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Properties

Label 2000.2
Level 2000
Weight 2
Dimension 61632
Nonzero newspaces 21
Sturm bound 480000
Trace bound 16

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Defining parameters

Level: \( N \) = \( 2000 = 2^{4} \cdot 5^{3} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 21 \)
Sturm bound: \(480000\)
Trace bound: \(16\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(2000))\).

Total New Old
Modular forms 122520 62784 59736
Cusp forms 117481 61632 55849
Eisenstein series 5039 1152 3887

Trace form

\( 61632 q - 128 q^{2} - 96 q^{3} - 128 q^{4} - 200 q^{5} - 232 q^{6} - 96 q^{7} - 128 q^{8} - 34 q^{9} + O(q^{10}) \) \( 61632 q - 128 q^{2} - 96 q^{3} - 128 q^{4} - 200 q^{5} - 232 q^{6} - 96 q^{7} - 128 q^{8} - 34 q^{9} - 160 q^{10} - 174 q^{11} - 128 q^{12} - 164 q^{13} - 128 q^{14} - 120 q^{15} - 232 q^{16} - 296 q^{17} - 128 q^{18} - 106 q^{19} - 160 q^{20} - 306 q^{21} - 128 q^{22} - 108 q^{23} - 128 q^{24} - 40 q^{25} - 312 q^{26} - 102 q^{27} - 96 q^{28} - 158 q^{29} - 160 q^{30} - 166 q^{31} - 88 q^{32} - 262 q^{33} - 64 q^{34} - 120 q^{35} - 168 q^{36} - 120 q^{37} - 72 q^{38} - 70 q^{39} - 160 q^{40} - 26 q^{41} - 48 q^{42} - 80 q^{43} - 72 q^{44} - 200 q^{45} - 168 q^{46} - 96 q^{47} - 64 q^{48} - 254 q^{49} - 160 q^{50} - 250 q^{51} - 96 q^{52} - 176 q^{53} - 80 q^{54} - 120 q^{55} - 232 q^{56} - 70 q^{57} - 128 q^{58} - 130 q^{59} - 160 q^{60} - 306 q^{61} - 128 q^{62} - 56 q^{63} - 152 q^{64} - 360 q^{65} - 296 q^{66} - 60 q^{67} - 184 q^{68} - 162 q^{69} - 160 q^{70} - 118 q^{71} - 248 q^{72} - 52 q^{73} - 216 q^{74} - 120 q^{75} - 376 q^{76} - 150 q^{77} - 264 q^{78} - 2 q^{79} - 160 q^{80} - 452 q^{81} - 192 q^{82} + 112 q^{83} - 280 q^{84} - 100 q^{85} - 296 q^{86} + 266 q^{87} - 216 q^{88} + 106 q^{89} - 160 q^{90} + 98 q^{91} - 168 q^{92} + 146 q^{93} - 184 q^{94} - 296 q^{96} + 4 q^{97} - 96 q^{98} + 318 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(2000))\)

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
2000.2.a \(\chi_{2000}(1, \cdot)\) 2000.2.a.a 2 1
2000.2.a.b 2
2000.2.a.c 2
2000.2.a.d 2
2000.2.a.e 2
2000.2.a.f 2
2000.2.a.g 2
2000.2.a.h 2
2000.2.a.i 2
2000.2.a.j 2
2000.2.a.k 2
2000.2.a.l 2
2000.2.a.m 4
2000.2.a.n 4
2000.2.a.o 4
2000.2.a.p 4
2000.2.a.q 4
2000.2.a.r 4
2000.2.c \(\chi_{2000}(1249, \cdot)\) 2000.2.c.a 4 1
2000.2.c.b 4
2000.2.c.c 4
2000.2.c.d 4
2000.2.c.e 4
2000.2.c.f 4
2000.2.c.g 4
2000.2.c.h 4
2000.2.c.i 8
2000.2.c.j 8
2000.2.d \(\chi_{2000}(1001, \cdot)\) None 0 1
2000.2.f \(\chi_{2000}(249, \cdot)\) None 0 1
2000.2.j \(\chi_{2000}(307, \cdot)\) n/a 384 2
2000.2.l \(\chi_{2000}(501, \cdot)\) n/a 384 2
2000.2.n \(\chi_{2000}(943, \cdot)\) 2000.2.n.a 8 2
2000.2.n.b 8
2000.2.n.c 8
2000.2.n.d 8
2000.2.n.e 32
2000.2.n.f 32
2000.2.o \(\chi_{2000}(807, \cdot)\) None 0 2
2000.2.q \(\chi_{2000}(749, \cdot)\) n/a 384 2
2000.2.s \(\chi_{2000}(1307, \cdot)\) n/a 384 2
2000.2.u \(\chi_{2000}(401, \cdot)\) n/a 168 4
2000.2.w \(\chi_{2000}(649, \cdot)\) None 0 4
2000.2.y \(\chi_{2000}(49, \cdot)\) n/a 168 4
2000.2.bb \(\chi_{2000}(201, \cdot)\) None 0 4
2000.2.bd \(\chi_{2000}(107, \cdot)\) n/a 1392 8
2000.2.be \(\chi_{2000}(101, \cdot)\) n/a 1392 8
2000.2.bh \(\chi_{2000}(7, \cdot)\) None 0 8
2000.2.bi \(\chi_{2000}(143, \cdot)\) n/a 360 8
2000.2.bl \(\chi_{2000}(149, \cdot)\) n/a 1392 8
2000.2.bm \(\chi_{2000}(43, \cdot)\) n/a 1392 8
2000.2.bo \(\chi_{2000}(81, \cdot)\) n/a 1480 20
2000.2.br \(\chi_{2000}(41, \cdot)\) None 0 20
2000.2.bt \(\chi_{2000}(9, \cdot)\) None 0 20
2000.2.bu \(\chi_{2000}(129, \cdot)\) n/a 1480 20
2000.2.bx \(\chi_{2000}(29, \cdot)\) n/a 11920 40
2000.2.by \(\chi_{2000}(67, \cdot)\) n/a 11920 40
2000.2.cb \(\chi_{2000}(23, \cdot)\) None 0 40
2000.2.cc \(\chi_{2000}(47, \cdot)\) n/a 3000 40
2000.2.cf \(\chi_{2000}(3, \cdot)\) n/a 11920 40
2000.2.cg \(\chi_{2000}(21, \cdot)\) n/a 11920 40

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

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(2000))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(2000)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 20}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(125))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(250))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(400))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(500))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(1000))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2000))\)\(^{\oplus 1}\)