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Properties

Label 504.3.g
Level $504$
Weight $3$
Character orbit 504.g
Rep. character $\chi_{504}(379,\cdot)$
Character field $\Q$
Dimension $60$
Newform subspaces $4$
Sturm bound $288$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 504.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(288\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 200 60 140
Cusp forms 184 60 124
Eisenstein series 16 0 16

Trace form

\( 60 q + q^{2} + q^{4} - 23 q^{8} + O(q^{10}) \) \( 60 q + q^{2} + q^{4} - 23 q^{8} + 12 q^{10} - 16 q^{11} + 7 q^{14} + 69 q^{16} - 8 q^{17} - 64 q^{19} + 32 q^{20} - 90 q^{22} - 324 q^{25} - 152 q^{26} + 7 q^{28} - 79 q^{32} - 6 q^{34} + 162 q^{38} + 4 q^{40} - 8 q^{41} + 176 q^{43} + 290 q^{44} - 12 q^{46} - 420 q^{49} - 191 q^{50} - 292 q^{52} - 49 q^{56} + 108 q^{58} - 288 q^{59} + 8 q^{62} + 121 q^{64} + 96 q^{65} + 16 q^{67} + 218 q^{68} - 84 q^{70} + 120 q^{73} + 64 q^{74} + 470 q^{76} - 272 q^{80} + 330 q^{82} + 160 q^{83} + 386 q^{86} + 102 q^{88} - 200 q^{89} + 64 q^{92} + 480 q^{94} - 72 q^{97} - 7 q^{98} + O(q^{100}) \)

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.g.a 504.g 8.d $4$ $13.733$ \(\Q(\sqrt{2}, \sqrt{-7})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(-3+\beta _{3})q^{4}+(-2\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
504.3.g.b 504.g 8.d $8$ $13.733$ 8.0.\(\cdots\).3 None \(-1\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
504.3.g.c 504.g 8.d $24$ $13.733$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
504.3.g.d 504.g 8.d $24$ $13.733$ None \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

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

\( S_{3}^{\mathrm{old}}(504, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)