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Properties

Label 504.2.a
Level $504$
Weight $2$
Character orbit 504.a
Rep. character $\chi_{504}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $8$
Sturm bound $192$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(192\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(504))\).

Total New Old
Modular forms 112 8 104
Cusp forms 81 8 73
Eisenstein series 31 0 31

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

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(3\)
Minus space\(-\)\(5\)

Trace form

\( 8 q - 2 q^{5} + O(q^{10}) \) \( 8 q - 2 q^{5} + 4 q^{11} - 2 q^{13} + 4 q^{17} + 6 q^{19} + 4 q^{25} - 4 q^{29} - 4 q^{31} + 6 q^{35} - 4 q^{37} + 12 q^{41} - 4 q^{43} + 12 q^{47} + 8 q^{49} + 8 q^{53} + 24 q^{55} - 22 q^{59} - 6 q^{61} - 12 q^{65} + 24 q^{67} + 16 q^{71} - 24 q^{73} - 4 q^{77} - 24 q^{79} - 30 q^{83} + 20 q^{85} - 8 q^{89} - 10 q^{91} - 24 q^{95} - 28 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(504))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
504.2.a.a $1$ $4.024$ \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) $+$ $+$ $+$ \(q-2q^{5}-q^{7}-2q^{11}+2q^{13}-6q^{17}+\cdots\)
504.2.a.b $1$ $4.024$ \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) $-$ $-$ $+$ \(q-2q^{5}-q^{7}-2q^{13}-6q^{17}-4q^{19}+\cdots\)
504.2.a.c $1$ $4.024$ \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) $+$ $-$ $+$ \(q-2q^{5}-q^{7}+4q^{11}+2q^{13}+6q^{17}+\cdots\)
504.2.a.d $1$ $4.024$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $-$ $+$ $-$ \(q-2q^{5}+q^{7}-6q^{11}-6q^{13}+2q^{17}+\cdots\)
504.2.a.e $1$ $4.024$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $-$ $-$ $-$ \(q-2q^{5}+q^{7}+6q^{13}+2q^{17}+4q^{19}+\cdots\)
504.2.a.f $1$ $4.024$ \(\Q\) None \(0\) \(0\) \(2\) \(-1\) $-$ $+$ $+$ \(q+2q^{5}-q^{7}+2q^{11}+2q^{13}+6q^{17}+\cdots\)
504.2.a.g $1$ $4.024$ \(\Q\) None \(0\) \(0\) \(2\) \(1\) $+$ $+$ $-$ \(q+2q^{5}+q^{7}+6q^{11}-6q^{13}-2q^{17}+\cdots\)
504.2.a.h $1$ $4.024$ \(\Q\) None \(0\) \(0\) \(4\) \(1\) $-$ $-$ $-$ \(q+4q^{5}+q^{7}+2q^{17}-2q^{19}-8q^{23}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(504))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(504)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)