Properties

Label 225.4
Level 225
Weight 4
Dimension 3848
Nonzero newspaces 12
Newform subspaces 54
Sturm bound 14400
Trace bound 4

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Defining parameters

Level: \( N \) = \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 54 \)
Sturm bound: \(14400\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(225))\).

Total New Old
Modular forms 5624 4033 1591
Cusp forms 5176 3848 1328
Eisenstein series 448 185 263

Trace form

\( 3848 q - 17 q^{2} - 27 q^{3} - 51 q^{4} - 21 q^{5} - 31 q^{6} - 21 q^{7} - 84 q^{8} - 107 q^{9} + O(q^{10}) \) \( 3848 q - 17 q^{2} - 27 q^{3} - 51 q^{4} - 21 q^{5} - 31 q^{6} - 21 q^{7} - 84 q^{8} - 107 q^{9} - 190 q^{10} - 104 q^{11} + 76 q^{12} + 379 q^{13} + 846 q^{14} + 136 q^{15} + 673 q^{16} + 188 q^{17} + 192 q^{18} - 668 q^{19} - 870 q^{20} - 1155 q^{21} - 2085 q^{22} - 1219 q^{23} - 691 q^{24} + 957 q^{25} + 1716 q^{26} + 648 q^{27} + 4066 q^{28} + 2977 q^{29} + 1128 q^{30} + 1811 q^{31} + 2833 q^{32} + 2398 q^{33} - 1665 q^{34} - 352 q^{35} + 1591 q^{36} - 3419 q^{37} - 6191 q^{38} - 2337 q^{39} - 4092 q^{40} - 4994 q^{41} - 7318 q^{42} + 110 q^{43} - 5226 q^{44} - 2292 q^{45} + 1066 q^{46} - 5401 q^{47} - 9373 q^{48} - 771 q^{49} + 1556 q^{50} - 743 q^{51} + 5610 q^{52} + 5587 q^{53} + 8623 q^{54} + 1758 q^{55} + 14556 q^{56} + 9219 q^{57} + 6636 q^{58} + 11504 q^{59} + 13940 q^{60} + 5945 q^{61} + 13472 q^{62} + 8571 q^{63} + 4666 q^{64} + 3373 q^{65} + 826 q^{66} - 2766 q^{67} - 8827 q^{68} - 5309 q^{69} - 14730 q^{70} - 4990 q^{71} - 10125 q^{72} - 16790 q^{73} - 17512 q^{74} - 6592 q^{75} - 14203 q^{76} - 9057 q^{77} - 1302 q^{78} - 4239 q^{79} - 15350 q^{80} - 2783 q^{81} - 3380 q^{82} - 12275 q^{83} - 17254 q^{84} - 1189 q^{85} - 20915 q^{86} - 13601 q^{87} + 4749 q^{88} - 11235 q^{89} - 1016 q^{90} - 660 q^{91} - 4184 q^{92} - 13589 q^{93} + 15078 q^{94} + 4330 q^{95} - 8912 q^{96} + 18464 q^{97} + 14200 q^{98} - 663 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(225))\)

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
225.4.a \(\chi_{225}(1, \cdot)\) 225.4.a.a 1 1
225.4.a.b 1
225.4.a.c 1
225.4.a.d 1
225.4.a.e 1
225.4.a.f 1
225.4.a.g 1
225.4.a.h 1
225.4.a.i 2
225.4.a.j 2
225.4.a.k 2
225.4.a.l 2
225.4.a.m 2
225.4.a.n 2
225.4.a.o 2
225.4.b \(\chi_{225}(199, \cdot)\) 225.4.b.a 2 1
225.4.b.b 2
225.4.b.c 2
225.4.b.d 2
225.4.b.e 2
225.4.b.f 2
225.4.b.g 2
225.4.b.h 4
225.4.b.i 4
225.4.e \(\chi_{225}(76, \cdot)\) 225.4.e.a 4 2
225.4.e.b 4
225.4.e.c 6
225.4.e.d 14
225.4.e.e 24
225.4.e.f 24
225.4.e.g 32
225.4.f \(\chi_{225}(107, \cdot)\) 225.4.f.a 4 2
225.4.f.b 4
225.4.f.c 12
225.4.f.d 16
225.4.h \(\chi_{225}(46, \cdot)\) 225.4.h.a 28 4
225.4.h.b 28
225.4.h.c 28
225.4.h.d 64
225.4.k \(\chi_{225}(49, \cdot)\) 225.4.k.a 8 2
225.4.k.b 8
225.4.k.c 12
225.4.k.d 28
225.4.k.e 48
225.4.m \(\chi_{225}(19, \cdot)\) 225.4.m.a 24 4
225.4.m.b 56
225.4.m.c 64
225.4.p \(\chi_{225}(32, \cdot)\) 225.4.p.a 48 4
225.4.p.b 64
225.4.p.c 96
225.4.q \(\chi_{225}(16, \cdot)\) 225.4.q.a 704 8
225.4.s \(\chi_{225}(8, \cdot)\) 225.4.s.a 240 8
225.4.u \(\chi_{225}(4, \cdot)\) 225.4.u.a 704 8
225.4.w \(\chi_{225}(2, \cdot)\) 225.4.w.a 1408 16

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(225))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(225)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(75))\)\(^{\oplus 2}\)