# Properties

 Label 84.4.k Level $84$ Weight $4$ Character orbit 84.k Rep. character $\chi_{84}(5,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $16$ Newform subspaces $3$ Sturm bound $64$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 84.k (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$21$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$3$$ Sturm bound: $$64$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$5$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(84, [\chi])$$.

Total New Old
Modular forms 108 16 92
Cusp forms 84 16 68
Eisenstein series 24 0 24

## Trace form

 $$16q - 22q^{7} - 30q^{9} + O(q^{10})$$ $$16q - 22q^{7} - 30q^{9} + 132q^{15} + 36q^{19} - 54q^{21} - 194q^{25} + 534q^{31} - 108q^{33} + 130q^{37} - 108q^{39} - 1040q^{43} - 342q^{45} - 932q^{49} - 300q^{51} + 1800q^{57} + 2148q^{61} + 1056q^{63} + 1100q^{67} + 486q^{73} - 3384q^{75} + 446q^{79} - 450q^{81} - 6144q^{85} - 2898q^{87} + 708q^{91} + 72q^{93} + 9216q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(84, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
84.4.k.a $$2$$ $$4.956$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$-9$$ $$0$$ $$-17$$ $$q+(-3-3\zeta_{6})q^{3}+(-18+19\zeta_{6})q^{7}+\cdots$$
84.4.k.b $$2$$ $$4.956$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$9$$ $$0$$ $$37$$ $$q+(3+3\zeta_{6})q^{3}+(18+\zeta_{6})q^{7}+3^{3}\zeta_{6}q^{9}+\cdots$$
84.4.k.c $$12$$ $$4.956$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-42$$ $$q-\beta _{5}q^{3}-\beta _{9}q^{5}+(-1+2\beta _{1}-2\beta _{4}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(84, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(84, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 2}$$