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

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

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

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 13
507.2.a.a 507.a 1.a $1$ $4.048$ \(\Q\) None \(-1\) \(-1\) \(-2\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}+4q^{7}+\cdots\)
507.2.a.b 507.a 1.a $1$ $4.048$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
507.2.a.c 507.a 1.a $1$ $4.048$ \(\Q\) None \(1\) \(-1\) \(-1\) \(2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
507.2.a.d 507.a 1.a $2$ $4.048$ \(\Q(\sqrt{17}) \) None \(-1\) \(2\) \(3\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+q^{3}+(2+\beta )q^{4}+(2-\beta )q^{5}+\cdots\)
507.2.a.e 507.a 1.a $2$ $4.048$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}+2\beta q^{5}+\beta q^{7}+q^{9}+\cdots\)
507.2.a.f 507.a 1.a $2$ $4.048$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-q^{3}+q^{4}-\beta q^{6}+2\beta q^{7}+\cdots\)
507.2.a.g 507.a 1.a $2$ $4.048$ \(\Q(\sqrt{17}) \) None \(1\) \(2\) \(-3\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+(2+\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
507.2.a.h 507.a 1.a $2$ $4.048$ \(\Q(\sqrt{2}) \) None \(2\) \(2\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}-2\beta q^{5}+\cdots\)
507.2.a.i 507.a 1.a $3$ $4.048$ \(\Q(\zeta_{14})^+\) None \(-3\) \(-3\) \(-6\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1}+\beta _{2})q^{2}-q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
507.2.a.j 507.a 1.a $3$ $4.048$ \(\Q(\zeta_{14})^+\) None \(-1\) \(3\) \(-4\) \(-10\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots\)
507.2.a.k 507.a 1.a $3$ $4.048$ \(\Q(\zeta_{14})^+\) None \(1\) \(3\) \(4\) \(10\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+(2-\beta _{1}+\beta _{2})q^{5}+\cdots\)
507.2.a.l 507.a 1.a $3$ $4.048$ \(\Q(\zeta_{14})^+\) None \(3\) \(-3\) \(6\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1}-\beta _{2})q^{2}-q^{3}+(4-\beta _{1})q^{4}+\cdots\)

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

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