# Properties

 Label 128.4.a Level $128$ Weight $4$ Character orbit 128.a Rep. character $\chi_{128}(1,\cdot)$ Character field $\Q$ Dimension $12$ Newform subspaces $8$ Sturm bound $64$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$128 = 2^{7}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 128.a (trivial) Character field: $$\Q$$ Newform subspaces: $$8$$ Sturm bound: $$64$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$3$$, $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(128))$$.

Total New Old
Modular forms 56 12 44
Cusp forms 40 12 28
Eisenstein series 16 0 16

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$Dim.
$$+$$$$7$$
$$-$$$$5$$

## Trace form

 $$12q + 108q^{9} + O(q^{10})$$ $$12q + 108q^{9} + 104q^{17} + 212q^{25} + 464q^{33} - 392q^{41} - 852q^{49} - 1360q^{57} - 2512q^{65} - 24q^{73} + 1596q^{81} + 1960q^{89} + 4328q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(128))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2
128.4.a.a $$1$$ $$7.552$$ $$\Q$$ None $$0$$ $$-2$$ $$-6$$ $$20$$ $$-$$ $$q-2q^{3}-6q^{5}+20q^{7}-23q^{9}-14q^{11}+\cdots$$
128.4.a.b $$1$$ $$7.552$$ $$\Q$$ None $$0$$ $$-2$$ $$6$$ $$-20$$ $$-$$ $$q-2q^{3}+6q^{5}-20q^{7}-23q^{9}-14q^{11}+\cdots$$
128.4.a.c $$1$$ $$7.552$$ $$\Q$$ None $$0$$ $$2$$ $$-6$$ $$-20$$ $$-$$ $$q+2q^{3}-6q^{5}-20q^{7}-23q^{9}+14q^{11}+\cdots$$
128.4.a.d $$1$$ $$7.552$$ $$\Q$$ None $$0$$ $$2$$ $$6$$ $$20$$ $$+$$ $$q+2q^{3}+6q^{5}+20q^{7}-23q^{9}+14q^{11}+\cdots$$
128.4.a.e $$2$$ $$7.552$$ $$\Q(\sqrt{3})$$ None $$0$$ $$-4$$ $$-4$$ $$-8$$ $$-$$ $$q+(-2+\beta )q^{3}+(-2-2\beta )q^{5}+(-4+\cdots)q^{7}+\cdots$$
128.4.a.f $$2$$ $$7.552$$ $$\Q(\sqrt{3})$$ None $$0$$ $$-4$$ $$4$$ $$8$$ $$+$$ $$q+(-2+\beta )q^{3}+(2+2\beta )q^{5}+(4+2\beta )q^{7}+\cdots$$
128.4.a.g $$2$$ $$7.552$$ $$\Q(\sqrt{3})$$ None $$0$$ $$4$$ $$-4$$ $$8$$ $$+$$ $$q+(2+\beta )q^{3}+(-2+2\beta )q^{5}+(4-2\beta )q^{7}+\cdots$$
128.4.a.h $$2$$ $$7.552$$ $$\Q(\sqrt{3})$$ None $$0$$ $$4$$ $$4$$ $$-8$$ $$+$$ $$q+(2+\beta )q^{3}+(2-2\beta )q^{5}+(-4+2\beta )q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(128))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(128)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(32))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(64))$$$$^{\oplus 2}$$