# Properties

 Label 80.4.a Level $80$ Weight $4$ Character orbit 80.a Rep. character $\chi_{80}(1,\cdot)$ Character field $\Q$ Dimension $6$ Newform subspaces $6$ Sturm bound $48$ Trace bound $7$

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## Defining parameters

 Level: $$N$$ $$=$$ $$80 = 2^{4} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 80.a (trivial) Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$48$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(80))$$.

Total New Old
Modular forms 42 6 36
Cusp forms 30 6 24
Eisenstein series 12 0 12

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

$$2$$$$5$$FrickeDim
$$+$$$$+$$$$+$$$$2$$
$$+$$$$-$$$$-$$$$1$$
$$-$$$$+$$$$-$$$$1$$
$$-$$$$-$$$$+$$$$2$$
Plus space$$+$$$$4$$
Minus space$$-$$$$2$$

## Trace form

 $$6 q - 6 q^{3} + 50 q^{7} + 74 q^{9} + O(q^{10})$$ $$6 q - 6 q^{3} + 50 q^{7} + 74 q^{9} - 20 q^{11} + 30 q^{15} - 76 q^{17} + 192 q^{19} + 68 q^{21} - 26 q^{23} + 150 q^{25} - 84 q^{27} - 228 q^{29} - 372 q^{31} - 384 q^{33} - 210 q^{35} - 8 q^{37} - 156 q^{39} + 128 q^{41} - 326 q^{43} - 220 q^{45} + 98 q^{47} - 14 q^{49} + 484 q^{51} + 768 q^{53} + 220 q^{55} - 552 q^{57} - 616 q^{59} - 256 q^{61} + 1906 q^{63} - 140 q^{65} + 1374 q^{67} + 268 q^{69} - 468 q^{71} + 1044 q^{73} - 150 q^{75} + 1424 q^{77} + 120 q^{79} + 1178 q^{81} - 2942 q^{83} + 120 q^{85} - 3540 q^{87} - 900 q^{89} + 3908 q^{91} - 2248 q^{93} + 760 q^{95} + 1244 q^{97} + 1052 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(80))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5
80.4.a.a $1$ $4.720$ $$\Q$$ None $$0$$ $$-10$$ $$-5$$ $$18$$ $+$ $+$ $$q-10q^{3}-5q^{5}+18q^{7}+73q^{9}+2^{4}q^{11}+\cdots$$
80.4.a.b $1$ $4.720$ $$\Q$$ None $$0$$ $$-4$$ $$5$$ $$-16$$ $+$ $-$ $$q-4q^{3}+5q^{5}-2^{4}q^{7}-11q^{9}-6^{2}q^{11}+\cdots$$
80.4.a.c $1$ $4.720$ $$\Q$$ None $$0$$ $$-4$$ $$5$$ $$16$$ $-$ $-$ $$q-4q^{3}+5q^{5}+2^{4}q^{7}-11q^{9}+60q^{11}+\cdots$$
80.4.a.d $1$ $4.720$ $$\Q$$ None $$0$$ $$-2$$ $$-5$$ $$-6$$ $-$ $+$ $$q-2q^{3}-5q^{5}-6q^{7}-23q^{9}-2^{5}q^{11}+\cdots$$
80.4.a.e $1$ $4.720$ $$\Q$$ None $$0$$ $$6$$ $$-5$$ $$34$$ $+$ $+$ $$q+6q^{3}-5q^{5}+34q^{7}+9q^{9}-2^{4}q^{11}+\cdots$$
80.4.a.f $1$ $4.720$ $$\Q$$ None $$0$$ $$8$$ $$5$$ $$4$$ $-$ $-$ $$q+8q^{3}+5q^{5}+4q^{7}+37q^{9}-12q^{11}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(80))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(80)) \simeq$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(5))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(10))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(20))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(40))$$$$^{\oplus 2}$$