# Properties

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

## 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

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

## 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 available 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

$$S_{2}^{\mathrm{old}}(\Gamma_1(588)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 18}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(21))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(42))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(49))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(98))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(294))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(588))$$$$^{\oplus 1}$$