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Properties

Label 45.12
Level 45
Weight 12
Dimension 561
Nonzero newspaces 6
Newform subspaces 17
Sturm bound 1728
Trace bound 1

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Defining parameters

Level: \( N \) = \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) = \( 12 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 17 \)
Sturm bound: \(1728\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_1(45))\).

Total New Old
Modular forms 824 587 237
Cusp forms 760 561 199
Eisenstein series 64 26 38

Trace form

\( 561 q + 136 q^{2} + 16 q^{3} + 14482 q^{4} + 8091 q^{5} - 41182 q^{6} - 153564 q^{7} + 376464 q^{8} - 203212 q^{9} + O(q^{10}) \) \( 561 q + 136 q^{2} + 16 q^{3} + 14482 q^{4} + 8091 q^{5} - 41182 q^{6} - 153564 q^{7} + 376464 q^{8} - 203212 q^{9} - 1107882 q^{10} + 969724 q^{11} + 2089424 q^{12} - 2344738 q^{13} - 10312068 q^{14} + 5015092 q^{15} + 53784714 q^{16} - 58239194 q^{17} - 39107840 q^{18} + 42735964 q^{19} + 74764868 q^{20} - 101769852 q^{21} + 29987942 q^{22} + 62160684 q^{23} + 199409946 q^{24} - 180345633 q^{25} - 420814556 q^{26} + 353644864 q^{27} - 142737816 q^{28} - 18452002 q^{29} - 499220176 q^{30} - 443937400 q^{31} + 1256012174 q^{32} + 742920476 q^{33} + 1639830894 q^{34} - 3298310584 q^{35} - 2638694486 q^{36} - 2998289114 q^{37} + 3477031942 q^{38} + 4986454720 q^{39} + 1309071916 q^{40} - 2552487298 q^{41} - 8017582848 q^{42} - 4914431944 q^{43} + 6729479348 q^{44} + 3492036482 q^{45} - 6503263688 q^{46} - 2850341432 q^{47} + 6690552986 q^{48} + 16662429365 q^{49} - 8589695330 q^{50} - 6273566920 q^{51} - 38908242848 q^{52} - 21331334906 q^{53} - 27551978110 q^{54} + 45395650348 q^{55} + 72050802636 q^{56} + 23914946152 q^{57} - 21080101400 q^{58} - 14816173604 q^{59} - 56768468104 q^{60} - 14060424254 q^{61} - 82825506948 q^{62} - 41929139700 q^{63} + 79535009292 q^{64} + 25744107352 q^{65} + 115757536420 q^{66} - 52569290736 q^{67} + 125486402462 q^{68} - 22350948840 q^{69} + 63173192298 q^{70} - 103189302016 q^{71} - 303190501686 q^{72} - 39547387462 q^{73} + 138945425732 q^{74} - 87888554060 q^{75} + 259857654502 q^{76} + 101172668616 q^{77} - 33211222252 q^{78} - 24203033932 q^{79} + 156834230648 q^{80} - 310892240104 q^{81} - 85864347536 q^{82} - 236909382408 q^{83} + 529108182156 q^{84} + 202516139658 q^{85} + 201798724414 q^{86} - 110782557548 q^{87} - 139755327978 q^{88} - 96664148982 q^{89} + 388701610372 q^{90} - 631279295664 q^{91} + 347617346604 q^{92} + 626122916136 q^{93} + 299293615620 q^{94} - 265823463992 q^{95} - 995328067280 q^{96} - 224650551446 q^{97} - 1317908990866 q^{98} - 45434414156 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_1(45))\)

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
45.12.a \(\chi_{45}(1, \cdot)\) 45.12.a.a 1 1
45.12.a.b 1
45.12.a.c 2
45.12.a.d 2
45.12.a.e 2
45.12.a.f 3
45.12.a.g 4
45.12.a.h 4
45.12.b \(\chi_{45}(19, \cdot)\) 45.12.b.a 2 1
45.12.b.b 4
45.12.b.c 8
45.12.b.d 12
45.12.e \(\chi_{45}(16, \cdot)\) 45.12.e.a 42 2
45.12.e.b 46
45.12.f \(\chi_{45}(8, \cdot)\) 45.12.f.a 44 2
45.12.j \(\chi_{45}(4, \cdot)\) 45.12.j.a 128 2
45.12.l \(\chi_{45}(2, \cdot)\) 45.12.l.a 256 4

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_1(45))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_1(45)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 2}\)