Properties

Label 48.9
Level 48
Weight 9
Dimension 211
Nonzero newspaces 4
Newform subspaces 10
Sturm bound 1152
Trace bound 1

Downloads

Learn more

Defining parameters

Level: \( N \) = \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 10 \)
Sturm bound: \(1152\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(48))\).

Total New Old
Modular forms 540 221 319
Cusp forms 484 211 273
Eisenstein series 56 10 46

Trace form

\( 211 q - q^{3} - 376 q^{4} + 1008 q^{5} - 3236 q^{6} - 1502 q^{7} + 17460 q^{8} - 15593 q^{9} + O(q^{10}) \) \( 211 q - q^{3} - 376 q^{4} + 1008 q^{5} - 3236 q^{6} - 1502 q^{7} + 17460 q^{8} - 15593 q^{9} + 3712 q^{10} + 39552 q^{11} - 14072 q^{12} + 59978 q^{13} + 116772 q^{14} - 91908 q^{15} - 278384 q^{16} + 115920 q^{17} - 316608 q^{18} - 399746 q^{19} + 690000 q^{20} + 55582 q^{21} - 1062216 q^{22} + 1691136 q^{23} - 789140 q^{24} + 934215 q^{25} - 1682100 q^{26} - 664609 q^{27} + 2069760 q^{28} - 1925712 q^{29} + 1739244 q^{30} - 622502 q^{31} + 1407840 q^{32} + 1658908 q^{33} - 951800 q^{34} - 2415744 q^{35} + 272464 q^{36} - 11102518 q^{37} + 2413320 q^{38} + 480802 q^{39} + 9033736 q^{40} + 13472208 q^{41} + 4285900 q^{42} - 7086530 q^{43} - 3982056 q^{44} - 4175124 q^{45} - 14814304 q^{46} + 17228128 q^{48} - 12668495 q^{49} + 17072676 q^{50} + 4935680 q^{51} + 1857808 q^{52} - 47554896 q^{53} - 43149836 q^{54} - 48096768 q^{55} - 9637152 q^{56} + 8344510 q^{57} + 100272776 q^{58} + 44938752 q^{59} + 109449720 q^{60} + 65705994 q^{61} - 17024868 q^{62} + 41480802 q^{63} - 179335936 q^{64} - 35887392 q^{65} + 22821916 q^{66} - 85944066 q^{67} - 12614400 q^{68} + 29440572 q^{69} + 48883224 q^{70} + 159664128 q^{71} - 119213628 q^{72} - 129476370 q^{73} + 140654028 q^{74} - 89766237 q^{75} + 53424848 q^{76} - 55698432 q^{77} + 172738568 q^{78} - 71371046 q^{79} + 38076648 q^{80} - 248187309 q^{81} - 162494856 q^{82} + 209328000 q^{83} - 120237320 q^{84} + 128322016 q^{85} - 123451824 q^{86} + 8110848 q^{87} + 157615168 q^{88} - 18967536 q^{89} + 228716856 q^{90} - 317038204 q^{91} + 351567216 q^{92} + 266945498 q^{93} + 323300232 q^{94} - 143939320 q^{96} - 120427482 q^{97} - 681890712 q^{98} + 327451900 q^{99} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(48))\)

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
48.9.b \(\chi_{48}(7, \cdot)\) None 0 1
48.9.e \(\chi_{48}(17, \cdot)\) 48.9.e.a 1 1
48.9.e.b 2
48.9.e.c 2
48.9.e.d 2
48.9.e.e 8
48.9.g \(\chi_{48}(31, \cdot)\) 48.9.g.a 2 1
48.9.g.b 2
48.9.g.c 4
48.9.h \(\chi_{48}(41, \cdot)\) None 0 1
48.9.i \(\chi_{48}(5, \cdot)\) 48.9.i.a 124 2
48.9.l \(\chi_{48}(19, \cdot)\) 48.9.l.a 64 2

Decomposition of \(S_{9}^{\mathrm{old}}(\Gamma_1(48))\) into lower level spaces

\( S_{9}^{\mathrm{old}}(\Gamma_1(48)) \cong \) \(S_{9}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 5}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)