# Properties

 Label 336.1 Level 336 Weight 1 Dimension 8 Nonzero newspaces 3 Newform subspaces 5 Sturm bound 6144 Trace bound 3

## Defining parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$3$$ Newform subspaces: $$5$$ Sturm bound: $$6144$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(336))$$.

Total New Old
Modular forms 366 52 314
Cusp forms 30 8 22
Eisenstein series 336 44 292

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

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

## Trace form

 $$8q + q^{3} + q^{7} - q^{9} + O(q^{10})$$ $$8q + q^{3} + q^{7} - q^{9} - 2q^{13} - q^{19} - 4q^{21} - q^{25} - 2q^{27} - q^{31} - 5q^{37} - q^{39} + 2q^{43} - q^{49} - 2q^{57} - 2q^{61} + q^{63} - q^{67} - 5q^{73} + q^{75} - q^{79} - q^{81} - q^{91} + 7q^{93} + 4q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(336))$$

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
336.1.d $$\chi_{336}(113, \cdot)$$ None 0 1
336.1.e $$\chi_{336}(167, \cdot)$$ None 0 1
336.1.f $$\chi_{336}(97, \cdot)$$ None 0 1
336.1.g $$\chi_{336}(295, \cdot)$$ None 0 1
336.1.l $$\chi_{336}(265, \cdot)$$ None 0 1
336.1.m $$\chi_{336}(127, \cdot)$$ None 0 1
336.1.n $$\chi_{336}(281, \cdot)$$ None 0 1
336.1.o $$\chi_{336}(335, \cdot)$$ 336.1.o.a 1 1
336.1.o.b 1
336.1.r $$\chi_{336}(13, \cdot)$$ None 0 2
336.1.t $$\chi_{336}(29, \cdot)$$ None 0 2
336.1.v $$\chi_{336}(83, \cdot)$$ None 0 2
336.1.x $$\chi_{336}(43, \cdot)$$ None 0 2
336.1.z $$\chi_{336}(47, \cdot)$$ 336.1.z.a 2 2
336.1.z.b 2
336.1.ba $$\chi_{336}(137, \cdot)$$ None 0 2
336.1.be $$\chi_{336}(79, \cdot)$$ None 0 2
336.1.bf $$\chi_{336}(73, \cdot)$$ None 0 2
336.1.bg $$\chi_{336}(151, \cdot)$$ None 0 2
336.1.bh $$\chi_{336}(145, \cdot)$$ None 0 2
336.1.bm $$\chi_{336}(215, \cdot)$$ None 0 2
336.1.bn $$\chi_{336}(65, \cdot)$$ 336.1.bn.a 2 2
336.1.bp $$\chi_{336}(67, \cdot)$$ None 0 4
336.1.br $$\chi_{336}(59, \cdot)$$ None 0 4
336.1.bt $$\chi_{336}(53, \cdot)$$ None 0 4
336.1.bv $$\chi_{336}(61, \cdot)$$ None 0 4

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(336))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(336)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 2}$$