## Defining parameters

 Level: $$N$$ = $$4000 = 2^{5} \cdot 5^{3}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$9$$ Newform subspaces: $$16$$ Sturm bound: $$960000$$ Trace bound: $$29$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(4000))$$.

Total New Old
Modular forms 6292 1408 4884
Cusp forms 532 128 404
Eisenstein series 5760 1280 4480

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 88 0 0 40

## Trace form

 $$128q + O(q^{10})$$ $$128q + 4q^{11} - 8q^{13} - 2q^{17} + 12q^{21} + 6q^{29} + 12q^{41} + 4q^{49} + 4q^{57} - 14q^{61} - 2q^{69} + 4q^{73} - 4q^{81} + 10q^{85} + 20q^{89} + 8q^{91} + 8q^{93} - 2q^{97} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(4000))$$

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
4000.1.b $$\chi_{4000}(2751, \cdot)$$ 4000.1.b.a 4 1
4000.1.b.b 4
4000.1.e $$\chi_{4000}(1999, \cdot)$$ 4000.1.e.a 2 1
4000.1.e.b 2
4000.1.g $$\chi_{4000}(751, \cdot)$$ 4000.1.g.a 4 1
4000.1.h $$\chi_{4000}(3999, \cdot)$$ 4000.1.h.a 4 1
4000.1.h.b 4
4000.1.i $$\chi_{4000}(57, \cdot)$$ None 0 2
4000.1.k $$\chi_{4000}(999, \cdot)$$ None 0 2
4000.1.m $$\chi_{4000}(2193, \cdot)$$ None 0 2
4000.1.p $$\chi_{4000}(193, \cdot)$$ 4000.1.p.a 8 2
4000.1.p.b 8
4000.1.r $$\chi_{4000}(1751, \cdot)$$ None 0 2
4000.1.t $$\chi_{4000}(2057, \cdot)$$ None 0 2
4000.1.w $$\chi_{4000}(1557, \cdot)$$ None 0 4
4000.1.x $$\chi_{4000}(251, \cdot)$$ None 0 4
4000.1.z $$\chi_{4000}(499, \cdot)$$ None 0 4
4000.1.bc $$\chi_{4000}(557, \cdot)$$ None 0 4
4000.1.bd $$\chi_{4000}(1551, \cdot)$$ None 0 4
4000.1.bf $$\chi_{4000}(799, \cdot)$$ 4000.1.bf.a 8 4
4000.1.bf.b 8
4000.1.bh $$\chi_{4000}(351, \cdot)$$ 4000.1.bh.a 8 4
4000.1.bi $$\chi_{4000}(399, \cdot)$$ None 0 4
4000.1.bk $$\chi_{4000}(457, \cdot)$$ None 0 8
4000.1.bn $$\chi_{4000}(199, \cdot)$$ None 0 8
4000.1.bo $$\chi_{4000}(257, \cdot)$$ 4000.1.bo.a 8 8
4000.1.bo.b 8
4000.1.bo.c 8
4000.1.br $$\chi_{4000}(593, \cdot)$$ None 0 8
4000.1.bs $$\chi_{4000}(151, \cdot)$$ None 0 8
4000.1.bv $$\chi_{4000}(393, \cdot)$$ None 0 8
4000.1.bx $$\chi_{4000}(93, \cdot)$$ None 0 16
4000.1.ca $$\chi_{4000}(99, \cdot)$$ None 0 16
4000.1.cc $$\chi_{4000}(51, \cdot)$$ None 0 16
4000.1.cd $$\chi_{4000}(157, \cdot)$$ None 0 16
4000.1.cf $$\chi_{4000}(159, \cdot)$$ None 0 20
4000.1.cg $$\chi_{4000}(79, \cdot)$$ None 0 20
4000.1.ci $$\chi_{4000}(111, \cdot)$$ None 0 20
4000.1.cl $$\chi_{4000}(31, \cdot)$$ None 0 20
4000.1.cm $$\chi_{4000}(71, \cdot)$$ None 0 40
4000.1.cp $$\chi_{4000}(73, \cdot)$$ None 0 40
4000.1.cq $$\chi_{4000}(33, \cdot)$$ 4000.1.cq.a 40 40
4000.1.ct $$\chi_{4000}(17, \cdot)$$ None 0 40
4000.1.cu $$\chi_{4000}(137, \cdot)$$ None 0 40
4000.1.cx $$\chi_{4000}(39, \cdot)$$ None 0 40
4000.1.cz $$\chi_{4000}(53, \cdot)$$ None 0 80
4000.1.db $$\chi_{4000}(11, \cdot)$$ None 0 80
4000.1.dc $$\chi_{4000}(19, \cdot)$$ None 0 80
4000.1.de $$\chi_{4000}(13, \cdot)$$ None 0 80

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(4000))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(4000)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(80))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(100))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(160))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(200))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(400))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(500))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(800))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1000))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(2000))$$$$^{\oplus 2}$$