Newform orbit 2600.1.o.f

• Label: 2600.1.o.f
• Level: $2600$
• Weight: $1$
• Character orbit: 2600.o
• Self dual: yes
• Analytic conductor: $1.298$
• Analytic rank: $0$
• Dimension: $2$
• Projective image: $D_{6}$
• CM discriminant: -104
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2600.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.29756903285\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{3}) \)
Defining polynomial:	\( x^{2} - 3 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 520)
Projective image:	\(D_{6}\)
Projective field:	Galois closure of 6.0.10816000.2

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times\).

\(n\)	\(1301\)	\(1601\)	\(1951\)	\(1977\)
\(\chi(n)\)	\(-1\)	\(-1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

51.1

1.73205
−1.73205

1.00000

−1.73205

1.00000

0

−1.73205

−1.00000

1.00000

2.00000

0

51.2

1.00000

1.73205

1.00000

0

1.73205

−1.00000

1.00000

2.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
104.h	odd	2	1	CM by \(\Q(\sqrt{-26}) \)
40.e	odd	2	1	inner
65.d	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2600.1.o.f		2
5.b	even	2	1		2600.1.o.e		2
5.c	odd	4	2		520.1.b.b	✓	4
8.d	odd	2	1		2600.1.o.e		2
13.b	even	2	1		2600.1.o.e		2
20.e	even	4	2		2080.1.b.b		4
40.e	odd	2	1	inner	2600.1.o.f		2
40.i	odd	4	2		2080.1.b.b		4
40.k	even	4	2		520.1.b.b	✓	4
65.d	even	2	1	inner	2600.1.o.f		2
65.h	odd	4	2		520.1.b.b	✓	4
104.h	odd	2	1	CM	2600.1.o.f		2
260.p	even	4	2		2080.1.b.b		4
520.b	odd	2	1		2600.1.o.e		2
520.bc	even	4	2		520.1.b.b	✓	4
520.bg	odd	4	2		2080.1.b.b		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
520.1.b.b	✓	4	5.c	odd	4	2
520.1.b.b	✓	4	40.k	even	4	2
520.1.b.b	✓	4	65.h	odd	4	2
520.1.b.b	✓	4	520.bc	even	4	2
2080.1.b.b		4	20.e	even	4	2
2080.1.b.b		4	40.i	odd	4	2
2080.1.b.b		4	260.p	even	4	2
2080.1.b.b		4	520.bg	odd	4	2
2600.1.o.e		2	5.b	even	2	1
2600.1.o.e		2	8.d	odd	2	1
2600.1.o.e		2	13.b	even	2	1
2600.1.o.e		2	520.b	odd	2	1
2600.1.o.f		2	1.a	even	1	1	trivial
2600.1.o.f		2	40.e	odd	2	1	inner
2600.1.o.f		2	65.d	even	2	1	inner
2600.1.o.f		2	104.h	odd	2	1	CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2600, [\chi])\):

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

\( T_{7} + 1 \)

Hecke characteristic polynomials

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