Properties

Label 126.2.a
Level 126
Weight 2
Character orbit 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

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

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
126.2.a.a \(1\) \(1.006\) \(\Q\) None \(-1\) \(0\) \(2\) \(-1\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{4}+2q^{5}-q^{7}-q^{8}-2q^{10}+\cdots\)
126.2.a.b \(1\) \(1.006\) \(\Q\) None \(1\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{4}+q^{7}+q^{8}-4q^{13}+q^{14}+\cdots\)

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))
$3$ 1
$5$ (\( 1 - 2 T + 5 T^{2} \))(\( 1 + 5 T^{2} \))
$7$ (\( 1 + T \))(\( 1 - T \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( 1 + 11 T^{2} \))
$13$ (\( 1 - 6 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))
$23$ (\( 1 + 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))
$29$ (\( 1 - 2 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 4 T + 31 T^{2} \))
$37$ (\( 1 + 10 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))
$41$ (\( 1 - 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))
$43$ (\( 1 + 4 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 12 T + 47 T^{2} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))
$59$ (\( 1 + 4 T + 59 T^{2} \))(\( 1 - 6 T + 59 T^{2} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 - 8 T + 61 T^{2} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 + 71 T^{2} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))
$79$ (\( 1 + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))
$89$ (\( 1 - 6 T + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))
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