# Properties

 Label 16.2 Level 16 Weight 2 Dimension 2 Nonzero newspaces 1 Newform subspaces 1 Sturm bound 32 Trace bound 0

## Defining parameters

 Level: $$N$$ = $$16 = 2^{4}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$1$$ Newform subspaces: $$1$$ Sturm bound: $$32$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(16))$$.

Total New Old
Modular forms 15 7 8
Cusp forms 2 2 0
Eisenstein series 13 5 8

## Trace form

 $$2q - 2q^{2} - 2q^{3} - 2q^{5} + 4q^{6} + 4q^{8} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} - 2q^{5} + 4q^{6} + 4q^{8} + 2q^{11} - 4q^{12} - 2q^{13} - 4q^{14} + 4q^{15} - 8q^{16} - 4q^{17} + 2q^{18} + 6q^{19} + 4q^{20} + 4q^{21} + 4q^{26} - 8q^{27} + 8q^{28} + 6q^{29} - 4q^{30} - 16q^{31} + 8q^{32} - 4q^{33} + 4q^{34} - 4q^{35} - 4q^{36} + 6q^{37} - 12q^{38} - 8q^{40} + 10q^{43} - 4q^{44} + 2q^{45} + 12q^{46} + 16q^{47} + 8q^{48} + 6q^{49} - 6q^{50} + 4q^{51} - 4q^{52} - 10q^{53} - 8q^{56} - 12q^{58} - 6q^{59} - 18q^{61} + 16q^{62} + 4q^{63} + 4q^{65} + 4q^{66} - 10q^{67} - 12q^{69} + 8q^{70} + 4q^{72} + 6q^{75} + 12q^{76} + 4q^{77} - 4q^{78} + 8q^{80} + 10q^{81} - 2q^{83} - 8q^{84} + 4q^{85} + 8q^{88} - 4q^{90} + 4q^{91} - 24q^{92} + 16q^{93} - 16q^{94} - 12q^{95} - 16q^{96} - 4q^{97} - 6q^{98} - 2q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$

We only show spaces with even 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
16.2.a $$\chi_{16}(1, \cdot)$$ None 0 1
16.2.b $$\chi_{16}(9, \cdot)$$ None 0 1
16.2.e $$\chi_{16}(5, \cdot)$$ 16.2.e.a 2 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 2 T^{2}$$
$3$ $$1 + 2 T + 2 T^{2} + 6 T^{3} + 9 T^{4}$$
$5$ $$( 1 - 2 T + 5 T^{2} )( 1 + 4 T + 5 T^{2} )$$
$7$ $$1 - 10 T^{2} + 49 T^{4}$$
$11$ $$1 - 2 T + 2 T^{2} - 22 T^{3} + 121 T^{4}$$
$13$ $$( 1 - 4 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$17$ $$( 1 + 2 T + 17 T^{2} )^{2}$$
$19$ $$1 - 6 T + 18 T^{2} - 114 T^{3} + 361 T^{4}$$
$23$ $$1 - 10 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 10 T + 29 T^{2} )( 1 + 4 T + 29 T^{2} )$$
$31$ $$( 1 + 8 T + 31 T^{2} )^{2}$$
$37$ $$1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4}$$
$41$ $$( 1 - 41 T^{2} )^{2}$$
$43$ $$1 - 10 T + 50 T^{2} - 430 T^{3} + 1849 T^{4}$$
$47$ $$( 1 - 8 T + 47 T^{2} )^{2}$$
$53$ $$( 1 - 4 T + 53 T^{2} )( 1 + 14 T + 53 T^{2} )$$
$59$ $$1 + 6 T + 18 T^{2} + 354 T^{3} + 3481 T^{4}$$
$61$ $$1 + 18 T + 162 T^{2} + 1098 T^{3} + 3721 T^{4}$$
$67$ $$1 + 10 T + 50 T^{2} + 670 T^{3} + 4489 T^{4}$$
$71$ $$1 - 42 T^{2} + 5041 T^{4}$$
$73$ $$1 - 130 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 79 T^{2} )^{2}$$
$83$ $$1 + 2 T + 2 T^{2} + 166 T^{3} + 6889 T^{4}$$
$89$ $$1 - 162 T^{2} + 7921 T^{4}$$
$97$ $$( 1 + 2 T + 97 T^{2} )^{2}$$