## Defining parameters

 Level: $$N$$ = $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ = $$1$$ Nonzero newspaces: $$9$$ Newform subspaces: $$18$$ Sturm bound: $$56448$$ Trace bound: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(980))$$.

Total New Old
Modular forms 1314 386 928
Cusp forms 114 96 18
Eisenstein series 1200 290 910

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 84 0 12 0

## Trace form

 $$96q + 2q^{4} + 2q^{5} + 4q^{6} - 6q^{9} + O(q^{10})$$ $$96q + 2q^{4} + 2q^{5} + 4q^{6} - 6q^{9} + 2q^{11} - 6q^{14} + 8q^{15} + 2q^{16} - 10q^{20} - 6q^{21} - 8q^{24} - 12q^{29} - 8q^{30} - 18q^{36} - 2q^{39} + 4q^{41} - 12q^{43} - 6q^{45} + 34q^{46} - 12q^{50} - 2q^{51} + 38q^{54} - 12q^{57} - 8q^{61} - 10q^{64} + 2q^{65} - 16q^{69} + 8q^{71} + 2q^{79} + 2q^{80} - 12q^{81} - 16q^{85} - 8q^{86} - 8q^{89} + 4q^{94} - 8q^{96} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(980))$$

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
980.1.b $$\chi_{980}(491, \cdot)$$ None 0 1
980.1.d $$\chi_{980}(881, \cdot)$$ None 0 1
980.1.f $$\chi_{980}(99, \cdot)$$ 980.1.f.a 1 1
980.1.f.b 1
980.1.f.c 1
980.1.f.d 1
980.1.f.e 4
980.1.h $$\chi_{980}(489, \cdot)$$ None 0 1
980.1.j $$\chi_{980}(587, \cdot)$$ 980.1.j.a 8 2
980.1.l $$\chi_{980}(197, \cdot)$$ 980.1.l.a 4 2
980.1.n $$\chi_{980}(129, \cdot)$$ 980.1.n.a 2 2
980.1.n.b 2
980.1.p $$\chi_{980}(79, \cdot)$$ 980.1.p.a 2 2
980.1.p.b 2
980.1.p.c 8
980.1.r $$\chi_{980}(521, \cdot)$$ None 0 2
980.1.t $$\chi_{980}(471, \cdot)$$ None 0 2
980.1.w $$\chi_{980}(177, \cdot)$$ 980.1.w.a 8 4
980.1.y $$\chi_{980}(227, \cdot)$$ 980.1.y.a 16 4
980.1.z $$\chi_{980}(69, \cdot)$$ None 0 6
980.1.ba $$\chi_{980}(239, \cdot)$$ 980.1.ba.a 6 6
980.1.ba.b 6
980.1.bc $$\chi_{980}(41, \cdot)$$ None 0 6
980.1.be $$\chi_{980}(71, \cdot)$$ None 0 6
980.1.bh $$\chi_{980}(57, \cdot)$$ None 0 12
980.1.bj $$\chi_{980}(27, \cdot)$$ None 0 12
980.1.bm $$\chi_{980}(11, \cdot)$$ None 0 12
980.1.bo $$\chi_{980}(61, \cdot)$$ None 0 12
980.1.bq $$\chi_{980}(39, \cdot)$$ 980.1.bq.a 12 12
980.1.bq.b 12
980.1.br $$\chi_{980}(89, \cdot)$$ None 0 12
980.1.bt $$\chi_{980}(3, \cdot)$$ None 0 24
980.1.bv $$\chi_{980}(37, \cdot)$$ None 0 24

## Decomposition of $$S_{1}^{\mathrm{old}}(\Gamma_1(980))$$ into lower level spaces

$$S_{1}^{\mathrm{old}}(\Gamma_1(980)) \cong$$ $$S_{1}^{\mathrm{new}}(\Gamma_1(140))$$$$^{\oplus 2}$$$$\oplus$$$$S_{1}^{\mathrm{new}}(\Gamma_1(196))$$$$^{\oplus 2}$$