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Properties

Label 51.4
Level 51
Weight 4
Dimension 200
Nonzero newspaces 5
Newform subspaces 9
Sturm bound 768
Trace bound 2

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Defining parameters

Level: \( N \) = \( 51 = 3 \cdot 17 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 5 \)
Newform subspaces: \( 9 \)
Sturm bound: \(768\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(51))\).

Total New Old
Modular forms 320 232 88
Cusp forms 256 200 56
Eisenstein series 64 32 32

Trace form

\( 200 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9} + O(q^{10}) \) \( 200 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9} + 192 q^{10} + 224 q^{11} + 88 q^{12} - 80 q^{13} - 320 q^{14} - 344 q^{15} - 896 q^{16} - 256 q^{17} - 464 q^{18} - 176 q^{19} - 224 q^{20} + 88 q^{21} + 432 q^{22} + 416 q^{23} + 1864 q^{24} + 2440 q^{25} + 2064 q^{26} - 8 q^{27} - 304 q^{28} - 520 q^{29} - 1352 q^{30} - 1744 q^{31} - 2480 q^{32} - 1056 q^{33} - 4240 q^{34} - 1984 q^{35} - 376 q^{36} - 1360 q^{37} - 1040 q^{38} - 1416 q^{39} + 1136 q^{40} + 920 q^{41} + 808 q^{42} + 2720 q^{43} + 4928 q^{44} + 1424 q^{45} + 3920 q^{46} + 3728 q^{48} - 16 q^{49} + 1592 q^{51} - 32 q^{52} + 3432 q^{53} + 6720 q^{54} + 5680 q^{55} + 5200 q^{56} + 2720 q^{57} + 2560 q^{58} + 320 q^{59} - 3712 q^{60} - 2416 q^{61} - 4144 q^{62} - 7064 q^{63} - 4272 q^{64} - 7320 q^{65} - 8352 q^{66} - 4288 q^{67} - 7280 q^{68} - 3760 q^{69} - 2560 q^{70} - 384 q^{71} - 3680 q^{72} - 616 q^{73} - 528 q^{74} - 872 q^{75} - 992 q^{76} + 416 q^{77} + 1776 q^{78} + 688 q^{79} - 272 q^{80} + 6432 q^{81} + 816 q^{82} + 7136 q^{83} - 8064 q^{84} - 2344 q^{85} - 5248 q^{86} - 8984 q^{87} - 13840 q^{88} - 3072 q^{89} - 3160 q^{90} - 4048 q^{91} - 4480 q^{92} + 1464 q^{93} + 752 q^{94} + 1664 q^{95} + 15584 q^{96} + 4592 q^{97} + 7056 q^{98} + 13040 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(51))\)

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
51.4.a \(\chi_{51}(1, \cdot)\) 51.4.a.a 1 1
51.4.a.b 1
51.4.a.c 1
51.4.a.d 2
51.4.a.e 3
51.4.d \(\chi_{51}(16, \cdot)\) 51.4.d.a 8 1
51.4.e \(\chi_{51}(4, \cdot)\) 51.4.e.a 16 2
51.4.h \(\chi_{51}(19, \cdot)\) 51.4.h.a 40 4
51.4.i \(\chi_{51}(5, \cdot)\) 51.4.i.a 128 8

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(51))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(51)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(51))\)\(^{\oplus 1}\)