## Defining parameters

 Level: $$N$$ = $$2700 = 2^{2} \cdot 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$10$$ Newform subspaces: $$16$$ Sturm bound: $$388800$$ Trace bound: $$16$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(2700))$$.

Total New Old
Modular forms 4604 747 3857
Cusp forms 404 91 313
Eisenstein series 4200 656 3544

The following table gives the dimensions of subspaces with specified projective image type.

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 83 0 0 8

## Trace form

 $$91 q + 6 q^{6} + q^{7} + O(q^{10})$$ $$91 q + 6 q^{6} + q^{7} - q^{13} + 8 q^{14} - 4 q^{16} - q^{19} + 2 q^{25} + 8 q^{29} - 8 q^{31} - 2 q^{34} - 18 q^{36} - 3 q^{37} - 2 q^{41} + 2 q^{43} - 22 q^{46} + 2 q^{49} + 6 q^{55} - 32 q^{56} + 35 q^{61} + 6 q^{64} - 3 q^{67} - 12 q^{69} - q^{73} - 18 q^{76} - 3 q^{79} - 12 q^{84} + 4 q^{86} - 4 q^{89} + 7 q^{91} - 14 q^{94} - q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(2700))$$

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
2700.1.b $$\chi_{2700}(1349, \cdot)$$ 2700.1.b.a 2 1
2700.1.b.b 2
2700.1.c $$\chi_{2700}(1351, \cdot)$$ 2700.1.c.a 2 1
2700.1.c.b 2
2700.1.f $$\chi_{2700}(1999, \cdot)$$ None 0 1
2700.1.g $$\chi_{2700}(701, \cdot)$$ 2700.1.g.a 1 1
2700.1.g.b 1
2700.1.g.c 1
2700.1.l $$\chi_{2700}(757, \cdot)$$ 2700.1.l.a 4 2
2700.1.l.b 4
2700.1.m $$\chi_{2700}(107, \cdot)$$ 2700.1.m.a 8 2
2700.1.m.b 8
2700.1.p $$\chi_{2700}(1601, \cdot)$$ None 0 2
2700.1.q $$\chi_{2700}(199, \cdot)$$ None 0 2
2700.1.t $$\chi_{2700}(451, \cdot)$$ 2700.1.t.a 4 2
2700.1.u $$\chi_{2700}(449, \cdot)$$ None 0 2
2700.1.y $$\chi_{2700}(271, \cdot)$$ None 0 4
2700.1.z $$\chi_{2700}(269, \cdot)$$ None 0 4
2700.1.bb $$\chi_{2700}(161, \cdot)$$ 2700.1.bb.a 8 4
2700.1.bc $$\chi_{2700}(379, \cdot)$$ None 0 4
2700.1.bd $$\chi_{2700}(793, \cdot)$$ None 0 4
2700.1.be $$\chi_{2700}(143, \cdot)$$ 2700.1.be.a 8 4
2700.1.bi $$\chi_{2700}(149, \cdot)$$ None 0 6
2700.1.bj $$\chi_{2700}(151, \cdot)$$ 2700.1.bj.a 12 6
2700.1.bl $$\chi_{2700}(101, \cdot)$$ None 0 6
2700.1.bo $$\chi_{2700}(499, \cdot)$$ None 0 6
2700.1.bp $$\chi_{2700}(323, \cdot)$$ None 0 8
2700.1.bq $$\chi_{2700}(217, \cdot)$$ None 0 8
2700.1.bt $$\chi_{2700}(19, \cdot)$$ None 0 8
2700.1.bu $$\chi_{2700}(341, \cdot)$$ None 0 8
2700.1.bw $$\chi_{2700}(89, \cdot)$$ None 0 8
2700.1.bx $$\chi_{2700}(91, \cdot)$$ None 0 8
2700.1.ca $$\chi_{2700}(407, \cdot)$$ 2700.1.ca.a 24 12
2700.1.cc $$\chi_{2700}(157, \cdot)$$ None 0 12
2700.1.ch $$\chi_{2700}(287, \cdot)$$ None 0 16
2700.1.ci $$\chi_{2700}(37, \cdot)$$ None 0 16
2700.1.cj $$\chi_{2700}(79, \cdot)$$ None 0 24
2700.1.cm $$\chi_{2700}(41, \cdot)$$ None 0 24
2700.1.co $$\chi_{2700}(31, \cdot)$$ None 0 24
2700.1.cp $$\chi_{2700}(29, \cdot)$$ None 0 24
2700.1.cr $$\chi_{2700}(13, \cdot)$$ None 0 48
2700.1.ct $$\chi_{2700}(23, \cdot)$$ None 0 48

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(2700))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(2700)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(108))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(180))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(300))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(540))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(675))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(900))$$$$^{\oplus 2}$$