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Properties

Label 1600.2
Level 1600
Weight 2
Dimension 39365
Nonzero newspaces 28
Sturm bound 307200
Trace bound 12

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Defining parameters

Level: \( N \) = \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 28 \)
Sturm bound: \(307200\)
Trace bound: \(12\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(1600))\).

Total New Old
Modular forms 78816 40267 38549
Cusp forms 74785 39365 35420
Eisenstein series 4031 902 3129

Trace form

\( 39365 q - 104 q^{2} - 78 q^{3} - 104 q^{4} - 128 q^{5} - 168 q^{6} - 80 q^{7} - 104 q^{8} - 133 q^{9} + O(q^{10}) \) \( 39365 q - 104 q^{2} - 78 q^{3} - 104 q^{4} - 128 q^{5} - 168 q^{6} - 80 q^{7} - 104 q^{8} - 133 q^{9} - 128 q^{10} - 130 q^{11} - 104 q^{12} - 112 q^{13} - 104 q^{14} - 96 q^{15} - 168 q^{16} - 190 q^{17} - 104 q^{18} - 86 q^{19} - 128 q^{20} - 164 q^{21} - 96 q^{22} - 80 q^{23} - 64 q^{24} - 160 q^{25} - 288 q^{26} - 72 q^{27} - 64 q^{28} - 88 q^{29} - 128 q^{30} - 104 q^{31} - 64 q^{32} - 52 q^{33} - 64 q^{34} - 96 q^{35} - 88 q^{36} - 96 q^{37} - 64 q^{38} - 64 q^{39} - 128 q^{40} - 210 q^{41} - 64 q^{42} - 90 q^{43} - 96 q^{44} - 128 q^{45} - 168 q^{46} - 96 q^{47} - 104 q^{48} - 191 q^{49} - 128 q^{50} - 220 q^{51} - 152 q^{52} - 128 q^{53} - 168 q^{54} - 96 q^{55} - 224 q^{56} - 88 q^{57} - 176 q^{58} - 10 q^{59} - 128 q^{60} - 176 q^{61} - 136 q^{62} + 100 q^{63} - 200 q^{64} - 320 q^{65} - 248 q^{66} + 106 q^{67} - 152 q^{68} + 76 q^{69} - 128 q^{70} + 64 q^{71} - 176 q^{72} - 2 q^{73} - 160 q^{74} + 16 q^{75} - 392 q^{76} + 92 q^{77} - 128 q^{78} + 96 q^{79} - 128 q^{80} - 35 q^{81} - 24 q^{82} + 82 q^{83} + 8 q^{84} - 32 q^{85} - 64 q^{86} + 144 q^{87} - 24 q^{88} + 46 q^{89} - 128 q^{90} - 4 q^{91} + 48 q^{92} + 128 q^{93} - 8 q^{94} - 48 q^{95} - 32 q^{96} + 26 q^{97} + 32 q^{98} + 90 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(1600))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
1600.2.a \(\chi_{1600}(1, \cdot)\) 1600.2.a.a 1 1
1600.2.a.b 1
1600.2.a.c 1
1600.2.a.d 1
1600.2.a.e 1
1600.2.a.f 1
1600.2.a.g 1
1600.2.a.h 1
1600.2.a.i 1
1600.2.a.j 1
1600.2.a.k 1
1600.2.a.l 1
1600.2.a.m 1
1600.2.a.n 1
1600.2.a.o 1
1600.2.a.p 1
1600.2.a.q 1
1600.2.a.r 1
1600.2.a.s 1
1600.2.a.t 1
1600.2.a.u 1
1600.2.a.v 1
1600.2.a.w 1
1600.2.a.x 1
1600.2.a.y 1
1600.2.a.z 2
1600.2.a.ba 2
1600.2.a.bb 2
1600.2.a.bc 2
1600.2.a.bd 2
1600.2.c \(\chi_{1600}(449, \cdot)\) 1600.2.c.a 2 1
1600.2.c.b 2
1600.2.c.c 2
1600.2.c.d 2
1600.2.c.e 2
1600.2.c.f 2
1600.2.c.g 2
1600.2.c.h 2
1600.2.c.i 2
1600.2.c.j 2
1600.2.c.k 2
1600.2.c.l 2
1600.2.c.m 2
1600.2.c.n 4
1600.2.c.o 4
1600.2.d \(\chi_{1600}(801, \cdot)\) 1600.2.d.a 2 1
1600.2.d.b 4
1600.2.d.c 4
1600.2.d.d 4
1600.2.d.e 4
1600.2.d.f 4
1600.2.d.g 4
1600.2.d.h 4
1600.2.d.i 8
1600.2.f \(\chi_{1600}(1249, \cdot)\) 1600.2.f.a 2 1
1600.2.f.b 2
1600.2.f.c 4
1600.2.f.d 4
1600.2.f.e 4
1600.2.f.f 4
1600.2.f.g 4
1600.2.f.h 4
1600.2.f.i 4
1600.2.f.j 4
1600.2.j \(\chi_{1600}(143, \cdot)\) 1600.2.j.a 2 2
1600.2.j.b 8
1600.2.j.c 16
1600.2.j.d 18
1600.2.j.e 24
1600.2.l \(\chi_{1600}(401, \cdot)\) 1600.2.l.a 2 2
1600.2.l.b 2
1600.2.l.c 2
1600.2.l.d 4
1600.2.l.e 4
1600.2.l.f 12
1600.2.l.g 12
1600.2.l.h 16
1600.2.l.i 16
1600.2.n \(\chi_{1600}(1343, \cdot)\) 1600.2.n.a 2 2
1600.2.n.b 2
1600.2.n.c 2
1600.2.n.d 2
1600.2.n.e 2
1600.2.n.f 2
1600.2.n.g 2
1600.2.n.h 2
1600.2.n.i 2
1600.2.n.j 2
1600.2.n.k 2
1600.2.n.l 2
1600.2.n.m 2
1600.2.n.n 2
1600.2.n.o 4
1600.2.n.p 4
1600.2.n.q 4
1600.2.n.r 4
1600.2.n.s 4
1600.2.n.t 4
1600.2.n.u 8
1600.2.n.v 8
1600.2.o \(\chi_{1600}(543, \cdot)\) 1600.2.o.a 2 2
1600.2.o.b 2
1600.2.o.c 2
1600.2.o.d 2
1600.2.o.e 8
1600.2.o.f 8
1600.2.o.g 8
1600.2.o.h 8
1600.2.o.i 8
1600.2.o.j 8
1600.2.o.k 8
1600.2.o.l 8
1600.2.q \(\chi_{1600}(49, \cdot)\) 1600.2.q.a 2 2
1600.2.q.b 2
1600.2.q.c 4
1600.2.q.d 4
1600.2.q.e 12
1600.2.q.f 12
1600.2.q.g 16
1600.2.q.h 16
1600.2.s \(\chi_{1600}(207, \cdot)\) 1600.2.s.a 2 2
1600.2.s.b 8
1600.2.s.c 16
1600.2.s.d 18
1600.2.s.e 24
1600.2.u \(\chi_{1600}(321, \cdot)\) n/a 232 4
1600.2.v \(\chi_{1600}(407, \cdot)\) None 0 4
1600.2.y \(\chi_{1600}(201, \cdot)\) None 0 4
1600.2.ba \(\chi_{1600}(249, \cdot)\) None 0 4
1600.2.bb \(\chi_{1600}(7, \cdot)\) None 0 4
1600.2.be \(\chi_{1600}(289, \cdot)\) n/a 240 4
1600.2.bg \(\chi_{1600}(129, \cdot)\) n/a 232 4
1600.2.bj \(\chi_{1600}(161, \cdot)\) n/a 240 4
1600.2.bl \(\chi_{1600}(43, \cdot)\) n/a 1136 8
1600.2.bm \(\chi_{1600}(101, \cdot)\) n/a 1192 8
1600.2.bn \(\chi_{1600}(149, \cdot)\) n/a 1136 8
1600.2.br \(\chi_{1600}(107, \cdot)\) n/a 1136 8
1600.2.bt \(\chi_{1600}(303, \cdot)\) n/a 464 8
1600.2.bu \(\chi_{1600}(81, \cdot)\) n/a 464 8
1600.2.bx \(\chi_{1600}(223, \cdot)\) n/a 480 8
1600.2.by \(\chi_{1600}(63, \cdot)\) n/a 464 8
1600.2.cb \(\chi_{1600}(209, \cdot)\) n/a 464 8
1600.2.cc \(\chi_{1600}(47, \cdot)\) n/a 464 8
1600.2.cf \(\chi_{1600}(87, \cdot)\) None 0 16
1600.2.cg \(\chi_{1600}(9, \cdot)\) None 0 16
1600.2.ci \(\chi_{1600}(41, \cdot)\) None 0 16
1600.2.cl \(\chi_{1600}(23, \cdot)\) None 0 16
1600.2.cm \(\chi_{1600}(3, \cdot)\) n/a 7616 32
1600.2.cq \(\chi_{1600}(29, \cdot)\) n/a 7616 32
1600.2.cr \(\chi_{1600}(21, \cdot)\) n/a 7616 32
1600.2.cs \(\chi_{1600}(67, \cdot)\) n/a 7616 32

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

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(1600))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(1600)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(80))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(160))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(320))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(400))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(800))\)\(^{\oplus 2}\)