Newform orbit 38.2.c.a

• Label: 38.2.c.a
• Level: $38$
• Weight: $2$
• Character orbit: 38.c
• Analytic conductor: $0.303$
• Analytic rank: $0$
• Dimension: $2$
• 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:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(-\zeta_{6}\)

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

0.500000	−	0.866025i
0.500000	+	0.866025i

0.500000

+

0.866025i

−0.500000

−

0.866025i

−0.500000

+

0.866025i

0

0.500000

−

0.866025i

−4.00000

−1.00000

1.00000

−

1.73205i

0

11.1

0.500000

−

0.866025i

−0.500000

+

0.866025i

−0.500000

−

0.866025i

0

0.500000

+

0.866025i

−4.00000

−1.00000

1.00000

+

1.73205i

0

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.a	✓	2
3.b	odd	2	1		342.2.g.b		2
4.b	odd	2	1		304.2.i.c		2
5.b	even	2	1		950.2.e.d		2
5.c	odd	4	2		950.2.j.e		4
8.b	even	2	1		1216.2.i.h		2
8.d	odd	2	1		1216.2.i.d		2
12.b	even	2	1		2736.2.s.m		2
19.b	odd	2	1		722.2.c.b		2
19.c	even	3	1	inner	38.2.c.a	✓	2
19.c	even	3	1		722.2.a.c		1
19.d	odd	6	1		722.2.a.d		1
19.d	odd	6	1		722.2.c.b		2
19.e	even	9	6		722.2.e.j		6
19.f	odd	18	6		722.2.e.i		6
57.f	even	6	1		6498.2.a.e		1
57.h	odd	6	1		342.2.g.b		2
57.h	odd	6	1		6498.2.a.s		1
76.f	even	6	1		5776.2.a.n		1
76.g	odd	6	1		304.2.i.c		2
76.g	odd	6	1		5776.2.a.g		1
95.i	even	6	1		950.2.e.d		2
95.m	odd	12	2		950.2.j.e		4
152.k	odd	6	1		1216.2.i.d		2
152.p	even	6	1		1216.2.i.h		2
228.m	even	6	1		2736.2.s.m		2

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 1 \)
$3$	\( T^{2} + T + 1 \)
$5$	\( T^{2} \)
$7$	\( (T + 4)^{2} \)
$11$	\( (T - 3)^{2} \)