Newform orbit 13.4.c.a

• Label: 13.4.c.a
• Level: $13$
• Weight: $4$
• Character orbit: 13.c
• Analytic conductor: $0.767$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	13.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.767024830075\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
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.

\(f(q)\)	\(=\)	\( q + (4 \zeta_{6} - 4) q^{2} + (2 \zeta_{6} - 2) q^{3} - 8 \zeta_{6} q^{4} + 17 q^{5} - 8 \zeta_{6} q^{6} - 20 \zeta_{6} q^{7} + 23 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} - 2 q^{3} - 8 q^{4} + 34 q^{5} - 8 q^{6} - 20 q^{7} + 23 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/13\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(-\zeta_{6}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

3.1

0.500000	−	0.866025i
0.500000	+	0.866025i

−2.00000

−

3.46410i

−1.00000

−

1.73205i

−4.00000

+

6.92820i

17.0000

−4.00000

+

6.92820i

−10.0000

+

17.3205i

0

11.5000

−

19.9186i

−34.0000

−

58.8897i

9.1

−2.00000

+

3.46410i

−1.00000

+

1.73205i

−4.00000

−

6.92820i

17.0000

−4.00000

−

6.92820i

−10.0000

−

17.3205i

0

11.5000

+

19.9186i

−34.0000

+

58.8897i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	13.4.c.a	✓	2
3.b	odd	2	1		117.4.g.c		2
4.b	odd	2	1		208.4.i.b		2
13.b	even	2	1		169.4.c.d		2
13.c	even	3	1	inner	13.4.c.a	✓	2
13.c	even	3	1		169.4.a.d		1
13.d	odd	4	2		169.4.e.c		4
13.e	even	6	1		169.4.a.a		1
13.e	even	6	1		169.4.c.d		2
13.f	odd	12	2		169.4.b.c		2
13.f	odd	12	2		169.4.e.c		4
39.h	odd	6	1		1521.4.a.k		1
39.i	odd	6	1		117.4.g.c		2
39.i	odd	6	1		1521.4.a.b		1
52.j	odd	6	1		208.4.i.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
13.4.c.a	✓	2	1.a	even	1	1	trivial
13.4.c.a	✓	2	13.c	even	3	1	inner
117.4.g.c		2	3.b	odd	2	1
117.4.g.c		2	39.i	odd	6	1
169.4.a.a		1	13.e	even	6	1
169.4.a.d		1	13.c	even	3	1
169.4.b.c		2	13.f	odd	12	2
169.4.c.d		2	13.b	even	2	1
169.4.c.d		2	13.e	even	6	1
169.4.e.c		4	13.d	odd	4	2
169.4.e.c		4	13.f	odd	12	2
208.4.i.b		2	4.b	odd	2	1
208.4.i.b		2	52.j	odd	6	1
1521.4.a.b		1	39.i	odd	6	1
1521.4.a.k		1	39.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4T_{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} + 4T + 16 \)
$3$	\( T^{2} + 2T + 4 \)
$5$	\( (T - 17)^{2} \)
$7$	\( T^{2} + 20T + 400 \)
$11$	\( T^{2} - 32T + 1024 \)