# Properties

 Label 76.6.a Level $76$ Weight $6$ Character orbit 76.a Rep. character $\chi_{76}(1,\cdot)$ Character field $\Q$ Dimension $7$ Newform subspaces $2$ Sturm bound $60$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 76.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$60$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{6}(\Gamma_0(76))$$.

Total New Old
Modular forms 53 7 46
Cusp forms 47 7 40
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.
$$-$$$$+$$$$-$$$$4$$
$$-$$$$-$$$$+$$$$3$$
Plus space$$+$$$$3$$
Minus space$$-$$$$4$$

## Trace form

 $$7q + 2q^{3} - 119q^{5} + 17q^{7} + 1067q^{9} + O(q^{10})$$ $$7q + 2q^{3} - 119q^{5} + 17q^{7} + 1067q^{9} - 523q^{11} + 732q^{13} + 1094q^{15} - 2909q^{17} - 361q^{19} + 942q^{21} + 2672q^{23} + 7352q^{25} - 7156q^{27} + 7754q^{29} - 11644q^{31} + 24286q^{33} + 23403q^{35} - 21258q^{37} + 5736q^{39} - 24450q^{41} - 12631q^{43} - 37151q^{45} - 1933q^{47} + 20390q^{49} + 81146q^{51} - 46884q^{53} - 100265q^{55} - 6498q^{57} + 21286q^{59} + 27413q^{61} - 7471q^{63} + 44516q^{65} + 1972q^{67} - 70616q^{69} + 87738q^{71} - 60673q^{73} - 125928q^{75} - 13529q^{77} + 142072q^{79} + 81155q^{81} - 127324q^{83} + 29301q^{85} + 125020q^{87} - 110132q^{89} + 60332q^{91} + 147392q^{93} + 36461q^{95} + 194176q^{97} + 146981q^{99} + O(q^{100})$$

## Decomposition of $$S_{6}^{\mathrm{new}}(\Gamma_0(76))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 19
76.6.a.a $$3$$ $$12.189$$ 3.3.272193.1 None $$0$$ $$-8$$ $$-9$$ $$-13$$ $$-$$ $$-$$ $$q+(-3-\beta _{1}-\beta _{2})q^{3}+(-2+3\beta _{1}+\cdots)q^{5}+\cdots$$
76.6.a.b $$4$$ $$12.189$$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$0$$ $$10$$ $$-110$$ $$30$$ $$-$$ $$+$$ $$q+(3+\beta _{3})q^{3}+(-26+\beta _{1}+2\beta _{2}+3\beta _{3})q^{5}+\cdots$$

## Decomposition of $$S_{6}^{\mathrm{old}}(\Gamma_0(76))$$ into lower level spaces

$$S_{6}^{\mathrm{old}}(\Gamma_0(76)) \cong$$ $$S_{6}^{\mathrm{new}}(\Gamma_0(4))$$$$^{\oplus 2}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(19))$$$$^{\oplus 3}$$$$\oplus$$$$S_{6}^{\mathrm{new}}(\Gamma_0(38))$$$$^{\oplus 2}$$