# Properties

 Label 588.4.i Level $588$ Weight $4$ Character orbit 588.i Rep. character $\chi_{588}(361,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $40$ Newform subspaces $12$ Sturm bound $448$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 588.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$12$$ Sturm bound: $$448$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(588, [\chi])$$.

Total New Old
Modular forms 720 40 680
Cusp forms 624 40 584
Eisenstein series 96 0 96

## Trace form

 $$40q + 8q^{5} - 180q^{9} + O(q^{10})$$ $$40q + 8q^{5} - 180q^{9} - 56q^{11} - 132q^{13} - 84q^{15} + 124q^{17} + 236q^{19} - 204q^{23} - 710q^{25} - 440q^{29} + 270q^{31} + 138q^{33} + 218q^{37} + 252q^{39} - 24q^{41} - 1568q^{43} + 72q^{45} - 36q^{47} + 288q^{51} - 428q^{53} + 740q^{55} + 1980q^{57} - 1096q^{59} - 204q^{61} - 1000q^{65} - 2116q^{67} - 1608q^{69} - 728q^{71} - 146q^{73} - 312q^{75} + 946q^{79} - 1620q^{81} + 3616q^{83} - 88q^{85} + 942q^{87} - 1152q^{89} - 258q^{93} + 1724q^{95} - 2804q^{97} + 1008q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(588, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
588.4.i.a $$2$$ $$34.693$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-14$$ $$0$$ $$q+(-3+3\zeta_{6})q^{3}-14\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots$$
588.4.i.b $$2$$ $$34.693$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$4$$ $$0$$ $$q+(-3+3\zeta_{6})q^{3}+4\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots$$
588.4.i.c $$2$$ $$34.693$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$6$$ $$0$$ $$q+(-3+3\zeta_{6})q^{3}+6\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots$$
588.4.i.d $$2$$ $$34.693$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$18$$ $$0$$ $$q+(-3+3\zeta_{6})q^{3}+18\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots$$
588.4.i.e $$2$$ $$34.693$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-18$$ $$0$$ $$q+(3-3\zeta_{6})q^{3}-18\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots$$
588.4.i.f $$2$$ $$34.693$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-6$$ $$0$$ $$q+(3-3\zeta_{6})q^{3}-6\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots$$
588.4.i.g $$2$$ $$34.693$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-4$$ $$0$$ $$q+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots$$
588.4.i.h $$2$$ $$34.693$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$14$$ $$0$$ $$q+(3-3\zeta_{6})q^{3}+14\zeta_{6}q^{5}-9\zeta_{6}q^{9}+\cdots$$
588.4.i.i $$4$$ $$34.693$$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$0$$ $$-6$$ $$-3$$ $$0$$ $$q+3\beta _{1}q^{3}+(-2-2\beta _{1}+\beta _{3})q^{5}+(-9+\cdots)q^{9}+\cdots$$
588.4.i.j $$4$$ $$34.693$$ $$\Q(\sqrt{-3}, \sqrt{193})$$ None $$0$$ $$6$$ $$11$$ $$0$$ $$q+(3-3\beta _{2})q^{3}+(-\beta _{1}+6\beta _{2})q^{5}-9\beta _{2}q^{9}+\cdots$$
588.4.i.k $$8$$ $$34.693$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$-12$$ $$0$$ $$0$$ $$q-3\beta _{1}q^{3}+\beta _{6}q^{5}+(-9+9\beta _{1})q^{9}+\cdots$$
588.4.i.l $$8$$ $$34.693$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$12$$ $$0$$ $$0$$ $$q+3\beta _{1}q^{3}-\beta _{6}q^{5}+(-9+9\beta _{1})q^{9}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(588, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(588, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(7, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(14, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(49, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(98, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(147, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(196, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(294, [\chi])$$$$^{\oplus 2}$$