Properties

Label 252.2.a
Level 252
Weight 2
Character orbit a
Rep. character \(\chi_{252}(1,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 2
Sturm bound 96
Trace bound 5

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Defining parameters

Level: \( N \) = \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 252.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(252))\).

Total New Old
Modular forms 60 2 58
Cusp forms 37 2 35
Eisenstein series 23 0 23

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\(+\)\(1\)
Minus space\(-\)\(1\)

Trace form

\( 2q - 4q^{5} + O(q^{10}) \) \( 2q - 4q^{5} + 4q^{11} - 4q^{13} + 4q^{17} - 8q^{19} + 4q^{23} + 6q^{25} - 4q^{29} + 8q^{31} + 4q^{35} + 4q^{37} - 12q^{41} - 8q^{43} - 24q^{47} + 2q^{49} + 12q^{53} + 8q^{55} + 8q^{59} - 4q^{61} + 24q^{65} - 20q^{71} - 12q^{73} + 8q^{77} + 8q^{79} + 16q^{83} - 16q^{85} - 12q^{89} + 8q^{91} + 16q^{95} - 12q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(252))\) 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
252.2.a.a \(1\) \(2.012\) \(\Q\) None \(0\) \(0\) \(-4\) \(-1\) \(-\) \(-\) \(+\) \(q-4q^{5}-q^{7}-2q^{11}-6q^{13}+4q^{17}+\cdots\)
252.2.a.b \(1\) \(2.012\) \(\Q\) None \(0\) \(0\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{7}+6q^{11}+2q^{13}-4q^{19}+6q^{23}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(252))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(252)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

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