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Properties

Label 250.4
Level 250
Weight 4
Dimension 1728
Nonzero newspaces 6
Newform subspaces 22
Sturm bound 15000
Trace bound 4

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Defining parameters

Level: \( N \) = \( 250 = 2 \cdot 5^{3} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 22 \)
Sturm bound: \(15000\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(250))\).

Total New Old
Modular forms 5805 1728 4077
Cusp forms 5445 1728 3717
Eisenstein series 360 0 360

Trace form

\( 1728 q + 2 q^{2} - 8 q^{3} - 4 q^{4} - 8 q^{6} - 4 q^{7} + 8 q^{8} + 83 q^{9} + O(q^{10}) \) \( 1728 q + 2 q^{2} - 8 q^{3} - 4 q^{4} - 8 q^{6} - 4 q^{7} + 8 q^{8} + 83 q^{9} - 44 q^{11} - 32 q^{12} - 58 q^{13} - 112 q^{14} + 48 q^{16} - 1214 q^{17} - 826 q^{18} - 460 q^{19} + 20 q^{20} + 1096 q^{21} + 1464 q^{22} + 1892 q^{23} + 864 q^{24} + 1440 q^{25} + 812 q^{26} + 2080 q^{27} + 464 q^{28} + 50 q^{29} - 240 q^{30} - 1544 q^{31} - 288 q^{32} - 3936 q^{33} - 2912 q^{34} - 1720 q^{35} - 36 q^{36} - 1104 q^{37} - 200 q^{38} - 4464 q^{39} - 874 q^{41} + 64 q^{42} + 592 q^{43} + 272 q^{44} + 2320 q^{45} + 32 q^{46} + 3236 q^{47} - 128 q^{48} + 3417 q^{49} + 4856 q^{51} - 232 q^{52} + 3552 q^{53} + 240 q^{54} + 760 q^{55} + 384 q^{56} - 320 q^{57} - 180 q^{58} - 9460 q^{59} - 2720 q^{60} - 9974 q^{61} - 9536 q^{62} - 22388 q^{63} - 64 q^{64} - 2555 q^{65} - 176 q^{66} + 1856 q^{67} + 4104 q^{68} + 11496 q^{69} + 4320 q^{70} + 8096 q^{71} + 296 q^{72} + 11882 q^{73} + 13788 q^{74} + 11840 q^{75} + 4880 q^{76} + 21712 q^{77} + 13328 q^{78} + 9920 q^{79} + 5603 q^{81} + 2004 q^{82} - 6288 q^{83} - 2208 q^{84} - 5655 q^{85} - 5448 q^{86} - 20240 q^{87} + 96 q^{88} - 25520 q^{89} - 9960 q^{90} - 12824 q^{91} - 8752 q^{92} - 7376 q^{93} - 1312 q^{94} + 3160 q^{95} - 128 q^{96} + 4826 q^{97} - 654 q^{98} + 11036 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(250))\)

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
250.4.a \(\chi_{250}(1, \cdot)\) 250.4.a.a 2 1
250.4.a.b 2
250.4.a.c 2
250.4.a.d 2
250.4.a.e 4
250.4.a.f 4
250.4.a.g 4
250.4.a.h 4
250.4.b \(\chi_{250}(249, \cdot)\) 250.4.b.a 4 1
250.4.b.b 4
250.4.b.c 8
250.4.b.d 8
250.4.d \(\chi_{250}(51, \cdot)\) 250.4.d.a 12 4
250.4.d.b 16
250.4.d.c 32
250.4.d.d 32
250.4.e \(\chi_{250}(49, \cdot)\) 250.4.e.a 24 4
250.4.e.b 32
250.4.e.c 32
250.4.g \(\chi_{250}(11, \cdot)\) 250.4.g.a 360 20
250.4.g.b 380
250.4.h \(\chi_{250}(9, \cdot)\) 250.4.h.a 760 20

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(250))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(250)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(125))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(250))\)\(^{\oplus 1}\)