## Defining parameters

 Level: $$N$$ = $$72\( 72 = 2^{3} \cdot 3^{2}$$ \) Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$288$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(72))$$.

Total New Old
Modular forms 50 11 39
Cusp forms 2 2 0
Eisenstein series 48 9 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

 $$2q - q^{2} - q^{3} - q^{4} - q^{6} + 2q^{8} - q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} - q^{4} - q^{6} + 2q^{8} - q^{9} + q^{11} + 2q^{12} - q^{16} - 2q^{17} + 2q^{18} - 2q^{19} + q^{22} - q^{24} - q^{25} + 2q^{27} - q^{32} + q^{33} + q^{34} - q^{36} + q^{38} + q^{41} + q^{43} - 2q^{44} - q^{48} - q^{49} - q^{50} + q^{51} - q^{54} + q^{57} + q^{59} + 2q^{64} - 2q^{66} + q^{67} + q^{68} - q^{72} - 2q^{73} - q^{75} + q^{76} - q^{81} - 2q^{82} - 2q^{83} + q^{86} + q^{88} + 4q^{89} + 2q^{96} + q^{97} + 2q^{98} - 2q^{99} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(72))$$

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
72.1.b $$\chi_{72}(19, \cdot)$$ None 0 1
72.1.e $$\chi_{72}(17, \cdot)$$ None 0 1
72.1.g $$\chi_{72}(55, \cdot)$$ None 0 1
72.1.h $$\chi_{72}(53, \cdot)$$ None 0 1
72.1.j $$\chi_{72}(5, \cdot)$$ None 0 2
72.1.k $$\chi_{72}(7, \cdot)$$ None 0 2
72.1.m $$\chi_{72}(41, \cdot)$$ None 0 2
72.1.p $$\chi_{72}(43, \cdot)$$ 72.1.p.a 2 2

## Hecke characteristic polynomials

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