Newform orbit 65.2.a.b

• Label: 65.2.a.b
• Level: $65$
• Weight: $2$
• Character orbit: 65.a
• Self dual: yes
• Analytic conductor: $0.519$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	65.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(0.519027613138\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{2}) \)
Defining polynomial:	\( x^{2} - 2 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

−1.41421
1.41421

−2.41421

−1.41421

3.82843

1.00000

3.41421

4.82843

−4.41421

−1.00000

−2.41421

1.2

0.414214

1.41421

−1.82843

1.00000

0.585786

−0.828427

−1.58579

−1.00000

0.414214

Atkin-Lehner signs

\( p \)	Sign
\(5\)	\(-1\)
\(13\)	\(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	65.2.a.b	✓	2
3.b	odd	2	1		585.2.a.m		2
4.b	odd	2	1		1040.2.a.j		2
5.b	even	2	1		325.2.a.i		2
5.c	odd	4	2		325.2.b.f		4
7.b	odd	2	1		3185.2.a.j		2
8.b	even	2	1		4160.2.a.bf		2
8.d	odd	2	1		4160.2.a.z		2
11.b	odd	2	1		7865.2.a.j		2
12.b	even	2	1		9360.2.a.cd		2
13.b	even	2	1		845.2.a.g		2
13.c	even	3	2		845.2.e.h		4
13.d	odd	4	2		845.2.c.b		4
13.e	even	6	2		845.2.e.c		4
13.f	odd	12	4		845.2.m.f		8
15.d	odd	2	1		2925.2.a.u		2
15.e	even	4	2		2925.2.c.r		4
20.d	odd	2	1		5200.2.a.bu		2
39.d	odd	2	1		7605.2.a.x		2
65.d	even	2	1		4225.2.a.r		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.2.a.b	✓	2	1.a	even	1	1	trivial
325.2.a.i		2	5.b	even	2	1
325.2.b.f		4	5.c	odd	4	2
585.2.a.m		2	3.b	odd	2	1
845.2.a.g		2	13.b	even	2	1
845.2.c.b		4	13.d	odd	4	2
845.2.e.c		4	13.e	even	6	2
845.2.e.h		4	13.c	even	3	2
845.2.m.f		8	13.f	odd	12	4
1040.2.a.j		2	4.b	odd	2	1
2925.2.a.u		2	15.d	odd	2	1
2925.2.c.r		4	15.e	even	4	2
3185.2.a.j		2	7.b	odd	2	1
4160.2.a.z		2	8.d	odd	2	1
4160.2.a.bf		2	8.b	even	2	1
4225.2.a.r		2	65.d	even	2	1
5200.2.a.bu		2	20.d	odd	2	1
7605.2.a.x		2	39.d	odd	2	1
7865.2.a.j		2	11.b	odd	2	1
9360.2.a.cd		2	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 2T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\).

Hecke characteristic polynomials

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