Properties

Label 2004.2.a
Level 2004
Weight 2
Character orbit a
Rep. character \(\chi_{2004}(1,\cdot)\)
Character field \(\Q\)
Dimension 28
Newforms 4
Sturm bound 672
Trace bound 3

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Defining parameters

Level: \( N \) = \( 2004 = 2^{2} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 2004.a (trivial)
Character field: \(\Q\)
Newforms: \( 4 \)
Sturm bound: \(672\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2004))\).

Total New Old
Modular forms 342 28 314
Cusp forms 331 28 303
Eisenstein series 11 0 11

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(167\)FrickeDim.
\(-\)\(+\)\(+\)\(-\)\(9\)
\(-\)\(+\)\(-\)\(+\)\(5\)
\(-\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(9\)
Plus space\(+\)\(10\)
Minus space\(-\)\(18\)

Trace form

\(28q \) \(\mathstrut +\mathstrut 28q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(28q \) \(\mathstrut +\mathstrut 28q^{9} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 16q^{23} \) \(\mathstrut +\mathstrut 40q^{25} \) \(\mathstrut +\mathstrut 4q^{29} \) \(\mathstrut +\mathstrut 8q^{33} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 16q^{37} \) \(\mathstrut -\mathstrut 8q^{39} \) \(\mathstrut +\mathstrut 4q^{41} \) \(\mathstrut +\mathstrut 4q^{47} \) \(\mathstrut +\mathstrut 20q^{49} \) \(\mathstrut +\mathstrut 8q^{53} \) \(\mathstrut -\mathstrut 8q^{55} \) \(\mathstrut +\mathstrut 8q^{57} \) \(\mathstrut -\mathstrut 12q^{59} \) \(\mathstrut +\mathstrut 16q^{61} \) \(\mathstrut +\mathstrut 36q^{65} \) \(\mathstrut +\mathstrut 24q^{67} \) \(\mathstrut -\mathstrut 4q^{69} \) \(\mathstrut -\mathstrut 12q^{71} \) \(\mathstrut -\mathstrut 24q^{73} \) \(\mathstrut +\mathstrut 8q^{75} \) \(\mathstrut +\mathstrut 16q^{77} \) \(\mathstrut -\mathstrut 12q^{79} \) \(\mathstrut +\mathstrut 28q^{81} \) \(\mathstrut -\mathstrut 8q^{83} \) \(\mathstrut +\mathstrut 36q^{85} \) \(\mathstrut +\mathstrut 20q^{89} \) \(\mathstrut -\mathstrut 16q^{91} \) \(\mathstrut +\mathstrut 52q^{97} \) \(\mathstrut -\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2004))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 167
2004.2.a.a \(5\) \(16.002\) 5.5.149169.1 None \(0\) \(-5\) \(-3\) \(-2\) \(-\) \(+\) \(-\) \(q-q^{3}+(-\beta _{1}+\beta _{3}-\beta _{4})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
2004.2.a.b \(5\) \(16.002\) 5.5.161121.1 None \(0\) \(5\) \(-7\) \(-2\) \(-\) \(-\) \(+\) \(q+q^{3}+(-1-\beta _{3}-\beta _{4})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
2004.2.a.c \(9\) \(16.002\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(-9\) \(1\) \(2\) \(-\) \(+\) \(+\) \(q-q^{3}+\beta _{1}q^{5}+(-\beta _{3}-\beta _{6})q^{7}+q^{9}+\cdots\)
2004.2.a.d \(9\) \(16.002\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(0\) \(9\) \(9\) \(2\) \(-\) \(-\) \(-\) \(q+q^{3}+(1-\beta _{1})q^{5}-\beta _{6}q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2004))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2004)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(167))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(334))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(501))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(668))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1002))\)\(^{\oplus 2}\)