# Properties

 Label 100.9.f Level $100$ Weight $9$ Character orbit 100.f Rep. character $\chi_{100}(57,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $24$ Newform subspaces $3$ Sturm bound $135$ Trace bound $1$

# Related objects

## Defining parameters

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

## Dimensions

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

Total New Old
Modular forms 258 24 234
Cusp forms 222 24 198
Eisenstein series 36 0 36

## Trace form

 $$24 q + 70 q^{3} + 2030 q^{7} + O(q^{10})$$ $$24 q + 70 q^{3} + 2030 q^{7} + 840 q^{11} - 33180 q^{13} - 43620 q^{17} - 627196 q^{21} + 663270 q^{23} - 1576040 q^{27} + 3166524 q^{31} + 944020 q^{33} - 5344080 q^{37} + 7246344 q^{41} + 10342710 q^{43} - 19232250 q^{47} + 42485848 q^{51} + 24320640 q^{53} - 88218320 q^{57} + 45976748 q^{61} + 77441350 q^{63} - 100675930 q^{67} + 16468572 q^{71} + 93528520 q^{73} - 134199660 q^{77} + 115768396 q^{81} + 10450350 q^{83} - 164801600 q^{87} + 347762196 q^{91} + 50183620 q^{93} + 179570760 q^{97} + O(q^{100})$$

## 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.f.a $4$ $40.738$ $$\Q(i, \sqrt{3309})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{3}+7\beta _{2}q^{7}-57\beta _{1}q^{9}+420q^{11}+\cdots$$
100.9.f.b $8$ $40.738$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$70$$ $$0$$ $$2030$$ $$q+(9+9\beta _{1}+\beta _{2})q^{3}+(254-254\beta _{1}+\cdots)q^{7}+\cdots$$
100.9.f.c $12$ $40.738$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{4}q^{3}+(7\beta _{7}-20\beta _{8}-\beta _{11})q^{7}+\cdots$$

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

$$S_{9}^{\mathrm{old}}(100, [\chi]) \simeq$$ $$S_{9}^{\mathrm{new}}(5, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 2}$$