## Defining parameters

 Level: $$N$$ = $$99\( 99 = 3^{2} \cdot 11$$ \) Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$720$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(99))$$.

Total New Old
Modular forms 82 43 39
Cusp forms 2 2 0
Eisenstein series 80 41 39

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

$$D_n$$ $$A_4$$ $$S_4$$ $$A_5$$
Dimension 2 0 0 0

## Trace form

 $$2q - q^{3} - q^{4} + q^{5} - q^{9} + O(q^{10})$$ $$2q - q^{3} - q^{4} + q^{5} - q^{9} - q^{11} - q^{12} + q^{15} - q^{16} + q^{20} - 2q^{23} + 2q^{27} + q^{31} + 2q^{33} + 2q^{36} - 2q^{37} + 2q^{44} - 2q^{45} + q^{47} + 2q^{48} - q^{49} - 2q^{53} - 2q^{55} + q^{59} - 2q^{60} + 2q^{64} + q^{67} - 2q^{69} - 2q^{71} - 2q^{80} - q^{81} + 4q^{89} - 2q^{92} + q^{93} + q^{97} - q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(99))$$

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
99.1.b $$\chi_{99}(89, \cdot)$$ None 0 1
99.1.c $$\chi_{99}(10, \cdot)$$ None 0 1
99.1.h $$\chi_{99}(43, \cdot)$$ 99.1.h.a 2 2
99.1.i $$\chi_{99}(23, \cdot)$$ None 0 2
99.1.k $$\chi_{99}(19, \cdot)$$ None 0 4
99.1.l $$\chi_{99}(26, \cdot)$$ None 0 4
99.1.n $$\chi_{99}(5, \cdot)$$ None 0 8
99.1.o $$\chi_{99}(7, \cdot)$$ None 0 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$3$ $$1 + T + T^{2}$$
$5$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
$7$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$11$ $$1 + T + T^{2}$$
$13$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$17$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$19$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$23$ $$( 1 + T + T^{2} )^{2}$$
$29$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$31$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
$37$ $$( 1 + T + T^{2} )^{2}$$
$41$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$43$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$47$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
$53$ $$( 1 + T + T^{2} )^{2}$$
$59$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
$61$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$67$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
$71$ $$( 1 + T + T^{2} )^{2}$$
$73$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$79$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$83$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
$89$ $$( 1 - T )^{4}$$
$97$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$