Embedded newform 38.4.c.a.7.1

• Label: 38.4.c.a.7.1
• Level: $38$
• Weight: $4$
• Character: 38.7
• Analytic conductor: $2.242$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	38.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.24207258022\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			7.1
Root			\(0.500000 - 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	38.7
Dual form			38.4.c.a.11.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.50000 - 4.33013i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-1.50000 - 2.59808i) q^{5} +(-5.00000 + 8.66025i) q^{6} -32.0000 q^{7} +8.00000 q^{8} +(1.00000 - 1.73205i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 5 q^{3} - 4 q^{4} - 3 q^{5} - 10 q^{6} - 64 q^{7} + 16 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−	1.73205i	−0.353553	−	0.612372i
							\(3\)	−2.50000	−	4.33013i	−0.481125	−	0.833333i	0.518640	−	0.854993i	\(-0.326438\pi\)
−0.999765	+	0.0216593i	\(0.993105\pi\)
\(5\)	−1.50000	−	2.59808i	−0.134164	−	0.232379i	0.791114	−	0.611669i	\(-0.209502\pi\)
							−0.925278	+	0.379290i	\(0.876168\pi\)
\(7\)	−32.0000			−1.72784			−0.863919	−	0.503631i	\(-0.831997\pi\)
							−0.863919	+	0.503631i	\(0.831997\pi\)
\(11\)	4.00000			0.109640			0.0548202	−	0.998496i	\(-0.482541\pi\)
							0.0548202	+	0.998496i	\(0.482541\pi\)
\(13\)	34.5000	−	59.7558i	0.736044	−	1.27487i	−0.218219	−	0.975900i	\(-0.570025\pi\)
							0.954264	−	0.298967i	\(-0.0966419\pi\)
\(17\)	−9.50000	−	16.4545i	−0.135535	−	0.234753i	0.790267	−	0.612763i	\(-0.209942\pi\)
							−0.925802	+	0.378010i	\(0.876609\pi\)
\(19\)	76.0000	−	32.9090i	0.917663	−	0.397360i
							\(23\)	−33.5000	+	58.0237i	−0.303706	+	0.526034i	−0.976972	−	0.213366i	\(-0.931557\pi\)
0.673267	+	0.739400i	\(0.264891\pi\)
\(29\)	−25.5000	+	44.1673i	−0.163284	+	0.282816i	−0.936045	−	0.351882i	\(-0.885542\pi\)
							0.772761	+	0.634698i	\(0.218875\pi\)
\(31\)	−132.000			−0.764771			−0.382385	−	0.924003i	\(-0.624897\pi\)
							−0.382385	+	0.924003i	\(0.624897\pi\)
\(37\)	−14.0000			−0.0622050			−0.0311025	−	0.999516i	\(-0.509902\pi\)
							−0.0311025	+	0.999516i	\(0.509902\pi\)
\(41\)	206.500	+	357.668i	0.786582	+	1.36240i	0.928049	+	0.372458i	\(0.121485\pi\)
							−0.141467	+	0.989943i	\(0.545182\pi\)
\(43\)	−64.5000	−	111.717i	−0.228748	−	0.396203i	0.728689	−	0.684844i	\(-0.240130\pi\)
							−0.957437	+	0.288641i	\(0.906796\pi\)
\(47\)	308.500	−	534.338i	0.957433	−	1.65832i	0.228733	−	0.973489i	\(-0.426542\pi\)
							0.728700	−	0.684833i	\(-0.240125\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.4.c.a.7.1	✓	2
3.2	odd	2		342.4.g.d.235.1		2
4.3	odd	2		304.4.i.b.273.1		2
19.7	even	3		722.4.a.e.1.1		1
19.11	even	3	inner	38.4.c.a.11.1	yes	2
19.12	odd	6		722.4.a.a.1.1		1
57.11	odd	6		342.4.g.d.163.1		2
76.11	odd	6		304.4.i.b.49.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.4.c.a.7.1	✓	2	1.1	even	1	trivial
38.4.c.a.11.1	yes	2	19.11	even	3	inner
304.4.i.b.49.1		2	76.11	odd	6
304.4.i.b.273.1		2	4.3	odd	2
342.4.g.d.163.1		2	57.11	odd	6
342.4.g.d.235.1		2	3.2	odd	2
722.4.a.a.1.1		1	19.12	odd	6
722.4.a.e.1.1		1	19.7	even	3