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Properties

Label 50.6
Level 50
Weight 6
Dimension 115
Nonzero newspaces 4
Newform subspaces 14
Sturm bound 900
Trace bound 1

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Defining parameters

Level: \( N \) = \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 14 \)
Sturm bound: \(900\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(50))\).

Total New Old
Modular forms 403 115 288
Cusp forms 347 115 232
Eisenstein series 56 0 56

Trace form

\( 115 q + 8 q^{2} - 8 q^{3} - 32 q^{4} - 85 q^{5} + 160 q^{6} + 624 q^{7} + 128 q^{8} - 1306 q^{9} + O(q^{10}) \) \( 115 q + 8 q^{2} - 8 q^{3} - 32 q^{4} - 85 q^{5} + 160 q^{6} + 624 q^{7} + 128 q^{8} - 1306 q^{9} + 180 q^{10} + 1480 q^{11} - 128 q^{12} - 228 q^{13} - 3136 q^{14} - 820 q^{15} - 2560 q^{16} + 1984 q^{17} - 316 q^{18} - 6400 q^{19} + 640 q^{20} - 2120 q^{21} + 2816 q^{22} + 12152 q^{23} + 6912 q^{24} + 25335 q^{25} + 1760 q^{26} + 15820 q^{27} + 3904 q^{28} - 34420 q^{29} - 30000 q^{30} - 22520 q^{31} - 3072 q^{32} - 63336 q^{33} - 13716 q^{34} + 60860 q^{35} - 14880 q^{36} + 64409 q^{37} + 34720 q^{38} + 60888 q^{39} + 320 q^{40} - 37120 q^{41} - 22784 q^{42} - 86288 q^{43} + 4736 q^{44} + 1835 q^{45} - 19840 q^{46} - 22456 q^{47} - 2048 q^{48} + 153531 q^{49} + 11300 q^{50} + 26780 q^{51} - 3648 q^{52} + 112257 q^{53} + 64320 q^{54} - 66180 q^{55} + 30720 q^{56} - 221840 q^{57} - 74640 q^{58} - 44860 q^{59} + 112640 q^{60} + 160880 q^{61} - 90704 q^{62} - 127508 q^{63} - 8192 q^{64} - 337865 q^{65} - 110480 q^{66} - 258296 q^{67} - 114816 q^{68} - 333372 q^{69} + 47360 q^{70} + 97280 q^{71} + 124544 q^{72} + 428212 q^{73} + 570624 q^{74} + 892460 q^{75} + 89600 q^{76} + 685888 q^{77} + 34448 q^{78} + 379120 q^{79} - 21760 q^{80} - 110410 q^{81} - 368624 q^{82} - 326088 q^{83} - 210624 q^{84} - 834505 q^{85} - 360240 q^{86} - 1636820 q^{87} - 105984 q^{88} - 509375 q^{89} + 244020 q^{90} - 186320 q^{91} + 379712 q^{92} + 525124 q^{93} + 316544 q^{94} + 379540 q^{95} + 40960 q^{96} + 614584 q^{97} - 5304 q^{98} + 1272548 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(50))\)

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
50.6.a \(\chi_{50}(1, \cdot)\) 50.6.a.a 1 1
50.6.a.b 1
50.6.a.c 1
50.6.a.d 1
50.6.a.e 1
50.6.a.f 1
50.6.a.g 1
50.6.b \(\chi_{50}(49, \cdot)\) 50.6.b.a 2 1
50.6.b.b 2
50.6.b.c 2
50.6.b.d 2
50.6.d \(\chi_{50}(11, \cdot)\) 50.6.d.a 24 4
50.6.d.b 28
50.6.e \(\chi_{50}(9, \cdot)\) 50.6.e.a 48 4

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(50))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(50)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 2}\)