Properties

Label 2004.1.g.c
Level 2004
Weight 1
Character orbit 2004.g
Self dual yes
Analytic conductor 1.000
Analytic rank 0
Dimension 2
Projective image \(D_{4}\)
CM discriminant -2004
Inner twists 4

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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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image \(D_{4}\)
Projective field Galois closure of 4.0.334668.2
Artin image $D_8$
Artin field Galois closure of 8.2.96577152768.2

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} -\beta q^{5} - q^{6} - q^{8} + q^{9} +O(q^{10})\) \( q - q^{2} + q^{3} + q^{4} -\beta q^{5} - q^{6} - q^{8} + q^{9} + \beta q^{10} + q^{12} -\beta q^{15} + q^{16} + \beta q^{17} - q^{18} -\beta q^{20} - q^{24} + q^{25} + q^{27} + \beta q^{30} - q^{32} -\beta q^{34} + q^{36} + \beta q^{40} + \beta q^{41} -\beta q^{43} -\beta q^{45} + q^{48} + q^{49} - q^{50} + \beta q^{51} + \beta q^{53} - q^{54} -\beta q^{60} + q^{64} + \beta q^{67} + \beta q^{68} - q^{72} + q^{75} + \beta q^{79} -\beta q^{80} + q^{81} -\beta q^{82} -2 q^{85} + \beta q^{86} + \beta q^{90} - q^{96} -2 q^{97} - q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} - 2q^{8} + 2q^{9} + 2q^{12} + 2q^{16} - 2q^{18} - 2q^{24} + 2q^{25} + 2q^{27} - 2q^{32} + 2q^{36} + 2q^{48} + 2q^{49} - 2q^{50} - 2q^{54} + 2q^{64} - 2q^{72} + 2q^{75} + 2q^{81} - 4q^{85} - 2q^{96} - 4q^{97} - 2q^{98} + O(q^{100}) \)

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
1.41421
−1.41421
−1.00000 1.00000 1.00000 −1.41421 −1.00000 0 −1.00000 1.00000 1.41421
2003.2 −1.00000 1.00000 1.00000 1.41421 −1.00000 0 −1.00000 1.00000 −1.41421
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
2004.g odd 2 1 CM by \(\Q(\sqrt{-501}) \)
12.b even 2 1 inner
167.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2004.1.g.c 2
3.b odd 2 1 2004.1.g.d yes 2
4.b odd 2 1 2004.1.g.d yes 2
12.b even 2 1 inner 2004.1.g.c 2
167.b odd 2 1 inner 2004.1.g.c 2
501.c even 2 1 2004.1.g.d yes 2
668.b even 2 1 2004.1.g.d yes 2
2004.g odd 2 1 CM 2004.1.g.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.1.g.c 2 1.a even 1 1 trivial
2004.1.g.c 2 12.b even 2 1 inner
2004.1.g.c 2 167.b odd 2 1 inner
2004.1.g.c 2 2004.g odd 2 1 CM
2004.1.g.d yes 2 3.b odd 2 1
2004.1.g.d yes 2 4.b odd 2 1
2004.1.g.d yes 2 501.c even 2 1
2004.1.g.d yes 2 668.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2004, [\chi])\):

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