# Properties

 Label 210.2.j Level $210$ Weight $2$ Character orbit 210.j Rep. character $\chi_{210}(113,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $24$ Newform subspaces $2$ Sturm bound $96$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$210 = 2 \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 210.j (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q(i)$$ Newform subspaces: $$2$$ Sturm bound: $$96$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(210, [\chi])$$.

Total New Old
Modular forms 112 24 88
Cusp forms 80 24 56
Eisenstein series 32 0 32

## Trace form

 $$24q + 8q^{3} + O(q^{10})$$ $$24q + 8q^{3} - 8q^{12} + 8q^{15} - 24q^{16} - 8q^{18} - 8q^{21} + 8q^{22} + 40q^{25} + 8q^{27} - 16q^{31} - 32q^{33} + 8q^{36} - 40q^{37} - 16q^{40} + 16q^{43} - 40q^{45} + 16q^{46} - 8q^{48} + 16q^{51} - 32q^{55} + 32q^{57} - 16q^{58} + 8q^{63} - 16q^{66} + 8q^{70} + 8q^{72} - 48q^{73} + 32q^{75} - 8q^{78} - 72q^{81} + 64q^{82} - 72q^{85} - 80q^{87} + 8q^{88} + 40q^{90} + 48q^{91} + 56q^{93} + 16q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(210, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
210.2.j.a $$12$$ $$1.677$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$4$$ $$-4$$ $$0$$ $$q+\beta _{2}q^{2}+\beta _{11}q^{3}-\beta _{7}q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots$$
210.2.j.b $$12$$ $$1.677$$ $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ None $$0$$ $$4$$ $$4$$ $$0$$ $$q-\beta _{1}q^{2}-\beta _{3}q^{3}+\beta _{7}q^{4}+(1-\beta _{1}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(210, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(210, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 2}$$