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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

\( 50 q + 2 q^{2} + q^{3} - 24 q^{4} + 8 q^{5} - q^{7} - 12 q^{8} - 25 q^{9} + 10 q^{10} - 6 q^{11} - 20 q^{12} + 8 q^{14} - 2 q^{15} - 22 q^{16} - 2 q^{17} - 4 q^{18} + 2 q^{20} + 10 q^{21} + 4 q^{22} + 2 q^{23}+ \cdots + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
507.2.e.a 507.e 13.c $2$ $4.048$ \(\Q(\sqrt{-3}) \) None 39.2.a.a \(-1\) \(1\) \(4\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.e.b 507.e 13.c $2$ $4.048$ \(\Q(\sqrt{-3}) \) None 39.2.a.a \(1\) \(1\) \(-4\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.e.c 507.e 13.c $2$ $4.048$ \(\Q(\sqrt{-3}) \) None 39.2.e.a \(1\) \(1\) \(2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.e.d 507.e 13.c $4$ $4.048$ \(\Q(\sqrt{2}, \sqrt{-3})\) None 39.2.a.b \(-2\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{2}+(-1-\beta _{2})q^{3}+\cdots\)
507.2.e.e 507.e 13.c $4$ $4.048$ \(\Q(\zeta_{12})\) None 39.2.b.a \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta_{2} q^{2}+\beta_1 q^{3}+(\beta_1-1)q^{4}+\cdots\)
507.2.e.f 507.e 13.c $4$ $4.048$ \(\Q(\zeta_{12})\) None 39.2.j.a \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta_1 q^{3}+(-2\beta_1+2)q^{4}-2\beta_{3} q^{5}+\cdots\)
507.2.e.g 507.e 13.c $4$ $4.048$ \(\Q(\sqrt{-3}, \sqrt{17})\) None 39.2.e.b \(1\) \(-2\) \(6\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}-\beta _{2}q^{3}+(-2+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
507.2.e.h 507.e 13.c $4$ $4.048$ \(\Q(\sqrt{2}, \sqrt{-3})\) None 39.2.a.b \(2\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{2})q^{2}+(-1-\beta _{2})q^{3}+\cdots\)
507.2.e.i 507.e 13.c $6$ $4.048$ 6.0.64827.1 None 507.2.a.i \(-3\) \(3\) \(12\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\beta _{1}+\beta _{4}+2\beta _{5})q^{2}+(1-\beta _{5})q^{3}+\cdots\)
507.2.e.j 507.e 13.c $6$ $4.048$ 6.0.64827.1 None 507.2.a.j \(-1\) \(-3\) \(8\) \(-10\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(-1+\beta _{5})q^{3}+(\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
507.2.e.k 507.e 13.c $6$ $4.048$ 6.0.64827.1 None 507.2.a.j \(1\) \(-3\) \(-8\) \(10\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(-1+\beta _{5})q^{3}+(\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
507.2.e.l 507.e 13.c $6$ $4.048$ 6.0.64827.1 None 507.2.a.i \(3\) \(3\) \(-12\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\beta _{1}-\beta _{4}-2\beta _{5})q^{2}+(1-\beta _{5})q^{3}+\cdots\)

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

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