# Properties

 Label 588.3.m Level $588$ Weight $3$ Character orbit 588.m Rep. character $\chi_{588}(313,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $26$ Newform subspaces $6$ Sturm bound $336$ Trace bound $5$

# Learn more

## Defining parameters

 Level: $$N$$ $$=$$ $$588 = 2^{2} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 588.m (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$6$$ Sturm bound: $$336$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(588, [\chi])$$.

Total New Old
Modular forms 496 26 470
Cusp forms 400 26 374
Eisenstein series 96 0 96

## Trace form

 $$26 q + 3 q^{3} + 6 q^{5} + 39 q^{9} + O(q^{10})$$ $$26 q + 3 q^{3} + 6 q^{5} + 39 q^{9} - 18 q^{11} - 24 q^{15} + 48 q^{17} + 63 q^{19} - 8 q^{23} + 13 q^{25} + 136 q^{29} - 27 q^{31} + 36 q^{33} - 97 q^{37} + 57 q^{39} - 286 q^{43} + 18 q^{45} - 66 q^{47} + 36 q^{51} + 156 q^{53} - 42 q^{57} - 372 q^{59} + 72 q^{61} + 162 q^{65} - 63 q^{67} + 316 q^{71} + 81 q^{73} + 249 q^{75} + 359 q^{79} - 117 q^{81} + 576 q^{85} + 162 q^{87} - 324 q^{89} - 117 q^{93} + 26 q^{95} - 108 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(588, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
588.3.m.a $2$ $16.022$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-3$$ $$0$$ $$q+(-1-\zeta_{6})q^{3}+(-2+\zeta_{6})q^{5}+3\zeta_{6}q^{9}+\cdots$$
588.3.m.b $2$ $16.022$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$12$$ $$0$$ $$q+(-1-\zeta_{6})q^{3}+(8-4\zeta_{6})q^{5}+3\zeta_{6}q^{9}+\cdots$$
588.3.m.c $2$ $16.022$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-12$$ $$0$$ $$q+(1+\zeta_{6})q^{3}+(-8+4\zeta_{6})q^{5}+3\zeta_{6}q^{9}+\cdots$$
588.3.m.d $4$ $16.022$ $$\Q(\sqrt{-3}, \sqrt{65})$$ None $$0$$ $$6$$ $$9$$ $$0$$ $$q+(2-\beta _{1})q^{3}+(2+\beta _{1}-\beta _{3})q^{5}+(3+\cdots)q^{9}+\cdots$$
588.3.m.e $8$ $16.022$ 8.0.339738624.1 None $$0$$ $$-12$$ $$0$$ $$0$$ $$q+(-1+\beta _{4})q^{3}+(-2\beta _{2}-\beta _{5}+\beta _{6}+\cdots)q^{5}+\cdots$$
588.3.m.f $8$ $16.022$ 8.0.339738624.1 None $$0$$ $$12$$ $$0$$ $$0$$ $$q+(1-\beta _{4})q^{3}+(2\beta _{2}+\beta _{5}+\beta _{6})q^{5}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(588, [\chi])$$ into lower level spaces

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