# Properties

 Label 252.4.k Level $252$ Weight $4$ Character orbit 252.k Rep. character $\chi_{252}(37,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $20$ Newform subspaces $6$ Sturm bound $192$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$252 = 2^{2} \cdot 3^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 252.k (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$6$$ Sturm bound: $$192$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(252, [\chi])$$.

Total New Old
Modular forms 312 20 292
Cusp forms 264 20 244
Eisenstein series 48 0 48

## Trace form

 $$20q - 6q^{5} + 4q^{7} + O(q^{10})$$ $$20q - 6q^{5} + 4q^{7} - 24q^{11} - 30q^{17} + 40q^{19} - 48q^{23} - 332q^{25} - 168q^{31} + 360q^{35} + 346q^{37} + 816q^{41} - 176q^{43} + 48q^{47} - 844q^{49} + 342q^{53} - 656q^{55} - 1152q^{59} + 918q^{61} - 1260q^{65} + 152q^{67} - 2184q^{71} + 1262q^{73} + 2154q^{77} + 192q^{79} + 2832q^{83} + 2572q^{85} - 858q^{89} - 1992q^{91} + 1272q^{95} - 976q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(252, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
252.4.k.a $$2$$ $$14.868$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-37$$ $$q+(-19+\zeta_{6})q^{7}+89q^{13}+(163-163\zeta_{6})q^{19}+\cdots$$
252.4.k.b $$2$$ $$14.868$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$17$$ $$q+(-1+19\zeta_{6})q^{7}-19q^{13}+(-107+\cdots)q^{19}+\cdots$$
252.4.k.c $$4$$ $$14.868$$ $$\Q(\sqrt{-3}, \sqrt{37})$$ None $$0$$ $$0$$ $$-14$$ $$24$$ $$q+(-7\beta _{1}-2\beta _{2})q^{5}+(10-8\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots$$
252.4.k.d $$4$$ $$14.868$$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$0$$ $$0$$ $$-3$$ $$-20$$ $$q+(-2+2\beta _{1}-\beta _{2}-\beta _{3})q^{5}+(-2+\cdots)q^{7}+\cdots$$
252.4.k.e $$4$$ $$14.868$$ $$\Q(\sqrt{-3}, \sqrt{385})$$ None $$0$$ $$0$$ $$0$$ $$14$$ $$q-\beta _{2}q^{5}+(-7+21\beta _{1})q^{7}+(-\beta _{2}+\cdots)q^{11}+\cdots$$
252.4.k.f $$4$$ $$14.868$$ $$\Q(\sqrt{-3}, \sqrt{193})$$ None $$0$$ $$0$$ $$11$$ $$6$$ $$q+(-\beta _{1}+6\beta _{2})q^{5}+(-1+2\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(252, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(252, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 2}$$