Newform orbit 2600.1.o.a

• Label: 2600.1.o.a
• Level: $2600$
• Weight: $1$
• Character orbit: 2600.o
• Self dual: yes
• Analytic conductor: $1.298$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{2}$
• CM/RM discs: -40, -104, 65
• 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:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 520)
Projective image:	\(D_{2}\)
Projective field:	Galois closure of \(\Q(\sqrt{-10}, \sqrt{-26})\)
Artin image:	$D_4$
Artin field:	Galois closure of 4.0.104000.1

$q$-expansion

\(f(q)\)

\(=\)

\( q - q^{2} + q^{4} - 2 q^{7} - q^{8} - q^{9}+O(q^{10}) \)

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

0

−1.00000

0

1.00000

0

0

−2.00000

−1.00000

−1.00000

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
40.e	odd	2	1	CM by \(\Q(\sqrt{-10}) \)
65.d	even	2	1	RM by \(\Q(\sqrt{65}) \)
104.h	odd	2	1	CM by \(\Q(\sqrt{-26}) \)

Twists

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

    

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

Hecke kernels

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

\( T_{3} \)

\( T_{7} + 2 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T + 1 \)
$3$	\( T \)
$5$	\( T \)
$7$	\( T + 2 \)
$11$	\( T \)