# Properties

 Label 429.2.i Level $429$ Weight $2$ Character orbit 429.i Rep. character $\chi_{429}(100,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $44$ Newform subspaces $6$ Sturm bound $112$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$429 = 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 429.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$6$$ Sturm bound: $$112$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$2$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(429, [\chi])$$.

Total New Old
Modular forms 120 44 76
Cusp forms 104 44 60
Eisenstein series 16 0 16

## Trace form

 $$44q + 4q^{2} + 2q^{3} - 20q^{4} + 16q^{5} - 2q^{7} - 24q^{8} - 22q^{9} + O(q^{10})$$ $$44q + 4q^{2} + 2q^{3} - 20q^{4} + 16q^{5} - 2q^{7} - 24q^{8} - 22q^{9} - 4q^{10} - 8q^{12} + 14q^{13} + 16q^{14} - 24q^{16} - 16q^{17} - 8q^{18} - 28q^{20} + 20q^{21} + 4q^{22} + 4q^{23} + 52q^{25} - 16q^{26} - 4q^{27} - 24q^{28} - 8q^{29} - 28q^{31} + 68q^{32} - 8q^{34} + 8q^{35} - 20q^{36} + 48q^{38} + 4q^{39} + 64q^{40} - 24q^{41} - 12q^{42} - 6q^{43} + 16q^{44} - 8q^{45} - 48q^{46} + 8q^{47} + 8q^{48} - 28q^{49} - 36q^{50} + 16q^{51} + 28q^{52} + 32q^{53} + 16q^{55} - 56q^{56} - 48q^{57} - 20q^{58} - 56q^{60} - 22q^{61} - 12q^{62} - 2q^{63} + 24q^{64} - 48q^{65} - 16q^{66} + 14q^{67} + 28q^{68} + 16q^{69} + 128q^{70} + 20q^{71} + 12q^{72} + 28q^{73} - 36q^{74} + 6q^{75} + 56q^{76} + 12q^{78} + 4q^{79} - 20q^{80} - 22q^{81} + 56q^{82} + 32q^{83} - 28q^{84} + 48q^{85} - 24q^{86} - 4q^{87} + 44q^{89} + 8q^{90} - 62q^{91} - 160q^{92} + 2q^{93} - 76q^{94} + 24q^{95} + 40q^{96} - 6q^{97} - 36q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(429, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
429.2.i.a $$2$$ $$3.426$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$4$$ $$-1$$ $$q+(1-\zeta_{6})q^{3}+2\zeta_{6}q^{4}+2q^{5}-\zeta_{6}q^{7}+\cdots$$
429.2.i.b $$2$$ $$3.426$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$4$$ $$3$$ $$q+(1-\zeta_{6})q^{3}+2\zeta_{6}q^{4}+2q^{5}+3\zeta_{6}q^{7}+\cdots$$
429.2.i.c $$10$$ $$3.426$$ 10.0.$$\cdots$$.1 None $$-2$$ $$-5$$ $$16$$ $$-9$$ $$q+\beta _{4}q^{2}+\beta _{3}q^{3}+(-\beta _{1}+\beta _{6})q^{4}+\cdots$$
429.2.i.d $$10$$ $$3.426$$ 10.0.$$\cdots$$.1 None $$0$$ $$5$$ $$-4$$ $$-7$$ $$q+(\beta _{2}+\beta _{7}-\beta _{8}+\beta _{9})q^{2}+(1-\beta _{4}+\cdots)q^{3}+\cdots$$
429.2.i.e $$10$$ $$3.426$$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$2$$ $$5$$ $$4$$ $$9$$ $$q+\beta _{1}q^{2}+(1+\beta _{5})q^{3}+(\beta _{1}-\beta _{3}+\beta _{5}+\cdots)q^{4}+\cdots$$
429.2.i.f $$10$$ $$3.426$$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$4$$ $$-5$$ $$-8$$ $$3$$ $$q+(1-\beta _{1}-\beta _{6})q^{2}+(-1+\beta _{6})q^{3}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(429, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(429, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(143, [\chi])$$$$^{\oplus 2}$$