Newform orbit 38.4.c.a

• Label: 38.4.c.a
• Level: $38$
• Weight: $4$
• Character orbit: 38.c
• 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:	\(\Q(\sqrt{-3}) \)
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:	$\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (2 \zeta_{6} - 2) q^{2} + (5 \zeta_{6} - 5) q^{3} - 4 \zeta_{6} q^{4} + (3 \zeta_{6} - 3) q^{5} - 10 \zeta_{6} q^{6} - 32 q^{7} + 8 q^{8} + 2 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 5 q^{3} - 4 q^{4} - 3 q^{5} - 10 q^{6} - 64 q^{7} + 16 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

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

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

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

−1.00000

−

1.73205i

−2.50000

−

4.33013i

−2.00000

+

3.46410i

−1.50000

−

2.59808i

−5.00000

+

8.66025i

−32.0000

8.00000

1.00000

−

1.73205i

−3.00000

+

5.19615i

11.1

−1.00000

+

1.73205i

−2.50000

+

4.33013i

−2.00000

−

3.46410i

−1.50000

+

2.59808i

−5.00000

−

8.66025i

−32.0000

8.00000

1.00000

+

1.73205i

−3.00000

−

5.19615i

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	38.4.c.a	✓	2
3.b	odd	2	1		342.4.g.d		2
4.b	odd	2	1		304.4.i.b		2
19.c	even	3	1	inner	38.4.c.a	✓	2
19.c	even	3	1		722.4.a.e		1
19.d	odd	6	1		722.4.a.a		1
57.h	odd	6	1		342.4.g.d		2
76.g	odd	6	1		304.4.i.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
38.4.c.a	✓	2	1.a	even	1	1	trivial
38.4.c.a	✓	2	19.c	even	3	1	inner
304.4.i.b		2	4.b	odd	2	1
304.4.i.b		2	76.g	odd	6	1
342.4.g.d		2	3.b	odd	2	1
342.4.g.d		2	57.h	odd	6	1
722.4.a.a		1	19.d	odd	6	1
722.4.a.e		1	19.c	even	3	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 5T_{3} + 25 \) acting on \(S_{4}^{\mathrm{new}}(38, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 2T + 4 \)
$3$	\( T^{2} + 5T + 25 \)
$5$	\( T^{2} + 3T + 9 \)
$7$	\( (T + 32)^{2} \)
$11$	\( (T - 4)^{2} \)