# Properties

 Label 1638.2.j Level $1638$ Weight $2$ Character orbit 1638.j Rep. character $\chi_{1638}(235,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $80$ Newform subspaces $20$ Sturm bound $672$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1638.j (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$20$$ Sturm bound: $$672$$ Trace bound: $$7$$ 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 80 624
Cusp forms 640 80 560
Eisenstein series 64 0 64

## Trace form

 $$80 q - 40 q^{4} - 8 q^{5} + O(q^{10})$$ $$80 q - 40 q^{4} - 8 q^{5} + 4 q^{10} - 4 q^{11} - 4 q^{13} + 10 q^{14} - 40 q^{16} + 18 q^{17} + 4 q^{19} + 16 q^{20} + 12 q^{22} + 12 q^{23} - 44 q^{25} - 6 q^{26} - 12 q^{28} + 4 q^{29} - 4 q^{31} - 32 q^{34} + 12 q^{35} - 10 q^{38} + 4 q^{40} - 32 q^{41} + 16 q^{43} - 4 q^{44} - 20 q^{46} - 12 q^{47} + 62 q^{49} - 16 q^{50} + 2 q^{52} - 22 q^{53} + 8 q^{55} - 2 q^{56} + 8 q^{59} - 34 q^{61} - 20 q^{62} + 80 q^{64} + 4 q^{65} - 8 q^{67} + 18 q^{68} + 8 q^{71} - 4 q^{73} - 8 q^{76} - 24 q^{77} - 8 q^{80} - 24 q^{82} + 24 q^{83} + 64 q^{85} + 12 q^{86} - 6 q^{88} - 12 q^{89} + 16 q^{91} - 24 q^{92} + 22 q^{94} + 8 q^{95} - 16 q^{97} + 72 q^{98} + O(q^{100})$$

## 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.j.a $2$ $13.079$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-2$$ $$-5$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots$$
1638.2.j.b $2$ $13.079$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$-2$$ $$-1$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots$$
1638.2.j.c $2$ $13.079$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$0$$ $$5$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(2+\zeta_{6})q^{7}+\cdots$$
1638.2.j.d $2$ $13.079$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$1$$ $$-5$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(-3+\cdots)q^{7}+\cdots$$
1638.2.j.e $2$ $13.079$ $$\Q(\sqrt{-3})$$ None $$-1$$ $$0$$ $$1$$ $$-1$$ $$q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots$$
1638.2.j.f $2$ $13.079$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$-4$$ $$-1$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-4\zeta_{6}q^{5}+\cdots$$
1638.2.j.g $2$ $13.079$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$-1$$ $$-5$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(-3+\cdots)q^{7}+\cdots$$
1638.2.j.h $2$ $13.079$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$-1$$ $$-1$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots$$
1638.2.j.i $2$ $13.079$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$0$$ $$-5$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-2-\zeta_{6})q^{7}+\cdots$$
1638.2.j.j $2$ $13.079$ $$\Q(\sqrt{-3})$$ None $$1$$ $$0$$ $$3$$ $$-1$$ $$q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots$$
1638.2.j.k $4$ $13.079$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$-2$$ $$0$$ $$-2$$ $$0$$ $$q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots$$
1638.2.j.l $4$ $13.079$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$-2$$ $$0$$ $$0$$ $$10$$ $$q-\beta _{1}q^{2}+(-1+\beta _{1})q^{4}-\beta _{2}q^{5}+(2+\cdots)q^{7}+\cdots$$
1638.2.j.m $4$ $13.079$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$-4$$ $$-2$$ $$q-\beta _{2}q^{2}+(-1-\beta _{2})q^{4}+(-\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots$$
1638.2.j.n $4$ $13.079$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$0$$ $$2$$ $$q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}+\beta _{1}q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots$$
1638.2.j.o $4$ $13.079$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$2$$ $$0$$ $$0$$ $$10$$ $$q+\beta _{1}q^{2}+(-1+\beta _{1})q^{4}-\beta _{2}q^{5}+(2+\cdots)q^{7}+\cdots$$
1638.2.j.p $6$ $13.079$ 6.0.309123.1 None $$-3$$ $$0$$ $$-2$$ $$-4$$ $$q-\beta _{4}q^{2}+(-1+\beta _{4})q^{4}+(-\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots$$
1638.2.j.q $6$ $13.079$ 6.0.21870000.1 None $$-3$$ $$0$$ $$3$$ $$3$$ $$q+\beta _{2}q^{2}+(-1-\beta _{2})q^{4}+(-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$$
1638.2.j.r $8$ $13.079$ 8.0.8681953329.1 None $$4$$ $$0$$ $$2$$ $$7$$ $$q+(1+\beta _{3})q^{2}+\beta _{3}q^{4}+(1+\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots$$
1638.2.j.s $10$ $13.079$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$-5$$ $$0$$ $$1$$ $$-3$$ $$q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(\beta _{6}-\beta _{8}+\cdots)q^{5}+\cdots$$
1638.2.j.t $10$ $13.079$ $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ None $$5$$ $$0$$ $$-1$$ $$-3$$ $$q-\beta _{2}q^{2}+(-1-\beta _{2})q^{4}-\beta _{8}q^{5}+(-1+\cdots)q^{7}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(1638, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(63, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(91, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(126, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(182, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(273, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(546, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(819, [\chi])$$$$^{\oplus 2}$$