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Properties

Label 40.10
Level 40
Weight 10
Dimension 215
Nonzero newspaces 5
Newform subspaces 8
Sturm bound 960
Trace bound 1

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Defining parameters

Level: \( N \) = \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) = \( 10 \)
Nonzero newspaces: \( 5 \)
Newform subspaces: \( 8 \)
Sturm bound: \(960\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_1(40))\).

Total New Old
Modular forms 456 227 229
Cusp forms 408 215 193
Eisenstein series 48 12 36

Trace form

\( 215 q + 32 q^{2} - 20 q^{3} + 852 q^{4} + 1763 q^{5} - 9376 q^{6} - 11656 q^{7} + 6764 q^{8} + 108207 q^{9} + O(q^{10}) \) \( 215 q + 32 q^{2} - 20 q^{3} + 852 q^{4} + 1763 q^{5} - 9376 q^{6} - 11656 q^{7} + 6764 q^{8} + 108207 q^{9} - 26396 q^{10} - 138204 q^{11} - 30792 q^{12} + 154430 q^{13} + 736160 q^{14} + 95540 q^{15} - 1445656 q^{16} + 116010 q^{17} + 1486172 q^{18} + 1671292 q^{19} + 471952 q^{20} - 1284376 q^{21} - 87936 q^{22} - 7183776 q^{23} + 9082136 q^{24} - 7794265 q^{25} - 405544 q^{26} + 1097800 q^{27} - 29566248 q^{28} - 13987566 q^{29} + 39775088 q^{30} + 21573312 q^{31} + 2565672 q^{32} + 29592792 q^{33} - 24145244 q^{34} - 64266808 q^{35} - 78550592 q^{36} - 2285818 q^{37} + 83705616 q^{38} + 161447208 q^{39} + 15314804 q^{40} + 6789702 q^{41} - 190172752 q^{42} - 185600476 q^{43} - 120024256 q^{44} + 16901227 q^{45} + 299797584 q^{46} + 126912024 q^{47} + 117911928 q^{48} - 60746605 q^{49} + 159565564 q^{50} - 171238744 q^{51} - 279236972 q^{52} - 49621434 q^{53} + 215422184 q^{54} + 65678252 q^{55} + 580371696 q^{56} - 314071728 q^{57} - 298909700 q^{58} - 164960668 q^{59} + 133213288 q^{60} + 7192786 q^{61} + 549830864 q^{62} + 578690712 q^{63} - 147767136 q^{64} - 439511554 q^{65} - 762232784 q^{66} - 911094948 q^{67} - 314301916 q^{68} + 416177096 q^{69} + 913752968 q^{70} - 1077356664 q^{71} + 1146587876 q^{72} - 374558646 q^{73} - 355036004 q^{74} + 1580010564 q^{75} + 1245867248 q^{76} + 480898640 q^{77} - 3055306280 q^{78} - 1921496464 q^{79} - 2320046424 q^{80} + 825643527 q^{81} + 1913759868 q^{82} + 1234872276 q^{83} + 3736782624 q^{84} - 236095390 q^{85} + 1594616240 q^{86} + 2190734512 q^{87} + 375939776 q^{88} - 1927718218 q^{89} - 1590813804 q^{90} - 2810083472 q^{91} - 291512080 q^{92} - 223844960 q^{93} - 9395872952 q^{94} + 2542232868 q^{95} - 1720360016 q^{96} - 1038686846 q^{97} + 8466029496 q^{98} + 6471816588 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_1(40))\)

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
40.10.a \(\chi_{40}(1, \cdot)\) 40.10.a.a 2 1
40.10.a.b 2
40.10.a.c 2
40.10.a.d 3
40.10.c \(\chi_{40}(9, \cdot)\) 40.10.c.a 14 1
40.10.d \(\chi_{40}(21, \cdot)\) 40.10.d.a 36 1
40.10.f \(\chi_{40}(29, \cdot)\) 40.10.f.a 52 1
40.10.j \(\chi_{40}(7, \cdot)\) None 0 2
40.10.k \(\chi_{40}(3, \cdot)\) 40.10.k.a 104 2

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_1(40))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_1(40)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 1}\)