# Properties

 Label 76.8.a Level $76$ Weight $8$ Character orbit 76.a Rep. character $\chi_{76}(1,\cdot)$ Character field $\Q$ Dimension $11$ Newform subspaces $2$ Sturm bound $80$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 76.a (trivial) Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$80$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_0(76))$$.

Total New Old
Modular forms 73 11 62
Cusp forms 67 11 56
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.
$$-$$$$+$$$$-$$$$5$$
$$-$$$$-$$$$+$$$$6$$
Plus space$$+$$$$6$$
Minus space$$-$$$$5$$

## Trace form

 $$11q + 26q^{3} - q^{5} - 1151q^{7} + 7679q^{9} + O(q^{10})$$ $$11q + 26q^{3} - q^{5} - 1151q^{7} + 7679q^{9} + 5321q^{11} + 648q^{13} + 5990q^{15} - 5631q^{17} + 6859q^{19} - 146310q^{21} + 33824q^{23} + 264190q^{25} + 356012q^{27} - 431054q^{29} + 132436q^{31} + 123730q^{33} - 144753q^{35} + 246150q^{37} - 205200q^{39} - 399770q^{41} + 393373q^{43} + 2069743q^{45} + 102323q^{47} + 1183852q^{49} - 1549438q^{51} + 998648q^{53} + 2281271q^{55} + 370386q^{57} + 933390q^{59} + 2445067q^{61} - 8177599q^{63} - 4404092q^{65} - 1091404q^{67} - 119816q^{69} - 6088678q^{71} - 6928883q^{73} + 12136824q^{75} - 8904315q^{77} + 3181880q^{79} + 17539343q^{81} + 8590644q^{83} - 767505q^{85} - 22198028q^{87} - 11720136q^{89} - 2351012q^{91} + 11190704q^{93} + 3834181q^{95} - 20085436q^{97} + 14272625q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_0(76))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 2 19
76.8.a.a $$5$$ $$23.741$$ $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ None $$0$$ $$-14$$ $$-280$$ $$414$$ $$-$$ $$+$$ $$q+(-3-\beta _{1})q^{3}+(-56+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$$
76.8.a.b $$6$$ $$23.741$$ $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ None $$0$$ $$40$$ $$279$$ $$-1565$$ $$-$$ $$-$$ $$q+(7-\beta _{1})q^{3}+(47-\beta _{1}-\beta _{2})q^{5}+(-262+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{8}^{\mathrm{old}}(\Gamma_0(76))$$ into lower level spaces

$$S_{8}^{\mathrm{old}}(\Gamma_0(76)) \cong$$ $$S_{8}^{\mathrm{new}}(\Gamma_0(2))$$$$^{\oplus 4}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(19))$$$$^{\oplus 3}$$$$\oplus$$$$S_{8}^{\mathrm{new}}(\Gamma_0(38))$$$$^{\oplus 2}$$