Properties

Label 126.2.a
Level $126$
Weight $2$
Character orbit 126.a
Rep. character $\chi_{126}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
Sturm bound $48$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 126.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(48\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(126))\).

Total New Old
Modular forms 32 2 30
Cusp forms 17 2 15
Eisenstein series 15 0 15

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

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(2\)

Trace form

\( 2 q + 2 q^{4} + 2 q^{5} + O(q^{10}) \) \( 2 q + 2 q^{4} + 2 q^{5} - 2 q^{10} + 4 q^{11} + 2 q^{13} + 2 q^{14} + 2 q^{16} - 8 q^{17} - 2 q^{19} + 2 q^{20} - 4 q^{22} - 8 q^{23} - 6 q^{25} - 10 q^{26} + 8 q^{29} - 4 q^{31} - 4 q^{34} - 2 q^{35} - 8 q^{37} + 6 q^{38} - 2 q^{40} + 4 q^{43} + 4 q^{44} + 8 q^{46} + 12 q^{47} + 2 q^{49} - 4 q^{50} + 2 q^{52} - 12 q^{53} + 8 q^{55} + 2 q^{56} + 4 q^{58} + 2 q^{59} + 14 q^{61} - 4 q^{62} + 2 q^{64} + 12 q^{65} - 8 q^{68} + 2 q^{70} - 8 q^{71} + 12 q^{73} + 12 q^{74} - 2 q^{76} - 4 q^{77} + 8 q^{79} + 2 q^{80} - 12 q^{82} + 10 q^{83} - 4 q^{85} + 12 q^{86} - 4 q^{88} + 12 q^{89} - 10 q^{91} - 8 q^{92} + 12 q^{94} - 8 q^{95} - 24 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(126))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
126.2.a.a 126.a 1.a $1$ $1.006$ \(\Q\) None \(-1\) \(0\) \(2\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{5}-q^{7}-q^{8}-2q^{10}+\cdots\)
126.2.a.b 126.a 1.a $1$ $1.006$ \(\Q\) None \(1\) \(0\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+q^{7}+q^{8}-4q^{13}+q^{14}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(126))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(126)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)