Newform orbit 65.2.e.a

• Label: 65.2.e.a
• Level: $65$
• Weight: $2$
• Character orbit: 65.e
• Analytic conductor: $0.519$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	65.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.519027613138\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial:	\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
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 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 2x^{2} + x + 1 \) :

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(1\)	\(\beta_{3}\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

16.1

0.809017	+	1.40126i
−0.309017	−	0.535233i
0.809017	−	1.40126i
−0.309017	+	0.535233i

−0.809017

−

1.40126i

1.11803

+

1.93649i

−0.309017

+

0.535233i

1.00000

1.80902

−

3.13331i

0.118034

−

0.204441i

−2.23607

−1.00000

+

1.73205i

−0.809017

−

1.40126i

16.2

0.309017

+

0.535233i

−1.11803

−

1.93649i

0.809017

−

1.40126i

1.00000

0.690983

−

1.19682i

−2.11803

+

3.66854i

2.23607

−1.00000

+

1.73205i

0.309017

+

0.535233i

61.1

−0.809017

+

1.40126i

1.11803

−

1.93649i

−0.309017

−

0.535233i

1.00000

1.80902

+

3.13331i

0.118034

+

0.204441i

−2.23607

−1.00000

−

1.73205i

−0.809017

+

1.40126i

61.2

0.309017

−

0.535233i

−1.11803

+

1.93649i

0.809017

+

1.40126i

1.00000

0.690983

+

1.19682i

−2.11803

−

3.66854i

2.23607

−1.00000

−

1.73205i

0.309017

−

0.535233i

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	65.2.e.a	✓	4
3.b	odd	2	1		585.2.j.e		4
4.b	odd	2	1		1040.2.q.n		4
5.b	even	2	1		325.2.e.b		4
5.c	odd	4	2		325.2.o.a		8
13.b	even	2	1		845.2.e.g		4
13.c	even	3	1	inner	65.2.e.a	✓	4
13.c	even	3	1		845.2.a.e		2
13.d	odd	4	2		845.2.m.e		8
13.e	even	6	1		845.2.a.b		2
13.e	even	6	1		845.2.e.g		4
13.f	odd	12	2		845.2.c.c		4
13.f	odd	12	2		845.2.m.e		8
39.h	odd	6	1		7605.2.a.bf		2
39.i	odd	6	1		585.2.j.e		4
39.i	odd	6	1		7605.2.a.ba		2
52.j	odd	6	1		1040.2.q.n		4
65.l	even	6	1		4225.2.a.y		2
65.n	even	6	1		325.2.e.b		4
65.n	even	6	1		4225.2.a.u		2
65.q	odd	12	2		325.2.o.a		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.e.a	✓	4	1.a	even	1	1	trivial
65.2.e.a	✓	4	13.c	even	3	1	inner
325.2.e.b		4	5.b	even	2	1
325.2.e.b		4	65.n	even	6	1
325.2.o.a		8	5.c	odd	4	2
325.2.o.a		8	65.q	odd	12	2
585.2.j.e		4	3.b	odd	2	1
585.2.j.e		4	39.i	odd	6	1
845.2.a.b		2	13.e	even	6	1
845.2.a.e		2	13.c	even	3	1
845.2.c.c		4	13.f	odd	12	2
845.2.e.g		4	13.b	even	2	1
845.2.e.g		4	13.e	even	6	1
845.2.m.e		8	13.d	odd	4	2
845.2.m.e		8	13.f	odd	12	2
1040.2.q.n		4	4.b	odd	2	1
1040.2.q.n		4	52.j	odd	6	1
4225.2.a.u		2	65.n	even	6	1
4225.2.a.y		2	65.l	even	6	1
7605.2.a.ba		2	39.i	odd	6	1
7605.2.a.bf		2	39.h	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} + 2T_{2}^{2} - T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(65, [\chi])\).

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{4} + T^{3} + 2 T^{2} - T + 1 \)
$3$	\( T^{4} + 5T^{2} + 25 \)
$5$	\( (T - 1)^{4} \)
$7$	\( T^{4} + 4 T^{3} + 17 T^{2} - 4 T + 1 \)
$11$	\( T^{4} + 4 T^{3} + 17 T^{2} - 4 T + 1 \)