Properties

Label 85.2
Level 85
Weight 2
Dimension 215
Nonzero newspaces 10
Newform subspaces 14
Sturm bound 1152
Trace bound 8

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Defining parameters

Level: \( N \) = \( 85 = 5 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 10 \)
Newform subspaces: \( 14 \)
Sturm bound: \(1152\)
Trace bound: \(8\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(85))\).

Total New Old
Modular forms 352 307 45
Cusp forms 225 215 10
Eisenstein series 127 92 35

Trace form

\( 215q - 19q^{2} - 20q^{3} - 23q^{4} - 25q^{5} - 60q^{6} - 24q^{7} - 31q^{8} - 29q^{9} + O(q^{10}) \) \( 215q - 19q^{2} - 20q^{3} - 23q^{4} - 25q^{5} - 60q^{6} - 24q^{7} - 31q^{8} - 29q^{9} - 23q^{10} - 44q^{11} + 4q^{12} - 14q^{13} - 8q^{14} - 4q^{15} - 7q^{16} - 17q^{17} - 7q^{18} - 20q^{19} - 3q^{20} - 32q^{21} - 20q^{22} - 24q^{23} + 4q^{24} + 3q^{25} - 18q^{26} - 8q^{27} + 24q^{28} - 6q^{29} + 28q^{30} - 16q^{31} + 33q^{32} + 16q^{33} + 61q^{34} - 24q^{35} + 21q^{36} + 10q^{37} + 20q^{38} + 24q^{39} + 57q^{40} - 2q^{41} + 80q^{42} + 20q^{43} + 92q^{44} + 15q^{45} - 24q^{46} + 16q^{47} + 68q^{48} - 9q^{49} - 3q^{50} - 52q^{51} - 2q^{52} + 18q^{53} + 104q^{54} + 28q^{55} + 72q^{56} + 96q^{57} + 70q^{58} + 52q^{59} + 116q^{60} + 18q^{61} + 96q^{62} + 120q^{63} - 47q^{64} + 38q^{65} + 64q^{66} - 4q^{67} + 17q^{68} + 32q^{69} - 40q^{70} - 88q^{71} - 59q^{72} - 50q^{73} - 106q^{74} - 28q^{75} - 204q^{76} - 80q^{77} - 104q^{78} - 96q^{79} - 199q^{80} - 153q^{81} - 198q^{82} - 68q^{83} - 288q^{84} - 141q^{85} - 132q^{86} - 136q^{87} - 260q^{88} - 58q^{89} - 107q^{90} - 128q^{91} - 120q^{92} - 80q^{93} - 96q^{94} - 60q^{95} - 28q^{96} - 114q^{97} + 13q^{98} - 28q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(85))\)

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
85.2.a \(\chi_{85}(1, \cdot)\) 85.2.a.a 1 1
85.2.a.b 2
85.2.a.c 2
85.2.b \(\chi_{85}(69, \cdot)\) 85.2.b.a 8 1
85.2.c \(\chi_{85}(84, \cdot)\) 85.2.c.a 8 1
85.2.d \(\chi_{85}(16, \cdot)\) 85.2.d.a 6 1
85.2.e \(\chi_{85}(21, \cdot)\) 85.2.e.a 12 2
85.2.j \(\chi_{85}(4, \cdot)\) 85.2.j.a 2 2
85.2.j.b 2
85.2.j.c 12
85.2.l \(\chi_{85}(26, \cdot)\) 85.2.l.a 24 4
85.2.m \(\chi_{85}(9, \cdot)\) 85.2.m.a 24 4
85.2.o \(\chi_{85}(3, \cdot)\) 85.2.o.a 56 8
85.2.r \(\chi_{85}(12, \cdot)\) 85.2.r.a 56 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(85))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(85)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 2}\)