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

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

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
507.2.f.a 507.f 39.f $4$ $4.048$ \(\Q(\zeta_{8})\) None \(0\) \(-4\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(-1-\zeta_{8}-\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{4}+\cdots\)
507.2.f.b 507.f 39.f $4$ $4.048$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-2\) $\mathrm{U}(1)[D_{4}]$ \(q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+2\zeta_{12}^{2}q^{4}+\cdots\)
507.2.f.c 507.f 39.f $4$ $4.048$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(2\) $\mathrm{U}(1)[D_{4}]$ \(q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+2\zeta_{12}^{2}q^{4}+\cdots\)
507.2.f.d 507.f 39.f $8$ $4.048$ 8.0.56070144.2 \(\Q(\sqrt{-39}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(-2\beta _{2}-\beta _{5})q^{4}+\cdots\)
507.2.f.e 507.f 39.f $8$ $4.048$ 8.0.56070144.2 None \(0\) \(4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\beta _{1}+\beta _{2}+\beta _{4})q^{2}+(\beta _{4}+\beta _{5})q^{3}+\cdots\)
507.2.f.f 507.f 39.f $8$ $4.048$ 8.0.56070144.2 None \(0\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\beta _{1}+\beta _{2}+\beta _{4})q^{2}+(1-\beta _{2}-\beta _{4}+\cdots)q^{3}+\cdots\)
507.2.f.g 507.f 39.f $48$ $4.048$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$

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

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