Newform orbit 65.2.f.a

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

Newspace parameters

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	65.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(0.519027613138\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-1}) \)
Defining polynomial:	\( x^{2} + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\)	\(27\)	\(41\)
\(\chi(n)\)	\(i\)	\(i\)

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

18.1

	−	1.00000i
		1.00000i

−

1.00000i

1.00000

+

1.00000i

1.00000

−2.00000

+

1.00000i

1.00000

−

1.00000i

−2.00000

−

3.00000i

−

1.00000i

1.00000

+

2.00000i

47.1

1.00000i

1.00000

−

1.00000i

1.00000

−2.00000

−

1.00000i

1.00000

+

1.00000i

−2.00000

3.00000i

1.00000i

1.00000

−

2.00000i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
65.f	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	65.2.f.a	✓	2
3.b	odd	2	1		585.2.n.c		2
4.b	odd	2	1		1040.2.cd.b		2
5.b	even	2	1		325.2.f.a		2
5.c	odd	4	1		65.2.k.a	yes	2
5.c	odd	4	1		325.2.k.a		2
13.b	even	2	1		845.2.f.a		2
13.c	even	3	2		845.2.t.a		4
13.d	odd	4	1		65.2.k.a	yes	2
13.d	odd	4	1		845.2.k.a		2
13.e	even	6	2		845.2.t.b		4
13.f	odd	12	2		845.2.o.a		4
13.f	odd	12	2		845.2.o.b		4
15.e	even	4	1		585.2.w.b		2
20.e	even	4	1		1040.2.bg.a		2
39.f	even	4	1		585.2.w.b		2
52.f	even	4	1		1040.2.bg.a		2
65.f	even	4	1	inner	65.2.f.a	✓	2
65.g	odd	4	1		325.2.k.a		2
65.h	odd	4	1		845.2.k.a		2
65.k	even	4	1		325.2.f.a		2
65.k	even	4	1		845.2.f.a		2
65.o	even	12	2		845.2.t.b		4
65.q	odd	12	2		845.2.o.a		4
65.r	odd	12	2		845.2.o.b		4
65.t	even	12	2		845.2.t.a		4
195.u	odd	4	1		585.2.n.c		2
260.l	odd	4	1		1040.2.cd.b		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.f.a	✓	2	1.a	even	1	1	trivial
65.2.f.a	✓	2	65.f	even	4	1	inner
65.2.k.a	yes	2	5.c	odd	4	1
65.2.k.a	yes	2	13.d	odd	4	1
325.2.f.a		2	5.b	even	2	1
325.2.f.a		2	65.k	even	4	1
325.2.k.a		2	5.c	odd	4	1
325.2.k.a		2	65.g	odd	4	1
585.2.n.c		2	3.b	odd	2	1
585.2.n.c		2	195.u	odd	4	1
585.2.w.b		2	15.e	even	4	1
585.2.w.b		2	39.f	even	4	1
845.2.f.a		2	13.b	even	2	1
845.2.f.a		2	65.k	even	4	1
845.2.k.a		2	13.d	odd	4	1
845.2.k.a		2	65.h	odd	4	1
845.2.o.a		4	13.f	odd	12	2
845.2.o.a		4	65.q	odd	12	2
845.2.o.b		4	13.f	odd	12	2
845.2.o.b		4	65.r	odd	12	2
845.2.t.a		4	13.c	even	3	2
845.2.t.a		4	65.t	even	12	2
845.2.t.b		4	13.e	even	6	2
845.2.t.b		4	65.o	even	12	2
1040.2.bg.a		2	20.e	even	4	1
1040.2.bg.a		2	52.f	even	4	1
1040.2.cd.b		2	4.b	odd	2	1
1040.2.cd.b		2	260.l	odd	4	1

Hecke kernels

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

Hecke characteristic polynomials

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