# Properties

 Label 21.2.a Level 21 Weight 2 Character orbit a Rep. character $$\chi_{21}(1,\cdot)$$ Character field $$\Q$$ Dimension 1 Newform subspaces 1 Sturm bound 5 Trace bound 0

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 21.a (trivial) Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$5$$ Trace bound: $$0$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(21))$$.

Total New Old
Modular forms 4 1 3
Cusp forms 1 1 0
Eisenstein series 3 0 3

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

$$3$$$$7$$FrickeDim.
$$-$$$$+$$$$-$$$$1$$
Plus space$$+$$$$0$$
Minus space$$-$$$$1$$

## Trace form

 $$q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} - q^{7} + 3q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} - q^{7} + 3q^{8} + q^{9} + 2q^{10} + 4q^{11} - q^{12} - 2q^{13} + q^{14} - 2q^{15} - q^{16} - 6q^{17} - q^{18} + 4q^{19} + 2q^{20} - q^{21} - 4q^{22} + 3q^{24} - q^{25} + 2q^{26} + q^{27} + q^{28} - 2q^{29} + 2q^{30} - 5q^{32} + 4q^{33} + 6q^{34} + 2q^{35} - q^{36} + 6q^{37} - 4q^{38} - 2q^{39} - 6q^{40} + 2q^{41} + q^{42} - 4q^{43} - 4q^{44} - 2q^{45} - q^{48} + q^{49} + q^{50} - 6q^{51} + 2q^{52} + 6q^{53} - q^{54} - 8q^{55} - 3q^{56} + 4q^{57} + 2q^{58} + 12q^{59} + 2q^{60} - 2q^{61} - q^{63} + 7q^{64} + 4q^{65} - 4q^{66} + 4q^{67} + 6q^{68} - 2q^{70} + 3q^{72} - 6q^{73} - 6q^{74} - q^{75} - 4q^{76} - 4q^{77} + 2q^{78} - 16q^{79} + 2q^{80} + q^{81} - 2q^{82} - 12q^{83} + q^{84} + 12q^{85} + 4q^{86} - 2q^{87} + 12q^{88} - 14q^{89} + 2q^{90} + 2q^{91} - 8q^{95} - 5q^{96} + 18q^{97} - q^{98} + 4q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(21))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 3 7
21.2.a.a $$1$$ $$0.168$$ $$\Q$$ None $$-1$$ $$1$$ $$-2$$ $$-1$$ $$-$$ $$+$$ $$q-q^{2}+q^{3}-q^{4}-2q^{5}-q^{6}-q^{7}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + 2 T^{2}$$
$3$ $$1 - T$$
$5$ $$1 + 2 T + 5 T^{2}$$
$7$ $$1 + T$$
$11$ $$1 - 4 T + 11 T^{2}$$
$13$ $$1 + 2 T + 13 T^{2}$$
$17$ $$1 + 6 T + 17 T^{2}$$
$19$ $$1 - 4 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 + 2 T + 29 T^{2}$$
$31$ $$1 + 31 T^{2}$$
$37$ $$1 - 6 T + 37 T^{2}$$
$41$ $$1 - 2 T + 41 T^{2}$$
$43$ $$1 + 4 T + 43 T^{2}$$
$47$ $$1 + 47 T^{2}$$
$53$ $$1 - 6 T + 53 T^{2}$$
$59$ $$1 - 12 T + 59 T^{2}$$
$61$ $$1 + 2 T + 61 T^{2}$$
$67$ $$1 - 4 T + 67 T^{2}$$
$71$ $$1 + 71 T^{2}$$
$73$ $$1 + 6 T + 73 T^{2}$$
$79$ $$1 + 16 T + 79 T^{2}$$
$83$ $$1 + 12 T + 83 T^{2}$$
$89$ $$1 + 14 T + 89 T^{2}$$
$97$ $$1 - 18 T + 97 T^{2}$$