# Properties

 Label 21.9.f Level $21$ Weight $9$ Character orbit 21.f Rep. character $\chi_{21}(10,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $22$ Newform subspaces $2$ Sturm bound $24$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 21.f (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$2$$ Sturm bound: $$24$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(21, [\chi])$$.

Total New Old
Modular forms 46 22 24
Cusp forms 38 22 16
Eisenstein series 8 0 8

## Trace form

 $$22 q + 6 q^{2} + 81 q^{3} - 1722 q^{4} + 1674 q^{5} + 3079 q^{7} - 12072 q^{8} + 24057 q^{9} + O(q^{10})$$ $$22 q + 6 q^{2} + 81 q^{3} - 1722 q^{4} + 1674 q^{5} + 3079 q^{7} - 12072 q^{8} + 24057 q^{9} - 29262 q^{10} - 14958 q^{11} - 41472 q^{12} - 123402 q^{14} + 59616 q^{15} - 258538 q^{16} + 219456 q^{17} - 13122 q^{18} + 309237 q^{19} - 325620 q^{21} - 1266260 q^{22} + 445164 q^{23} + 777114 q^{24} + 1391731 q^{25} + 3302586 q^{26} - 2828538 q^{28} - 4261728 q^{29} - 275400 q^{30} + 3651243 q^{31} + 1066812 q^{32} + 4140720 q^{33} + 1514130 q^{35} - 7532028 q^{36} - 456001 q^{37} - 5131890 q^{38} - 81891 q^{39} + 12437250 q^{40} - 8004744 q^{42} - 8459182 q^{43} + 14140548 q^{44} + 3661038 q^{45} - 15299828 q^{46} - 18612774 q^{47} - 9241595 q^{49} - 16810740 q^{50} - 2506788 q^{51} + 19482372 q^{52} + 30723528 q^{53} + 90608088 q^{56} + 34737822 q^{57} - 28968902 q^{58} - 98837676 q^{59} - 51843078 q^{60} - 50014056 q^{61} - 5460939 q^{63} + 229866388 q^{64} + 6899514 q^{65} + 85243428 q^{66} - 29819311 q^{67} - 109292364 q^{68} - 133048638 q^{70} + 126221796 q^{71} - 13200732 q^{72} - 111587493 q^{73} + 55158942 q^{74} + 15634539 q^{75} + 229050804 q^{77} + 25862652 q^{78} + 9466451 q^{79} - 429558480 q^{80} - 52612659 q^{81} + 197507376 q^{82} - 234741240 q^{84} - 184335936 q^{85} + 77931870 q^{86} + 226959570 q^{87} - 241689634 q^{88} + 25623972 q^{89} + 81002931 q^{91} - 319270728 q^{92} - 59852601 q^{93} + 713523336 q^{94} + 355741938 q^{95} + 58602366 q^{96} - 371209056 q^{98} - 65426292 q^{99} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(21, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
21.9.f.a $10$ $8.555$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$3$$ $$-405$$ $$285$$ $$4305$$ $$q+(1+\beta _{1}-\beta _{2}+\beta _{3})q^{2}+(-54-3^{3}\beta _{3}+\cdots)q^{3}+\cdots$$
21.9.f.b $12$ $8.555$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$3$$ $$486$$ $$1389$$ $$-1226$$ $$q+(1+\beta _{2}-\beta _{3})q^{2}+(54+3^{3}\beta _{2})q^{3}+\cdots$$

## Decomposition of $$S_{9}^{\mathrm{old}}(21, [\chi])$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(21, [\chi]) \cong$$ $$S_{9}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 2}$$