Properties

Label 507.2.e
Level $507$
Weight $2$
Character orbit 507.e
Rep. character $\chi_{507}(22,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $50$
Newform subspaces $12$
Sturm bound $121$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(121\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

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

Total New Old
Modular forms 150 50 100
Cusp forms 94 50 44
Eisenstein series 56 0 56

Trace form

\( 50q + 2q^{2} + q^{3} - 24q^{4} + 8q^{5} - q^{7} - 12q^{8} - 25q^{9} + O(q^{10}) \) \( 50q + 2q^{2} + q^{3} - 24q^{4} + 8q^{5} - q^{7} - 12q^{8} - 25q^{9} + 10q^{10} - 6q^{11} - 20q^{12} + 8q^{14} - 2q^{15} - 22q^{16} - 2q^{17} - 4q^{18} + 2q^{20} + 10q^{21} + 4q^{22} + 2q^{23} + 12q^{24} + 50q^{25} - 2q^{27} - 18q^{28} + 8q^{29} + 12q^{30} - 10q^{31} + 14q^{32} - 2q^{33} - 4q^{34} - 2q^{35} - 24q^{36} + 12q^{37} + 32q^{38} - 20q^{40} + 10q^{41} - 12q^{42} + 23q^{43} + 16q^{44} - 4q^{45} - 4q^{46} - 12q^{47} + 12q^{48} - 24q^{49} - 28q^{50} - 24q^{51} - 36q^{53} + 20q^{55} - 16q^{56} - 24q^{57} - 26q^{58} - 14q^{59} + 33q^{61} + 12q^{62} - q^{63} - 12q^{64} - 8q^{66} + 3q^{67} + 42q^{68} + 14q^{69} + 8q^{70} + 34q^{71} + 6q^{72} + 2q^{73} + 10q^{74} + 15q^{75} + 4q^{76} - 36q^{77} - 46q^{79} + 22q^{80} - 25q^{81} + 38q^{82} + 48q^{83} - 14q^{84} + 14q^{85} + 6q^{87} + 12q^{88} + 4q^{89} - 20q^{90} - 88q^{92} - 3q^{93} - 16q^{94} - 8q^{96} - 15q^{97} + 22q^{98} + 12q^{99} + O(q^{100}) \)

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.e.a \(2\) \(4.048\) \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(4\) \(4\) \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.e.b \(2\) \(4.048\) \(\Q(\sqrt{-3}) \) None \(1\) \(1\) \(-4\) \(-4\) \(q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.e.c \(2\) \(4.048\) \(\Q(\sqrt{-3}) \) None \(1\) \(1\) \(2\) \(2\) \(q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.e.d \(4\) \(4.048\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-2\) \(-2\) \(0\) \(0\) \(q+(-1+\beta _{1}-\beta _{2})q^{2}+(-1-\beta _{2})q^{3}+\cdots\)
507.2.e.e \(4\) \(4.048\) \(\Q(\zeta_{12})\) None \(0\) \(2\) \(0\) \(0\) \(q-\zeta_{12}^{2}q^{2}+\zeta_{12}q^{3}+(-1+\zeta_{12}+\cdots)q^{4}+\cdots\)
507.2.e.f \(4\) \(4.048\) \(\Q(\zeta_{12})\) None \(0\) \(2\) \(0\) \(0\) \(q+\zeta_{12}q^{3}+(2-2\zeta_{12})q^{4}-2\zeta_{12}^{3}q^{5}+\cdots\)
507.2.e.g \(4\) \(4.048\) \(\Q(\sqrt{-3}, \sqrt{17})\) None \(1\) \(-2\) \(6\) \(-3\) \(q+\beta _{1}q^{2}-\beta _{2}q^{3}+(-2+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
507.2.e.h \(4\) \(4.048\) \(\Q(\sqrt{2}, \sqrt{-3})\) None \(2\) \(-2\) \(0\) \(0\) \(q+(1+\beta _{1}+\beta _{2})q^{2}+(-1-\beta _{2})q^{3}+\cdots\)
507.2.e.i \(6\) \(4.048\) 6.0.64827.1 None \(-3\) \(3\) \(12\) \(-2\) \(q+(-2+2\beta _{1}+\beta _{4}+2\beta _{5})q^{2}+(1-\beta _{5})q^{3}+\cdots\)
507.2.e.j \(6\) \(4.048\) 6.0.64827.1 None \(-1\) \(-3\) \(8\) \(-10\) \(q-\beta _{1}q^{2}+(-1+\beta _{5})q^{3}+(\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
507.2.e.k \(6\) \(4.048\) 6.0.64827.1 None \(1\) \(-3\) \(-8\) \(10\) \(q+\beta _{1}q^{2}+(-1+\beta _{5})q^{3}+(\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
507.2.e.l \(6\) \(4.048\) 6.0.64827.1 None \(3\) \(3\) \(-12\) \(2\) \(q+(2-2\beta _{1}-\beta _{4}-2\beta _{5})q^{2}+(1-\beta _{5})q^{3}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(507, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)