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 disc. -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 \(2\) and degree \(1\))

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: \( 1 \)
Projective image \(D_{4}\)
Projective field Galois closure of 4.2.24048.2
Artin image size \(16\)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
2004.g Odd 1 CM by \(\Q(\sqrt{-501}) \) yes
12.b Even 1 yes
167.b Odd 1 yes

Hecke kernels

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

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