# Properties

 Label 1638.2.r Level $1638$ Weight $2$ Character orbit 1638.r Rep. character $\chi_{1638}(757,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $68$ Newform subspaces $27$ Sturm bound $672$ Trace bound $17$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.r (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$13$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$27$$ Sturm bound: $$672$$ Trace bound: $$17$$ Distinguishing $$T_p$$: $$5$$, $$11$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1638, [\chi])$$.

Total New Old
Modular forms 704 68 636
Cusp forms 640 68 572
Eisenstein series 64 0 64

## Trace form

 $$68q + 2q^{2} - 34q^{4} + 4q^{5} - 4q^{8} + O(q^{10})$$ $$68q + 2q^{2} - 34q^{4} + 4q^{5} - 4q^{8} - 2q^{10} - 12q^{11} + 22q^{13} + 8q^{14} - 34q^{16} - 6q^{17} - 16q^{19} - 2q^{20} + 4q^{22} + 40q^{25} + 10q^{26} + 2q^{29} - 24q^{31} + 2q^{32} + 44q^{34} - 2q^{37} - 16q^{38} + 4q^{40} + 18q^{41} + 20q^{43} + 24q^{44} + 12q^{46} - 34q^{49} + 12q^{50} - 8q^{52} + 28q^{53} - 12q^{55} - 4q^{56} - 14q^{58} + 16q^{59} - 14q^{61} + 12q^{62} + 68q^{64} - 26q^{65} + 12q^{67} - 6q^{68} + 8q^{71} - 44q^{73} - 42q^{74} - 16q^{76} + 32q^{77} + 40q^{79} - 2q^{80} - 22q^{82} + 88q^{83} - 22q^{85} - 8q^{86} + 4q^{88} - 28q^{89} + 20q^{91} + 44q^{95} + 20q^{97} + 2q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1638, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1638.2.r.a $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-8$$ $$-1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-4q^{5}-\zeta_{6}q^{7}+\cdots$$
1638.2.r.b $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-8$$ $$1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-4q^{5}+\zeta_{6}q^{7}+\cdots$$
1638.2.r.c $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-6$$ $$-1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-3q^{5}-\zeta_{6}q^{7}+\cdots$$
1638.2.r.d $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-4$$ $$1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-2q^{5}+\zeta_{6}q^{7}+\cdots$$
1638.2.r.e $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-2$$ $$1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-q^{5}+\zeta_{6}q^{7}+\cdots$$
1638.2.r.f $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$-1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{7}+q^{8}+\cdots$$
1638.2.r.g $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$-1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{7}+q^{8}+\cdots$$
1638.2.r.h $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$-1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{7}+q^{8}+\cdots$$
1638.2.r.i $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$-1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{7}+q^{8}+\cdots$$
1638.2.r.j $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$4$$ $$-1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+2q^{5}-\zeta_{6}q^{7}+\cdots$$
1638.2.r.k $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$4$$ $$1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+2q^{5}+\zeta_{6}q^{7}+\cdots$$
1638.2.r.l $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$4$$ $$1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+2q^{5}+\zeta_{6}q^{7}+\cdots$$
1638.2.r.m $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$6$$ $$-1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+3q^{5}-\zeta_{6}q^{7}+\cdots$$
1638.2.r.n $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$6$$ $$-1$$ $$q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+3q^{5}-\zeta_{6}q^{7}+\cdots$$
1638.2.r.o $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$-6$$ $$1$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-3q^{5}+\zeta_{6}q^{7}+\cdots$$
1638.2.r.p $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$-1$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{7}-q^{8}+\cdots$$
1638.2.r.q $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$-1$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{7}-q^{8}+\cdots$$
1638.2.r.r $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$-1$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{7}-q^{8}+\cdots$$
1638.2.r.s $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$-1$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{7}-q^{8}+\cdots$$
1638.2.r.t $$2$$ $$13.079$$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$8$$ $$1$$ $$q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+4q^{5}+\zeta_{6}q^{7}+\cdots$$
1638.2.r.u $$4$$ $$13.079$$ $$\Q(\sqrt{-3}, \sqrt{13})$$ None $$-2$$ $$0$$ $$8$$ $$2$$ $$q+(-1+\beta _{1})q^{2}-\beta _{1}q^{4}+2q^{5}+\beta _{1}q^{7}+\cdots$$
1638.2.r.v $$4$$ $$13.079$$ $$\Q(\zeta_{12})$$ None $$2$$ $$0$$ $$-8$$ $$2$$ $$q+\zeta_{12}q^{2}+(-1+\zeta_{12})q^{4}+(-2-\zeta_{12}^{3})q^{5}+\cdots$$
1638.2.r.w $$4$$ $$13.079$$ $$\Q(\sqrt{-3}, \sqrt{13})$$ None $$2$$ $$0$$ $$-8$$ $$2$$ $$q+(1-\beta _{1})q^{2}-\beta _{1}q^{4}-2q^{5}+\beta _{1}q^{7}+\cdots$$
1638.2.r.x $$4$$ $$13.079$$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$2$$ $$0$$ $$-2$$ $$-2$$ $$q-\beta _{1}q^{2}+(-1-\beta _{1})q^{4}+(-1+\beta _{2}+\cdots)q^{5}+\cdots$$
1638.2.r.y $$4$$ $$13.079$$ $$\Q(\sqrt{-3}, \sqrt{17})$$ None $$2$$ $$0$$ $$2$$ $$-2$$ $$q+(1-\beta _{2})q^{2}-\beta _{2}q^{4}+(1+\beta _{3})q^{5}+\cdots$$
1638.2.r.z $$4$$ $$13.079$$ $$\Q(\sqrt{-3}, \sqrt{17})$$ None $$2$$ $$0$$ $$6$$ $$2$$ $$q+\beta _{2}q^{2}+(-1+\beta _{2})q^{4}+(1-\beta _{3})q^{5}+\cdots$$
1638.2.r.ba $$4$$ $$13.079$$ $$\Q(\sqrt{-3}, \sqrt{-43})$$ None $$2$$ $$0$$ $$8$$ $$2$$ $$q+\beta _{2}q^{2}+(-1+\beta _{2})q^{4}+2q^{5}+(1+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1638, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1638, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(26, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(39, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(78, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(91, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(117, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(182, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(234, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(273, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(546, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(819, [\chi])$$$$^{\oplus 2}$$