Embedded newform 38.3.f.a.13.4

• Label: 38.3.f.a.13.4
• 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			13.4
Character	\(\chi\)	\(=\)	38.13
Dual form			38.3.f.a.3.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.909039 + 1.08335i) q^{2} +(1.41923 - 3.89929i) q^{3} +(-0.347296 + 1.96962i) q^{4} +(-0.197003 - 1.11726i) q^{5} +(5.51443 - 2.00709i) q^{6} +(-5.55599 + 9.62326i) q^{7} +(-2.44949 + 1.41421i) q^{8} +(-6.29588 - 5.28287i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 6 q^{3} + 12 q^{6} - 18 q^{7} + 6 q^{9}+O(q^{10}) \)

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
							\(3\)	1.41923	−	3.89929i	0.473075	−	1.29976i	−0.442193	−	0.896920i	\(-0.645799\pi\)
0.915269	−	0.402844i	\(-0.131978\pi\)
\(5\)	−0.197003	−	1.11726i	−0.0394006	−	0.223452i	0.958749	−	0.284253i	\(-0.0917456\pi\)
							−0.998150	+	0.0608012i	\(0.980634\pi\)
\(7\)	−5.55599	+	9.62326i	−0.793713	+	1.37475i	0.129940	+	0.991522i	\(0.458522\pi\)
							−0.923653	+	0.383230i	\(0.874812\pi\)
\(11\)	3.53059	+	6.11516i	0.320962	+	0.555923i	0.980687	−	0.195584i	\(-0.0626603\pi\)
							−0.659724	+	0.751508i	\(0.729327\pi\)
\(13\)	−6.32302	−	17.3723i	−0.486386	−	1.33633i	−0.903931	−	0.427678i	\(-0.859332\pi\)
							0.417545	−	0.908656i	\(-0.362891\pi\)
\(17\)	0.827932	−	0.694717i	0.0487019	−	0.0408657i	−0.618112	−	0.786090i	\(-0.712102\pi\)
							0.666814	+	0.745224i	\(0.267658\pi\)
\(19\)	0.856354	−	18.9807i	0.0450713	−	0.998984i
							\(23\)	−7.48990	+	42.4773i	−0.325648	+	1.84684i	0.179436	+	0.983770i	\(0.442573\pi\)
−0.505084	+	0.863070i	\(0.668538\pi\)
\(29\)	17.7183	−	21.1159i	0.610977	−	0.728134i	−0.368514	−	0.929622i	\(-0.620133\pi\)
							0.979491	+	0.201488i	\(0.0645778\pi\)
\(31\)	−30.5074	−	17.6135i	−0.984110	−	0.568176i	−0.0806016	−	0.996746i	\(-0.525684\pi\)
							−0.903508	+	0.428570i	\(0.859017\pi\)
\(37\)			31.2690i			0.845109i	0.906338	+	0.422554i	\(0.138866\pi\)
							−0.906338	+	0.422554i	\(0.861134\pi\)
\(41\)	−1.56100	+	4.28880i	−0.0380731	+	0.104605i	−0.957272	−	0.289187i	\(-0.906615\pi\)
							0.919199	+	0.393792i	\(0.128837\pi\)
\(43\)	−4.60962	−	26.1425i	−0.107201	−	0.607965i	−0.990319	−	0.138814i	\(-0.955671\pi\)
							0.883118	−	0.469151i	\(-0.155440\pi\)
\(47\)	19.3721	+	16.2551i	0.412172	+	0.345853i	0.825176	−	0.564876i	\(-0.191076\pi\)
							−0.413004	+	0.910729i	\(0.635520\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.3.f.a.13.4	yes	24
3.2	odd	2		342.3.z.b.127.2		24
4.3	odd	2		304.3.z.c.241.1		24
19.3	odd	18	inner	38.3.f.a.3.4	✓	24
19.4	even	9		722.3.b.f.721.22		24
19.15	odd	18		722.3.b.f.721.3		24
57.41	even	18		342.3.z.b.307.2		24
76.3	even	18		304.3.z.c.193.1		24

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.3.f.a.3.4	✓	24	19.3	odd	18	inner
38.3.f.a.13.4	yes	24	1.1	even	1	trivial
304.3.z.c.193.1		24	76.3	even	18
304.3.z.c.241.1		24	4.3	odd	2
342.3.z.b.127.2		24	3.2	odd	2
342.3.z.b.307.2		24	57.41	even	18
722.3.b.f.721.3		24	19.15	odd	18
722.3.b.f.721.22		24	19.4	even	9