## Defining parameters

 Level: $$N$$ = $$735 = 3 \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$2$$ Newform subspaces: $$8$$ Sturm bound: $$37632$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(735))$$.

Total New Old
Modular forms 990 308 682
Cusp forms 30 18 12
Eisenstein series 960 290 670

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 18 0 0 0

## Trace form

 $$18 q + O(q^{10})$$ $$18 q - 6 q^{15} + 6 q^{36} - 18 q^{64} - 12 q^{85} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(735))$$

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
735.1.c $$\chi_{735}(491, \cdot)$$ None 0 1
735.1.e $$\chi_{735}(244, \cdot)$$ None 0 1
735.1.f $$\chi_{735}(344, \cdot)$$ 735.1.f.a 1 1
735.1.f.b 1
735.1.f.c 2
735.1.f.d 2
735.1.h $$\chi_{735}(391, \cdot)$$ None 0 1
735.1.k $$\chi_{735}(293, \cdot)$$ None 0 2
735.1.l $$\chi_{735}(148, \cdot)$$ None 0 2
735.1.n $$\chi_{735}(31, \cdot)$$ None 0 2
735.1.o $$\chi_{735}(569, \cdot)$$ 735.1.o.a 2 2
735.1.o.b 2
735.1.o.c 4
735.1.o.d 4
735.1.r $$\chi_{735}(19, \cdot)$$ None 0 2
735.1.t $$\chi_{735}(116, \cdot)$$ None 0 2
735.1.w $$\chi_{735}(67, \cdot)$$ None 0 4
735.1.x $$\chi_{735}(68, \cdot)$$ None 0 4
735.1.z $$\chi_{735}(76, \cdot)$$ None 0 6
735.1.bb $$\chi_{735}(29, \cdot)$$ None 0 6
735.1.bc $$\chi_{735}(34, \cdot)$$ None 0 6
735.1.be $$\chi_{735}(71, \cdot)$$ None 0 6
735.1.bh $$\chi_{735}(22, \cdot)$$ None 0 12
735.1.bk $$\chi_{735}(62, \cdot)$$ None 0 12
735.1.bl $$\chi_{735}(11, \cdot)$$ None 0 12
735.1.bn $$\chi_{735}(94, \cdot)$$ None 0 12
735.1.bq $$\chi_{735}(44, \cdot)$$ None 0 12
735.1.br $$\chi_{735}(61, \cdot)$$ None 0 12
735.1.bs $$\chi_{735}(17, \cdot)$$ None 0 24
735.1.bv $$\chi_{735}(37, \cdot)$$ None 0 24

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(735))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(735)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(147))$$$$^{\oplus 2}$$