# Properties

 Label 76.4.a Level $76$ Weight $4$ Character orbit 76.a Rep. character $\chi_{76}(1,\cdot)$ Character field $\Q$ Dimension $5$ Newform subspaces $2$ Sturm bound $40$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 76.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$40$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(76))$$.

Total New Old
Modular forms 33 5 28
Cusp forms 27 5 22
Eisenstein series 6 0 6

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

$$2$$$$19$$FrickeDim.
$$-$$$$+$$$$-$$$$2$$
$$-$$$$-$$$$+$$$$3$$
Plus space$$+$$$$3$$
Minus space$$-$$$$2$$

## Trace form

 $$5q - 4q^{3} + 4q^{5} + 14q^{7} + 35q^{9} + O(q^{10})$$ $$5q - 4q^{3} + 4q^{5} + 14q^{7} + 35q^{9} + 8q^{11} - 46q^{13} + 20q^{15} + 82q^{17} + 19q^{19} - 72q^{21} + 98q^{23} - 35q^{25} + 8q^{27} + 62q^{29} - 204q^{31} - 404q^{33} - 348q^{35} - 86q^{37} + 606q^{39} - 330q^{41} + 144q^{43} + 88q^{45} - 292q^{47} + 97q^{49} + 104q^{51} - 78q^{53} - 4q^{55} + 114q^{57} + 508q^{59} + 40q^{61} + 1592q^{63} + 368q^{65} + 1588q^{67} - 2300q^{69} - 660q^{71} - 1570q^{73} - 156q^{75} + 3250q^{77} - 2676q^{79} + 437q^{81} + 1712q^{83} - 750q^{85} - 2474q^{87} + 1078q^{89} - 472q^{91} + 1316q^{93} + 266q^{95} + 758q^{97} + 20q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(76))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 19
76.4.a.a $$2$$ $$4.484$$ $$\Q(\sqrt{33})$$ None $$0$$ $$-5$$ $$-5$$ $$-30$$ $$-$$ $$+$$ $$q+(-2-\beta )q^{3}+(-5+5\beta )q^{5}+(-13+\cdots)q^{7}+\cdots$$
76.4.a.b $$3$$ $$4.484$$ 3.3.35529.1 None $$0$$ $$1$$ $$9$$ $$44$$ $$-$$ $$-$$ $$q+(\beta _{1}+\beta _{2})q^{3}+(3+\beta _{2})q^{5}+(15-\beta _{1}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(76))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(76)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(19))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(38))$$$$^{\oplus 2}$$