Properties

Label 588.2
Level 588
Weight 2
Dimension 3852
Nonzero newspaces 16
Newform subspaces 61
Sturm bound 37632
Trace bound 3

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Defining parameters

Level: \( N \) = \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 16 \)
Newform subspaces: \( 61 \)
Sturm bound: \(37632\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(588))\).

Total New Old
Modular forms 10008 4048 5960
Cusp forms 8809 3852 4957
Eisenstein series 1199 196 1003

Trace form

\( 3852q - 2q^{3} - 30q^{4} - 12q^{5} - 9q^{6} - 8q^{7} + 12q^{8} - 20q^{9} + O(q^{10}) \) \( 3852q - 2q^{3} - 30q^{4} - 12q^{5} - 9q^{6} - 8q^{7} + 12q^{8} - 20q^{9} + 6q^{10} + 12q^{11} + 15q^{12} - 16q^{13} + 24q^{14} + 36q^{15} + 18q^{16} + 24q^{17} + 15q^{18} + 32q^{19} - 7q^{21} - 102q^{22} + 24q^{23} - 33q^{24} - 60q^{26} + 4q^{27} - 84q^{28} + 72q^{29} - 81q^{30} + 8q^{31} - 60q^{32} - 48q^{33} - 90q^{34} - 6q^{35} - 99q^{36} - 36q^{38} - 27q^{39} - 90q^{40} + 36q^{41} - 69q^{42} + 4q^{43} - 48q^{44} - 78q^{45} - 90q^{46} + 48q^{47} - 150q^{48} + 66q^{49} - 84q^{50} + 42q^{51} - 90q^{52} + 36q^{53} - 117q^{54} + 210q^{55} - 18q^{56} - 16q^{57} - 42q^{58} + 108q^{59} - 45q^{60} + 94q^{61} + 24q^{63} + 6q^{64} + 12q^{65} + 15q^{66} + 120q^{67} + 72q^{68} - 18q^{69} + 30q^{70} + 39q^{72} + 32q^{73} + 132q^{74} - 20q^{75} + 126q^{76} + 93q^{78} + 24q^{79} + 18q^{80} - 272q^{81} - 84q^{82} - 144q^{83} - 138q^{84} - 252q^{85} - 162q^{86} - 192q^{87} - 264q^{88} - 192q^{89} - 180q^{90} - 188q^{91} - 126q^{92} - 304q^{93} - 372q^{94} - 300q^{95} - 246q^{96} - 448q^{97} - 546q^{98} - 60q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(588))\)

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
588.2.a \(\chi_{588}(1, \cdot)\) 588.2.a.a 1 1
588.2.a.b 1
588.2.a.c 1
588.2.a.d 1
588.2.a.e 1
588.2.a.f 1
588.2.b \(\chi_{588}(391, \cdot)\) 588.2.b.a 8 1
588.2.b.b 8
588.2.b.c 12
588.2.b.d 12
588.2.e \(\chi_{588}(491, \cdot)\) 588.2.e.a 4 1
588.2.e.b 8
588.2.e.c 12
588.2.e.d 12
588.2.e.e 12
588.2.e.f 24
588.2.f \(\chi_{588}(293, \cdot)\) 588.2.f.a 2 1
588.2.f.b 2
588.2.f.c 2
588.2.f.d 8
588.2.i \(\chi_{588}(361, \cdot)\) 588.2.i.a 2 2
588.2.i.b 2
588.2.i.c 2
588.2.i.d 2
588.2.i.e 2
588.2.i.f 2
588.2.i.g 2
588.2.k \(\chi_{588}(509, \cdot)\) 588.2.k.a 2 2
588.2.k.b 2
588.2.k.c 2
588.2.k.d 2
588.2.k.e 2
588.2.k.f 16
588.2.n \(\chi_{588}(263, \cdot)\) 588.2.n.a 4 2
588.2.n.b 4
588.2.n.c 16
588.2.n.d 24
588.2.n.e 24
588.2.n.f 24
588.2.n.g 24
588.2.n.h 24
588.2.o \(\chi_{588}(19, \cdot)\) 588.2.o.a 8 2
588.2.o.b 8
588.2.o.c 8
588.2.o.d 8
588.2.o.e 24
588.2.o.f 24
588.2.q \(\chi_{588}(85, \cdot)\) 588.2.q.a 30 6
588.2.q.b 30
588.2.t \(\chi_{588}(41, \cdot)\) 588.2.t.a 12 6
588.2.t.b 96
588.2.u \(\chi_{588}(71, \cdot)\) 588.2.u.a 648 6
588.2.x \(\chi_{588}(55, \cdot)\) 588.2.x.a 168 6
588.2.x.b 168
588.2.y \(\chi_{588}(25, \cdot)\) 588.2.y.a 48 12
588.2.y.b 60
588.2.ba \(\chi_{588}(103, \cdot)\) 588.2.ba.a 336 12
588.2.ba.b 336
588.2.bb \(\chi_{588}(11, \cdot)\) 588.2.bb.a 1296 12
588.2.be \(\chi_{588}(5, \cdot)\) 588.2.be.a 12 12
588.2.be.b 216

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(588))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(588)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(294))\)\(^{\oplus 2}\)