Properties

Label 40.2.a
Level $40$
Weight $2$
Character orbit 40.a
Rep. character $\chi_{40}(1,\cdot)$
Character field $\Q$
Dimension $1$
Newform subspaces $1$
Sturm bound $12$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 40.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 \(M_{2}(\Gamma_0(40))\).

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.

\(2\)\(5\)FrickeDim.
\(+\)\(-\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(1\)

Trace form

\( q + q^{5} - 4q^{7} - 3q^{9} + O(q^{10}) \) \( q + q^{5} - 4q^{7} - 3q^{9} + 4q^{11} - 2q^{13} + 2q^{17} + 4q^{19} + 4q^{23} + q^{25} - 2q^{29} - 8q^{31} - 4q^{35} + 6q^{37} - 6q^{41} - 8q^{43} - 3q^{45} + 4q^{47} + 9q^{49} + 6q^{53} + 4q^{55} - 4q^{59} - 2q^{61} + 12q^{63} - 2q^{65} + 8q^{67} - 6q^{73} - 16q^{77} + 9q^{81} - 16q^{83} + 2q^{85} - 6q^{89} + 8q^{91} + 4q^{95} - 14q^{97} - 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(40))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
40.2.a.a \(1\) \(0.319\) \(\Q\) None \(0\) \(0\) \(1\) \(-4\) \(+\) \(-\) \(q+q^{5}-4q^{7}-3q^{9}+4q^{11}-2q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(40))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(40)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T^{2} \)
$5$ \( 1 - T \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 - 4 T + 11 T^{2} \)
$13$ \( 1 + 2 T + 13 T^{2} \)
$17$ \( 1 - 2 T + 17 T^{2} \)
$19$ \( 1 - 4 T + 19 T^{2} \)
$23$ \( 1 - 4 T + 23 T^{2} \)
$29$ \( 1 + 2 T + 29 T^{2} \)
$31$ \( 1 + 8 T + 31 T^{2} \)
$37$ \( 1 - 6 T + 37 T^{2} \)
$41$ \( 1 + 6 T + 41 T^{2} \)
$43$ \( 1 + 8 T + 43 T^{2} \)
$47$ \( 1 - 4 T + 47 T^{2} \)
$53$ \( 1 - 6 T + 53 T^{2} \)
$59$ \( 1 + 4 T + 59 T^{2} \)
$61$ \( 1 + 2 T + 61 T^{2} \)
$67$ \( 1 - 8 T + 67 T^{2} \)
$71$ \( 1 + 71 T^{2} \)
$73$ \( 1 + 6 T + 73 T^{2} \)
$79$ \( 1 + 79 T^{2} \)
$83$ \( 1 + 16 T + 83 T^{2} \)
$89$ \( 1 + 6 T + 89 T^{2} \)
$97$ \( 1 + 14 T + 97 T^{2} \)
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