# Properties

 Label 16.4.a Level 16 Weight 4 Character orbit a Rep. character $$\chi_{16}(1,\cdot)$$ Character field $$\Q$$ Dimension 1 Newform subspaces 1 Sturm bound 8 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 16.a (trivial) Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$8$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(16))$$.

Total New Old
Modular forms 9 2 7
Cusp forms 3 1 2
Eisenstein series 6 1 5

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$Dim.
$$+$$$$1$$

## Trace form

 $$q + 4q^{3} - 2q^{5} - 24q^{7} - 11q^{9} + O(q^{10})$$ $$q + 4q^{3} - 2q^{5} - 24q^{7} - 11q^{9} + 44q^{11} + 22q^{13} - 8q^{15} + 50q^{17} - 44q^{19} - 96q^{21} + 56q^{23} - 121q^{25} - 152q^{27} + 198q^{29} + 160q^{31} + 176q^{33} + 48q^{35} - 162q^{37} + 88q^{39} - 198q^{41} - 52q^{43} + 22q^{45} - 528q^{47} + 233q^{49} + 200q^{51} - 242q^{53} - 88q^{55} - 176q^{57} + 668q^{59} + 550q^{61} + 264q^{63} - 44q^{65} - 188q^{67} + 224q^{69} - 728q^{71} + 154q^{73} - 484q^{75} - 1056q^{77} + 656q^{79} - 311q^{81} - 236q^{83} - 100q^{85} + 792q^{87} + 714q^{89} - 528q^{91} + 640q^{93} + 88q^{95} - 478q^{97} - 484q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(16))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2
16.4.a.a $$1$$ $$0.944$$ $$\Q$$ None $$0$$ $$4$$ $$-2$$ $$-24$$ $$+$$ $$q+4q^{3}-2q^{5}-24q^{7}-11q^{9}+44q^{11}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(16))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(16)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 4 T + 27 T^{2}$$
$5$ $$1 + 2 T + 125 T^{2}$$
$7$ $$1 + 24 T + 343 T^{2}$$
$11$ $$1 - 44 T + 1331 T^{2}$$
$13$ $$1 - 22 T + 2197 T^{2}$$
$17$ $$1 - 50 T + 4913 T^{2}$$
$19$ $$1 + 44 T + 6859 T^{2}$$
$23$ $$1 - 56 T + 12167 T^{2}$$
$29$ $$1 - 198 T + 24389 T^{2}$$
$31$ $$1 - 160 T + 29791 T^{2}$$
$37$ $$1 + 162 T + 50653 T^{2}$$
$41$ $$1 + 198 T + 68921 T^{2}$$
$43$ $$1 + 52 T + 79507 T^{2}$$
$47$ $$1 + 528 T + 103823 T^{2}$$
$53$ $$1 + 242 T + 148877 T^{2}$$
$59$ $$1 - 668 T + 205379 T^{2}$$
$61$ $$1 - 550 T + 226981 T^{2}$$
$67$ $$1 + 188 T + 300763 T^{2}$$
$71$ $$1 + 728 T + 357911 T^{2}$$
$73$ $$1 - 154 T + 389017 T^{2}$$
$79$ $$1 - 656 T + 493039 T^{2}$$
$83$ $$1 + 236 T + 571787 T^{2}$$
$89$ $$1 - 714 T + 704969 T^{2}$$
$97$ $$1 + 478 T + 912673 T^{2}$$