Properties

Label 600.2
Level 600
Weight 2
Dimension 3651
Nonzero newspaces 18
Newforms 81
Sturm bound 38400
Trace bound 8

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Defining parameters

Level: \( N \) = \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 18 \)
Newforms: \( 81 \)
Sturm bound: \(38400\)
Trace bound: \(8\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(600))\).

Total New Old
Modular forms 10272 3815 6457
Cusp forms 8929 3651 5278
Eisenstein series 1343 164 1179

Trace form

\(3651q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 15q^{3} \) \(\mathstrut -\mathstrut 28q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 18q^{6} \) \(\mathstrut -\mathstrut 44q^{7} \) \(\mathstrut +\mathstrut 4q^{8} \) \(\mathstrut -\mathstrut 31q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3651q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 15q^{3} \) \(\mathstrut -\mathstrut 28q^{4} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 18q^{6} \) \(\mathstrut -\mathstrut 44q^{7} \) \(\mathstrut +\mathstrut 4q^{8} \) \(\mathstrut -\mathstrut 31q^{9} \) \(\mathstrut -\mathstrut 32q^{10} \) \(\mathstrut -\mathstrut 12q^{11} \) \(\mathstrut +\mathstrut 20q^{12} \) \(\mathstrut -\mathstrut 10q^{13} \) \(\mathstrut +\mathstrut 52q^{14} \) \(\mathstrut -\mathstrut 8q^{15} \) \(\mathstrut +\mathstrut 24q^{16} \) \(\mathstrut +\mathstrut 6q^{17} \) \(\mathstrut +\mathstrut 38q^{18} \) \(\mathstrut +\mathstrut 40q^{19} \) \(\mathstrut +\mathstrut 40q^{20} \) \(\mathstrut +\mathstrut 32q^{21} \) \(\mathstrut +\mathstrut 48q^{22} \) \(\mathstrut +\mathstrut 48q^{23} \) \(\mathstrut +\mathstrut 32q^{24} \) \(\mathstrut -\mathstrut 58q^{25} \) \(\mathstrut +\mathstrut 56q^{26} \) \(\mathstrut +\mathstrut 45q^{27} \) \(\mathstrut -\mathstrut 40q^{28} \) \(\mathstrut +\mathstrut 30q^{29} \) \(\mathstrut -\mathstrut 12q^{30} \) \(\mathstrut +\mathstrut 36q^{31} \) \(\mathstrut -\mathstrut 72q^{32} \) \(\mathstrut +\mathstrut 28q^{33} \) \(\mathstrut -\mathstrut 132q^{34} \) \(\mathstrut +\mathstrut 48q^{35} \) \(\mathstrut -\mathstrut 32q^{36} \) \(\mathstrut +\mathstrut 72q^{37} \) \(\mathstrut -\mathstrut 104q^{38} \) \(\mathstrut +\mathstrut 78q^{39} \) \(\mathstrut -\mathstrut 112q^{40} \) \(\mathstrut +\mathstrut 46q^{41} \) \(\mathstrut -\mathstrut 96q^{42} \) \(\mathstrut +\mathstrut 136q^{43} \) \(\mathstrut -\mathstrut 112q^{44} \) \(\mathstrut +\mathstrut 70q^{45} \) \(\mathstrut -\mathstrut 176q^{46} \) \(\mathstrut +\mathstrut 120q^{47} \) \(\mathstrut -\mathstrut 140q^{48} \) \(\mathstrut +\mathstrut 55q^{49} \) \(\mathstrut -\mathstrut 40q^{50} \) \(\mathstrut -\mathstrut 62q^{51} \) \(\mathstrut -\mathstrut 72q^{52} \) \(\mathstrut +\mathstrut 64q^{53} \) \(\mathstrut -\mathstrut 110q^{54} \) \(\mathstrut +\mathstrut 64q^{55} \) \(\mathstrut -\mathstrut 8q^{56} \) \(\mathstrut +\mathstrut 16q^{57} \) \(\mathstrut -\mathstrut 36q^{58} \) \(\mathstrut +\mathstrut 20q^{59} \) \(\mathstrut -\mathstrut 92q^{60} \) \(\mathstrut -\mathstrut 50q^{61} \) \(\mathstrut -\mathstrut 4q^{62} \) \(\mathstrut -\mathstrut 120q^{63} \) \(\mathstrut -\mathstrut 88q^{64} \) \(\mathstrut +\mathstrut 6q^{65} \) \(\mathstrut -\mathstrut 188q^{66} \) \(\mathstrut -\mathstrut 160q^{67} \) \(\mathstrut -\mathstrut 112q^{68} \) \(\mathstrut -\mathstrut 24q^{69} \) \(\mathstrut -\mathstrut 104q^{70} \) \(\mathstrut -\mathstrut 128q^{71} \) \(\mathstrut -\mathstrut 200q^{72} \) \(\mathstrut +\mathstrut 74q^{73} \) \(\mathstrut -\mathstrut 160q^{74} \) \(\mathstrut -\mathstrut 88q^{75} \) \(\mathstrut -\mathstrut 232q^{76} \) \(\mathstrut -\mathstrut 188q^{78} \) \(\mathstrut -\mathstrut 204q^{79} \) \(\mathstrut -\mathstrut 80q^{80} \) \(\mathstrut +\mathstrut 57q^{81} \) \(\mathstrut -\mathstrut 348q^{82} \) \(\mathstrut -\mathstrut 268q^{83} \) \(\mathstrut -\mathstrut 300q^{84} \) \(\mathstrut -\mathstrut 82q^{85} \) \(\mathstrut -\mathstrut 184q^{86} \) \(\mathstrut -\mathstrut 206q^{87} \) \(\mathstrut -\mathstrut 472q^{88} \) \(\mathstrut +\mathstrut 128q^{89} \) \(\mathstrut -\mathstrut 172q^{90} \) \(\mathstrut -\mathstrut 184q^{91} \) \(\mathstrut -\mathstrut 384q^{92} \) \(\mathstrut -\mathstrut 32q^{93} \) \(\mathstrut -\mathstrut 496q^{94} \) \(\mathstrut -\mathstrut 256q^{95} \) \(\mathstrut -\mathstrut 84q^{96} \) \(\mathstrut -\mathstrut 86q^{97} \) \(\mathstrut -\mathstrut 378q^{98} \) \(\mathstrut -\mathstrut 180q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(600))\)

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
600.2.a \(\chi_{600}(1, \cdot)\) 600.2.a.a 1 1
600.2.a.b 1
600.2.a.c 1
600.2.a.d 1
600.2.a.e 1
600.2.a.f 1
600.2.a.g 1
600.2.a.h 1
600.2.a.i 1
600.2.b \(\chi_{600}(251, \cdot)\) 600.2.b.a 2 1
600.2.b.b 4
600.2.b.c 4
600.2.b.d 4
600.2.b.e 8
600.2.b.f 8
600.2.b.g 12
600.2.b.h 12
600.2.b.i 16
600.2.d \(\chi_{600}(349, \cdot)\) 600.2.d.a 2 1
600.2.d.b 2
600.2.d.c 2
600.2.d.d 2
600.2.d.e 6
600.2.d.f 6
600.2.d.g 8
600.2.d.h 8
600.2.f \(\chi_{600}(49, \cdot)\) 600.2.f.a 2 1
600.2.f.b 2
600.2.f.c 2
600.2.f.d 2
600.2.f.e 2
600.2.h \(\chi_{600}(551, \cdot)\) None 0 1
600.2.k \(\chi_{600}(301, \cdot)\) 600.2.k.a 2 1
600.2.k.b 2
600.2.k.c 6
600.2.k.d 8
600.2.k.e 8
600.2.k.f 12
600.2.m \(\chi_{600}(299, \cdot)\) 600.2.m.a 4 1
600.2.m.b 8
600.2.m.c 16
600.2.m.d 16
600.2.m.e 24
600.2.o \(\chi_{600}(599, \cdot)\) None 0 1
600.2.r \(\chi_{600}(257, \cdot)\) 600.2.r.a 4 2
600.2.r.b 4
600.2.r.c 4
600.2.r.d 4
600.2.r.e 4
600.2.r.f 16
600.2.s \(\chi_{600}(7, \cdot)\) None 0 2
600.2.v \(\chi_{600}(43, \cdot)\) 600.2.v.a 16 2
600.2.v.b 24
600.2.v.c 32
600.2.w \(\chi_{600}(293, \cdot)\) 600.2.w.a 4 2
600.2.w.b 4
600.2.w.c 4
600.2.w.d 4
600.2.w.e 4
600.2.w.f 4
600.2.w.g 4
600.2.w.h 4
600.2.w.i 8
600.2.w.j 32
600.2.w.k 64
600.2.y \(\chi_{600}(121, \cdot)\) 600.2.y.a 4 4
600.2.y.b 4
600.2.y.c 12
600.2.y.d 12
600.2.y.e 16
600.2.y.f 16
600.2.ba \(\chi_{600}(71, \cdot)\) None 0 4
600.2.bc \(\chi_{600}(169, \cdot)\) 600.2.bc.a 8 4
600.2.bc.b 24
600.2.bc.c 24
600.2.be \(\chi_{600}(109, \cdot)\) 600.2.be.a 120 4
600.2.be.b 120
600.2.bg \(\chi_{600}(11, \cdot)\) 600.2.bg.a 464 4
600.2.bi \(\chi_{600}(119, \cdot)\) None 0 4
600.2.bk \(\chi_{600}(59, \cdot)\) 600.2.bk.a 464 4
600.2.bm \(\chi_{600}(61, \cdot)\) 600.2.bm.a 240 4
600.2.bp \(\chi_{600}(53, \cdot)\) 600.2.bp.a 16 8
600.2.bp.b 16
600.2.bp.c 896
600.2.bq \(\chi_{600}(67, \cdot)\) 600.2.bq.a 480 8
600.2.bt \(\chi_{600}(103, \cdot)\) None 0 8
600.2.bu \(\chi_{600}(17, \cdot)\) 600.2.bu.a 240 8

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(600))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(600)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(100))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(200))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(300))\)\(^{\oplus 2}\)