# Properties

 Label 2016.1 Level 2016 Weight 1 Dimension 71 Nonzero newspaces 12 Newforms 20 Sturm bound 221184 Trace bound 25

## Defining parameters

 Level: $$N$$ = $$2016 = 2^{5} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$12$$ Newforms: $$20$$ Sturm bound: $$221184$$ Trace bound: $$25$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(2016))$$.

Total New Old
Modular forms 3524 521 3003
Cusp forms 452 71 381
Eisenstein series 3072 450 2622

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 51 20 0 0

## Trace form

 $$71q$$ $$\mathstrut +\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 2q^{5}$$ $$\mathstrut +\mathstrut q^{7}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$71q$$ $$\mathstrut +\mathstrut 2q^{3}$$ $$\mathstrut +\mathstrut 2q^{5}$$ $$\mathstrut +\mathstrut q^{7}$$ $$\mathstrut +\mathstrut 2q^{11}$$ $$\mathstrut -\mathstrut 4q^{13}$$ $$\mathstrut -\mathstrut 8q^{15}$$ $$\mathstrut +\mathstrut 4q^{16}$$ $$\mathstrut +\mathstrut 10q^{17}$$ $$\mathstrut +\mathstrut 4q^{19}$$ $$\mathstrut -\mathstrut 2q^{21}$$ $$\mathstrut -\mathstrut 4q^{22}$$ $$\mathstrut +\mathstrut 2q^{23}$$ $$\mathstrut -\mathstrut 13q^{25}$$ $$\mathstrut +\mathstrut 8q^{27}$$ $$\mathstrut -\mathstrut 4q^{29}$$ $$\mathstrut -\mathstrut 6q^{31}$$ $$\mathstrut -\mathstrut 4q^{33}$$ $$\mathstrut +\mathstrut 4q^{35}$$ $$\mathstrut +\mathstrut 6q^{37}$$ $$\mathstrut +\mathstrut 4q^{39}$$ $$\mathstrut +\mathstrut 8q^{41}$$ $$\mathstrut -\mathstrut 4q^{44}$$ $$\mathstrut -\mathstrut 19q^{49}$$ $$\mathstrut -\mathstrut 2q^{51}$$ $$\mathstrut -\mathstrut 2q^{53}$$ $$\mathstrut -\mathstrut 4q^{56}$$ $$\mathstrut -\mathstrut 4q^{57}$$ $$\mathstrut +\mathstrut 4q^{61}$$ $$\mathstrut +\mathstrut 4q^{63}$$ $$\mathstrut +\mathstrut 2q^{65}$$ $$\mathstrut +\mathstrut 4q^{67}$$ $$\mathstrut -\mathstrut 16q^{69}$$ $$\mathstrut -\mathstrut 2q^{71}$$ $$\mathstrut +\mathstrut 2q^{73}$$ $$\mathstrut -\mathstrut 4q^{74}$$ $$\mathstrut +\mathstrut 2q^{77}$$ $$\mathstrut +\mathstrut 8q^{79}$$ $$\mathstrut +\mathstrut 12q^{81}$$ $$\mathstrut -\mathstrut 4q^{83}$$ $$\mathstrut -\mathstrut 4q^{85}$$ $$\mathstrut -\mathstrut 2q^{89}$$ $$\mathstrut +\mathstrut 4q^{91}$$ $$\mathstrut +\mathstrut 4q^{92}$$ $$\mathstrut +\mathstrut 4q^{93}$$ $$\mathstrut +\mathstrut 8q^{95}$$ $$\mathstrut -\mathstrut 4q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(2016))$$

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
2016.1.d $$\chi_{2016}(449, \cdot)$$ None 0 1
2016.1.e $$\chi_{2016}(1007, \cdot)$$ 2016.1.e.a 4 1
2016.1.f $$\chi_{2016}(1441, \cdot)$$ 2016.1.f.a 4 1
2016.1.g $$\chi_{2016}(1135, \cdot)$$ None 0 1
2016.1.l $$\chi_{2016}(433, \cdot)$$ 2016.1.l.a 1 1
2016.1.l.b 2
2016.1.m $$\chi_{2016}(127, \cdot)$$ None 0 1
2016.1.n $$\chi_{2016}(1457, \cdot)$$ None 0 1
2016.1.o $$\chi_{2016}(2015, \cdot)$$ None 0 1
2016.1.u $$\chi_{2016}(937, \cdot)$$ None 0 2
2016.1.w $$\chi_{2016}(953, \cdot)$$ None 0 2
2016.1.y $$\chi_{2016}(503, \cdot)$$ None 0 2
2016.1.ba $$\chi_{2016}(631, \cdot)$$ None 0 2
2016.1.bc $$\chi_{2016}(47, \cdot)$$ None 0 2
2016.1.bd $$\chi_{2016}(65, \cdot)$$ None 0 2
2016.1.bi $$\chi_{2016}(79, \cdot)$$ 2016.1.bi.a 4 2
2016.1.bj $$\chi_{2016}(1825, \cdot)$$ None 0 2
2016.1.bk $$\chi_{2016}(1151, \cdot)$$ None 0 2
2016.1.bl $$\chi_{2016}(305, \cdot)$$ None 0 2
2016.1.bo $$\chi_{2016}(113, \cdot)$$ None 0 2
2016.1.bp $$\chi_{2016}(383, \cdot)$$ None 0 2
2016.1.bq $$\chi_{2016}(401, \cdot)$$ None 0 2
2016.1.br $$\chi_{2016}(671, \cdot)$$ None 0 2
2016.1.bv $$\chi_{2016}(1105, \cdot)$$ 2016.1.bv.a 2 2
2016.1.bv.b 2
2016.1.bv.c 4
2016.1.bw $$\chi_{2016}(1663, \cdot)$$ 2016.1.bw.a 4 2
2016.1.bx $$\chi_{2016}(241, \cdot)$$ None 0 2
2016.1.by $$\chi_{2016}(799, \cdot)$$ None 0 2
2016.1.cd $$\chi_{2016}(415, \cdot)$$ 2016.1.cd.a 4 2
2016.1.ce $$\chi_{2016}(145, \cdot)$$ 2016.1.ce.a 4 2
2016.1.cf $$\chi_{2016}(1423, \cdot)$$ None 0 2
2016.1.cg $$\chi_{2016}(577, \cdot)$$ None 0 2
2016.1.cl $$\chi_{2016}(97, \cdot)$$ None 0 2
2016.1.cm $$\chi_{2016}(655, \cdot)$$ 2016.1.cm.a 4 2
2016.1.cn $$\chi_{2016}(481, \cdot)$$ None 0 2
2016.1.co $$\chi_{2016}(463, \cdot)$$ None 0 2
2016.1.ct $$\chi_{2016}(1121, \cdot)$$ None 0 2
2016.1.cu $$\chi_{2016}(1391, \cdot)$$ None 0 2
2016.1.cv $$\chi_{2016}(641, \cdot)$$ None 0 2
2016.1.cw $$\chi_{2016}(335, \cdot)$$ None 0 2
2016.1.db $$\chi_{2016}(143, \cdot)$$ None 0 2
2016.1.dc $$\chi_{2016}(737, \cdot)$$ None 0 2
2016.1.dd $$\chi_{2016}(319, \cdot)$$ 2016.1.dd.a 4 2
2016.1.de $$\chi_{2016}(817, \cdot)$$ None 0 2
2016.1.di $$\chi_{2016}(1055, \cdot)$$ None 0 2
2016.1.dj $$\chi_{2016}(977, \cdot)$$ None 0 2
2016.1.dl $$\chi_{2016}(379, \cdot)$$ None 0 4
2016.1.dn $$\chi_{2016}(251, \cdot)$$ 2016.1.dn.a 4 4
2016.1.dn.b 4
2016.1.dn.c 4
2016.1.dn.d 4
2016.1.dp $$\chi_{2016}(181, \cdot)$$ 2016.1.dp.a 4 4
2016.1.dp.b 4
2016.1.dp.c 4
2016.1.dr $$\chi_{2016}(197, \cdot)$$ None 0 4
2016.1.dt $$\chi_{2016}(281, \cdot)$$ None 0 4
2016.1.dv $$\chi_{2016}(265, \cdot)$$ None 0 4
2016.1.dx $$\chi_{2016}(311, \cdot)$$ None 0 4
2016.1.dy $$\chi_{2016}(151, \cdot)$$ None 0 4
2016.1.eb $$\chi_{2016}(487, \cdot)$$ None 0 4
2016.1.ed $$\chi_{2016}(215, \cdot)$$ None 0 4
2016.1.ee $$\chi_{2016}(887, \cdot)$$ None 0 4
2016.1.eh $$\chi_{2016}(583, \cdot)$$ None 0 4
2016.1.ej $$\chi_{2016}(313, \cdot)$$ None 0 4
2016.1.el $$\chi_{2016}(233, \cdot)$$ None 0 4
2016.1.em $$\chi_{2016}(137, \cdot)$$ None 0 4
2016.1.eo $$\chi_{2016}(745, \cdot)$$ None 0 4
2016.1.er $$\chi_{2016}(73, \cdot)$$ None 0 4
2016.1.et $$\chi_{2016}(473, \cdot)$$ None 0 4
2016.1.ev $$\chi_{2016}(295, \cdot)$$ None 0 4
2016.1.ex $$\chi_{2016}(167, \cdot)$$ None 0 4
2016.1.ey $$\chi_{2016}(83, \cdot)$$ None 0 8
2016.1.fa $$\chi_{2016}(43, \cdot)$$ None 0 8
2016.1.fd $$\chi_{2016}(229, \cdot)$$ None 0 8
2016.1.fe $$\chi_{2016}(221, \cdot)$$ None 0 8
2016.1.ff $$\chi_{2016}(53, \cdot)$$ None 0 8
2016.1.fi $$\chi_{2016}(325, \cdot)$$ None 0 8
2016.1.fj $$\chi_{2016}(61, \cdot)$$ None 0 8
2016.1.fn $$\chi_{2016}(149, \cdot)$$ None 0 8
2016.1.fp $$\chi_{2016}(403, \cdot)$$ None 0 8
2016.1.fq $$\chi_{2016}(395, \cdot)$$ None 0 8
2016.1.fr $$\chi_{2016}(59, \cdot)$$ None 0 8
2016.1.fu $$\chi_{2016}(67, \cdot)$$ None 0 8
2016.1.fv $$\chi_{2016}(163, \cdot)$$ None 0 8
2016.1.fz $$\chi_{2016}(131, \cdot)$$ None 0 8
2016.1.ga $$\chi_{2016}(29, \cdot)$$ None 0 8
2016.1.gc $$\chi_{2016}(13, \cdot)$$ None 0 8

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(2016))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(2016)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 9}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(63))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(84))$$$$^{\oplus 8}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(144))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(168))$$$$^{\oplus 6}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(224))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(252))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(288))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(336))$$$$^{\oplus 4}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(504))$$$$^{\oplus 3}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(672))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(1008))$$$$^{\oplus 2}$$