# Properties

 Label 637.2.h Level $637$ Weight $2$ Character orbit 637.h Rep. character $\chi_{637}(165,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $86$ Newform subspaces $13$ Sturm bound $130$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$637 = 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 637.h (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$91$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$13$$ Sturm bound: $$130$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$2$$, $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(637, [\chi])$$.

Total New Old
Modular forms 146 102 44
Cusp forms 114 86 28
Eisenstein series 32 16 16

## Trace form

 $$86q + 2q^{2} - 4q^{3} + 82q^{4} + 2q^{5} + 6q^{6} - 12q^{8} - 33q^{9} + O(q^{10})$$ $$86q + 2q^{2} - 4q^{3} + 82q^{4} + 2q^{5} + 6q^{6} - 12q^{8} - 33q^{9} - q^{10} - 11q^{11} - 2q^{12} + 4q^{13} - 19q^{15} + 82q^{16} + 14q^{17} - 5q^{18} - 2q^{20} - 4q^{22} - 6q^{23} + 20q^{24} - 31q^{25} + 18q^{26} + 26q^{27} - 2q^{29} + 36q^{30} - 13q^{31} - 42q^{32} - 7q^{33} - 28q^{34} - 3q^{36} - 2q^{37} + 16q^{38} - 19q^{39} - 4q^{40} + 11q^{41} - 14q^{43} - 58q^{44} - 50q^{45} - 24q^{46} + 2q^{47} - 18q^{48} - 24q^{50} + 16q^{51} - 43q^{52} - 17q^{53} - 18q^{54} - 18q^{55} - 68q^{57} - q^{58} + 34q^{59} + 33q^{60} - 8q^{61} - 2q^{62} - 16q^{64} + 58q^{65} - 9q^{66} + 44q^{67} + 54q^{68} - 23q^{69} + q^{71} - 47q^{72} + 17q^{73} - 46q^{74} + 18q^{75} + 10q^{76} + 5q^{78} + 22q^{79} + 4q^{80} + 41q^{81} + 2q^{82} + 54q^{83} + 39q^{85} + 8q^{86} + 74q^{87} + 9q^{88} - 4q^{89} - 20q^{90} - 114q^{92} - 32q^{93} - 44q^{94} + 58q^{95} - 34q^{96} + 30q^{97} + 36q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(637, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
637.2.h.a $$2$$ $$5.086$$ $$\Q(\sqrt{-3})$$ None $$2$$ $$-3$$ $$3$$ $$0$$ $$q+q^{2}-3\zeta_{6}q^{3}-q^{4}+3\zeta_{6}q^{5}-3\zeta_{6}q^{6}+\cdots$$
637.2.h.b $$4$$ $$5.086$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$-4$$ $$0$$ $$-2$$ $$0$$ $$q+(-1+\beta _{3})q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(1+\cdots)q^{4}+\cdots$$
637.2.h.c $$4$$ $$5.086$$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$-4$$ $$0$$ $$2$$ $$0$$ $$q+(-1+\beta _{3})q^{2}+(\beta _{1}+\beta _{3})q^{3}+(1-2\beta _{3})q^{4}+\cdots$$
637.2.h.d $$4$$ $$5.086$$ $$\Q(\zeta_{12})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $$q-\zeta_{12}^{3}q^{2}+(-1+\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots$$
637.2.h.e $$4$$ $$5.086$$ $$\Q(\zeta_{12})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $$q-\zeta_{12}^{3}q^{2}+(1-\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{3}+\cdots$$
637.2.h.f $$4$$ $$5.086$$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$6$$ $$-3$$ $$3$$ $$0$$ $$q+(1-\beta _{2})q^{2}+(\beta _{1}+\beta _{2}+\beta _{3})q^{3}-3\beta _{2}q^{4}+\cdots$$
637.2.h.g $$4$$ $$5.086$$ $$\Q(\sqrt{-3}, \sqrt{5})$$ None $$6$$ $$3$$ $$-3$$ $$0$$ $$q+(1-\beta _{2})q^{2}+(-\beta _{1}-\beta _{2}-\beta _{3})q^{3}+\cdots$$
637.2.h.h $$8$$ $$5.086$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$-2$$ $$-1$$ $$7$$ $$0$$ $$q+\beta _{3}q^{2}-\beta _{5}q^{3}+(1-\beta _{2}-\beta _{3})q^{4}+\cdots$$
637.2.h.i $$8$$ $$5.086$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$-2$$ $$1$$ $$-7$$ $$0$$ $$q+\beta _{3}q^{2}+\beta _{5}q^{3}+(1-\beta _{2}-\beta _{3})q^{4}+\cdots$$
637.2.h.j $$8$$ $$5.086$$ 8.0.$$\cdots$$.6 None $$4$$ $$0$$ $$0$$ $$0$$ $$q+(1-\beta _{5}+\beta _{6})q^{2}+(-\beta _{3}+\beta _{7})q^{3}+\cdots$$
637.2.h.k $$8$$ $$5.086$$ 8.0.$$\cdots$$.7 None $$8$$ $$0$$ $$0$$ $$0$$ $$q+q^{2}+\beta _{5}q^{3}-q^{4}+\beta _{1}q^{5}+\beta _{5}q^{6}+\cdots$$
637.2.h.l $$12$$ $$5.086$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-4$$ $$-1$$ $$-1$$ $$0$$ $$q+(\beta _{3}+\beta _{5}-\beta _{11})q^{2}-\beta _{11}q^{3}+(1+\cdots)q^{4}+\cdots$$
637.2.h.m $$16$$ $$5.086$$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$-8$$ $$0$$ $$0$$ $$0$$ $$q+(-1-\beta _{4})q^{2}+(-\beta _{1}+\beta _{5})q^{3}+(1+\cdots)q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(637, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(637, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(91, [\chi])$$$$^{\oplus 2}$$