Properties

Label 36.9
Level 36
Weight 9
Dimension 129
Nonzero newspaces 4
Newform subspaces 7
Sturm bound 648
Trace bound 1

Downloads

Learn more

Defining parameters

Level: \( N \) = \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 7 \)
Sturm bound: \(648\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(36))\).

Total New Old
Modular forms 308 139 169
Cusp forms 268 129 139
Eisenstein series 40 10 30

Trace form

\( 129 q - 3 q^{2} + 21 q^{3} - 333 q^{4} + 609 q^{5} + 1359 q^{6} + 1539 q^{7} - 8454 q^{8} + 7203 q^{9} + O(q^{10}) \) \( 129 q - 3 q^{2} + 21 q^{3} - 333 q^{4} + 609 q^{5} + 1359 q^{6} + 1539 q^{7} - 8454 q^{8} + 7203 q^{9} + 13416 q^{10} - 5490 q^{11} - 25668 q^{12} + 78585 q^{13} - 27132 q^{14} + 33975 q^{15} + 127119 q^{16} + 137922 q^{17} - 558168 q^{18} + 383622 q^{19} + 28812 q^{20} + 226113 q^{21} + 421143 q^{22} + 379071 q^{23} + 36633 q^{24} - 438750 q^{25} - 2749068 q^{26} + 726408 q^{27} + 116940 q^{28} - 1029315 q^{29} + 3910260 q^{30} + 554757 q^{31} - 2235513 q^{32} + 3414972 q^{33} + 1662885 q^{34} + 2094867 q^{36} - 4119174 q^{37} + 875835 q^{38} + 695721 q^{39} - 6834996 q^{40} - 1856508 q^{41} - 1253550 q^{42} - 3794808 q^{43} + 11353002 q^{44} + 9234039 q^{45} - 3317736 q^{46} - 2770281 q^{47} - 6410637 q^{48} - 5130960 q^{49} - 31088223 q^{50} + 5772195 q^{51} - 14681034 q^{52} + 6502818 q^{53} + 1445961 q^{54} - 18209142 q^{55} + 67235886 q^{56} + 4830321 q^{57} + 13315308 q^{58} - 43273584 q^{59} - 19327032 q^{60} + 27682665 q^{61} - 17520060 q^{62} + 19287969 q^{63} - 72722238 q^{64} + 28459641 q^{65} - 62514462 q^{66} + 17420298 q^{67} + 178779147 q^{68} - 22528035 q^{69} + 165094674 q^{70} - 94972077 q^{72} + 70913136 q^{73} - 223456152 q^{74} - 17457507 q^{75} - 110277411 q^{76} - 87881421 q^{77} - 58314714 q^{78} + 107620779 q^{79} + 332136432 q^{80} - 7747869 q^{81} + 228622638 q^{82} + 52482087 q^{83} + 48970518 q^{84} - 58141026 q^{85} - 328132311 q^{86} + 149643225 q^{87} - 270199797 q^{88} + 40130466 q^{89} - 119152692 q^{90} - 44160390 q^{91} + 439656330 q^{92} - 99873789 q^{93} + 580388148 q^{94} - 401097996 q^{95} - 151076052 q^{96} - 494446146 q^{97} - 724463724 q^{98} + 80448723 q^{99} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(36))\)

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

Label \(\chi\) Newforms Dimension \(\chi\) degree
36.9.c \(\chi_{36}(17, \cdot)\) 36.9.c.a 2 1
36.9.d \(\chi_{36}(19, \cdot)\) 36.9.d.a 1 1
36.9.d.b 2
36.9.d.c 8
36.9.d.d 8
36.9.f \(\chi_{36}(7, \cdot)\) 36.9.f.a 92 2
36.9.g \(\chi_{36}(5, \cdot)\) 36.9.g.a 16 2

Decomposition of \(S_{9}^{\mathrm{old}}(\Gamma_1(36))\) into lower level spaces

\( S_{9}^{\mathrm{old}}(\Gamma_1(36)) \cong \) \(S_{9}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)