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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(40))\) 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 5
40.2.a.a 40.a 1.a $1$ $0.319$ \(\Q\) None \(0\) \(0\) \(1\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-4q^{7}-3q^{9}+4q^{11}-2q^{13}+\cdots\)

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

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