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Properties

Label 1000.2
Level 1000
Weight 2
Dimension 15296
Nonzero newspaces 15
Newform subspaces 56
Sturm bound 120000
Trace bound 9

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Defining parameters

Level: \( N \) = \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 15 \)
Newform subspaces: \( 56 \)
Sturm bound: \(120000\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(1000))\).

Total New Old
Modular forms 31080 15808 15272
Cusp forms 28921 15296 13625
Eisenstein series 2159 512 1647

Trace form

\( 15296 q - 64 q^{2} - 64 q^{3} - 64 q^{4} - 116 q^{6} - 68 q^{7} - 64 q^{8} - 133 q^{9} + O(q^{10}) \) \( 15296 q - 64 q^{2} - 64 q^{3} - 64 q^{4} - 116 q^{6} - 68 q^{7} - 64 q^{8} - 133 q^{9} - 80 q^{10} - 120 q^{11} - 52 q^{12} - 2 q^{13} - 52 q^{14} - 80 q^{15} - 100 q^{16} - 126 q^{17} - 36 q^{18} - 48 q^{19} - 80 q^{20} + 8 q^{21} - 44 q^{22} - 60 q^{23} - 36 q^{24} - 160 q^{25} - 140 q^{26} - 52 q^{27} - 68 q^{28} - 6 q^{29} - 80 q^{30} - 108 q^{31} - 84 q^{32} - 128 q^{33} - 96 q^{34} - 80 q^{35} - 148 q^{36} + 16 q^{37} - 84 q^{38} + 28 q^{39} - 80 q^{40} - 194 q^{41} - 84 q^{42} + 40 q^{43} - 76 q^{44} + 80 q^{45} - 140 q^{46} + 44 q^{47} - 68 q^{48} - 23 q^{49} - 80 q^{50} - 20 q^{51} - 40 q^{52} + 96 q^{53} - 36 q^{54} - 40 q^{55} - 60 q^{56} + 56 q^{57} - 16 q^{58} + 40 q^{59} - 80 q^{60} + 18 q^{61} - 4 q^{62} + 12 q^{63} - 28 q^{64} - 155 q^{65} - 92 q^{66} - 80 q^{67} - 40 q^{68} + 8 q^{69} - 80 q^{70} - 132 q^{71} - 192 q^{72} - 94 q^{73} - 80 q^{74} - 80 q^{75} - 276 q^{76} - 16 q^{77} - 124 q^{78} - 140 q^{79} - 80 q^{80} - 213 q^{81} - 256 q^{82} - 240 q^{83} - 412 q^{84} - 75 q^{85} - 340 q^{86} - 348 q^{87} - 540 q^{88} - 136 q^{89} - 380 q^{90} - 364 q^{91} - 476 q^{92} - 240 q^{93} - 668 q^{94} - 200 q^{95} - 460 q^{96} - 254 q^{97} - 608 q^{98} - 464 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(1000))\)

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
1000.2.a \(\chi_{1000}(1, \cdot)\) 1000.2.a.a 2 1
1000.2.a.b 2
1000.2.a.c 2
1000.2.a.d 2
1000.2.a.e 4
1000.2.a.f 4
1000.2.a.g 4
1000.2.a.h 4
1000.2.c \(\chi_{1000}(249, \cdot)\) 1000.2.c.a 4 1
1000.2.c.b 4
1000.2.c.c 8
1000.2.c.d 8
1000.2.d \(\chi_{1000}(501, \cdot)\) 1000.2.d.a 4 1
1000.2.d.b 4
1000.2.d.c 40
1000.2.d.d 48
1000.2.f \(\chi_{1000}(749, \cdot)\) 1000.2.f.a 4 1
1000.2.f.b 4
1000.2.f.c 20
1000.2.f.d 20
1000.2.f.e 48
1000.2.j \(\chi_{1000}(807, \cdot)\) None 0 2
1000.2.k \(\chi_{1000}(307, \cdot)\) 1000.2.k.a 8 2
1000.2.k.b 8
1000.2.k.c 16
1000.2.k.d 16
1000.2.k.e 16
1000.2.k.f 64
1000.2.k.g 64
1000.2.m \(\chi_{1000}(201, \cdot)\) 1000.2.m.a 4 4
1000.2.m.b 8
1000.2.m.c 16
1000.2.m.d 32
1000.2.m.e 32
1000.2.o \(\chi_{1000}(149, \cdot)\) 1000.2.o.a 112 4
1000.2.o.b 224
1000.2.q \(\chi_{1000}(49, \cdot)\) 1000.2.q.a 8 4
1000.2.q.b 16
1000.2.q.c 32
1000.2.q.d 32
1000.2.t \(\chi_{1000}(101, \cdot)\) 1000.2.t.a 112 4
1000.2.t.b 224
1000.2.v \(\chi_{1000}(43, \cdot)\) 1000.2.v.a 8 8
1000.2.v.b 8
1000.2.v.c 8
1000.2.v.d 8
1000.2.v.e 8
1000.2.v.f 8
1000.2.v.g 208
1000.2.v.h 208
1000.2.v.i 208
1000.2.w \(\chi_{1000}(7, \cdot)\) None 0 8
1000.2.y \(\chi_{1000}(41, \cdot)\) 1000.2.y.a 360 20
1000.2.y.b 380
1000.2.bb \(\chi_{1000}(21, \cdot)\) 1000.2.bb.a 2960 20
1000.2.bd \(\chi_{1000}(29, \cdot)\) 1000.2.bd.a 2960 20
1000.2.be \(\chi_{1000}(9, \cdot)\) 1000.2.be.a 760 20
1000.2.bh \(\chi_{1000}(3, \cdot)\) 1000.2.bh.a 5920 40
1000.2.bi \(\chi_{1000}(23, \cdot)\) None 0 40

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(1000))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(1000)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(125))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(250))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(500))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1000))\)\(^{\oplus 1}\)