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Properties

Label 99.8
Level 99
Weight 8
Dimension 1920
Nonzero newspaces 8
Newform subspaces 21
Sturm bound 5760
Trace bound 2

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Defining parameters

Level: \( N \) = \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 21 \)
Sturm bound: \(5760\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(99))\).

Total New Old
Modular forms 2600 2000 600
Cusp forms 2440 1920 520
Eisenstein series 160 80 80

Trace form

\( 1920 q + 15 q^{2} - 68 q^{3} - 117 q^{4} + 1125 q^{5} + 2446 q^{6} - 1744 q^{7} - 18979 q^{8} - 2000 q^{9} + O(q^{10}) \) \( 1920 q + 15 q^{2} - 68 q^{3} - 117 q^{4} + 1125 q^{5} + 2446 q^{6} - 1744 q^{7} - 18979 q^{8} - 2000 q^{9} + 27806 q^{10} + 10792 q^{11} - 16144 q^{12} - 25648 q^{13} - 13014 q^{14} + 2356 q^{15} + 45791 q^{16} - 24126 q^{17} - 85772 q^{18} - 169608 q^{19} + 154196 q^{20} + 374428 q^{21} + 506664 q^{22} + 412835 q^{23} - 1091438 q^{24} - 973547 q^{25} - 1192150 q^{26} - 17054 q^{27} + 1785662 q^{28} + 1684916 q^{29} + 1760044 q^{30} - 162283 q^{31} - 2355064 q^{32} - 1329852 q^{33} - 1549156 q^{34} - 3191372 q^{35} - 2499322 q^{36} + 1309265 q^{37} + 7507160 q^{38} + 5233882 q^{39} + 1891278 q^{40} + 209148 q^{41} - 5703992 q^{42} - 2896468 q^{43} + 493592 q^{44} - 1747110 q^{45} + 483242 q^{46} + 1900450 q^{47} + 15032870 q^{48} - 1507444 q^{49} - 192403 q^{50} + 1544414 q^{51} - 5348940 q^{52} - 16131002 q^{53} - 29389638 q^{54} - 1029369 q^{55} + 17674392 q^{56} + 18720240 q^{57} + 30187558 q^{58} + 26643267 q^{59} + 20554512 q^{60} + 3396960 q^{61} - 17303940 q^{62} - 8303772 q^{63} - 38872829 q^{64} - 28335030 q^{65} - 39774352 q^{66} - 40620179 q^{67} - 29108462 q^{68} + 9550404 q^{69} + 14778966 q^{70} + 31597179 q^{71} + 79897180 q^{72} + 72470010 q^{73} + 79551864 q^{74} - 24130534 q^{75} + 21506 q^{76} - 44374392 q^{77} - 35055816 q^{78} - 81494648 q^{79} - 69251214 q^{80} + 2057200 q^{81} + 8819637 q^{82} + 53125150 q^{83} + 92060350 q^{84} + 15059048 q^{85} - 33165073 q^{86} - 44513136 q^{87} + 110808214 q^{88} - 3311857 q^{89} - 132747628 q^{90} + 35202868 q^{91} - 141443656 q^{92} - 76461744 q^{93} - 88013268 q^{94} - 24803814 q^{95} + 221604362 q^{96} - 7415949 q^{97} + 239728024 q^{98} + 195300576 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(99))\)

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
99.8.a \(\chi_{99}(1, \cdot)\) 99.8.a.a 1 1
99.8.a.b 2
99.8.a.c 2
99.8.a.d 2
99.8.a.e 3
99.8.a.f 4
99.8.a.g 4
99.8.a.h 5
99.8.a.i 5
99.8.d \(\chi_{99}(98, \cdot)\) 99.8.d.a 28 1
99.8.e \(\chi_{99}(34, \cdot)\) 99.8.e.a 66 2
99.8.e.b 74
99.8.f \(\chi_{99}(37, \cdot)\) 99.8.f.a 24 4
99.8.f.b 28
99.8.f.c 28
99.8.f.d 56
99.8.g \(\chi_{99}(32, \cdot)\) 99.8.g.a 4 2
99.8.g.b 160
99.8.j \(\chi_{99}(8, \cdot)\) 99.8.j.a 112 4
99.8.m \(\chi_{99}(4, \cdot)\) 99.8.m.a 656 8
99.8.p \(\chi_{99}(2, \cdot)\) 99.8.p.a 656 8

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(99))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(99)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(33))\)\(^{\oplus 2}\)