# Properties

 Label 1380.2.r Level $1380$ Weight $2$ Character orbit 1380.r Rep. character $\chi_{1380}(737,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $88$ Newform subspaces $3$ Sturm bound $576$ Trace bound $7$

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## Defining parameters

 Level: $$N$$ $$=$$ $$1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1380.r (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q(i)$$ Newform subspaces: $$3$$ Sturm bound: $$576$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1380, [\chi])$$.

Total New Old
Modular forms 600 88 512
Cusp forms 552 88 464
Eisenstein series 48 0 48

## Trace form

 $$88q + 8q^{7} + O(q^{10})$$ $$88q + 8q^{7} + 16q^{13} + 12q^{15} + 8q^{21} - 36q^{27} - 16q^{31} - 44q^{33} + 16q^{37} + 40q^{45} + 24q^{51} - 8q^{55} + 24q^{57} + 16q^{61} - 4q^{63} + 16q^{67} - 8q^{73} - 28q^{75} - 8q^{81} - 32q^{85} + 16q^{87} - 8q^{93} - 72q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1380, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
1380.2.r.a $$4$$ $$11.019$$ $$\Q(\zeta_{8})$$ None $$0$$ $$-4$$ $$0$$ $$-4$$ $$q+(-1-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+(-2\zeta_{8}+\zeta_{8}^{3})q^{5}+\cdots$$
1380.2.r.b $$4$$ $$11.019$$ $$\Q(\zeta_{8})$$ None $$0$$ $$-4$$ $$0$$ $$12$$ $$q+(-1-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+(-\zeta_{8}+2\zeta_{8}^{3})q^{5}+\cdots$$
1380.2.r.c $$80$$ $$11.019$$ None $$0$$ $$8$$ $$0$$ $$0$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1380, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1380, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(345, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(690, [\chi])$$$$^{\oplus 2}$$