# Properties

 Label 312.1 Level 312 Weight 1 Dimension 4 Nonzero newspaces 1 Newform subspaces 1 Sturm bound 5376 Trace bound 0

## Defining parameters

 Level: $$N$$ = $$312 = 2^{3} \cdot 3 \cdot 13$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$5376$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(312))$$.

Total New Old
Modular forms 316 48 268
Cusp forms 28 4 24
Eisenstein series 288 44 244

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 4 0 0 0

## Trace form

 $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} - 4q^{10} + 4q^{12} - 4q^{16} + 4q^{22} - 4q^{25} + 4q^{30} - 4q^{39} - 4q^{40} + 4q^{49} + 4q^{52} + 8q^{55} - 4q^{66} + 4q^{81} - 4q^{82} + 4q^{88} + 4q^{90} - 4q^{94} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(312))$$

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
312.1.b $$\chi_{312}(77, \cdot)$$ 312.1.b.a 4 1
312.1.e $$\chi_{312}(235, \cdot)$$ None 0 1
312.1.f $$\chi_{312}(209, \cdot)$$ None 0 1
312.1.i $$\chi_{312}(103, \cdot)$$ None 0 1
312.1.k $$\chi_{312}(79, \cdot)$$ None 0 1
312.1.l $$\chi_{312}(233, \cdot)$$ None 0 1
312.1.o $$\chi_{312}(259, \cdot)$$ None 0 1
312.1.p $$\chi_{312}(53, \cdot)$$ None 0 1
312.1.r $$\chi_{312}(109, \cdot)$$ None 0 2
312.1.s $$\chi_{312}(73, \cdot)$$ None 0 2
312.1.v $$\chi_{312}(47, \cdot)$$ None 0 2
312.1.w $$\chi_{312}(83, \cdot)$$ None 0 2
312.1.z $$\chi_{312}(127, \cdot)$$ None 0 2
312.1.bc $$\chi_{312}(113, \cdot)$$ None 0 2
312.1.bd $$\chi_{312}(139, \cdot)$$ None 0 2
312.1.bg $$\chi_{312}(101, \cdot)$$ None 0 2
312.1.bh $$\chi_{312}(29, \cdot)$$ None 0 2
312.1.bi $$\chi_{312}(43, \cdot)$$ None 0 2
312.1.bl $$\chi_{312}(17, \cdot)$$ None 0 2
312.1.bm $$\chi_{312}(55, \cdot)$$ None 0 2
312.1.bq $$\chi_{312}(11, \cdot)$$ None 0 4
312.1.br $$\chi_{312}(71, \cdot)$$ None 0 4
312.1.bu $$\chi_{312}(97, \cdot)$$ None 0 4
312.1.bv $$\chi_{312}(37, \cdot)$$ None 0 4

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(312))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(312)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(104))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(156))$$$$^{\oplus 2}$$
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