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Properties

Label 504.2.r
Level $504$
Weight $2$
Character orbit 504.r
Rep. character $\chi_{504}(169,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $36$
Newform subspaces $6$
Sturm bound $192$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.r (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 6 \)
Sturm bound: \(192\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(504, [\chi])\).

Total New Old
Modular forms 208 36 172
Cusp forms 176 36 140
Eisenstein series 32 0 32

Trace form

\( 36 q - 2 q^{3} + 6 q^{9} + O(q^{10}) \) \( 36 q - 2 q^{3} + 6 q^{9} + 14 q^{11} + 20 q^{15} + 12 q^{17} - 12 q^{19} + 4 q^{23} - 18 q^{25} - 20 q^{27} - 12 q^{29} - 18 q^{33} - 24 q^{35} - 28 q^{39} - 6 q^{41} + 6 q^{43} + 36 q^{45} - 12 q^{47} - 18 q^{49} + 38 q^{51} + 48 q^{53} + 30 q^{57} + 14 q^{59} - 4 q^{63} - 16 q^{65} + 6 q^{67} - 16 q^{69} + 24 q^{71} + 36 q^{73} - 22 q^{75} - 8 q^{77} + 6 q^{81} + 16 q^{83} - 24 q^{85} + 40 q^{87} - 48 q^{89} - 44 q^{93} + 12 q^{95} - 42 q^{97} + 36 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(504, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
504.2.r.a 504.r 9.c $2$ $4.024$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+2\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}-\zeta_{6}q^{7}+\cdots\)
504.2.r.b 504.r 9.c $2$ $4.024$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-2\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{5}-\zeta_{6}q^{7}+\cdots\)
504.2.r.c 504.r 9.c $6$ $4.024$ \(\Q(\zeta_{18})\) None \(0\) \(0\) \(3\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\zeta_{18}^{2}-\zeta_{18}^{5})q^{3}+(1-\zeta_{18}+\zeta_{18}^{2}+\cdots)q^{5}+\cdots\)
504.2.r.d 504.r 9.c $8$ $4.024$ 8.0.508277025.1 None \(0\) \(-4\) \(4\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{4}-\beta _{6})q^{3}+(1+\beta _{1}+2\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)
504.2.r.e 504.r 9.c $8$ $4.024$ 8.0.2091141441.1 None \(0\) \(-1\) \(-3\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{4}q^{3}+(\beta _{2}+\beta _{4}+\beta _{6})q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
504.2.r.f 504.r 9.c $10$ $4.024$ 10.0.\(\cdots\).1 None \(0\) \(0\) \(-3\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{6}q^{3}+(-\beta _{2}-\beta _{7})q^{5}+(1-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(504, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(504, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)