# Properties

 Label 8.4.b Level 8 Weight 4 Character orbit b Rep. character $$\chi_{8}(5,\cdot)$$ Character field $$\Q$$ Dimension 2 Newform subspaces 1 Sturm bound 4 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 8.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$8$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$4$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(8, [\chi])$$.

Total New Old
Modular forms 4 4 0
Cusp forms 2 2 0
Eisenstein series 2 2 0

## Trace form

 $$2q - 2q^{2} - 12q^{4} + 28q^{6} - 16q^{7} + 40q^{8} - 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 12q^{4} + 28q^{6} - 16q^{7} + 40q^{8} - 2q^{9} - 56q^{10} - 56q^{12} + 16q^{14} + 112q^{15} + 16q^{16} - 28q^{17} + 2q^{18} + 112q^{20} - 84q^{22} - 304q^{23} - 112q^{24} + 26q^{25} + 280q^{26} + 96q^{28} - 112q^{30} + 448q^{31} - 352q^{32} + 168q^{33} + 28q^{34} + 12q^{36} - 196q^{38} - 560q^{39} + 224q^{40} - 140q^{41} - 224q^{42} + 168q^{44} + 304q^{46} + 672q^{47} + 672q^{48} - 558q^{49} - 26q^{50} - 560q^{52} + 728q^{54} - 336q^{55} - 320q^{56} + 392q^{57} - 840q^{58} - 672q^{60} - 448q^{62} + 16q^{63} + 576q^{64} + 1120q^{65} - 168q^{66} + 168q^{68} + 448q^{70} - 144q^{71} - 40q^{72} - 588q^{73} + 1288q^{74} + 392q^{76} + 560q^{78} - 928q^{79} - 1344q^{80} - 1510q^{81} + 140q^{82} + 448q^{84} - 2324q^{86} + 1680q^{87} + 336q^{88} + 532q^{89} + 56q^{90} + 1824q^{92} - 672q^{94} - 784q^{95} - 448q^{96} + 1988q^{97} + 558q^{98} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(8, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
8.4.b.a $$2$$ $$0.472$$ $$\Q(\sqrt{-7})$$ None $$-2$$ $$0$$ $$0$$ $$-16$$ $$q+(-1-\beta )q^{2}+2\beta q^{3}+(-6+2\beta )q^{4}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T + 8 T^{2}$$
$3$ $$1 - 26 T^{2} + 729 T^{4}$$
$5$ $$1 - 138 T^{2} + 15625 T^{4}$$
$7$ $$( 1 + 8 T + 343 T^{2} )^{2}$$
$11$ $$1 - 2410 T^{2} + 1771561 T^{4}$$
$13$ $$1 - 1594 T^{2} + 4826809 T^{4}$$
$17$ $$( 1 + 14 T + 4913 T^{2} )^{2}$$
$19$ $$1 - 12346 T^{2} + 47045881 T^{4}$$
$23$ $$( 1 + 152 T + 12167 T^{2} )^{2}$$
$29$ $$1 - 23578 T^{2} + 594823321 T^{4}$$
$31$ $$( 1 - 224 T + 29791 T^{2} )^{2}$$
$37$ $$1 - 42058 T^{2} + 2565726409 T^{4}$$
$41$ $$( 1 + 70 T + 68921 T^{2} )^{2}$$
$43$ $$1 + 33878 T^{2} + 6321363049 T^{4}$$
$47$ $$( 1 - 336 T + 103823 T^{2} )^{2}$$
$53$ $$1 - 296746 T^{2} + 22164361129 T^{4}$$
$59$ $$1 - 125130 T^{2} + 42180533641 T^{4}$$
$61$ $$1 - 444890 T^{2} + 51520374361 T^{4}$$
$67$ $$1 - 571034 T^{2} + 90458382169 T^{4}$$
$71$ $$( 1 + 72 T + 357911 T^{2} )^{2}$$
$73$ $$( 1 + 294 T + 389017 T^{2} )^{2}$$
$79$ $$( 1 + 464 T + 493039 T^{2} )^{2}$$
$83$ $$1 - 846522 T^{2} + 326940373369 T^{4}$$
$89$ $$( 1 - 266 T + 704969 T^{2} )^{2}$$
$97$ $$( 1 - 994 T + 912673 T^{2} )^{2}$$