# Properties

 Label 162.13.d Level $162$ Weight $13$ Character orbit 162.d Rep. character $\chi_{162}(53,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $96$ Newform subspaces $8$ Sturm bound $351$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$162 = 2 \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$13$$ Character orbit: $$[\chi]$$ $$=$$ 162.d (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$8$$ Sturm bound: $$351$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{13}(162, [\chi])$$.

Total New Old
Modular forms 672 96 576
Cusp forms 624 96 528
Eisenstein series 48 0 48

## Trace form

 $$96 q + 98304 q^{4} + 343200 q^{7} + O(q^{10})$$ $$96 q + 98304 q^{4} + 343200 q^{7} - 4250400 q^{13} - 201326592 q^{16} + 308198640 q^{19} + 134668800 q^{22} + 1794454512 q^{25} + 1405747200 q^{28} - 3001349040 q^{31} + 755573760 q^{34} + 27365822400 q^{37} - 4733380680 q^{43} - 33186650112 q^{46} - 34079455152 q^{49} + 8704819200 q^{52} + 301214327184 q^{55} - 7326144000 q^{58} + 6311563104 q^{61} - 824633720832 q^{64} + 173066025720 q^{67} - 64914186240 q^{70} - 284072089200 q^{73} + 315595407360 q^{76} - 22027137144 q^{79} + 753291878400 q^{82} - 31677547824 q^{85} - 275801702400 q^{88} - 1979828714496 q^{91} - 311763336192 q^{94} + 522094640520 q^{97} + O(q^{100})$$

## Decomposition of $$S_{13}^{\mathrm{new}}(162, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
162.13.d.a $4$ $148.067$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-67144$$ $$q+(-2^{5}\beta _{1}+2^{5}\beta _{3})q^{2}+(2^{11}-2^{11}\beta _{2}+\cdots)q^{4}+\cdots$$
162.13.d.b $4$ $148.067$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$98744$$ $$q+(-2^{5}\beta _{1}+2^{5}\beta _{3})q^{2}+(2^{11}-2^{11}\beta _{2}+\cdots)q^{4}+\cdots$$
162.13.d.c $8$ $148.067$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-271484$$ $$q+\beta _{3}q^{2}-2^{11}\beta _{1}q^{4}+(\beta _{2}+139\beta _{3}+\cdots)q^{5}+\cdots$$
162.13.d.d $8$ $148.067$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-153080$$ $$q+(-\beta _{3}+\beta _{6})q^{2}+(2^{11}+2^{11}\beta _{1})q^{4}+\cdots$$
162.13.d.e $8$ $148.067$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$68284$$ $$q+\beta _{3}q^{2}+(2^{11}-2^{11}\beta _{1})q^{4}+(60\beta _{2}+\cdots)q^{5}+\cdots$$
162.13.d.f $16$ $148.067$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$393320$$ $$q+\beta _{8}q^{2}-2^{11}\beta _{1}q^{4}+(-10^{2}\beta _{8}+10^{2}\beta _{9}+\cdots)q^{5}+\cdots$$
162.13.d.g $24$ $148.067$ None $$0$$ $$0$$ $$0$$ $$54336$$
162.13.d.h $24$ $148.067$ None $$0$$ $$0$$ $$0$$ $$220224$$

## Decomposition of $$S_{13}^{\mathrm{old}}(162, [\chi])$$ into lower level spaces

$$S_{13}^{\mathrm{old}}(162, [\chi]) \cong$$ $$S_{13}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{13}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{13}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{13}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{13}^{\mathrm{new}}(81, [\chi])$$$$^{\oplus 2}$$