Properties

Label 180.2
Level 180
Weight 2
Dimension 337
Nonzero newspaces 12
Newform subspaces 21
Sturm bound 3456
Trace bound 4

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Defining parameters

Level: \( N \) = \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 21 \)
Sturm bound: \(3456\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(180))\).

Total New Old
Modular forms 1024 393 631
Cusp forms 705 337 368
Eisenstein series 319 56 263

Trace form

\( 337q + 6q^{4} - 4q^{5} - 10q^{6} + 8q^{7} + 12q^{9} + O(q^{10}) \) \( 337q + 6q^{4} - 4q^{5} - 10q^{6} + 8q^{7} + 12q^{9} - 6q^{10} + 22q^{11} - 20q^{12} + 22q^{13} - 36q^{14} + 3q^{15} - 30q^{16} - 8q^{17} - 52q^{18} + 12q^{19} - 58q^{20} - 26q^{21} - 42q^{22} - 24q^{23} - 42q^{24} - 14q^{25} - 68q^{26} - 24q^{27} - 56q^{28} - 72q^{29} - 30q^{30} - 18q^{31} - 10q^{32} - 78q^{33} - 42q^{34} - 56q^{35} + 18q^{36} - 32q^{37} + 10q^{38} - 50q^{39} - 42q^{40} - 100q^{41} + 16q^{42} - 12q^{43} - 111q^{45} - 40q^{46} - 60q^{47} - 34q^{48} - 81q^{49} + 26q^{50} - 48q^{51} + 8q^{52} - 104q^{53} - 34q^{54} - 26q^{55} + 68q^{56} - 100q^{57} + 20q^{58} - 38q^{59} + 46q^{60} - 96q^{61} + 100q^{62} + 2q^{63} + 60q^{64} - 37q^{65} + 88q^{66} + 20q^{67} + 134q^{68} + 34q^{69} + 42q^{70} + 84q^{71} + 138q^{72} + 40q^{73} + 132q^{74} + 69q^{75} - 26q^{76} + 30q^{77} + 152q^{78} + 30q^{79} + 112q^{80} + 96q^{81} - 76q^{82} + 84q^{83} + 100q^{84} - 54q^{85} + 22q^{86} + 130q^{87} - 42q^{88} + 98q^{89} + 138q^{90} + 48q^{91} + 56q^{92} - 2q^{93} - 48q^{94} + 96q^{95} + 104q^{96} - 34q^{97} + 126q^{98} + 94q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(180))\)

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
180.2.a \(\chi_{180}(1, \cdot)\) 180.2.a.a 1 1
180.2.d \(\chi_{180}(109, \cdot)\) 180.2.d.a 2 1
180.2.e \(\chi_{180}(71, \cdot)\) 180.2.e.a 8 1
180.2.h \(\chi_{180}(179, \cdot)\) 180.2.h.a 4 1
180.2.h.b 8
180.2.i \(\chi_{180}(61, \cdot)\) 180.2.i.a 2 2
180.2.i.b 6
180.2.j \(\chi_{180}(17, \cdot)\) 180.2.j.a 4 2
180.2.k \(\chi_{180}(127, \cdot)\) 180.2.k.a 2 2
180.2.k.b 2
180.2.k.c 2
180.2.k.d 8
180.2.k.e 12
180.2.n \(\chi_{180}(59, \cdot)\) 180.2.n.a 4 2
180.2.n.b 4
180.2.n.c 8
180.2.n.d 48
180.2.q \(\chi_{180}(11, \cdot)\) 180.2.q.a 48 2
180.2.r \(\chi_{180}(49, \cdot)\) 180.2.r.a 12 2
180.2.w \(\chi_{180}(77, \cdot)\) 180.2.w.a 24 4
180.2.x \(\chi_{180}(7, \cdot)\) 180.2.x.a 128 4

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(180))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(180)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(90))\)\(^{\oplus 2}\)