## Defining parameters

 Level: $$N$$ = $$10 = 2 \cdot 5$$ Weight: $$k$$ = $$5$$ Nonzero newspaces: $$1$$ Newforms: $$2$$ Sturm bound: $$30$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(\Gamma_1(10))$$.

Total New Old
Modular forms 16 4 12
Cusp forms 8 4 4
Eisenstein series 8 0 8

## Trace form

 $$4q + 20q^{3} - 60q^{5} - 64q^{6} + 20q^{7} + O(q^{10})$$ $$4q + 20q^{3} - 60q^{5} - 64q^{6} + 20q^{7} + 160q^{10} + 168q^{11} + 160q^{12} - 60q^{13} + 20q^{15} - 256q^{16} - 1020q^{17} - 640q^{18} + 968q^{21} + 1280q^{22} + 1620q^{23} - 700q^{25} - 1344q^{26} + 320q^{27} - 160q^{28} + 960q^{30} - 1112q^{31} - 1720q^{33} - 2220q^{35} + 32q^{36} - 1020q^{37} + 960q^{38} + 1280q^{40} + 5448q^{41} - 2240q^{42} + 660q^{43} + 6400q^{45} + 2176q^{46} + 1620q^{47} - 1280q^{48} - 4800q^{50} - 10712q^{51} - 480q^{52} - 4860q^{53} - 2520q^{55} + 3072q^{56} + 5120q^{57} + 3520q^{58} - 4960q^{60} - 7032q^{61} - 3840q^{62} + 7700q^{63} + 7620q^{65} + 10112q^{66} + 8660q^{67} + 8160q^{68} + 1600q^{70} + 5928q^{71} - 5120q^{72} - 12860q^{73} - 13100q^{75} - 5120q^{76} - 14520q^{77} - 5760q^{78} + 3840q^{80} + 964q^{81} - 2560q^{82} - 300q^{83} + 16580q^{85} - 14784q^{86} + 11840q^{87} - 10240q^{88} + 9440q^{90} + 15528q^{91} + 12960q^{92} + 2120q^{93} - 9600q^{95} + 4096q^{96} - 13500q^{97} + 3840q^{98} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(\Gamma_1(10))$$

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
10.5.c $$\chi_{10}(3, \cdot)$$ 10.5.c.a 2 2
10.5.c.b 2

## Decomposition of $$S_{5}^{\mathrm{old}}(\Gamma_1(10))$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(\Gamma_1(10)) \cong$$ $$S_{5}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 2}$$