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Properties

Label 500.4
Level 500
Weight 4
Dimension 11968
Nonzero newspaces 9
Sturm bound 60000
Trace bound 9

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Defining parameters

Level: \( N \) = \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 9 \)
Sturm bound: \(60000\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(500))\).

Total New Old
Modular forms 22950 12224 10726
Cusp forms 22050 11968 10082
Eisenstein series 900 256 644

Trace form

\( 11968 q - 32 q^{2} + 4 q^{3} - 30 q^{4} - 80 q^{5} - 46 q^{6} - 16 q^{7} - 74 q^{8} - 169 q^{9} + O(q^{10}) \) \( 11968 q - 32 q^{2} + 4 q^{3} - 30 q^{4} - 80 q^{5} - 46 q^{6} - 16 q^{7} - 74 q^{8} - 169 q^{9} - 40 q^{10} - 20 q^{11} - 110 q^{12} + 68 q^{13} - 30 q^{14} + 130 q^{16} + 464 q^{17} + 276 q^{18} + 116 q^{19} - 40 q^{20} - 644 q^{21} + 330 q^{22} - 832 q^{23} - 50 q^{24} - 800 q^{25} - 534 q^{26} - 1232 q^{27} - 910 q^{28} - 394 q^{29} - 40 q^{30} + 448 q^{31} - 662 q^{32} + 1700 q^{33} - 30 q^{34} + 860 q^{35} - 186 q^{36} - 550 q^{37} - 3410 q^{38} - 3624 q^{39} - 2090 q^{40} - 22 q^{41} - 270 q^{42} + 460 q^{43} + 2630 q^{44} + 2240 q^{45} + 2954 q^{46} + 3608 q^{47} + 9470 q^{48} + 4757 q^{49} + 4370 q^{50} + 4496 q^{51} + 6186 q^{52} + 2074 q^{53} + 7510 q^{54} + 760 q^{55} + 2834 q^{56} - 1964 q^{57} - 538 q^{58} - 268 q^{59} - 6140 q^{60} - 634 q^{61} - 8290 q^{62} - 4664 q^{63} - 9690 q^{64} - 235 q^{65} - 5190 q^{66} - 2476 q^{67} - 2458 q^{68} - 7092 q^{69} - 40 q^{70} - 3976 q^{71} - 1662 q^{72} - 3372 q^{73} - 50 q^{74} - 5920 q^{75} + 1750 q^{76} - 6860 q^{77} + 3150 q^{78} - 3952 q^{79} - 40 q^{80} - 9415 q^{81} + 17066 q^{82} - 6252 q^{83} + 22530 q^{84} - 6965 q^{85} + 7354 q^{86} + 2896 q^{87} + 5490 q^{88} + 6504 q^{89} - 1390 q^{90} + 7792 q^{91} - 10470 q^{92} + 16404 q^{93} - 22110 q^{94} + 3160 q^{95} - 16366 q^{96} + 34100 q^{97} - 27024 q^{98} + 10580 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(500))\)

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
500.4.a \(\chi_{500}(1, \cdot)\) 500.4.a.a 4 1
500.4.a.b 6
500.4.a.c 6
500.4.a.d 8
500.4.c \(\chi_{500}(249, \cdot)\) 500.4.c.a 4 1
500.4.c.b 8
500.4.c.c 12
500.4.e \(\chi_{500}(307, \cdot)\) n/a 288 2
500.4.g \(\chi_{500}(101, \cdot)\) 500.4.g.a 28 4
500.4.g.b 64
500.4.i \(\chi_{500}(49, \cdot)\) 500.4.i.a 32 4
500.4.i.b 56
500.4.l \(\chi_{500}(7, \cdot)\) n/a 1032 8
500.4.m \(\chi_{500}(21, \cdot)\) n/a 740 20
500.4.o \(\chi_{500}(9, \cdot)\) n/a 760 20
500.4.r \(\chi_{500}(3, \cdot)\) n/a 8920 40

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(500))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(500)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(125))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(250))\)\(^{\oplus 2}\)