Properties

Label 40.4.d
Level $40$
Weight $4$
Character orbit 40.d
Rep. character $\chi_{40}(21,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $1$
Sturm bound $24$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 40.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(24\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(40, [\chi])\).

Total New Old
Modular forms 20 12 8
Cusp forms 16 12 4
Eisenstein series 4 0 4

Trace form

\( 12 q + 2 q^{2} + 16 q^{4} - 36 q^{6} + 28 q^{7} - 40 q^{8} - 108 q^{9} + O(q^{10}) \) \( 12 q + 2 q^{2} + 16 q^{4} - 36 q^{6} + 28 q^{7} - 40 q^{8} - 108 q^{9} + 30 q^{10} + 188 q^{12} + 68 q^{14} - 60 q^{15} - 56 q^{16} - 206 q^{18} + 20 q^{20} - 164 q^{22} + 604 q^{23} + 360 q^{24} - 300 q^{25} - 308 q^{26} - 436 q^{28} + 40 q^{30} - 264 q^{31} + 72 q^{32} - 232 q^{33} - 180 q^{34} + 440 q^{36} + 820 q^{38} + 600 q^{39} + 120 q^{40} + 40 q^{41} + 884 q^{42} - 472 q^{44} - 1268 q^{46} - 940 q^{47} + 424 q^{48} + 1308 q^{49} - 50 q^{50} + 1024 q^{52} - 1512 q^{54} + 440 q^{55} - 728 q^{56} - 680 q^{57} - 360 q^{58} - 820 q^{60} + 592 q^{62} - 1300 q^{63} - 2048 q^{64} + 2928 q^{66} - 2344 q^{68} + 1160 q^{70} - 1592 q^{71} - 152 q^{72} + 432 q^{73} - 420 q^{74} + 2256 q^{76} + 3320 q^{78} + 2016 q^{79} + 1600 q^{80} + 2508 q^{81} + 88 q^{82} + 1048 q^{84} - 244 q^{86} - 1968 q^{87} + 4080 q^{88} - 424 q^{89} - 2250 q^{90} - 900 q^{92} + 292 q^{94} - 1520 q^{95} - 5920 q^{96} - 1584 q^{97} - 7266 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(40, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
40.4.d.a 40.d 8.b $12$ $2.360$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(2\) \(0\) \(0\) \(28\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(1-\beta _{4})q^{4}-\beta _{8}q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(40, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(40, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 2}\)