Properties

Label 15.8
Level 15
Weight 8
Dimension 36
Nonzero newspaces 3
Newforms 5
Sturm bound 128
Trace bound 1

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Defining parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 3 \)
Newforms: \( 5 \)
Sturm bound: \(128\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(15))\).

Total New Old
Modular forms 64 44 20
Cusp forms 48 36 12
Eisenstein series 16 8 8

Trace form

\(36q \) \(\mathstrut -\mathstrut 28q^{2} \) \(\mathstrut +\mathstrut 78q^{3} \) \(\mathstrut -\mathstrut 200q^{4} \) \(\mathstrut -\mathstrut 444q^{5} \) \(\mathstrut +\mathstrut 1356q^{6} \) \(\mathstrut +\mathstrut 3608q^{7} \) \(\mathstrut -\mathstrut 2436q^{8} \) \(\mathstrut -\mathstrut 2916q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(36q \) \(\mathstrut -\mathstrut 28q^{2} \) \(\mathstrut +\mathstrut 78q^{3} \) \(\mathstrut -\mathstrut 200q^{4} \) \(\mathstrut -\mathstrut 444q^{5} \) \(\mathstrut +\mathstrut 1356q^{6} \) \(\mathstrut +\mathstrut 3608q^{7} \) \(\mathstrut -\mathstrut 2436q^{8} \) \(\mathstrut -\mathstrut 2916q^{9} \) \(\mathstrut -\mathstrut 4616q^{10} \) \(\mathstrut +\mathstrut 7952q^{11} \) \(\mathstrut +\mathstrut 8340q^{12} \) \(\mathstrut +\mathstrut 5360q^{13} \) \(\mathstrut -\mathstrut 34488q^{14} \) \(\mathstrut +\mathstrut 26814q^{15} \) \(\mathstrut +\mathstrut 38000q^{16} \) \(\mathstrut -\mathstrut 32032q^{17} \) \(\mathstrut -\mathstrut 126972q^{18} \) \(\mathstrut -\mathstrut 64176q^{19} \) \(\mathstrut +\mathstrut 46444q^{20} \) \(\mathstrut +\mathstrut 134904q^{21} \) \(\mathstrut +\mathstrut 146376q^{22} \) \(\mathstrut +\mathstrut 170088q^{23} \) \(\mathstrut -\mathstrut 237168q^{24} \) \(\mathstrut -\mathstrut 85644q^{25} \) \(\mathstrut -\mathstrut 445816q^{26} \) \(\mathstrut -\mathstrut 224802q^{27} \) \(\mathstrut +\mathstrut 436280q^{28} \) \(\mathstrut +\mathstrut 717688q^{29} \) \(\mathstrut +\mathstrut 774336q^{30} \) \(\mathstrut +\mathstrut 71568q^{31} \) \(\mathstrut -\mathstrut 1063916q^{32} \) \(\mathstrut -\mathstrut 774984q^{33} \) \(\mathstrut +\mathstrut 238760q^{34} \) \(\mathstrut +\mathstrut 636520q^{35} \) \(\mathstrut +\mathstrut 569016q^{36} \) \(\mathstrut -\mathstrut 789232q^{37} \) \(\mathstrut -\mathstrut 769432q^{38} \) \(\mathstrut -\mathstrut 1224396q^{39} \) \(\mathstrut -\mathstrut 2054392q^{40} \) \(\mathstrut +\mathstrut 883240q^{41} \) \(\mathstrut +\mathstrut 1622112q^{42} \) \(\mathstrut +\mathstrut 976328q^{43} \) \(\mathstrut +\mathstrut 537976q^{44} \) \(\mathstrut +\mathstrut 893916q^{45} \) \(\mathstrut +\mathstrut 4371848q^{46} \) \(\mathstrut +\mathstrut 1210520q^{47} \) \(\mathstrut +\mathstrut 1149684q^{48} \) \(\mathstrut -\mathstrut 2687956q^{49} \) \(\mathstrut -\mathstrut 5555236q^{50} \) \(\mathstrut -\mathstrut 4088820q^{51} \) \(\mathstrut -\mathstrut 8572528q^{52} \) \(\mathstrut +\mathstrut 1575488q^{53} \) \(\mathstrut -\mathstrut 393660q^{54} \) \(\mathstrut +\mathstrut 7062576q^{55} \) \(\mathstrut +\mathstrut 6479040q^{56} \) \(\mathstrut +\mathstrut 5320296q^{57} \) \(\mathstrut -\mathstrut 1200264q^{58} \) \(\mathstrut -\mathstrut 1197184q^{59} \) \(\mathstrut +\mathstrut 6515256q^{60} \) \(\mathstrut +\mathstrut 1179144q^{61} \) \(\mathstrut +\mathstrut 5573928q^{62} \) \(\mathstrut -\mathstrut 323496q^{63} \) \(\mathstrut +\mathstrut 9724488q^{64} \) \(\mathstrut -\mathstrut 3366592q^{65} \) \(\mathstrut -\mathstrut 30638208q^{66} \) \(\mathstrut -\mathstrut 21756808q^{67} \) \(\mathstrut -\mathstrut 9434504q^{68} \) \(\mathstrut -\mathstrut 5890968q^{69} \) \(\mathstrut -\mathstrut 1513560q^{70} \) \(\mathstrut -\mathstrut 4007216q^{71} \) \(\mathstrut +\mathstrut 34470036q^{72} \) \(\mathstrut +\mathstrut 6676736q^{73} \) \(\mathstrut +\mathstrut 20919712q^{74} \) \(\mathstrut +\mathstrut 19838094q^{75} \) \(\mathstrut +\mathstrut 37710128q^{76} \) \(\mathstrut +\mathstrut 6733248q^{77} \) \(\mathstrut -\mathstrut 10796952q^{78} \) \(\mathstrut -\mathstrut 10874880q^{79} \) \(\mathstrut -\mathstrut 39917876q^{80} \) \(\mathstrut -\mathstrut 24912684q^{81} \) \(\mathstrut -\mathstrut 40630392q^{82} \) \(\mathstrut -\mathstrut 17321256q^{83} \) \(\mathstrut -\mathstrut 43954272q^{84} \) \(\mathstrut -\mathstrut 33002768q^{85} \) \(\mathstrut -\mathstrut 6212848q^{86} \) \(\mathstrut +\mathstrut 50204484q^{87} \) \(\mathstrut +\mathstrut 97866504q^{88} \) \(\mathstrut +\mathstrut 58210824q^{89} \) \(\mathstrut +\mathstrut 80324424q^{90} \) \(\mathstrut +\mathstrut 88309648q^{91} \) \(\mathstrut +\mathstrut 15479424q^{92} \) \(\mathstrut -\mathstrut 54464928q^{93} \) \(\mathstrut -\mathstrut 78524776q^{94} \) \(\mathstrut -\mathstrut 33956656q^{95} \) \(\mathstrut -\mathstrut 104433168q^{96} \) \(\mathstrut -\mathstrut 63625168q^{97} \) \(\mathstrut -\mathstrut 21248492q^{98} \) \(\mathstrut -\mathstrut 9879408q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(15))\)

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
15.8.a \(\chi_{15}(1, \cdot)\) 15.8.a.a 1 1
15.8.a.b 1
15.8.a.c 2
15.8.b \(\chi_{15}(4, \cdot)\) 15.8.b.a 8 1
15.8.e \(\chi_{15}(2, \cdot)\) 15.8.e.a 24 2

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(15))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(15)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 2}\)