## Defining parameters

 Level: $$N$$ = $$39 = 3 \cdot 13$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$6$$ Newform subspaces: $$14$$ Sturm bound: $$448$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(39))$$.

Total New Old
Modular forms 192 138 54
Cusp forms 144 114 30
Eisenstein series 48 24 24

## Trace form

 $$114 q - 6 q^{3} - 12 q^{4} - 6 q^{6} + 60 q^{7} + 144 q^{8} - 6 q^{9} + O(q^{10})$$ $$114 q - 6 q^{3} - 12 q^{4} - 6 q^{6} + 60 q^{7} + 144 q^{8} - 6 q^{9} - 192 q^{10} - 120 q^{11} - 240 q^{12} - 300 q^{13} - 240 q^{14} - 78 q^{15} - 12 q^{16} + 378 q^{17} + 660 q^{18} + 876 q^{19} + 1032 q^{20} - 6 q^{21} - 792 q^{22} - 456 q^{23} - 1158 q^{24} - 750 q^{25} - 1500 q^{26} - 840 q^{27} - 1056 q^{28} - 390 q^{29} - 1062 q^{30} + 108 q^{31} + 1020 q^{32} + 882 q^{33} + 2244 q^{34} + 1560 q^{35} + 3060 q^{36} + 630 q^{37} + 1746 q^{39} + 3864 q^{40} + 2478 q^{41} + 2496 q^{42} + 1332 q^{43} + 300 q^{44} - 288 q^{45} - 2532 q^{46} - 1488 q^{47} - 3366 q^{48} - 3972 q^{49} - 4416 q^{50} - 2448 q^{51} - 1656 q^{52} - 1896 q^{53} + 1698 q^{54} - 2820 q^{55} - 2772 q^{56} - 1170 q^{57} - 2064 q^{58} - 984 q^{59} - 2664 q^{60} + 2250 q^{61} - 636 q^{62} - 5154 q^{63} + 1800 q^{64} + 2430 q^{65} - 2472 q^{66} + 228 q^{67} - 4680 q^{68} - 1674 q^{69} - 960 q^{70} - 528 q^{71} + 3084 q^{72} + 1140 q^{73} + 5772 q^{74} + 7896 q^{75} + 12468 q^{76} + 6624 q^{77} + 14604 q^{78} + 5496 q^{79} + 9312 q^{80} + 5610 q^{81} - 288 q^{82} - 1224 q^{83} + 3600 q^{84} - 30 q^{85} + 6384 q^{86} + 2646 q^{87} - 3888 q^{88} + 3480 q^{89} - 2160 q^{90} + 396 q^{91} - 6360 q^{92} - 7554 q^{93} - 10620 q^{94} - 7080 q^{95} - 17244 q^{96} - 11028 q^{97} - 9516 q^{98} - 11394 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(39))$$

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
39.4.a $$\chi_{39}(1, \cdot)$$ 39.4.a.a 1 1
39.4.a.b 2
39.4.a.c 3
39.4.b $$\chi_{39}(25, \cdot)$$ 39.4.b.a 4 1
39.4.b.b 4
39.4.e $$\chi_{39}(16, \cdot)$$ 39.4.e.a 2 2
39.4.e.b 2
39.4.e.c 8
39.4.f $$\chi_{39}(5, \cdot)$$ 39.4.f.a 4 2
39.4.f.b 20
39.4.j $$\chi_{39}(4, \cdot)$$ 39.4.j.a 2 2
39.4.j.b 4
39.4.j.c 10
39.4.k $$\chi_{39}(2, \cdot)$$ 39.4.k.a 48 4

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(39))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(39)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 2}$$