# Properties

 Label 45.3.g Level $45$ Weight $3$ Character orbit 45.g Rep. character $\chi_{45}(28,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $8$ Newform subspaces $2$ Sturm bound $18$ Trace bound $2$

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## Defining parameters

 Level: $$N$$ $$=$$ $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 45.g (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q(i)$$ Newform subspaces: $$2$$ Sturm bound: $$18$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(45, [\chi])$$.

Total New Old
Modular forms 32 12 20
Cusp forms 16 8 8
Eisenstein series 16 4 12

## Trace form

 $$8 q + 4 q^{2} + 4 q^{5} - 16 q^{7} - 12 q^{8} + O(q^{10})$$ $$8 q + 4 q^{2} + 4 q^{5} - 16 q^{7} - 12 q^{8} - 16 q^{10} - 16 q^{11} + 8 q^{13} + 56 q^{16} + 40 q^{17} + 36 q^{20} - 80 q^{22} - 56 q^{23} - 64 q^{25} - 88 q^{26} + 64 q^{28} + 16 q^{31} + 76 q^{32} + 40 q^{35} + 104 q^{37} + 96 q^{38} + 168 q^{40} + 56 q^{41} + 32 q^{43} - 176 q^{46} - 128 q^{47} - 164 q^{50} - 40 q^{52} - 56 q^{53} - 224 q^{55} - 312 q^{58} - 32 q^{61} - 88 q^{62} + 112 q^{65} + 80 q^{67} + 104 q^{68} + 240 q^{70} + 272 q^{71} + 296 q^{73} + 240 q^{76} - 88 q^{77} - 164 q^{80} + 328 q^{82} + 16 q^{83} + 272 q^{85} + 224 q^{86} - 288 q^{88} - 416 q^{91} - 104 q^{92} - 144 q^{95} - 40 q^{97} + 188 q^{98} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(45, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
45.3.g.a $4$ $1.226$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$-20$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(2\beta _{1}+\beta _{3})q^{5}+(-5+\cdots)q^{7}+\cdots$$
45.3.g.b $4$ $1.226$ $$\Q(i, \sqrt{6})$$ None $$4$$ $$0$$ $$4$$ $$4$$ $$q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}+\beta _{2}+2\beta _{3})q^{4}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(45, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(45, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(15, [\chi])$$$$^{\oplus 2}$$