Properties

Label 48.2
Level 48
Weight 2
Dimension 23
Nonzero newspaces 4
Newforms 4
Sturm bound 256
Trace bound 1

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Defining parameters

Level: \( N \) = \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 4 \)
Newforms: \( 4 \)
Sturm bound: \(256\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(48))\).

Total New Old
Modular forms 92 31 61
Cusp forms 37 23 14
Eisenstein series 55 8 47

Trace form

\(23q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut -\mathstrut 5q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(23q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 8q^{6} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut -\mathstrut 12q^{8} \) \(\mathstrut -\mathstrut 5q^{9} \) \(\mathstrut -\mathstrut 8q^{10} \) \(\mathstrut -\mathstrut 12q^{11} \) \(\mathstrut -\mathstrut 10q^{13} \) \(\mathstrut +\mathstrut 12q^{14} \) \(\mathstrut -\mathstrut 10q^{15} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 8q^{18} \) \(\mathstrut -\mathstrut 16q^{19} \) \(\mathstrut +\mathstrut 16q^{20} \) \(\mathstrut +\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 16q^{22} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 28q^{24} \) \(\mathstrut +\mathstrut 9q^{25} \) \(\mathstrut +\mathstrut 20q^{26} \) \(\mathstrut +\mathstrut 11q^{27} \) \(\mathstrut -\mathstrut 10q^{29} \) \(\mathstrut +\mathstrut 20q^{30} \) \(\mathstrut +\mathstrut 16q^{31} \) \(\mathstrut -\mathstrut 8q^{33} \) \(\mathstrut -\mathstrut 8q^{34} \) \(\mathstrut +\mathstrut 24q^{35} \) \(\mathstrut +\mathstrut 16q^{36} \) \(\mathstrut -\mathstrut 34q^{37} \) \(\mathstrut -\mathstrut 8q^{38} \) \(\mathstrut +\mathstrut 18q^{39} \) \(\mathstrut -\mathstrut 24q^{40} \) \(\mathstrut -\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 44q^{42} \) \(\mathstrut -\mathstrut 40q^{44} \) \(\mathstrut -\mathstrut 14q^{45} \) \(\mathstrut -\mathstrut 48q^{46} \) \(\mathstrut -\mathstrut 64q^{48} \) \(\mathstrut -\mathstrut 45q^{49} \) \(\mathstrut -\mathstrut 36q^{50} \) \(\mathstrut +\mathstrut 34q^{51} \) \(\mathstrut -\mathstrut 32q^{52} \) \(\mathstrut +\mathstrut 14q^{53} \) \(\mathstrut -\mathstrut 48q^{54} \) \(\mathstrut +\mathstrut 32q^{55} \) \(\mathstrut -\mathstrut 8q^{57} \) \(\mathstrut +\mathstrut 16q^{58} \) \(\mathstrut +\mathstrut 28q^{59} \) \(\mathstrut +\mathstrut 8q^{60} \) \(\mathstrut +\mathstrut 54q^{61} \) \(\mathstrut -\mathstrut 12q^{62} \) \(\mathstrut +\mathstrut 8q^{63} \) \(\mathstrut +\mathstrut 64q^{64} \) \(\mathstrut -\mathstrut 12q^{65} \) \(\mathstrut +\mathstrut 52q^{66} \) \(\mathstrut +\mathstrut 16q^{67} \) \(\mathstrut +\mathstrut 32q^{68} \) \(\mathstrut +\mathstrut 28q^{69} \) \(\mathstrut +\mathstrut 72q^{70} \) \(\mathstrut -\mathstrut 8q^{71} \) \(\mathstrut +\mathstrut 36q^{72} \) \(\mathstrut +\mathstrut 30q^{73} \) \(\mathstrut +\mathstrut 52q^{74} \) \(\mathstrut -\mathstrut 19q^{75} \) \(\mathstrut +\mathstrut 64q^{76} \) \(\mathstrut +\mathstrut 16q^{77} \) \(\mathstrut +\mathstrut 48q^{78} \) \(\mathstrut -\mathstrut 16q^{79} \) \(\mathstrut +\mathstrut 8q^{80} \) \(\mathstrut +\mathstrut 7q^{81} \) \(\mathstrut +\mathstrut 24q^{82} \) \(\mathstrut -\mathstrut 36q^{83} \) \(\mathstrut -\mathstrut 8q^{84} \) \(\mathstrut +\mathstrut 12q^{85} \) \(\mathstrut -\mathstrut 16q^{86} \) \(\mathstrut -\mathstrut 54q^{87} \) \(\mathstrut -\mathstrut 32q^{88} \) \(\mathstrut -\mathstrut 6q^{89} \) \(\mathstrut -\mathstrut 24q^{90} \) \(\mathstrut -\mathstrut 64q^{91} \) \(\mathstrut -\mathstrut 16q^{92} \) \(\mathstrut -\mathstrut 16q^{93} \) \(\mathstrut -\mathstrut 40q^{94} \) \(\mathstrut -\mathstrut 56q^{95} \) \(\mathstrut -\mathstrut 56q^{96} \) \(\mathstrut -\mathstrut 34q^{97} \) \(\mathstrut -\mathstrut 40q^{98} \) \(\mathstrut -\mathstrut 64q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(48))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
48.2.a \(\chi_{48}(1, \cdot)\) 48.2.a.a 1 1
48.2.c \(\chi_{48}(47, \cdot)\) 48.2.c.a 2 1
48.2.d \(\chi_{48}(25, \cdot)\) None 0 1
48.2.f \(\chi_{48}(23, \cdot)\) None 0 1
48.2.j \(\chi_{48}(13, \cdot)\) 48.2.j.a 8 2
48.2.k \(\chi_{48}(11, \cdot)\) 48.2.k.a 12 2

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(48))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(48)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 2}\)