Properties

Label 2004.1.g.a
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 disc. -167, -2004, 12
Inner twists 4

Related objects

Downloads

Learn more about

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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\sqrt{3}, \sqrt{-167})\)
Artin image size \(8\)
Artin image $D_4$
Artin field Galois closure of 4.2.24048.1

$q$-expansion

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

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
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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
12.b Even 1 RM by \(\Q(\sqrt{3}) \) yes
167.b Odd 1 CM by \(\Q(\sqrt{-167}) \) yes
2004.g Odd 1 CM by \(\Q(\sqrt{-501}) \) yes

Hecke kernels

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

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