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Properties

Label 1500.4
Level 1500
Weight 4
Dimension 72320
Nonzero newspaces 18
Sturm bound 480000
Trace bound 16

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Defining parameters

Level: \( N \) = \( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 18 \)
Sturm bound: \(480000\)
Trace bound: \(16\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(1500))\).

Total New Old
Modular forms 181800 72832 108968
Cusp forms 178200 72320 105880
Eisenstein series 3600 512 3088

Trace form

\( 72320 q - 4 q^{3} - 68 q^{4} - 70 q^{6} + 32 q^{7} + 84 q^{8} - 50 q^{9} + O(q^{10}) \) \( 72320 q - 4 q^{3} - 68 q^{4} - 70 q^{6} + 32 q^{7} + 84 q^{8} - 50 q^{9} - 80 q^{10} + 56 q^{11} + 46 q^{12} - 296 q^{13} - 492 q^{16} - 960 q^{17} - 254 q^{18} - 280 q^{19} + 532 q^{21} - 356 q^{22} + 1640 q^{23} + 14 q^{24} + 1280 q^{25} + 368 q^{26} - 292 q^{27} + 828 q^{28} - 28 q^{29} - 40 q^{30} - 1328 q^{31} + 340 q^{32} - 2588 q^{33} - 356 q^{34} - 1720 q^{35} + 810 q^{36} + 988 q^{37} + 7928 q^{38} + 4880 q^{39} + 4020 q^{40} + 1148 q^{41} + 962 q^{42} + 424 q^{43} - 5320 q^{44} - 2400 q^{45} - 7124 q^{46} - 5680 q^{47} - 11470 q^{48} - 7778 q^{49} - 8820 q^{50} - 4256 q^{51} - 12380 q^{52} - 5788 q^{53} - 6066 q^{54} - 1520 q^{55} - 3152 q^{56} - 2052 q^{57} + 3748 q^{58} + 4432 q^{59} + 6060 q^{60} + 4068 q^{61} + 19728 q^{62} + 13544 q^{63} + 21124 q^{64} + 1910 q^{65} + 6026 q^{66} + 2312 q^{67} + 4432 q^{68} - 2388 q^{69} - 80 q^{70} + 1680 q^{71} - 882 q^{72} + 2000 q^{73} - 5920 q^{75} - 308 q^{76} - 1104 q^{77} - 1874 q^{78} + 1304 q^{79} - 5974 q^{81} - 37428 q^{82} + 10120 q^{83} - 27334 q^{84} + 16490 q^{85} - 20368 q^{86} + 6120 q^{87} - 20852 q^{88} + 4360 q^{89} + 500 q^{90} - 9728 q^{91} + 14312 q^{92} - 11148 q^{93} + 36364 q^{94} - 6320 q^{95} + 12866 q^{96} - 69416 q^{97} + 51032 q^{98} + 1008 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(1500))\)

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
1500.4.a \(\chi_{1500}(1, \cdot)\) 1500.4.a.a 6 1
1500.4.a.b 6
1500.4.a.c 6
1500.4.a.d 6
1500.4.a.e 6
1500.4.a.f 6
1500.4.a.g 6
1500.4.a.h 6
1500.4.d \(\chi_{1500}(1249, \cdot)\) 1500.4.d.a 12 1
1500.4.d.b 12
1500.4.d.c 12
1500.4.d.d 12
1500.4.e \(\chi_{1500}(251, \cdot)\) n/a 576 1
1500.4.h \(\chi_{1500}(1499, \cdot)\) n/a 576 1
1500.4.i \(\chi_{1500}(557, \cdot)\) n/a 192 2
1500.4.j \(\chi_{1500}(307, \cdot)\) n/a 576 2
1500.4.m \(\chi_{1500}(301, \cdot)\) n/a 176 4
1500.4.n \(\chi_{1500}(551, \cdot)\) n/a 2112 4
1500.4.o \(\chi_{1500}(49, \cdot)\) n/a 184 4
1500.4.r \(\chi_{1500}(299, \cdot)\) n/a 2112 4
1500.4.w \(\chi_{1500}(7, \cdot)\) n/a 2160 8
1500.4.x \(\chi_{1500}(257, \cdot)\) n/a 720 8
1500.4.y \(\chi_{1500}(61, \cdot)\) n/a 1520 20
1500.4.bb \(\chi_{1500}(59, \cdot)\) n/a 17920 20
1500.4.bc \(\chi_{1500}(109, \cdot)\) n/a 1480 20
1500.4.bf \(\chi_{1500}(11, \cdot)\) n/a 17920 20
1500.4.bh \(\chi_{1500}(67, \cdot)\) n/a 18000 40
1500.4.bi \(\chi_{1500}(17, \cdot)\) n/a 6000 40

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(1500))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(1500)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 24}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(125))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(150))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(250))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(300))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(375))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(500))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(750))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(1500))\)\(^{\oplus 1}\)