# Properties

 Label 21.3.f Level $21$ Weight $3$ Character orbit 21.f Rep. character $\chi_{21}(10,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $6$ Newform subspaces $3$ Sturm bound $8$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 21.f (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$8$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(21, [\chi])$$.

Total New Old
Modular forms 14 6 8
Cusp forms 6 6 0
Eisenstein series 8 0 8

## Trace form

 $$6 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 6 q^{5} - q^{7} + 8 q^{8} + 9 q^{9} + O(q^{10})$$ $$6 q - 2 q^{2} - 3 q^{3} - 2 q^{4} - 6 q^{5} - q^{7} + 8 q^{8} + 9 q^{9} - 30 q^{10} - 14 q^{11} + 24 q^{12} + 22 q^{14} - 24 q^{15} + 22 q^{16} + 48 q^{17} + 6 q^{18} + 33 q^{19} - 36 q^{21} + 28 q^{22} - 68 q^{23} - 54 q^{24} - 9 q^{25} - 126 q^{26} - 34 q^{28} + 64 q^{29} + 48 q^{30} - 69 q^{31} + 12 q^{32} + 108 q^{33} + 114 q^{35} - 12 q^{36} + 43 q^{37} + 174 q^{38} + 9 q^{39} + 42 q^{40} - 48 q^{42} - 134 q^{43} - 108 q^{44} - 18 q^{45} + 52 q^{46} - 222 q^{47} - 27 q^{49} - 68 q^{50} + 36 q^{51} + 84 q^{52} - 32 q^{53} + 32 q^{56} - 90 q^{57} + 34 q^{58} + 84 q^{59} + 18 q^{60} + 216 q^{61} - 15 q^{63} - 28 q^{64} - 78 q^{65} - 108 q^{66} + 69 q^{67} + 36 q^{68} - 198 q^{70} + 244 q^{71} + 12 q^{72} - 129 q^{73} + 38 q^{74} + 39 q^{75} + 52 q^{77} + 204 q^{78} - 21 q^{79} + 48 q^{80} - 27 q^{81} - 144 q^{82} + 120 q^{84} - 96 q^{85} + 158 q^{86} + 54 q^{87} - 202 q^{88} - 60 q^{89} + 303 q^{91} - 168 q^{92} - 45 q^{93} - 48 q^{94} - 150 q^{95} - 234 q^{96} - 176 q^{98} - 84 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(21, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
21.3.f.a $2$ $0.572$ $$\Q(\sqrt{-3})$$ None $$-3$$ $$-3$$ $$9$$ $$13$$ $$q-3\zeta_{6}q^{2}+(-1-\zeta_{6})q^{3}+(-5+5\zeta_{6})q^{4}+\cdots$$
21.3.f.b $2$ $0.572$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$3$$ $$-9$$ $$-7$$ $$q-\zeta_{6}q^{2}+(1+\zeta_{6})q^{3}+(3-3\zeta_{6})q^{4}+\cdots$$
21.3.f.c $2$ $0.572$ $$\Q(\sqrt{-3})$$ None $$2$$ $$-3$$ $$-6$$ $$-7$$ $$q+2\zeta_{6}q^{2}+(-1-\zeta_{6})q^{3}+(-4+2\zeta_{6})q^{5}+\cdots$$