# Properties

 Label 405.3.h Level $405$ Weight $3$ Character orbit 405.h Rep. character $\chi_{405}(134,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $92$ Newform subspaces $11$ Sturm bound $162$ Trace bound $5$

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## Defining parameters

 Level: $$N$$ $$=$$ $$405 = 3^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 405.h (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$45$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$11$$ Sturm bound: $$162$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(405, [\chi])$$.

Total New Old
Modular forms 240 100 140
Cusp forms 192 92 100
Eisenstein series 48 8 40

## Trace form

 $$92 q - 84 q^{4} + O(q^{10})$$ $$92 q - 84 q^{4} - 20 q^{10} - 140 q^{16} - 8 q^{19} - 4 q^{25} - 56 q^{31} - 10 q^{34} + 92 q^{40} - 508 q^{46} + 290 q^{49} - 156 q^{55} + 100 q^{61} - 316 q^{64} + 102 q^{70} + 474 q^{76} + 616 q^{79} + 76 q^{85} + 552 q^{91} - 748 q^{94} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(405, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
405.3.h.a $2$ $11.035$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-15})$$ $$-1$$ $$0$$ $$-5$$ $$0$$ $$q+(-1+\zeta_{6})q^{2}+3\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots$$
405.3.h.b $2$ $11.035$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-15})$$ $$1$$ $$0$$ $$5$$ $$0$$ $$q+(1-\zeta_{6})q^{2}+3\zeta_{6}q^{4}+5\zeta_{6}q^{5}+7q^{8}+\cdots$$
405.3.h.c $4$ $11.035$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$-2$$ $$0$$ $$-1$$ $$0$$ $$q+\beta _{2}q^{2}+(3+3\beta _{2})q^{4}+(-1-\beta _{2}+\cdots)q^{5}+\cdots$$
405.3.h.d $4$ $11.035$ $$\Q(\zeta_{12})$$ None $$-2$$ $$0$$ $$8$$ $$0$$ $$q+(-1+\zeta_{12}^{2})q^{2}+3\zeta_{12}^{2}q^{4}+(\zeta_{12}+\cdots)q^{5}+\cdots$$
405.3.h.e $4$ $11.035$ $$\Q(\sqrt{-3}, \sqrt{5})$$ $$\Q(\sqrt{-15})$$ $$-1$$ $$0$$ $$10$$ $$0$$ $$q-\beta _{2}q^{2}+(-1+7\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots$$
405.3.h.f $4$ $11.035$ $$\Q(\sqrt{-3}, \sqrt{5})$$ $$\Q(\sqrt{-15})$$ $$1$$ $$0$$ $$-10$$ $$0$$ $$q+\beta _{2}q^{2}+(-1+7\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots$$
405.3.h.g $4$ $11.035$ $$\Q(\zeta_{12})$$ None $$2$$ $$0$$ $$-8$$ $$0$$ $$q+(1-\zeta_{12}^{2})q^{2}+3\zeta_{12}^{2}q^{4}+(\zeta_{12}+\cdots)q^{5}+\cdots$$
405.3.h.h $4$ $11.035$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$2$$ $$0$$ $$1$$ $$0$$ $$q+(1+\beta _{2})q^{2}-3\beta _{2}q^{4}+(-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$$
405.3.h.i $8$ $11.035$ 8.0.3317760000.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{7}q^{2}-6\beta _{2}q^{4}+(-\beta _{4}+\beta _{6})q^{5}+\cdots$$
405.3.h.j $8$ $11.035$ 8.0.$$\cdots$$.5 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{5}q^{2}+(-3+3\beta _{2})q^{4}+(-\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots$$
405.3.h.k $48$ $11.035$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{3}^{\mathrm{old}}(405, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(405, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(135, [\chi])$$$$^{\oplus 2}$$