# Properties

 Label 100.9.d Level $100$ Weight $9$ Character orbit 100.d Rep. character $\chi_{100}(99,\cdot)$ Character field $\Q$ Dimension $70$ Newform subspaces $4$ Sturm bound $135$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$100 = 2^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 100.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$20$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$135$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(100, [\chi])$$.

Total New Old
Modular forms 126 74 52
Cusp forms 114 70 44
Eisenstein series 12 4 8

## Trace form

 $$70 q + 478 q^{4} + 2778 q^{6} + 144346 q^{9} + O(q^{10})$$ $$70 q + 478 q^{4} + 2778 q^{6} + 144346 q^{9} - 107772 q^{14} - 14110 q^{16} + 367936 q^{21} + 1859618 q^{24} + 40352 q^{26} - 1333964 q^{29} - 7090802 q^{34} - 3068292 q^{36} + 2071340 q^{41} - 14424090 q^{44} + 6929908 q^{46} + 49848826 q^{49} + 30719506 q^{54} + 47382228 q^{56} + 64259660 q^{61} + 8910478 q^{64} - 10514590 q^{66} - 16422144 q^{69} + 77566828 q^{74} + 91340010 q^{76} + 16585926 q^{81} + 97196996 q^{84} - 100634352 q^{86} + 19314196 q^{89} - 141340392 q^{94} - 19366962 q^{96} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(100, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
100.9.d.a $2$ $40.738$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-8iq^{2}-2^{8}q^{4}+2^{11}iq^{8}-3^{8}q^{9}+\cdots$$
100.9.d.b $4$ $40.738$ $$\Q(i, \sqrt{39})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{2})q^{2}-8\beta _{1}q^{3}+(56-2\beta _{3})q^{4}+\cdots$$
100.9.d.c $32$ $40.738$ None $$0$$ $$0$$ $$0$$ $$0$$
100.9.d.d $32$ $40.738$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{9}^{\mathrm{old}}(100, [\chi])$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(100, [\chi]) \simeq$$ $$S_{9}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 2}$$