# Properties

 Label 714.2.i Level $714$ Weight $2$ Character orbit 714.i Rep. character $\chi_{714}(205,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $40$ Newform subspaces $15$ Sturm bound $288$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$714 = 2 \cdot 3 \cdot 7 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 714.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$15$$ Sturm bound: $$288$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(714, [\chi])$$.

Total New Old
Modular forms 304 40 264
Cusp forms 272 40 232
Eisenstein series 32 0 32

## Trace form

 $$40 q - 20 q^{4} + 8 q^{6} - 4 q^{7} - 20 q^{9} + O(q^{10})$$ $$40 q - 20 q^{4} + 8 q^{6} - 4 q^{7} - 20 q^{9} - 4 q^{10} + 8 q^{11} - 8 q^{14} - 8 q^{15} - 20 q^{16} + 16 q^{19} - 8 q^{21} - 8 q^{22} - 16 q^{23} - 4 q^{24} - 24 q^{25} + 8 q^{26} - 4 q^{28} + 32 q^{29} - 12 q^{31} + 4 q^{33} + 16 q^{34} + 32 q^{35} + 40 q^{36} + 8 q^{37} + 8 q^{38} - 8 q^{39} - 4 q^{40} - 64 q^{41} - 4 q^{42} + 48 q^{43} + 8 q^{44} + 8 q^{46} - 40 q^{47} - 4 q^{49} - 16 q^{50} - 8 q^{53} - 4 q^{54} - 40 q^{55} + 16 q^{56} + 32 q^{57} + 4 q^{58} - 24 q^{59} + 4 q^{60} - 16 q^{61} - 48 q^{62} + 8 q^{63} + 40 q^{64} + 40 q^{65} - 8 q^{67} + 44 q^{70} - 48 q^{71} + 24 q^{74} - 32 q^{76} + 48 q^{77} - 20 q^{79} - 20 q^{81} + 96 q^{83} + 16 q^{84} + 40 q^{86} + 4 q^{87} + 4 q^{88} + 8 q^{89} + 8 q^{90} + 80 q^{91} + 32 q^{92} + 8 q^{94} - 32 q^{95} - 4 q^{96} - 56 q^{97} - 8 q^{98} - 16 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(714, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
714.2.i.a $$2$$ $$5.701$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$-1$$ $$1$$ $$-4$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
714.2.i.b $$2$$ $$5.701$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$-1$$ $$1$$ $$5$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
714.2.i.c $$2$$ $$5.701$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$1$$ $$-3$$ $$4$$ $$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
714.2.i.d $$2$$ $$5.701$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$1$$ $$-1$$ $$-4$$ $$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
714.2.i.e $$2$$ $$5.701$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$1$$ $$-1$$ $$1$$ $$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
714.2.i.f $$2$$ $$5.701$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$1$$ $$3$$ $$5$$ $$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
714.2.i.g $$2$$ $$5.701$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-1$$ $$-3$$ $$-4$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
714.2.i.h $$2$$ $$5.701$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$-1$$ $$3$$ $$-4$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
714.2.i.i $$2$$ $$5.701$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$1$$ $$-1$$ $$-4$$ $$q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
714.2.i.j $$2$$ $$5.701$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$1$$ $$-1$$ $$1$$ $$q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
714.2.i.k $$2$$ $$5.701$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$1$$ $$-1$$ $$5$$ $$q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-1+\zeta_{6})q^{4}+\cdots$$
714.2.i.l $$4$$ $$5.701$$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$-2$$ $$-2$$ $$-2$$ $$-3$$ $$q-\beta _{2}q^{2}+(-1+\beta _{2})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots$$
714.2.i.m $$4$$ $$5.701$$ $$\Q(\sqrt{-3}, \sqrt{17})$$ None $$-2$$ $$-2$$ $$0$$ $$2$$ $$q-\beta _{2}q^{2}+(-1+\beta _{2})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots$$
714.2.i.n $$4$$ $$5.701$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$-2$$ $$2$$ $$2$$ $$q+(1+\beta _{2})q^{2}+\beta _{2}q^{3}+\beta _{2}q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots$$
714.2.i.o $$6$$ $$5.701$$ 6.0.11337408.1 None $$3$$ $$3$$ $$3$$ $$-6$$ $$q+\beta _{2}q^{2}+(1-\beta _{2})q^{3}+(-1+\beta _{2})q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(714, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(714, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(119, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(238, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(357, [\chi])$$$$^{\oplus 2}$$