Properties

Label 1999.1.b.a
Level $1999$
Weight $1$
Character orbit 1999.b
Self dual yes
Analytic conductor $0.998$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -1999
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1999 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1999.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.997630960253\)
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 3.1.1999.1
Artin image: $S_3$
Artin field: Galois closure of 3.1.1999.1

$q$-expansion

\(f(q)\) \(=\) \( q + 2q^{2} + 3q^{4} - q^{5} + 4q^{8} + q^{9} + O(q^{10}) \) \( q + 2q^{2} + 3q^{4} - q^{5} + 4q^{8} + q^{9} - 2q^{10} - q^{11} - q^{13} + 5q^{16} + 2q^{18} - 3q^{20} - 2q^{22} - q^{23} - 2q^{26} - q^{31} + 6q^{32} + 3q^{36} - q^{37} - 4q^{40} + 2q^{41} - 3q^{44} - q^{45} - 2q^{46} + q^{49} - 3q^{52} - q^{53} + q^{55} - q^{59} - q^{61} - 2q^{62} + 7q^{64} + q^{65} - q^{71} + 4q^{72} - 2q^{74} + 2q^{79} - 5q^{80} + q^{81} + 4q^{82} - 4q^{88} - 2q^{90} - 3q^{92} + 2q^{98} - q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1999\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1998.1
0
2.00000 0 3.00000 −1.00000 0 0 4.00000 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
1999.b odd 2 1 CM by \(\Q(\sqrt{-1999}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1999.1.b.a 1
1999.b odd 2 1 CM 1999.1.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1999.1.b.a 1 1.a even 1 1 trivial
1999.1.b.a 1 1999.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 2 \) acting on \(S_{1}^{\mathrm{new}}(1999, [\chi])\).

Hecke characteristic polynomials

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