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Properties

Label 40.6
Level 40
Weight 6
Dimension 117
Nonzero newspaces 5
Newform subspaces 8
Sturm bound 576
Trace bound 1

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Defining parameters

Level: \( N \) = \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 5 \)
Newform subspaces: \( 8 \)
Sturm bound: \(576\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(40))\).

Total New Old
Modular forms 264 129 135
Cusp forms 216 117 99
Eisenstein series 48 12 36

Trace form

\( 117 q - 44 q^{3} - 44 q^{4} + 33 q^{5} + 224 q^{6} - 72 q^{7} + 492 q^{8} - 399 q^{9} + O(q^{10}) \) \( 117 q - 44 q^{3} - 44 q^{4} + 33 q^{5} + 224 q^{6} - 72 q^{7} + 492 q^{8} - 399 q^{9} - 636 q^{10} - 276 q^{11} - 2184 q^{12} + 222 q^{13} + 2016 q^{14} - 580 q^{15} + 5096 q^{16} + 1170 q^{17} - 1924 q^{18} - 1756 q^{19} + 2192 q^{20} + 1400 q^{21} + 10688 q^{22} - 3600 q^{23} - 1192 q^{24} - 13075 q^{25} - 23144 q^{26} - 4952 q^{27} - 7912 q^{28} + 9318 q^{29} + 11888 q^{30} + 18144 q^{31} + 14760 q^{32} - 11544 q^{33} + 34148 q^{34} + 8552 q^{35} + 7360 q^{36} + 454 q^{37} - 69104 q^{38} - 37800 q^{39} - 75276 q^{40} + 43482 q^{41} + 24944 q^{42} + 31676 q^{43} + 83008 q^{44} - 13403 q^{45} + 72272 q^{46} + 15256 q^{47} + 4920 q^{48} - 63987 q^{49} - 49636 q^{50} - 53608 q^{51} - 106860 q^{52} - 17306 q^{53} - 104152 q^{54} - 71868 q^{55} + 13936 q^{56} + 8880 q^{57} + 145596 q^{58} + 159068 q^{59} + 198568 q^{60} + 102654 q^{61} + 66064 q^{62} + 273288 q^{63} - 58208 q^{64} + 31366 q^{65} - 233936 q^{66} - 41596 q^{67} - 146588 q^{68} - 226072 q^{69} + 33928 q^{70} - 356744 q^{71} + 237092 q^{72} - 221774 q^{73} + 113052 q^{74} + 70044 q^{75} - 126224 q^{76} + 228896 q^{77} - 65576 q^{78} + 395664 q^{79} - 198424 q^{80} + 601413 q^{81} - 199620 q^{82} - 113428 q^{83} - 462816 q^{84} - 279310 q^{85} - 238928 q^{86} - 666752 q^{87} + 187072 q^{88} - 568670 q^{89} + 20916 q^{90} - 135504 q^{91} + 333936 q^{92} - 4400 q^{93} + 463560 q^{94} + 638508 q^{95} + 517168 q^{96} + 649786 q^{97} + 282776 q^{98} + 709860 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(40))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
40.6.a \(\chi_{40}(1, \cdot)\) 40.6.a.a 1 1
40.6.a.b 1
40.6.a.c 1
40.6.a.d 2
40.6.c \(\chi_{40}(9, \cdot)\) 40.6.c.a 8 1
40.6.d \(\chi_{40}(21, \cdot)\) 40.6.d.a 20 1
40.6.f \(\chi_{40}(29, \cdot)\) 40.6.f.a 28 1
40.6.j \(\chi_{40}(7, \cdot)\) None 0 2
40.6.k \(\chi_{40}(3, \cdot)\) 40.6.k.a 56 2

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(40))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(40)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 2}\)