## Defining parameters

 Level: $$N$$ = $$416 = 2^{5} \cdot 13$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$4$$ Newform subspaces: $$5$$ Sturm bound: $$10752$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(416))$$.

Total New Old
Modular forms 416 122 294
Cusp forms 32 12 20
Eisenstein series 384 110 274

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 8 4 0 0

## Trace form

 $$12 q + 2 q^{3} + O(q^{10})$$ $$12 q + 2 q^{3} + 4 q^{13} - 4 q^{17} - 4 q^{21} - 4 q^{25} - 2 q^{27} + 2 q^{29} + 2 q^{33} - 2 q^{35} - 2 q^{37} - 8 q^{41} + 2 q^{43} - 6 q^{45} - 2 q^{51} - 4 q^{57} - 4 q^{61} - 8 q^{65} - 2 q^{69} - 4 q^{77} + 6 q^{85} + 2 q^{89} + 2 q^{91} + 2 q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(416))$$

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
416.1.c $$\chi_{416}(415, \cdot)$$ None 0 1
416.1.d $$\chi_{416}(287, \cdot)$$ None 0 1
416.1.g $$\chi_{416}(79, \cdot)$$ None 0 1
416.1.h $$\chi_{416}(207, \cdot)$$ 416.1.h.a 1 1
416.1.h.b 1
416.1.j $$\chi_{416}(177, \cdot)$$ None 0 2
416.1.m $$\chi_{416}(265, \cdot)$$ None 0 2
416.1.o $$\chi_{416}(103, \cdot)$$ None 0 2
416.1.q $$\chi_{416}(183, \cdot)$$ None 0 2
416.1.r $$\chi_{416}(57, \cdot)$$ None 0 2
416.1.t $$\chi_{416}(161, \cdot)$$ 416.1.t.a 2 2
416.1.v $$\chi_{416}(367, \cdot)$$ None 0 2
416.1.x $$\chi_{416}(303, \cdot)$$ None 0 2
416.1.y $$\chi_{416}(95, \cdot)$$ None 0 2
416.1.bb $$\chi_{416}(159, \cdot)$$ 416.1.bb.a 4 2
416.1.bc $$\chi_{416}(5, \cdot)$$ None 0 4
416.1.be $$\chi_{416}(51, \cdot)$$ None 0 4
416.1.bh $$\chi_{416}(27, \cdot)$$ None 0 4
416.1.bj $$\chi_{416}(21, \cdot)$$ None 0 4
416.1.bl $$\chi_{416}(33, \cdot)$$ 416.1.bl.a 4 4
416.1.bm $$\chi_{416}(41, \cdot)$$ None 0 4
416.1.bo $$\chi_{416}(55, \cdot)$$ None 0 4
416.1.bq $$\chi_{416}(23, \cdot)$$ None 0 4
416.1.bt $$\chi_{416}(137, \cdot)$$ None 0 4
416.1.bv $$\chi_{416}(145, \cdot)$$ None 0 4
416.1.bw $$\chi_{416}(37, \cdot)$$ None 0 8
416.1.by $$\chi_{416}(3, \cdot)$$ None 0 8
416.1.cb $$\chi_{416}(43, \cdot)$$ None 0 8
416.1.cd $$\chi_{416}(141, \cdot)$$ None 0 8

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(416))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(416)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(104))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(208))$$$$^{\oplus 2}$$