## 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

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

## 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

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