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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

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

Display 100 coefficients 10 coefficients

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 available 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

\( S_{2}^{\mathrm{old}}(\Gamma_1(180)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(180))\)\(^{\oplus 1}\)