# Properties

 Label 570.4.i Level $570$ Weight $4$ Character orbit 570.i Rep. character $\chi_{570}(121,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $80$ Newform subspaces $15$ Sturm bound $480$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$570 = 2 \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 570.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$15$$ Sturm bound: $$480$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(570, [\chi])$$.

Total New Old
Modular forms 736 80 656
Cusp forms 704 80 624
Eisenstein series 32 0 32

## Trace form

 $$80q - 12q^{3} - 160q^{4} - 56q^{7} - 360q^{9} + O(q^{10})$$ $$80q - 12q^{3} - 160q^{4} - 56q^{7} - 360q^{9} + 40q^{10} + 56q^{11} + 96q^{12} + 156q^{13} + 168q^{14} - 640q^{16} - 136q^{17} + 164q^{19} + 84q^{21} + 288q^{22} + 320q^{23} - 1000q^{25} + 768q^{26} + 216q^{27} + 112q^{28} - 40q^{29} - 88q^{31} + 264q^{33} - 240q^{34} + 260q^{35} - 1440q^{36} - 1688q^{37} + 192q^{38} + 168q^{39} + 160q^{40} - 348q^{41} - 48q^{42} + 28q^{43} - 112q^{44} - 624q^{46} + 1016q^{47} - 192q^{48} + 6432q^{49} + 624q^{52} - 568q^{53} - 1344q^{56} - 420q^{57} - 2912q^{58} - 216q^{59} + 404q^{61} + 464q^{62} + 252q^{63} + 5120q^{64} + 560q^{65} + 240q^{66} + 1236q^{67} + 1088q^{68} - 1200q^{69} + 560q^{70} + 1632q^{71} - 1500q^{73} + 264q^{74} + 600q^{75} + 560q^{76} - 10528q^{77} + 624q^{78} + 3452q^{79} - 3240q^{81} - 2496q^{82} + 2176q^{83} - 672q^{84} + 2064q^{86} + 816q^{87} - 2304q^{88} + 3204q^{89} + 360q^{90} - 2532q^{91} + 1280q^{92} - 324q^{93} - 4448q^{94} + 1080q^{95} + 3480q^{97} - 1504q^{98} - 252q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(570, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
570.4.i.a $$2$$ $$33.631$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$-3$$ $$-5$$ $$26$$ $$q+(-2+2\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}+\cdots$$
570.4.i.b $$2$$ $$33.631$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$3$$ $$-5$$ $$-68$$ $$q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$$
570.4.i.c $$2$$ $$33.631$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$3$$ $$-5$$ $$-28$$ $$q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$$
570.4.i.d $$2$$ $$33.631$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$3$$ $$-5$$ $$2$$ $$q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$$
570.4.i.e $$2$$ $$33.631$$ $$\Q(\sqrt{-3})$$ None $$-2$$ $$3$$ $$-5$$ $$58$$ $$q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$$
570.4.i.f $$2$$ $$33.631$$ $$\Q(\sqrt{-3})$$ None $$2$$ $$3$$ $$5$$ $$-38$$ $$q+(2-2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$$
570.4.i.g $$2$$ $$33.631$$ $$\Q(\sqrt{-3})$$ None $$2$$ $$3$$ $$5$$ $$40$$ $$q+(2-2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots$$
570.4.i.h $$4$$ $$33.631$$ $$\Q(\sqrt{-3}, \sqrt{385})$$ None $$-4$$ $$6$$ $$10$$ $$36$$ $$q+(-2+2\beta _{2})q^{2}+(3-3\beta _{2})q^{3}-4\beta _{2}q^{4}+\cdots$$
570.4.i.i $$4$$ $$33.631$$ $$\Q(\sqrt{-3}, \sqrt{481})$$ None $$4$$ $$6$$ $$10$$ $$6$$ $$q+2\beta _{2}q^{2}+3\beta _{2}q^{3}+(-4+4\beta _{2})q^{4}+\cdots$$
570.4.i.j $$6$$ $$33.631$$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$-6$$ $$9$$ $$15$$ $$-46$$ $$q+(-2+2\beta _{4})q^{2}+(3-3\beta _{4})q^{3}-4\beta _{4}q^{4}+\cdots$$
570.4.i.k $$8$$ $$33.631$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$-8$$ $$-12$$ $$-20$$ $$-100$$ $$q+(-2-2\beta _{2})q^{2}+(-3-3\beta _{2})q^{3}+\cdots$$
570.4.i.l $$10$$ $$33.631$$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$10$$ $$-15$$ $$25$$ $$18$$ $$q+2\beta _{2}q^{2}-3\beta _{2}q^{3}+(-4+4\beta _{2})q^{4}+\cdots$$
570.4.i.m $$10$$ $$33.631$$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$10$$ $$15$$ $$-25$$ $$38$$ $$q+(2-2\beta _{3})q^{2}+(3-3\beta _{3})q^{3}-4\beta _{3}q^{4}+\cdots$$
570.4.i.n $$12$$ $$33.631$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-12$$ $$-18$$ $$30$$ $$8$$ $$q-2\beta _{3}q^{2}-3\beta _{3}q^{3}+(-4+4\beta _{3})q^{4}+\cdots$$
570.4.i.o $$12$$ $$33.631$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$12$$ $$-18$$ $$-30$$ $$-8$$ $$q-2\beta _{4}q^{2}+3\beta _{4}q^{3}+(-4-4\beta _{4})q^{4}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(570, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(570, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(19, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(38, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(57, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(95, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(114, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(190, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(285, [\chi])$$$$^{\oplus 2}$$