Properties

Label 507.2.a
Level $507$
Weight $2$
Character orbit 507.a
Rep. character $\chi_{507}(1,\cdot)$
Character field $\Q$
Dimension $25$
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.a (trivial)
Character field: \(\Q\)
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 \(M_{2}(\Gamma_0(507))\).

Total New Old
Modular forms 74 25 49
Cusp forms 47 25 22
Eisenstein series 27 0 27

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(7\)
\(-\)\(+\)\(-\)\(9\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(9\)
Minus space\(-\)\(16\)

Trace form

\( 25q + q^{2} - q^{3} + 27q^{4} - 2q^{5} + 3q^{6} + 4q^{7} + 9q^{8} + 25q^{9} + O(q^{10}) \) \( 25q + q^{2} - q^{3} + 27q^{4} - 2q^{5} + 3q^{6} + 4q^{7} + 9q^{8} + 25q^{9} + 2q^{10} - 7q^{12} - 8q^{14} + 2q^{15} + 19q^{16} - 10q^{17} + q^{18} - 14q^{20} - 4q^{21} - 16q^{22} + 4q^{23} + 3q^{24} + 19q^{25} - q^{27} + 12q^{28} - 2q^{29} + 6q^{30} + 4q^{31} - 11q^{32} + 8q^{33} - 14q^{34} + 20q^{35} + 27q^{36} + 6q^{37} + 4q^{38} + 2q^{40} - 22q^{41} - 12q^{42} - 8q^{43} + 8q^{44} - 2q^{45} - 8q^{46} + 12q^{47} - 15q^{48} + 9q^{49} + 7q^{50} - 6q^{51} - 18q^{53} + 3q^{54} - 20q^{55} - 32q^{56} + 14q^{58} - 16q^{59} - 18q^{60} - 18q^{61} - 36q^{62} + 4q^{63} + 15q^{64} - 4q^{66} + 6q^{68} + 4q^{69} - 8q^{70} - 4q^{71} + 9q^{72} - 14q^{73} + 2q^{74} - 15q^{75} - 16q^{76} - 20q^{79} + 2q^{80} + 25q^{81} - 2q^{82} + 20q^{84} + 28q^{85} + 36q^{86} - 18q^{87} - 24q^{88} - 22q^{89} + 2q^{90} - 20q^{92} + 12q^{93} - 8q^{94} - 36q^{95} - q^{96} - 6q^{97} - 7q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(507))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 13
507.2.a.a \(1\) \(4.048\) \(\Q\) None \(-1\) \(-1\) \(-2\) \(4\) \(+\) \(+\) \(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}+4q^{7}+\cdots\)
507.2.a.b \(1\) \(4.048\) \(\Q\) None \(-1\) \(-1\) \(1\) \(-2\) \(+\) \(+\) \(q-q^{2}-q^{3}-q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
507.2.a.c \(1\) \(4.048\) \(\Q\) None \(1\) \(-1\) \(-1\) \(2\) \(+\) \(+\) \(q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
507.2.a.d \(2\) \(4.048\) \(\Q(\sqrt{17}) \) None \(-1\) \(2\) \(3\) \(3\) \(-\) \(+\) \(q-\beta q^{2}+q^{3}+(2+\beta )q^{4}+(2-\beta )q^{5}+\cdots\)
507.2.a.e \(2\) \(4.048\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(q-q^{3}-2q^{4}+2\beta q^{5}+\beta q^{7}+q^{9}+\cdots\)
507.2.a.f \(2\) \(4.048\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(-\) \(q+\beta q^{2}-q^{3}+q^{4}-\beta q^{6}+2\beta q^{7}+\cdots\)
507.2.a.g \(2\) \(4.048\) \(\Q(\sqrt{17}) \) None \(1\) \(2\) \(-3\) \(-3\) \(-\) \(+\) \(q+\beta q^{2}+q^{3}+(2+\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
507.2.a.h \(2\) \(4.048\) \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(0\) \(-\) \(+\) \(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}-2\beta q^{5}+\cdots\)
507.2.a.i \(3\) \(4.048\) \(\Q(\zeta_{14})^+\) None \(-3\) \(-3\) \(-6\) \(-2\) \(+\) \(+\) \(q+(-1+\beta _{1}+\beta _{2})q^{2}-q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
507.2.a.j \(3\) \(4.048\) \(\Q(\zeta_{14})^+\) None \(-1\) \(3\) \(-4\) \(-10\) \(-\) \(-\) \(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots\)
507.2.a.k \(3\) \(4.048\) \(\Q(\zeta_{14})^+\) None \(1\) \(3\) \(4\) \(10\) \(-\) \(+\) \(q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(2-\beta _{1}+\beta _{2})q^{5}+\cdots\)
507.2.a.l \(3\) \(4.048\) \(\Q(\zeta_{14})^+\) None \(3\) \(-3\) \(6\) \(2\) \(+\) \(-\) \(q+(1-\beta _{1}-\beta _{2})q^{2}-q^{3}+(4-\beta _{1})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(507))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(507)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)