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Properties

Label 126.2.f
Level $126$
Weight $2$
Character orbit 126.f
Rep. character $\chi_{126}(43,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $12$
Newform subspaces $4$
Sturm bound $48$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.f (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(48\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(126, [\chi])\).

Total New Old
Modular forms 56 12 44
Cusp forms 40 12 28
Eisenstein series 16 0 16

Trace form

\( 12 q + 2 q^{2} + 6 q^{3} - 6 q^{4} - 4 q^{5} + 2 q^{6} - 4 q^{8} - 2 q^{9} + O(q^{10}) \) \( 12 q + 2 q^{2} + 6 q^{3} - 6 q^{4} - 4 q^{5} + 2 q^{6} - 4 q^{8} - 2 q^{9} - 2 q^{11} + 4 q^{14} - 8 q^{15} - 6 q^{16} + 4 q^{17} + 12 q^{19} - 4 q^{20} + 4 q^{21} - 6 q^{22} + 4 q^{23} + 2 q^{24} - 18 q^{25} - 36 q^{27} + 8 q^{29} + 16 q^{30} - 12 q^{31} + 2 q^{32} + 14 q^{33} - 6 q^{34} - 8 q^{35} - 2 q^{36} + 24 q^{37} - 14 q^{38} + 28 q^{39} - 18 q^{41} - 6 q^{43} + 4 q^{44} - 40 q^{45} - 6 q^{48} - 6 q^{49} + 10 q^{50} - 22 q^{51} + 2 q^{54} + 24 q^{55} + 4 q^{56} - 6 q^{57} + 14 q^{59} + 16 q^{60} + 24 q^{62} + 12 q^{63} + 12 q^{64} + 36 q^{65} + 32 q^{66} - 18 q^{67} - 2 q^{68} + 32 q^{69} + 16 q^{71} - 6 q^{72} + 12 q^{73} + 4 q^{74} + 22 q^{75} - 6 q^{76} + 8 q^{77} + 20 q^{78} - 12 q^{79} + 8 q^{80} - 26 q^{81} - 36 q^{82} - 44 q^{83} - 8 q^{84} + 12 q^{85} - 2 q^{86} + 16 q^{87} - 6 q^{88} - 24 q^{89} - 52 q^{90} + 24 q^{91} + 4 q^{92} - 8 q^{93} - 44 q^{95} - 4 q^{96} + 6 q^{97} - 4 q^{98} + 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(126, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
126.2.f.a 126.f 9.c $2$ $1.006$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(3\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(1-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
126.2.f.b 126.f 9.c $2$ $1.006$ \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(-2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(2-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots\)
126.2.f.c 126.f 9.c $4$ $1.006$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(-2\) \(4\) \(-2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{2})q^{2}+(1-\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots\)
126.2.f.d 126.f 9.c $4$ $1.006$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(2\) \(-1\) \(-3\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(-1+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(126, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(126, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)