# Properties

 Label 21.9.d Level $21$ Weight $9$ Character orbit 21.d Rep. character $\chi_{21}(13,\cdot)$ Character field $\Q$ Dimension $10$ Newform subspaces $1$ Sturm bound $24$ Trace bound $0$

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## Defining parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 21.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q$$ Newform subspaces: $$1$$ Sturm bound: $$24$$ Trace bound: $$0$$

## Dimensions

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

Total New Old
Modular forms 24 10 14
Cusp forms 20 10 10
Eisenstein series 4 0 4

## Trace form

 $$10 q - 6 q^{2} + 1210 q^{4} - 4002 q^{7} + 20154 q^{8} - 21870 q^{9} + O(q^{10})$$ $$10 q - 6 q^{2} + 1210 q^{4} - 4002 q^{7} + 20154 q^{8} - 21870 q^{9} + 29412 q^{11} - 94974 q^{14} - 59616 q^{15} + 66546 q^{16} + 13122 q^{18} + 21060 q^{21} + 1114748 q^{22} - 900204 q^{23} + 1073026 q^{25} - 3421102 q^{28} - 110940 q^{29} - 2665872 q^{30} + 8720778 q^{32} - 1603464 q^{35} - 2646270 q^{36} - 8108292 q^{37} + 4817880 q^{39} + 3774600 q^{42} + 7375212 q^{43} + 12971124 q^{44} - 10934356 q^{46} + 2729458 q^{49} - 12200166 q^{50} - 9033768 q^{51} - 158124 q^{53} - 10193862 q^{56} + 11936160 q^{57} - 5552620 q^{58} - 42295608 q^{60} + 8752374 q^{63} + 3537138 q^{64} + 134354064 q^{65} - 42107908 q^{67} + 148996104 q^{70} - 71162172 q^{71} - 44076798 q^{72} - 171840396 q^{74} - 122623068 q^{77} + 112354776 q^{78} + 22383884 q^{79} + 47829690 q^{81} + 85279392 q^{84} - 169548552 q^{85} - 25858716 q^{86} + 262117924 q^{88} - 153506400 q^{91} + 34290228 q^{92} - 80374032 q^{93} + 496786896 q^{95} - 164217126 q^{98} - 64324044 q^{99} + O(q^{100})$$

## 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.d.a $10$ $8.555$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$-6$$ $$0$$ $$0$$ $$-4002$$ $$q+(-1-\beta _{1})q^{2}-\beta _{4}q^{3}+(119-5\beta _{1}+\cdots)q^{4}+\cdots$$

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

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