Properties

Label 45.2.a
Level 45
Weight 2
Character orbit a
Rep. character \(\chi_{45}(1,\cdot)\)
Character field \(\Q\)
Dimension 1
Newform subspaces 1
Sturm bound 12
Trace bound 0

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Defining parameters

Level: \( N \) = \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 45.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(45))\).

Total New Old
Modular forms 10 1 9
Cusp forms 3 1 2
Eisenstein series 7 0 7

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

\(3\)\(5\)FrickeDim.
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(1\)

Trace form

\( q + q^{2} - q^{4} - q^{5} - 3q^{8} + O(q^{10}) \) \( q + q^{2} - q^{4} - q^{5} - 3q^{8} - q^{10} + 4q^{11} - 2q^{13} - q^{16} - 2q^{17} + 4q^{19} + q^{20} + 4q^{22} + q^{25} - 2q^{26} + 2q^{29} + 5q^{32} - 2q^{34} - 10q^{37} + 4q^{38} + 3q^{40} - 10q^{41} + 4q^{43} - 4q^{44} - 8q^{47} - 7q^{49} + q^{50} + 2q^{52} + 10q^{53} - 4q^{55} + 2q^{58} + 4q^{59} - 2q^{61} + 7q^{64} + 2q^{65} + 12q^{67} + 2q^{68} + 8q^{71} + 10q^{73} - 10q^{74} - 4q^{76} + q^{80} - 10q^{82} - 12q^{83} + 2q^{85} + 4q^{86} - 12q^{88} + 6q^{89} - 8q^{94} - 4q^{95} + 2q^{97} - 7q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(45))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5
45.2.a.a \(1\) \(0.359\) \(\Q\) None \(1\) \(0\) \(-1\) \(0\) \(-\) \(+\) \(q+q^{2}-q^{4}-q^{5}-3q^{8}-q^{10}+4q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(45))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(45)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

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