# Properties

 Label 3600.2.w Level $3600$ Weight $2$ Character orbit 3600.w Rep. character $\chi_{3600}(593,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $72$ Newform subspaces $13$ Sturm bound $1440$ Trace bound $31$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$3600 = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3600.w (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$15$$ Character field: $$\Q(i)$$ Newform subspaces: $$13$$ Sturm bound: $$1440$$ Trace bound: $$31$$ Distinguishing $$T_p$$: $$7$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(3600, [\chi])$$.

Total New Old
Modular forms 1584 72 1512
Cusp forms 1296 72 1224
Eisenstein series 288 0 288

## Trace form

 $$72q + O(q^{10})$$ $$72q - 32q^{31} - 8q^{37} - 32q^{43} - 64q^{61} - 48q^{67} + 16q^{73} - 48q^{91} + 16q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(3600, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
3600.2.w.a $$4$$ $$28.746$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$-12$$ $$q+(-3+3\zeta_{8})q^{7}-\zeta_{8}^{2}q^{11}+(3+3\zeta_{8}+\cdots)q^{13}+\cdots$$
3600.2.w.b $$4$$ $$28.746$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+(-2+2\zeta_{8})q^{7}-2\zeta_{8}^{2}q^{11}+(-1+\cdots)q^{13}+\cdots$$
3600.2.w.c $$4$$ $$28.746$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+(-1+\zeta_{8})q^{7}+\zeta_{8}^{2}q^{11}+(-3-3\zeta_{8}+\cdots)q^{13}+\cdots$$
3600.2.w.d $$4$$ $$28.746$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-4\zeta_{8}^{2}q^{11}+(-3+3\zeta_{8})q^{13}+4\zeta_{8}q^{19}+\cdots$$
3600.2.w.e $$4$$ $$28.746$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+(1-\zeta_{8})q^{7}+\zeta_{8}^{2}q^{11}+(3+3\zeta_{8}+\cdots)q^{13}+\cdots$$
3600.2.w.f $$4$$ $$28.746$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+(2-2\zeta_{8})q^{7}+2\zeta_{8}^{2}q^{11}+(-3-3\zeta_{8}+\cdots)q^{13}+\cdots$$
3600.2.w.g $$4$$ $$28.746$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$8$$ $$q+(2-2\zeta_{8})q^{7}+2\zeta_{8}^{2}q^{11}+(3+3\zeta_{8}+\cdots)q^{13}+\cdots$$
3600.2.w.h $$4$$ $$28.746$$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$12$$ $$q+(3-3\zeta_{8})q^{7}+\zeta_{8}^{2}q^{11}+(-3-3\zeta_{8}+\cdots)q^{13}+\cdots$$
3600.2.w.i $$8$$ $$28.746$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$-16$$ $$q+(-2+2\zeta_{24}^{2}-\zeta_{24}^{3})q^{7}+\zeta_{24}^{6}q^{11}+\cdots$$
3600.2.w.j $$8$$ $$28.746$$ 8.0.40960000.1 None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+(-1+\beta _{1}+\beta _{4})q^{7}+(-\beta _{3}-\beta _{6}+\cdots)q^{11}+\cdots$$
3600.2.w.k $$8$$ $$28.746$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{3}q^{7}+\zeta_{24}^{5}q^{11}+\zeta_{24}q^{13}+\cdots$$
3600.2.w.l $$8$$ $$28.746$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{3}q^{7}+\zeta_{24}^{7}q^{11}+3\zeta_{24}q^{13}+\cdots$$
3600.2.w.m $$8$$ $$28.746$$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$0$$ $$16$$ $$q+(2-\zeta_{24}+2\zeta_{24}^{2})q^{7}-\zeta_{24}^{6}q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(3600, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(3600, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(30, [\chi])$$$$^{\oplus 16}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(75, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(150, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(225, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(240, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(300, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(360, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(450, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(600, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(720, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(900, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1200, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(1800, [\chi])$$$$^{\oplus 2}$$