## Defining parameters

 Level: $$N$$ = $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$3$$ Newform subspaces: $$4$$ Sturm bound: $$18816$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(588))$$.

Total New Old
Modular forms 644 112 532
Cusp forms 44 16 28
Eisenstein series 600 96 504

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 16 0 0 0

## Trace form

 $$16q + 2q^{3} + q^{7} + 2q^{9} + O(q^{10})$$ $$16q + 2q^{3} + q^{7} + 2q^{9} + 4q^{13} - 2q^{19} - 2q^{21} + 2q^{25} - 4q^{27} - 2q^{31} - 9q^{37} - 9q^{39} - 2q^{43} + q^{49} - 2q^{57} - 3q^{61} - 6q^{63} - 2q^{67} - 2q^{73} + 2q^{75} - 2q^{79} + 2q^{81} - q^{91} - 2q^{93} - 8q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(588))$$

We only show spaces with odd 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
588.1.c $$\chi_{588}(197, \cdot)$$ 588.1.c.a 1 1
588.1.c.b 1
588.1.d $$\chi_{588}(97, \cdot)$$ None 0 1
588.1.g $$\chi_{588}(295, \cdot)$$ None 0 1
588.1.h $$\chi_{588}(587, \cdot)$$ None 0 1
588.1.j $$\chi_{588}(215, \cdot)$$ None 0 2
588.1.l $$\chi_{588}(67, \cdot)$$ None 0 2
588.1.m $$\chi_{588}(313, \cdot)$$ None 0 2
588.1.p $$\chi_{588}(557, \cdot)$$ 588.1.p.a 2 2
588.1.r $$\chi_{588}(83, \cdot)$$ None 0 6
588.1.s $$\chi_{588}(43, \cdot)$$ None 0 6
588.1.v $$\chi_{588}(13, \cdot)$$ None 0 6
588.1.w $$\chi_{588}(29, \cdot)$$ None 0 6
588.1.z $$\chi_{588}(53, \cdot)$$ 588.1.z.a 12 12
588.1.bc $$\chi_{588}(61, \cdot)$$ None 0 12
588.1.bd $$\chi_{588}(151, \cdot)$$ None 0 12
588.1.bf $$\chi_{588}(47, \cdot)$$ None 0 12

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(588))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(588)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 2}$$