## Defining parameters

 Level: $$N$$ = $$12 = 2^{2} \cdot 3$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$2$$ Newform subspaces: $$2$$ Sturm bound: $$32$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(12))$$.

Total New Old
Modular forms 17 9 8
Cusp forms 7 5 2
Eisenstein series 10 4 6

## Trace form

 $$5q + 3q^{3} - 8q^{4} - 18q^{5} - 24q^{6} + 8q^{7} - 3q^{9} + O(q^{10})$$ $$5q + 3q^{3} - 8q^{4} - 18q^{5} - 24q^{6} + 8q^{7} - 3q^{9} + 80q^{10} + 36q^{11} + 120q^{12} - 50q^{13} - 54q^{15} - 224q^{16} + 18q^{17} - 240q^{18} - 100q^{19} + 144q^{21} + 240q^{22} + 72q^{23} + 288q^{24} + 379q^{25} + 27q^{27} - 240q^{28} - 234q^{29} - 240q^{30} - 16q^{31} - 372q^{33} + 320q^{34} - 144q^{35} + 24q^{36} - 746q^{37} - 30q^{39} + 320q^{40} + 90q^{41} + 240q^{42} + 452q^{43} + 798q^{45} - 672q^{46} + 432q^{47} - 480q^{48} + 853q^{49} + 54q^{51} + 80q^{52} + 414q^{53} + 792q^{54} - 648q^{55} - 1380q^{57} - 1360q^{58} - 684q^{59} - 960q^{60} - 1346q^{61} + 72q^{63} + 1408q^{64} + 180q^{65} + 1200q^{66} + 332q^{67} + 1560q^{69} + 480q^{70} - 360q^{71} - 960q^{72} + 1666q^{73} + 597q^{75} + 2160q^{76} + 288q^{77} + 240q^{78} + 512q^{79} - 2763q^{81} - 1120q^{82} - 1188q^{83} - 240q^{84} - 1604q^{85} - 702q^{87} - 2880q^{88} - 630q^{89} - 240q^{90} - 80q^{91} + 3432q^{93} - 1344q^{94} + 1800q^{95} + 384q^{96} + 2026q^{97} + 324q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$

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
12.4.a $$\chi_{12}(1, \cdot)$$ 12.4.a.a 1 1
12.4.b $$\chi_{12}(11, \cdot)$$ 12.4.b.a 4 1

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(12))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(12)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 2}$$