Properties

Label 225.2
Level 225
Weight 2
Dimension 1221
Nonzero newspaces 12
Newforms 34
Sturm bound 7200
Trace bound 2

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Defining parameters

Level: \( N \) = \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 12 \)
Newforms: \( 34 \)
Sturm bound: \(7200\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(225))\).

Total New Old
Modular forms 2024 1406 618
Cusp forms 1577 1221 356
Eisenstein series 447 185 262

Trace form

\(1221q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut -\mathstrut 24q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 21q^{5} \) \(\mathstrut -\mathstrut 40q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 24q^{8} \) \(\mathstrut -\mathstrut 32q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(1221q \) \(\mathstrut -\mathstrut 16q^{2} \) \(\mathstrut -\mathstrut 24q^{3} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 21q^{5} \) \(\mathstrut -\mathstrut 40q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 24q^{8} \) \(\mathstrut -\mathstrut 32q^{9} \) \(\mathstrut -\mathstrut 75q^{10} \) \(\mathstrut -\mathstrut 46q^{11} \) \(\mathstrut -\mathstrut 56q^{12} \) \(\mathstrut -\mathstrut 26q^{13} \) \(\mathstrut -\mathstrut 54q^{14} \) \(\mathstrut -\mathstrut 44q^{15} \) \(\mathstrut -\mathstrut 56q^{16} \) \(\mathstrut -\mathstrut 44q^{17} \) \(\mathstrut -\mathstrut 72q^{18} \) \(\mathstrut -\mathstrut 70q^{19} \) \(\mathstrut -\mathstrut 90q^{20} \) \(\mathstrut -\mathstrut 72q^{21} \) \(\mathstrut -\mathstrut 86q^{22} \) \(\mathstrut -\mathstrut 74q^{23} \) \(\mathstrut -\mathstrut 112q^{24} \) \(\mathstrut -\mathstrut 63q^{25} \) \(\mathstrut -\mathstrut 132q^{26} \) \(\mathstrut -\mathstrut 72q^{27} \) \(\mathstrut -\mathstrut 166q^{28} \) \(\mathstrut -\mathstrut 94q^{29} \) \(\mathstrut -\mathstrut 72q^{30} \) \(\mathstrut -\mathstrut 58q^{31} \) \(\mathstrut -\mathstrut 91q^{32} \) \(\mathstrut -\mathstrut 56q^{33} \) \(\mathstrut -\mathstrut 83q^{34} \) \(\mathstrut -\mathstrut 32q^{35} \) \(\mathstrut -\mathstrut 8q^{36} \) \(\mathstrut -\mathstrut 99q^{37} \) \(\mathstrut -\mathstrut 28q^{38} \) \(\mathstrut +\mathstrut 24q^{39} \) \(\mathstrut -\mathstrut 57q^{40} \) \(\mathstrut +\mathstrut 26q^{41} \) \(\mathstrut +\mathstrut 32q^{42} \) \(\mathstrut -\mathstrut 6q^{43} \) \(\mathstrut +\mathstrut 36q^{44} \) \(\mathstrut -\mathstrut 12q^{45} \) \(\mathstrut -\mathstrut 142q^{46} \) \(\mathstrut +\mathstrut 10q^{47} \) \(\mathstrut +\mathstrut 80q^{48} \) \(\mathstrut -\mathstrut 91q^{49} \) \(\mathstrut -\mathstrut 89q^{50} \) \(\mathstrut -\mathstrut 80q^{51} \) \(\mathstrut -\mathstrut 144q^{52} \) \(\mathstrut -\mathstrut 79q^{53} \) \(\mathstrut -\mathstrut 32q^{54} \) \(\mathstrut -\mathstrut 142q^{55} \) \(\mathstrut -\mathstrut 66q^{56} \) \(\mathstrut -\mathstrut 72q^{57} \) \(\mathstrut -\mathstrut 152q^{58} \) \(\mathstrut -\mathstrut 128q^{59} \) \(\mathstrut +\mathstrut 20q^{60} \) \(\mathstrut -\mathstrut 66q^{61} \) \(\mathstrut -\mathstrut 68q^{62} \) \(\mathstrut -\mathstrut 60q^{63} \) \(\mathstrut +\mathstrut 22q^{64} \) \(\mathstrut +\mathstrut 53q^{65} \) \(\mathstrut -\mathstrut 128q^{66} \) \(\mathstrut +\mathstrut 42q^{67} \) \(\mathstrut +\mathstrut 190q^{68} \) \(\mathstrut -\mathstrut 20q^{69} \) \(\mathstrut +\mathstrut 150q^{70} \) \(\mathstrut -\mathstrut 62q^{71} \) \(\mathstrut +\mathstrut 192q^{72} \) \(\mathstrut +\mathstrut 34q^{73} \) \(\mathstrut +\mathstrut 304q^{74} \) \(\mathstrut +\mathstrut 128q^{75} \) \(\mathstrut +\mathstrut 52q^{76} \) \(\mathstrut +\mathstrut 234q^{77} \) \(\mathstrut +\mathstrut 240q^{78} \) \(\mathstrut +\mathstrut 114q^{79} \) \(\mathstrut +\mathstrut 425q^{80} \) \(\mathstrut +\mathstrut 88q^{81} \) \(\mathstrut +\mathstrut 170q^{82} \) \(\mathstrut +\mathstrut 284q^{83} \) \(\mathstrut +\mathstrut 344q^{84} \) \(\mathstrut +\mathstrut 21q^{85} \) \(\mathstrut +\mathstrut 266q^{86} \) \(\mathstrut +\mathstrut 148q^{87} \) \(\mathstrut +\mathstrut 90q^{88} \) \(\mathstrut +\mathstrut 177q^{89} \) \(\mathstrut +\mathstrut 184q^{90} \) \(\mathstrut -\mathstrut 86q^{91} \) \(\mathstrut +\mathstrut 182q^{92} \) \(\mathstrut +\mathstrut 100q^{93} \) \(\mathstrut -\mathstrut 182q^{94} \) \(\mathstrut -\mathstrut 70q^{95} \) \(\mathstrut +\mathstrut 136q^{96} \) \(\mathstrut -\mathstrut 174q^{97} \) \(\mathstrut +\mathstrut 56q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(225))\)

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
225.2.a \(\chi_{225}(1, \cdot)\) 225.2.a.a 1 1
225.2.a.b 1
225.2.a.c 1
225.2.a.d 1
225.2.a.e 1
225.2.a.f 2
225.2.b \(\chi_{225}(199, \cdot)\) 225.2.b.a 2 1
225.2.b.b 2
225.2.b.c 2
225.2.e \(\chi_{225}(76, \cdot)\) 225.2.e.a 2 2
225.2.e.b 6
225.2.e.c 8
225.2.e.d 8
225.2.e.e 8
225.2.f \(\chi_{225}(107, \cdot)\) 225.2.f.a 4 2
225.2.f.b 8
225.2.h \(\chi_{225}(46, \cdot)\) 225.2.h.a 4 4
225.2.h.b 4
225.2.h.c 8
225.2.h.d 12
225.2.h.e 16
225.2.k \(\chi_{225}(49, \cdot)\) 225.2.k.a 4 2
225.2.k.b 12
225.2.k.c 16
225.2.m \(\chi_{225}(19, \cdot)\) 225.2.m.a 8 4
225.2.m.b 16
225.2.m.c 24
225.2.p \(\chi_{225}(32, \cdot)\) 225.2.p.a 16 4
225.2.p.b 16
225.2.p.c 32
225.2.q \(\chi_{225}(16, \cdot)\) 225.2.q.a 224 8
225.2.s \(\chi_{225}(8, \cdot)\) 225.2.s.a 80 8
225.2.u \(\chi_{225}(4, \cdot)\) 225.2.u.a 224 8
225.2.w \(\chi_{225}(2, \cdot)\) 225.2.w.a 448 16

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(225))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(225)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(75))\)\(^{\oplus 2}\)