## Defining parameters

 Level: $$N$$ = $$54 = 2 \cdot 3^{3}$$ Weight: $$k$$ = $$7$$ Nonzero newspaces: $$3$$ Newform subspaces: $$5$$ Sturm bound: $$1134$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{7}(\Gamma_1(54))$$.

Total New Old
Modular forms 516 128 388
Cusp forms 456 128 328
Eisenstein series 60 0 60

## Trace form

 $$128 q - 64 q^{4} - 864 q^{5} + 336 q^{6} + 1444 q^{7} - 2496 q^{9} + O(q^{10})$$ $$128 q - 64 q^{4} - 864 q^{5} + 336 q^{6} + 1444 q^{7} - 2496 q^{9} - 1824 q^{10} - 756 q^{11} + 960 q^{12} + 868 q^{13} + 9504 q^{14} + 5112 q^{15} + 2048 q^{16} - 22080 q^{18} - 18644 q^{19} + 13824 q^{20} + 16080 q^{21} + 14784 q^{22} + 16416 q^{23} + 57458 q^{25} + 3078 q^{27} - 23168 q^{28} + 58536 q^{29} + 149184 q^{30} + 52276 q^{31} - 57294 q^{33} + 42912 q^{34} - 536544 q^{35} - 120576 q^{36} - 197060 q^{37} + 148176 q^{38} + 81696 q^{39} + 58368 q^{40} + 353916 q^{41} + 222528 q^{42} + 177916 q^{43} + 327132 q^{45} - 154176 q^{46} - 1069524 q^{47} - 98304 q^{48} - 111162 q^{49} - 622080 q^{50} - 1066716 q^{51} - 27776 q^{52} + 372240 q^{54} + 693396 q^{55} + 304128 q^{56} + 1674126 q^{57} + 420576 q^{58} + 486378 q^{59} + 354816 q^{60} - 51908 q^{61} - 1675032 q^{63} + 1114112 q^{64} - 3695112 q^{65} - 786240 q^{66} + 928156 q^{67} - 997056 q^{68} + 3773592 q^{69} + 741552 q^{70} + 855360 q^{71} - 233472 q^{72} + 2309404 q^{73} + 1340928 q^{74} - 1831308 q^{75} - 454592 q^{76} - 6203088 q^{77} - 2385792 q^{78} - 3686756 q^{79} + 3791448 q^{81} + 523776 q^{82} + 3042360 q^{83} - 283392 q^{84} + 4627800 q^{85} - 1734048 q^{86} + 5122692 q^{87} + 563712 q^{88} - 964710 q^{89} + 469440 q^{90} + 1235240 q^{91} + 1914624 q^{92} - 4105752 q^{93} - 1735824 q^{94} + 231552 q^{95} - 491520 q^{96} + 2565580 q^{97} + 4393440 q^{98} + 3499164 q^{99} + O(q^{100})$$

## Decomposition of $$S_{7}^{\mathrm{new}}(\Gamma_1(54))$$

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
54.7.b $$\chi_{54}(53, \cdot)$$ 54.7.b.a 2 1
54.7.b.b 2
54.7.b.c 4
54.7.d $$\chi_{54}(17, \cdot)$$ 54.7.d.a 12 2
54.7.f $$\chi_{54}(5, \cdot)$$ 54.7.f.a 108 6

## Decomposition of $$S_{7}^{\mathrm{old}}(\Gamma_1(54))$$ into lower level spaces

$$S_{7}^{\mathrm{old}}(\Gamma_1(54)) \cong$$ $$S_{7}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 6}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 3}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$$$\oplus$$$$S_{7}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 2}$$