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Properties

Label 507.2.j
Level $507$
Weight $2$
Character orbit 507.j
Rep. character $\chi_{507}(316,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $54$
Newform subspaces $9$
Sturm bound $121$
Trace bound $10$

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Defining parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.j (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 9 \)
Sturm bound: \(121\)
Trace bound: \(10\)
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 54 96
Cusp forms 94 54 40
Eisenstein series 56 0 56

Trace form

\( 54 q - q^{3} + 30 q^{4} + 3 q^{7} - 27 q^{9} + O(q^{10}) \) \( 54 q - q^{3} + 30 q^{4} + 3 q^{7} - 27 q^{9} - 4 q^{10} + 6 q^{11} + 12 q^{12} - 8 q^{14} - 6 q^{15} - 20 q^{16} - 4 q^{17} - 6 q^{19} - 12 q^{20} + 8 q^{22} - 10 q^{23} - 34 q^{25} + 2 q^{27} - 6 q^{28} + 14 q^{29} - 4 q^{30} + 6 q^{33} + 10 q^{35} + 30 q^{36} + 8 q^{38} - 24 q^{40} + 12 q^{41} - 13 q^{43} - 6 q^{45} + 4 q^{48} + 24 q^{49} + 8 q^{51} - 56 q^{53} - 12 q^{56} + 6 q^{59} + 17 q^{61} - 16 q^{62} - 3 q^{63} - 32 q^{64} - 8 q^{66} - 15 q^{67} + 24 q^{68} + 2 q^{69} + 18 q^{71} + 12 q^{74} - q^{75} + 12 q^{76} + 20 q^{77} - 2 q^{79} + 24 q^{80} - 27 q^{81} - 20 q^{82} - 6 q^{84} - 2 q^{87} - 24 q^{88} - 12 q^{89} + 8 q^{90} - 32 q^{92} - 3 q^{93} + 12 q^{94} - 32 q^{95} + 9 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.j.a 507.j 13.e $2$ $4.048$ \(\Q(\sqrt{-3}) \) None 39.2.b.a \(-3\) \(1\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.j.b 507.j 13.e $2$ $4.048$ \(\Q(\sqrt{-3}) \) None 39.2.j.a \(0\) \(1\) \(0\) \(3\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(2-4\zeta_{6})q^{5}+\cdots\)
507.2.j.c 507.j 13.e $2$ $4.048$ \(\Q(\sqrt{-3}) \) None 39.2.b.a \(3\) \(1\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
507.2.j.d 507.j 13.e $4$ $4.048$ \(\Q(\zeta_{12})\) None 39.2.e.a \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}-\zeta_{12}^{2}q^{4}+\cdots\)
507.2.j.e 507.j 13.e $4$ $4.048$ \(\Q(\zeta_{12})\) None 39.2.a.a \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}-\zeta_{12}^{2}q^{4}+\cdots\)
507.2.j.f 507.j 13.e $8$ $4.048$ \(\Q(\zeta_{24})\) None 39.2.a.b \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{24}^{3}q^{2}-\zeta_{24}q^{3}+(1-\zeta_{24}-\zeta_{24}^{4}+\cdots)q^{4}+\cdots\)
507.2.j.g 507.j 13.e $8$ $4.048$ 8.0.1731891456.1 None 39.2.e.b \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(2+3\beta _{2}-\beta _{4}+\beta _{5}+\cdots)q^{4}+\cdots\)
507.2.j.h 507.j 13.e $12$ $4.048$ 12.0.\(\cdots\).1 None 507.2.a.j \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+(-1+\beta _{7})q^{3}+(1-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
507.2.j.i 507.j 13.e $12$ $4.048$ 12.0.\(\cdots\).1 None 507.2.a.i \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+\beta _{8}+2\beta _{11})q^{2}+\beta _{7}q^{3}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(507, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)