Properties

Label 2004.1.g.b
Level 2004
Weight 1
Character orbit 2004.g
Self dual yes
Analytic conductor 1.000
Analytic rank 0
Dimension 1
Projective image \(D_{2}\)
CM/RM discs -167, -2004, 12
Inner twists 4

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2004.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(1.00012628532\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{3}, \sqrt{-167})\)
Artin image $D_4$
Artin field Galois closure of 4.2.24048.2

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - 2q^{11} + q^{12} + q^{16} + q^{18} - 2q^{22} + q^{24} - q^{25} + q^{27} + q^{32} - 2q^{33} + q^{36} - 2q^{44} - 2q^{47} + q^{48} + q^{49} - q^{50} + q^{54} - 2q^{61} + q^{64} - 2q^{66} + q^{72} - q^{75} + q^{81} - 2q^{88} - 2q^{94} + q^{96} + 2q^{97} + q^{98} - 2q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2004\mathbb{Z}\right)^\times\).

\(n\) \(673\) \(1003\) \(1337\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2003.1
0
1.00000 1.00000 1.00000 0 1.00000 0 1.00000 1.00000 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
12.b even 2 1 RM by \(\Q(\sqrt{3}) \)
167.b odd 2 1 CM by \(\Q(\sqrt{-167}) \)
2004.g odd 2 1 CM by \(\Q(\sqrt{-501}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2004.1.g.b yes 1
3.b odd 2 1 2004.1.g.a 1
4.b odd 2 1 2004.1.g.a 1
12.b even 2 1 RM 2004.1.g.b yes 1
167.b odd 2 1 CM 2004.1.g.b yes 1
501.c even 2 1 2004.1.g.a 1
668.b even 2 1 2004.1.g.a 1
2004.g odd 2 1 CM 2004.1.g.b yes 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.1.g.a 1 3.b odd 2 1
2004.1.g.a 1 4.b odd 2 1
2004.1.g.a 1 501.c even 2 1
2004.1.g.a 1 668.b even 2 1
2004.1.g.b yes 1 1.a even 1 1 trivial
2004.1.g.b yes 1 12.b even 2 1 RM
2004.1.g.b yes 1 167.b odd 2 1 CM
2004.1.g.b yes 1 2004.g 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 \(S_{1}^{\mathrm{new}}(2004, [\chi])\):

\( T_{5} \)
\( T_{7} \)
\( T_{11} + 2 \)
\( T_{179} - 2 \)

Hecke characteristic polynomials

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