# Properties

 Label 47.1 Level 47 Weight 1 Dimension 2 Nonzero newspaces 1 Newform subspaces 1 Sturm bound 184 Trace bound 0

## Defining parameters

 Level: $$N$$ = $$47\( 47$$ \) Weight: $$k$$ = $$1$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$184$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{1}(\Gamma_1(47))$$.

Total New Old
Modular forms 25 25 0
Cusp forms 2 2 0
Eisenstein series 23 23 0

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

 $$2q - q^{2} - q^{3} + q^{4} - 2q^{6} - q^{7} - 2q^{8} + q^{9} + O(q^{10})$$ $$2q - q^{2} - q^{3} + q^{4} - 2q^{6} - q^{7} - 2q^{8} + q^{9} + 2q^{12} + 3q^{14} - q^{17} + 2q^{18} - 2q^{21} + q^{24} + 2q^{25} - 2q^{27} - 3q^{28} + 2q^{32} - 2q^{34} - 2q^{36} - q^{37} + q^{42} + 2q^{47} + q^{49} - q^{50} + 3q^{51} - q^{53} + q^{54} + q^{56} - q^{59} - q^{61} + 2q^{63} - q^{64} + 2q^{68} - q^{71} - q^{72} - 2q^{74} - q^{75} - q^{79} + 4q^{83} - q^{84} - q^{89} - q^{94} - q^{96} - q^{97} - 3q^{98} + O(q^{100})$$

## Decomposition of $$S_{1}^{\mathrm{new}}(\Gamma_1(47))$$

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
47.1.b $$\chi_{47}(46, \cdot)$$ 47.1.b.a 2 1
47.1.d $$\chi_{47}(5, \cdot)$$ None 0 22

## Hecke characteristic polynomials

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