# Properties

 Label 503.2.a Level $503$ Weight $2$ Character orbit 503.a Rep. character $\chi_{503}(1,\cdot)$ Character field $\Q$ Dimension $42$ Newform subspaces $6$ Sturm bound $84$ Trace bound $3$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$503$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 503.a (trivial) Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$84$$ Trace bound: $$3$$ Distinguishing $$T_p$$: $$2$$, $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(503))$$.

Total New Old
Modular forms 43 43 0
Cusp forms 42 42 0
Eisenstein series 1 1 0

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

$$503$$Dim.
$$+$$$$11$$
$$-$$$$31$$

## Trace form

 $$42q + q^{2} + 2q^{3} + 41q^{4} - 2q^{6} + 4q^{7} + 3q^{8} + 42q^{9} + O(q^{10})$$ $$42q + q^{2} + 2q^{3} + 41q^{4} - 2q^{6} + 4q^{7} + 3q^{8} + 42q^{9} + 16q^{12} + 8q^{13} - 4q^{14} + 4q^{15} + 43q^{16} + 10q^{17} + 5q^{18} - 2q^{19} - 14q^{20} - 4q^{21} + 18q^{22} + 14q^{23} - 10q^{24} + 60q^{25} + 6q^{26} + 26q^{27} - 10q^{28} - 4q^{29} - 20q^{30} - 3q^{32} + 6q^{33} - 30q^{34} - 16q^{35} + 43q^{36} + 24q^{37} - 30q^{38} + 2q^{39} - 26q^{40} - 38q^{42} + 14q^{43} - 20q^{44} - 6q^{45} - 12q^{46} - 8q^{47} + 14q^{48} + 40q^{49} - 13q^{50} - 2q^{51} + 26q^{52} + 14q^{53} - 48q^{54} - 14q^{55} - 6q^{56} + 6q^{57} + 44q^{58} - 2q^{59} - 56q^{60} + 4q^{61} - 12q^{62} + 12q^{63} + 61q^{64} - 12q^{65} - 10q^{66} + 18q^{67} + 30q^{68} - 42q^{69} + 16q^{70} - 26q^{71} - 35q^{72} + 36q^{73} - 54q^{74} + 34q^{75} - 14q^{76} - 16q^{77} - 40q^{78} + 20q^{79} - 68q^{80} + 58q^{81} - 28q^{82} - 6q^{83} - 94q^{84} + 16q^{85} + 2q^{86} - 10q^{87} + 72q^{88} - 6q^{89} - 132q^{90} - 2q^{91} + 36q^{92} - 8q^{94} + 50q^{95} - 66q^{96} + 76q^{97} - 17q^{98} - 10q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(503))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 503
503.2.a.a $$1$$ $$4.016$$ $$\Q$$ None $$-1$$ $$1$$ $$-4$$ $$-3$$ $$-$$ $$q-q^{2}+q^{3}-q^{4}-4q^{5}-q^{6}-3q^{7}+\cdots$$
503.2.a.b $$1$$ $$4.016$$ $$\Q$$ None $$1$$ $$1$$ $$-2$$ $$-3$$ $$+$$ $$q+q^{2}+q^{3}-q^{4}-2q^{5}+q^{6}-3q^{7}+\cdots$$
503.2.a.c $$1$$ $$4.016$$ $$\Q$$ None $$1$$ $$3$$ $$-2$$ $$3$$ $$-$$ $$q+q^{2}+3q^{3}-q^{4}-2q^{5}+3q^{6}+3q^{7}+\cdots$$
503.2.a.d $$3$$ $$4.016$$ 3.3.257.1 None $$0$$ $$1$$ $$0$$ $$1$$ $$-$$ $$q+\beta _{2}q^{2}+\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots$$
503.2.a.e $$10$$ $$4.016$$ $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ None $$-4$$ $$-8$$ $$-1$$ $$-5$$ $$+$$ $$q-\beta _{9}q^{2}+(-1+\beta _{1})q^{3}+\beta _{5}q^{4}-\beta _{2}q^{5}+\cdots$$
503.2.a.f $$26$$ $$4.016$$ None $$4$$ $$4$$ $$9$$ $$11$$ $$-$$