Properties

Label 2003.1.b.a
Level 2003
Weight 1
Character orbit 2003.b
Self dual yes
Analytic conductor 1.000
Analytic rank 0
Dimension 1
Projective image \(D_{3}\)
CM discriminant -2003
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 2003 \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 2003.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.999627220304\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{3}\)
Projective field Galois closure of 3.1.2003.1
Artin image $S_3$
Artin field Galois closure of 3.1.2003.1

$q$-expansion

\(f(q)\) \(=\) \( q - q^{3} + q^{4} + O(q^{10}) \) \( q - q^{3} + q^{4} - q^{12} - q^{13} + q^{16} + 2q^{19} + q^{25} + q^{27} + q^{39} - q^{47} - q^{48} + q^{49} - q^{52} + 2q^{53} - 2q^{57} - q^{59} + q^{64} - q^{73} - q^{75} + 2q^{76} - q^{79} - q^{81} + 2q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2003\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2002.1
0
0 −1.00000 1.00000 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
2003.b odd 2 1 CM by \(\Q(\sqrt{-2003}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2003.1.b.a 1
2003.b odd 2 1 CM 2003.1.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2003.1.b.a 1 1.a even 1 1 trivial
2003.1.b.a 1 2003.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2003, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )( 1 + T ) \)
$3$ \( 1 + T + T^{2} \)
$5$ \( ( 1 - T )( 1 + T ) \)
$7$ \( ( 1 - T )( 1 + T ) \)
$11$ \( ( 1 - T )( 1 + T ) \)
$13$ \( 1 + T + T^{2} \)
$17$ \( ( 1 - T )( 1 + T ) \)
$19$ \( ( 1 - T )^{2} \)
$23$ \( ( 1 - T )( 1 + T ) \)
$29$ \( ( 1 - T )( 1 + T ) \)
$31$ \( ( 1 - T )( 1 + T ) \)
$37$ \( ( 1 - T )( 1 + T ) \)
$41$ \( ( 1 - T )( 1 + T ) \)
$43$ \( ( 1 - T )( 1 + T ) \)
$47$ \( 1 + T + T^{2} \)
$53$ \( ( 1 - T )^{2} \)
$59$ \( 1 + T + T^{2} \)
$61$ \( ( 1 - T )( 1 + T ) \)
$67$ \( ( 1 - T )( 1 + T ) \)
$71$ \( ( 1 - T )( 1 + T ) \)
$73$ \( 1 + T + T^{2} \)
$79$ \( 1 + T + T^{2} \)
$83$ \( ( 1 - T )( 1 + T ) \)
$89$ \( ( 1 - T )^{2} \)
$97$ \( ( 1 - T )( 1 + T ) \)
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