745\pi\)
\(47\)	−13.4551	+	76.3075i	−0.286278	+	1.62356i	0.414406	+	0.910092i	\(0.363989\pi\)
							−0.700684	+	0.713472i	\(0.747122\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.3.f.a.29.2	yes	24
3.2	odd	2		342.3.z.b.181.3		24
4.3	odd	2		304.3.z.c.257.2		24
19.2	odd	18	inner	38.3.f.a.21.2	✓	24
19.6	even	9		722.3.b.f.721.20		24
19.13	odd	18		722.3.b.f.721.5		24
57.2	even	18		342.3.z.b.325.3		24
76.59	even	18		304.3.z.c.97.2		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.f.a.21.2	✓	24	19.2	odd	18	inner
38.3.f.a.29.2	yes	24	1.1	even	1	trivial
304.3.z.c.97.2		24	76.59	even	18
304.3.z.c.257.2		24	4.3	odd	2
342.3.z.b.181.3		24	3.2	odd	2
342.3.z.b.325.3		24	57.2	even	18
722.3.b.f.721.5		24	19.13	odd	18
722.3.b.f.721.20		24	19.6	even	9