# Properties

 Label 420.2.l Level $420$ Weight $2$ Character orbit 420.l Rep. character $\chi_{420}(239,\cdot)$ Character field $\Q$ Dimension $72$ Newform subspaces $8$ Sturm bound $192$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$420 = 2^{2} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 420.l (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$60$$ Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$192$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$11$$, $$17$$, $$43$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(420, [\chi])$$.

Total New Old
Modular forms 104 72 32
Cusp forms 88 72 16
Eisenstein series 16 0 16

## Trace form

 $$72 q + 4 q^{4} + O(q^{10})$$ $$72 q + 4 q^{4} + 8 q^{10} - 12 q^{16} - 44 q^{24} - 24 q^{30} - 32 q^{34} + 4 q^{36} - 8 q^{40} - 24 q^{45} - 24 q^{46} + 72 q^{49} + 68 q^{54} + 52 q^{60} - 64 q^{61} + 52 q^{64} - 76 q^{66} - 48 q^{69} + 12 q^{70} - 24 q^{76} - 16 q^{81} + 24 q^{85} - 56 q^{90} - 152 q^{94} + 60 q^{96} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(420, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
420.2.l.a $4$ $3.354$ $$\Q(\zeta_{8})$$ None $$0$$ $$-4$$ $$0$$ $$4$$ $$q-\zeta_{8}^{2}q^{2}+(-1+\zeta_{8}^{2})q^{3}-2q^{4}+\cdots$$
420.2.l.b $4$ $3.354$ $$\Q(\zeta_{8})$$ None $$0$$ $$4$$ $$0$$ $$-4$$ $$q+\zeta_{8}^{2}q^{2}+(1-\zeta_{8}^{2})q^{3}-2q^{4}+(\zeta_{8}^{2}+\cdots)q^{5}+\cdots$$
420.2.l.c $8$ $3.354$ 8.0.386672896.3 None $$0$$ $$-2$$ $$-8$$ $$-8$$ $$q-\beta _{3}q^{2}+\beta _{2}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots$$
420.2.l.d $8$ $3.354$ 8.0.386672896.3 None $$0$$ $$-2$$ $$8$$ $$-8$$ $$q+\beta _{3}q^{2}-\beta _{6}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots$$
420.2.l.e $8$ $3.354$ 8.0.386672896.3 None $$0$$ $$2$$ $$-8$$ $$8$$ $$q-\beta _{3}q^{2}+\beta _{6}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots$$
420.2.l.f $8$ $3.354$ 8.0.386672896.3 None $$0$$ $$2$$ $$8$$ $$8$$ $$q+\beta _{3}q^{2}-\beta _{2}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots$$
420.2.l.g $16$ $3.354$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-16$$ $$q+\beta _{3}q^{2}+\beta _{4}q^{3}+\beta _{11}q^{4}+\beta _{5}q^{5}+\cdots$$
420.2.l.h $16$ $3.354$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$16$$ $$q+\beta _{3}q^{2}+\beta _{6}q^{3}+\beta _{11}q^{4}+(-\beta _{4}+\cdots)q^{5}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(420, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(420, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 2}$$