Properties

Label 40.2.f
Level $40$
Weight $2$
Character orbit 40.f
Rep. character $\chi_{40}(29,\cdot)$
Character field $\Q$
Dimension $4$
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.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 40 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(40, [\chi])\).

Total New Old
Modular forms 8 8 0
Cusp forms 4 4 0
Eisenstein series 4 4 0

Trace form

\( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 4 q^{10} + 12 q^{14} - 8 q^{15} - 8 q^{16} + 12 q^{20} + 16 q^{24} - 4 q^{25} - 12 q^{30} + 16 q^{31} - 24 q^{34} + 4 q^{36} - 16 q^{40} - 24 q^{44} - 12 q^{46} + 4 q^{49} + 24 q^{50} + 16 q^{54} + 24 q^{55} + 8 q^{60} + 32 q^{64} + 24 q^{66} - 12 q^{70} - 48 q^{71} - 24 q^{74} + 24 q^{76} - 16 q^{79} - 24 q^{80} - 20 q^{81} - 24 q^{84} + 12 q^{86} + 24 q^{89} - 4 q^{90} + 36 q^{94} - 24 q^{95} - 16 q^{96} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(40, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
40.2.f.a 40.f 40.f $4$ $0.319$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(-1+\beta _{3})q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)