# Properties

 Label 50.2.b Level 50 Weight 2 Character orbit b Rep. character $$\chi_{50}(49,\cdot)$$ Character field $$\Q$$ Dimension 2 Newform subspaces 1 Sturm bound 15 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 50.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$15$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(50, [\chi])$$.

Total New Old
Modular forms 14 2 12
Cusp forms 2 2 0
Eisenstein series 12 0 12

## Trace form

 $$2q - 2q^{4} - 2q^{6} + 4q^{9} + O(q^{10})$$ $$2q - 2q^{4} - 2q^{6} + 4q^{9} - 6q^{11} + 4q^{14} + 2q^{16} - 10q^{19} + 4q^{21} + 2q^{24} + 8q^{26} + 4q^{31} - 6q^{34} - 4q^{36} + 8q^{39} - 6q^{41} + 6q^{44} - 12q^{46} + 6q^{49} - 6q^{51} - 10q^{54} - 4q^{56} + 4q^{61} - 2q^{64} + 6q^{66} - 12q^{69} + 24q^{71} + 4q^{74} + 10q^{76} + 20q^{79} + 2q^{81} - 4q^{84} + 8q^{86} - 30q^{89} - 16q^{91} + 24q^{94} - 2q^{96} - 12q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(50, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
50.2.b.a $$2$$ $$0.399$$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{2}+iq^{3}-q^{4}-q^{6}-2iq^{7}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$1 - 5 T^{2} + 9 T^{4}$$
$5$ 1
$7$ $$1 - 10 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 3 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} )$$
$17$ $$1 - 25 T^{2} + 289 T^{4}$$
$19$ $$( 1 + 5 T + 19 T^{2} )^{2}$$
$23$ $$1 - 10 T^{2} + 529 T^{4}$$
$29$ $$( 1 + 29 T^{2} )^{2}$$
$31$ $$( 1 - 2 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 + 3 T + 41 T^{2} )^{2}$$
$43$ $$1 - 70 T^{2} + 1849 T^{4}$$
$47$ $$1 + 50 T^{2} + 2209 T^{4}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 - 2 T + 61 T^{2} )^{2}$$
$67$ $$1 + 35 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 12 T + 71 T^{2} )^{2}$$
$73$ $$1 - 25 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 10 T + 79 T^{2} )^{2}$$
$83$ $$1 - 85 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 15 T + 89 T^{2} )^{2}$$
$97$ $$1 - 190 T^{2} + 9409 T^{4}$$