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Properties

Label 504.3.f
Level $504$
Weight $3$
Character orbit 504.f
Rep. character $\chi_{504}(433,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $3$
Sturm bound $288$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 504.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(288\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(504, [\chi])\).

Total New Old
Modular forms 208 20 188
Cusp forms 176 20 156
Eisenstein series 32 0 32

Trace form

\( 20 q - 4 q^{7} + O(q^{10}) \) \( 20 q - 4 q^{7} + 8 q^{11} - 72 q^{23} - 60 q^{25} - 40 q^{29} + 24 q^{35} - 40 q^{37} - 104 q^{43} + 52 q^{49} + 40 q^{53} + 176 q^{65} - 8 q^{67} + 232 q^{71} - 168 q^{77} + 56 q^{79} + 80 q^{85} - 144 q^{91} - 240 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(504, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
504.3.f.a 504.f 7.b $4$ $13.733$ 4.0.2048.2 None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+\beta _{3})q^{5}+(1-\beta _{1}+\beta _{2}+\beta _{3})q^{7}+\cdots\)
504.3.f.b 504.f 7.b $8$ $13.733$ 8.0.\(\cdots\).3 None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{6}q^{5}+(-1-\beta _{3})q^{7}-\beta _{5}q^{11}+\cdots\)
504.3.f.c 504.f 7.b $8$ $13.733$ 8.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{4}q^{5}+(-1+\beta _{3}-\beta _{4})q^{7}+(3+2\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(504, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(504, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)