# Properties

 Label 529.2.a Level $529$ Weight $2$ Character orbit 529.a Rep. character $\chi_{529}(1,\cdot)$ Character field $\Q$ Dimension $31$ Newform subspaces $10$ Sturm bound $92$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$529 = 23^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 529.a (trivial) Character field: $$\Q$$ Newform subspaces: $$10$$ Sturm bound: $$92$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$2$$, $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(529))$$.

Total New Old
Modular forms 58 52 6
Cusp forms 35 31 4
Eisenstein series 23 21 2

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

$$23$$Dim.
$$+$$$$13$$
$$-$$$$18$$

## Trace form

 $$31q + q^{2} + 23q^{4} + 2q^{5} + 8q^{6} - 2q^{7} + 3q^{8} + 7q^{9} + O(q^{10})$$ $$31q + q^{2} + 23q^{4} + 2q^{5} + 8q^{6} - 2q^{7} + 3q^{8} + 7q^{9} - 6q^{10} + 6q^{11} - 2q^{12} - 6q^{13} - 4q^{14} + 10q^{15} + 3q^{16} - 6q^{17} + 5q^{18} + 4q^{19} + 4q^{20} + 10q^{21} + 2q^{22} - 26q^{24} - 13q^{25} + 6q^{26} + 6q^{27} + 6q^{28} + 8q^{29} - 10q^{30} + 2q^{31} - 3q^{32} - 10q^{33} + 8q^{34} - 6q^{35} - 17q^{36} - 2q^{37} - 2q^{38} + 6q^{39} + 10q^{40} - 8q^{44} + 4q^{45} - 18q^{47} + 24q^{48} - 29q^{49} - 29q^{50} - 10q^{51} + 12q^{52} + 8q^{53} + 4q^{54} + 8q^{55} + 10q^{56} + 6q^{58} + 4q^{59} - 4q^{61} - 8q^{62} - 4q^{63} - 39q^{64} + 6q^{65} - 10q^{66} + 10q^{67} - 2q^{68} - 10q^{70} - 14q^{71} + 15q^{72} - 14q^{73} + 6q^{74} - 12q^{75} - 2q^{76} + 24q^{77} - 40q^{78} + 4q^{79} - 18q^{80} - 25q^{81} + 24q^{82} + 22q^{83} - 10q^{84} + 26q^{85} + 14q^{87} - 10q^{88} + 12q^{89} - 12q^{90} - 6q^{91} + 20q^{93} + 22q^{94} + 8q^{95} - 24q^{96} - 22q^{97} + 3q^{98} + 12q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(529))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 23
529.2.a.a $$2$$ $$4.224$$ $$\Q(\sqrt{5})$$ None $$-1$$ $$0$$ $$2$$ $$-2$$ $$-$$ $$q-\beta q^{2}+(-1+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots$$
529.2.a.b $$2$$ $$4.224$$ $$\Q(\sqrt{3})$$ None $$0$$ $$2$$ $$0$$ $$-6$$ $$-$$ $$q+\beta q^{2}+(1+\beta )q^{3}+q^{4}+\beta q^{5}+(3+\cdots)q^{6}+\cdots$$
529.2.a.c $$2$$ $$4.224$$ $$\Q(\sqrt{3})$$ None $$0$$ $$2$$ $$0$$ $$6$$ $$-$$ $$q+\beta q^{2}+(1+\beta )q^{3}+q^{4}-\beta q^{5}+(3+\cdots)q^{6}+\cdots$$
529.2.a.d $$2$$ $$4.224$$ $$\Q(\sqrt{2})$$ None $$2$$ $$0$$ $$-2$$ $$-4$$ $$-$$ $$q+(1+\beta )q^{2}+\beta q^{3}+(1+2\beta )q^{4}+(-1+\cdots)q^{5}+\cdots$$
529.2.a.e $$2$$ $$4.224$$ $$\Q(\sqrt{2})$$ None $$2$$ $$0$$ $$2$$ $$4$$ $$-$$ $$q+(1+\beta )q^{2}+\beta q^{3}+(1+2\beta )q^{4}+(1+\cdots)q^{5}+\cdots$$
529.2.a.f $$3$$ $$4.224$$ 3.3.621.1 $$\Q(\sqrt{-23})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$-$$ $$q+\beta _{1}q^{2}-\beta _{2}q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots$$
529.2.a.g $$4$$ $$4.224$$ $$\Q(\zeta_{24})^+$$ None $$-4$$ $$-4$$ $$0$$ $$0$$ $$+$$ $$q-q^{2}+(-1+\beta _{2})q^{3}-q^{4}-\beta _{3}q^{5}+\cdots$$
529.2.a.h $$4$$ $$4.224$$ $$\Q(\sqrt{2}, \sqrt{13})$$ None $$-2$$ $$-4$$ $$0$$ $$0$$ $$+$$ $$q+(-1+\beta _{3})q^{2}-q^{3}+(2-\beta _{3})q^{4}+\cdots$$
529.2.a.i $$5$$ $$4.224$$ $$\Q(\zeta_{22})^+$$ None $$2$$ $$2$$ $$-7$$ $$-8$$ $$+$$ $$q+(\beta _{3}-\beta _{4})q^{2}+(1+\beta _{2}-\beta _{3}+\beta _{4})q^{3}+\cdots$$
529.2.a.j $$5$$ $$4.224$$ $$\Q(\zeta_{22})^+$$ None $$2$$ $$2$$ $$7$$ $$8$$ $$-$$ $$q+(\beta _{3}-\beta _{4})q^{2}+(1+\beta _{2}-\beta _{3}+\beta _{4})q^{3}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(529))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(529)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(23))$$$$^{\oplus 2}$$