# Properties

 Label 85.2.a Level $85$ Weight $2$ Character orbit 85.a Rep. character $\chi_{85}(1,\cdot)$ Character field $\Q$ Dimension $5$ Newform subspaces $3$ Sturm bound $18$ Trace bound $2$

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## Defining parameters

 Level: $$N$$ $$=$$ $$85 = 5 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 85.a (trivial) Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$18$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(85))$$.

Total New Old
Modular forms 10 5 5
Cusp forms 7 5 2
Eisenstein series 3 0 3

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

$$5$$$$17$$FrickeDim.
$$+$$$$+$$$$+$$$$2$$
$$+$$$$-$$$$-$$$$1$$
$$-$$$$+$$$$-$$$$2$$
Plus space$$+$$$$2$$
Minus space$$-$$$$3$$

## Trace form

 $$5q - q^{2} + 3q^{4} - q^{5} - 4q^{6} - 8q^{7} - 9q^{8} + 9q^{9} + O(q^{10})$$ $$5q - q^{2} + 3q^{4} - q^{5} - 4q^{6} - 8q^{7} - 9q^{8} + 9q^{9} + q^{10} + 4q^{12} - 6q^{13} + 12q^{14} + 4q^{15} - 5q^{16} - 3q^{17} - q^{18} + 4q^{19} + q^{20} - 8q^{21} + 8q^{22} - 4q^{23} + 8q^{24} + 5q^{25} - 6q^{26} - 12q^{27} - 12q^{28} - 10q^{29} - 8q^{30} + 11q^{32} + 28q^{33} + 3q^{34} + 4q^{35} - 25q^{36} - 10q^{37} + 4q^{38} + 4q^{39} + 9q^{40} + 14q^{41} + 4q^{42} - 12q^{44} - 5q^{45} + 24q^{46} + 20q^{47} - 24q^{48} - 11q^{49} - q^{50} + 4q^{51} + 6q^{52} + 14q^{53} - 20q^{54} + 12q^{55} + 16q^{56} + 2q^{58} - 4q^{59} - 6q^{61} + 8q^{62} - 12q^{63} - 5q^{64} - 10q^{65} - 24q^{66} - 24q^{67} - 5q^{68} + 4q^{71} + 7q^{72} - 26q^{73} + 18q^{74} + 20q^{76} + 4q^{77} + 36q^{78} - 8q^{79} - 15q^{80} + 37q^{81} - 6q^{82} + 24q^{83} - q^{85} + 4q^{86} - 8q^{87} + 28q^{88} - 22q^{89} - 23q^{90} - 4q^{91} - 8q^{92} - 28q^{93} - 16q^{94} + 4q^{95} + 12q^{96} + 2q^{97} - 29q^{98} + 12q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(85))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 5 17
85.2.a.a $$1$$ $$0.679$$ $$\Q$$ None $$1$$ $$2$$ $$-1$$ $$-2$$ $$+$$ $$-$$ $$q+q^{2}+2q^{3}-q^{4}-q^{5}+2q^{6}-2q^{7}+\cdots$$
85.2.a.b $$2$$ $$0.679$$ $$\Q(\sqrt{2})$$ None $$-2$$ $$-4$$ $$-2$$ $$-4$$ $$+$$ $$+$$ $$q+(-1+\beta )q^{2}+(-2-\beta )q^{3}+(1-2\beta )q^{4}+\cdots$$
85.2.a.c $$2$$ $$0.679$$ $$\Q(\sqrt{3})$$ None $$0$$ $$2$$ $$2$$ $$-2$$ $$-$$ $$+$$ $$q+\beta q^{2}+(1-\beta )q^{3}+q^{4}+q^{5}+(-3+\cdots)q^{6}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(85))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(85)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(17))$$$$^{\oplus 2}$$