Properties

Label 507.2.f
Level $507$
Weight $2$
Character orbit 507.f
Rep. character $\chi_{507}(239,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $84$
Newform subspaces $7$
Sturm bound $121$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 39 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 7 \)
Sturm bound: \(121\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(507, [\chi])\).

Total New Old
Modular forms 148 124 24
Cusp forms 92 84 8
Eisenstein series 56 40 16

Trace form

\( 84q + 4q^{3} + 4q^{6} - 4q^{7} + 4q^{9} + O(q^{10}) \) \( 84q + 4q^{3} + 4q^{6} - 4q^{7} + 4q^{9} - 8q^{15} - 12q^{16} - 8q^{18} - 4q^{19} + 4q^{21} + 24q^{22} - 12q^{24} - 56q^{27} - 4q^{28} + 20q^{31} + 16q^{33} - 4q^{37} - 16q^{40} + 16q^{45} - 24q^{46} - 88q^{48} + 4q^{54} - 40q^{55} + 4q^{57} - 8q^{58} + 8q^{60} - 24q^{61} + 4q^{63} - 36q^{66} + 20q^{67} - 8q^{70} + 24q^{72} - 4q^{73} + 4q^{76} - 8q^{79} + 28q^{81} + 4q^{84} + 40q^{87} - 20q^{93} + 64q^{94} - 20q^{96} - 28q^{97} - 32q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(507, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
507.2.f.a \(4\) \(4.048\) \(\Q(\zeta_{8})\) None \(0\) \(-4\) \(0\) \(-4\) \(q+\zeta_{8}q^{2}+(-1-\zeta_{8}-\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{4}+\cdots\)
507.2.f.b \(4\) \(4.048\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-2\) \(q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+2\zeta_{12}^{2}q^{4}+\cdots\)
507.2.f.c \(4\) \(4.048\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(2\) \(q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+2\zeta_{12}^{2}q^{4}+\cdots\)
507.2.f.d \(8\) \(4.048\) 8.0.56070144.2 \(\Q(\sqrt{-39}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(-2\beta _{2}-\beta _{5})q^{4}+\cdots\)
507.2.f.e \(8\) \(4.048\) 8.0.56070144.2 None \(0\) \(4\) \(0\) \(-8\) \(q+(\beta _{1}+\beta _{2}+\beta _{4})q^{2}+(\beta _{4}+\beta _{5})q^{3}+\cdots\)
507.2.f.f \(8\) \(4.048\) 8.0.56070144.2 None \(0\) \(4\) \(0\) \(8\) \(q+(\beta _{1}+\beta _{2}+\beta _{4})q^{2}+(1-\beta _{2}-\beta _{4}+\cdots)q^{3}+\cdots\)
507.2.f.g \(48\) \(4.048\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{2}^{\mathrm{old}}(507, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(507, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)