# Properties

 Label 501.2.a Level $501$ Weight $2$ Character orbit 501.a Rep. character $\chi_{501}(1,\cdot)$ Character field $\Q$ Dimension $27$ Newform subspaces $5$ Sturm bound $112$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$501 = 3 \cdot 167$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 501.a (trivial) Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$112$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(501))$$.

Total New Old
Modular forms 58 27 31
Cusp forms 55 27 28
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.

$$3$$$$167$$FrickeDim.
$$+$$$$+$$$$+$$$$5$$
$$+$$$$-$$$$-$$$$9$$
$$-$$$$+$$$$-$$$$8$$
$$-$$$$-$$$$+$$$$5$$
Plus space$$+$$$$10$$
Minus space$$-$$$$17$$

## Trace form

 $$27q - 3q^{2} - q^{3} + 25q^{4} - 6q^{5} + q^{6} - 8q^{7} - 15q^{8} + 27q^{9} + O(q^{10})$$ $$27q - 3q^{2} - q^{3} + 25q^{4} - 6q^{5} + q^{6} - 8q^{7} - 15q^{8} + 27q^{9} - 2q^{10} + q^{12} - 2q^{13} + 8q^{14} + 2q^{15} + 33q^{16} - 2q^{17} - 3q^{18} + 4q^{19} - 6q^{20} - 8q^{21} - 8q^{23} - 3q^{24} + 9q^{25} + 22q^{26} - q^{27} - 20q^{28} - 10q^{29} + 10q^{30} - 8q^{31} - 23q^{32} - 4q^{33} + 2q^{34} - 4q^{35} + 25q^{36} - 14q^{37} + 20q^{38} + 2q^{39} + 42q^{40} + 10q^{41} - 4q^{42} + 16q^{43} - 6q^{45} + 8q^{46} + 12q^{47} + q^{48} + 11q^{49} - 13q^{50} + 2q^{51} - 26q^{52} + 2q^{53} + q^{54} + 24q^{55} - 4q^{56} - 12q^{57} - 46q^{58} + 32q^{59} + 6q^{60} + 2q^{61} - 20q^{62} - 8q^{63} + q^{64} - 16q^{65} - 20q^{66} - 8q^{67} + 46q^{68} + 4q^{69} - 8q^{70} + 36q^{71} - 15q^{72} - 34q^{73} - 6q^{74} - 7q^{75} - 20q^{76} - 2q^{78} + 24q^{79} - 42q^{80} + 27q^{81} + 38q^{82} + 36q^{83} - 16q^{84} - 24q^{85} - 76q^{86} + 10q^{87} - 4q^{88} - 6q^{89} - 2q^{90} - 8q^{91} - 60q^{92} - 32q^{93} - 8q^{94} + 32q^{95} + 25q^{96} - 46q^{97} - 59q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(501))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 3 167
501.2.a.a $$1$$ $$4.001$$ $$\Q$$ None $$1$$ $$-1$$ $$-4$$ $$4$$ $$+$$ $$-$$ $$q+q^{2}-q^{3}-q^{4}-4q^{5}-q^{6}+4q^{7}+\cdots$$
501.2.a.b $$5$$ $$4.001$$ 5.5.36497.1 None $$-4$$ $$5$$ $$-9$$ $$-4$$ $$-$$ $$-$$ $$q+(-1+\beta _{3})q^{2}+q^{3}+(1+\beta _{1}-2\beta _{3}+\cdots)q^{4}+\cdots$$
501.2.a.c $$5$$ $$4.001$$ 5.5.38569.1 None $$0$$ $$-5$$ $$-1$$ $$-4$$ $$+$$ $$+$$ $$q-\beta _{1}q^{2}-q^{3}+(\beta _{2}+\beta _{3})q^{4}+(\beta _{1}+\beta _{4})q^{5}+\cdots$$
501.2.a.d $$8$$ $$4.001$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$-3$$ $$-8$$ $$1$$ $$0$$ $$+$$ $$-$$ $$q-\beta _{3}q^{2}-q^{3}+(1-\beta _{2}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots$$
501.2.a.e $$8$$ $$4.001$$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$3$$ $$8$$ $$7$$ $$-4$$ $$-$$ $$+$$ $$q+\beta _{1}q^{2}+q^{3}+(1+\beta _{3}+\beta _{5}+\beta _{7})q^{4}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(501))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(501)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(167))$$$$^{\oplus 2}$$