Newform orbit 38.2.c.b

• Label: 38.2.c.b
• Level: $38$
• Weight: $2$
• Character orbit: 38.c
• Analytic conductor: $0.303$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(0.303431527681\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{7})\)
Defining polynomial:	\( x^{4} + 7x^{2} + 49 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + \beta_{2} q^{2} + (\beta_{3} + \beta_1) q^{3} + ( - \beta_{2} - 1) q^{4} + ( - \beta_{3} + \beta_{2} - \beta_1) q^{5} - \beta_1 q^{6} + ( - \beta_{3} + 1) q^{7} + q^{8} + ( - 4 \beta_{2} - 4) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{4} - 2 q^{5} + 4 q^{7} + 4 q^{8} - 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 7x^{2} + 49 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{2} ) / 7 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{3} ) / 7 \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(-1 - \beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

7.1

1.32288	−	2.29129i
−1.32288	+	2.29129i
1.32288	+	2.29129i
−1.32288	−	2.29129i

−0.500000

−

0.866025i

−1.32288

−

2.29129i

−0.500000

+

0.866025i

0.822876

+

1.42526i

−1.32288

+

2.29129i

3.64575

1.00000

−2.00000

+

3.46410i

0.822876

−

1.42526i

7.2

−0.500000

−

0.866025i

1.32288

+

2.29129i

−0.500000

+

0.866025i

−1.82288

−

3.15731i

1.32288

−

2.29129i

−1.64575

1.00000

−2.00000

+

3.46410i

−1.82288

+

3.15731i

11.1

−0.500000

+

0.866025i

−1.32288

+

2.29129i

−0.500000

−

0.866025i

0.822876

−

1.42526i

−1.32288

−

2.29129i

3.64575

1.00000

−2.00000

−

3.46410i

0.822876

+

1.42526i

11.2

−0.500000

+

0.866025i

1.32288

−

2.29129i

−0.500000

−

0.866025i

−1.82288

+

3.15731i

1.32288

+

2.29129i

−1.64575

1.00000

−2.00000

−

3.46410i

−1.82288

−

3.15731i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	38.2.c.b	✓	4
3.b	odd	2	1		342.2.g.f		4
4.b	odd	2	1		304.2.i.e		4
5.b	even	2	1		950.2.e.k		4
5.c	odd	4	2		950.2.j.g		8
8.b	even	2	1		1216.2.i.l		4
8.d	odd	2	1		1216.2.i.k		4
12.b	even	2	1		2736.2.s.v		4
19.b	odd	2	1		722.2.c.j		4
19.c	even	3	1	inner	38.2.c.b	✓	4
19.c	even	3	1		722.2.a.j		2
19.d	odd	6	1		722.2.a.g		2
19.d	odd	6	1		722.2.c.j		4
19.e	even	9	6		722.2.e.n		12
19.f	odd	18	6		722.2.e.o		12
57.f	even	6	1		6498.2.a.bg		2
57.h	odd	6	1		342.2.g.f		4
57.h	odd	6	1		6498.2.a.ba		2
76.f	even	6	1		5776.2.a.z		2
76.g	odd	6	1		304.2.i.e		4
76.g	odd	6	1		5776.2.a.ba		2
95.i	even	6	1		950.2.e.k		4
95.m	odd	12	2		950.2.j.g		8
152.k	odd	6	1		1216.2.i.k		4
152.p	even	6	1		1216.2.i.l		4
228.m	even	6	1		2736.2.s.v		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
38.2.c.b	✓	4	1.a	even	1	1	trivial
38.2.c.b	✓	4	19.c	even	3	1	inner
304.2.i.e		4	4.b	odd	2	1
304.2.i.e		4	76.g	odd	6	1
342.2.g.f		4	3.b	odd	2	1
342.2.g.f		4	57.h	odd	6	1
722.2.a.g		2	19.d	odd	6	1
722.2.a.j		2	19.c	even	3	1
722.2.c.j		4	19.b	odd	2	1
722.2.c.j		4	19.d	odd	6	1
722.2.e.n		12	19.e	even	9	6
722.2.e.o		12	19.f	odd	18	6
950.2.e.k		4	5.b	even	2	1
950.2.e.k		4	95.i	even	6	1
950.2.j.g		8	5.c	odd	4	2
950.2.j.g		8	95.m	odd	12	2
1216.2.i.k		4	8.d	odd	2	1
1216.2.i.k		4	152.k	odd	6	1
1216.2.i.l		4	8.b	even	2	1
1216.2.i.l		4	152.p	even	6	1
2736.2.s.v		4	12.b	even	2	1
2736.2.s.v		4	228.m	even	6	1
5776.2.a.z		2	76.f	even	6	1
5776.2.a.ba		2	76.g	odd	6	1
6498.2.a.ba		2	57.h	odd	6	1
6498.2.a.bg		2	57.f	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 7T_{3}^{2} + 49 \) acting on \(S_{2}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} + T + 1)^{2} \)
$3$	\( T^{4} + 7T^{2} + 49 \)
$5$	\( T^{4} + 2 T^{3} + 10 T^{2} - 12 T + 36 \)
$7$	\( (T^{2} - 2 T - 6)^{2} \)
$11$	\( (T^{2} + 4 T - 3)^{2} \)