Newform orbit 13.4.e.b

• Label: 13.4.e.b
• Level: $13$
• Weight: $4$
• Character orbit: 13.e
• 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.e (of order \(6\), 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:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\)	\(=\)	\( q + (\zeta_{6} + 1) q^{2} + (2 \zeta_{6} - 2) q^{3} - 5 \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 1) q^{5} + (2 \zeta_{6} - 4) q^{6} + (8 \zeta_{6} - 16) q^{7} + ( - 26 \zeta_{6} + 13) q^{8} + 23 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} - 2 q^{3} - 5 q^{4} - 6 q^{6} - 24 q^{7} + 23 q^{9}+O(q^{10}) \)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

4.1

0.500000	+	0.866025i
0.500000	−	0.866025i

1.50000

+

0.866025i

−1.00000

+

1.73205i

−2.50000

−

4.33013i

−

1.73205i

−3.00000

+

1.73205i

−12.0000

+

6.92820i

−

22.5167i

11.5000

+

19.9186i

1.50000

−

2.59808i

10.1

1.50000

−

0.866025i

−1.00000

−

1.73205i

−2.50000

+

4.33013i

1.73205i

−3.00000

−

1.73205i

−12.0000

−

6.92820i

22.5167i

11.5000

−

19.9186i

1.50000

+

2.59808i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.e	even	6	1	inner

Twists

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

    

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} + 3 \) acting on \(S_{4}^{\mathrm{new}}(13, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - 3T + 3 \)
$3$	\( T^{2} + 2T + 4 \)
$5$	\( T^{2} + 3 \)
$7$	\( T^{2} + 24T + 192 \)
$11$	\( T^{2} - 24T + 192 \)