Newform orbit 2600.2.a.p

• Label: 2600.2.a.p
• Level: $2600$
• Weight: $2$
• Character orbit: 2600.a
• Self dual: yes
• Analytic conductor: $20.761$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(20.7611045255\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{17}) \)
Defining polynomial:	\( x^{2} - x - 4 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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

2.56155
−1.56155

0

−2.56155

0

0

0

2.56155

0

3.56155

0

1.2

0

1.56155

0

0

0

−1.56155

0

−0.561553

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\(-1\)
\(5\)	\(1\)
\(13\)	\(1\)

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	2600.2.a.p		2
4.b	odd	2	1		5200.2.a.bw		2
5.b	even	2	1		104.2.a.b	✓	2
5.c	odd	4	2		2600.2.d.k		4
15.d	odd	2	1		936.2.a.j		2
20.d	odd	2	1		208.2.a.e		2
35.c	odd	2	1		5096.2.a.m		2
40.e	odd	2	1		832.2.a.n		2
40.f	even	2	1		832.2.a.k		2
60.h	even	2	1		1872.2.a.u		2
65.d	even	2	1		1352.2.a.g		2
65.g	odd	4	2		1352.2.f.c		4
65.l	even	6	2		1352.2.i.d		4
65.n	even	6	2		1352.2.i.f		4
65.s	odd	12	4		1352.2.o.d		8
80.k	odd	4	2		3328.2.b.w		4
80.q	even	4	2		3328.2.b.y		4
120.i	odd	2	1		7488.2.a.cu		2
120.m	even	2	1		7488.2.a.cv		2
260.g	odd	2	1		2704.2.a.p		2
260.u	even	4	2		2704.2.f.k		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
104.2.a.b	✓	2	5.b	even	2	1
208.2.a.e		2	20.d	odd	2	1
832.2.a.k		2	40.f	even	2	1
832.2.a.n		2	40.e	odd	2	1
936.2.a.j		2	15.d	odd	2	1
1352.2.a.g		2	65.d	even	2	1
1352.2.f.c		4	65.g	odd	4	2
1352.2.i.d		4	65.l	even	6	2
1352.2.i.f		4	65.n	even	6	2
1352.2.o.d		8	65.s	odd	12	4
1872.2.a.u		2	60.h	even	2	1
2600.2.a.p		2	1.a	even	1	1	trivial
2600.2.d.k		4	5.c	odd	4	2
2704.2.a.p		2	260.g	odd	2	1
2704.2.f.k		4	260.u	even	4	2
3328.2.b.w		4	80.k	odd	4	2
3328.2.b.y		4	80.q	even	4	2
5096.2.a.m		2	35.c	odd	2	1
5200.2.a.bw		2	4.b	odd	2	1
7488.2.a.cu		2	120.i	odd	2	1
7488.2.a.cv		2	120.m	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2600))\):

\( T_{3}^{2} + T_{3} - 4 \)

\( T_{7}^{2} - T_{7} - 4 \)

\( T_{11}^{2} + 2T_{11} - 16 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} + T - 4 \)
$5$	\( T^{2} \)
$7$	\( T^{2} - T - 4 \)
$11$	\( T^{2} + 2T - 16 \)