# Properties

 Label 401.2.a Level 401 Weight 2 Character orbit a Rep. character $$\chi_{401}(1,\cdot)$$ Character field $$\Q$$ Dimension 33 Newforms 2 Sturm bound 67 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ = $$401$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 401.a (trivial) Character field: $$\Q$$ Newforms: $$2$$ Sturm bound: $$67$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(401))$$.

Total New Old
Modular forms 34 34 0
Cusp forms 33 33 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.

$$401$$Dim.
$$+$$$$12$$
$$-$$$$21$$

## Trace form

 $$33q - 3q^{2} - 2q^{3} + 33q^{4} - 4q^{5} + 4q^{7} - 3q^{8} + 23q^{9} + O(q^{10})$$ $$33q - 3q^{2} - 2q^{3} + 33q^{4} - 4q^{5} + 4q^{7} - 3q^{8} + 23q^{9} - 4q^{10} - 10q^{11} - 16q^{12} - 2q^{13} - 14q^{14} - 10q^{15} + 21q^{16} - 11q^{18} + 4q^{19} - 4q^{20} - 2q^{21} + 2q^{22} - 4q^{23} + 12q^{24} + 33q^{25} + 4q^{26} + 4q^{27} + 2q^{28} - 10q^{29} + 2q^{30} + 18q^{31} + q^{32} - 4q^{33} + 8q^{34} + 24q^{35} + 31q^{36} + 6q^{37} + 12q^{38} + 12q^{39} - 32q^{40} - 18q^{41} + 26q^{42} + 8q^{43} - 28q^{44} - 44q^{45} - 20q^{46} + 6q^{47} - 10q^{48} + 45q^{49} + 3q^{50} - 32q^{51} - 6q^{52} + 2q^{53} + 10q^{54} + 2q^{55} - 58q^{56} - 10q^{57} - 16q^{58} + 6q^{59} - 36q^{60} + 8q^{61} + 30q^{62} + 30q^{63} + 21q^{64} - 26q^{65} - 52q^{66} + 12q^{67} - 14q^{68} + 22q^{69} + 14q^{70} - 41q^{72} - 2q^{73} - 14q^{74} + 22q^{75} + 6q^{76} + 12q^{77} - 24q^{78} - 4q^{79} - 28q^{80} - 47q^{81} - 22q^{82} + 14q^{83} - 46q^{84} + 6q^{85} - 36q^{86} + 10q^{87} - 8q^{88} - 36q^{89} - 10q^{90} + 28q^{91} - 8q^{92} + 4q^{93} - 8q^{94} + 28q^{95} + 14q^{96} + 6q^{97} - 63q^{98} - 36q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(401))$$ into irreducible Hecke orbits

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 401
401.2.a.a $$12$$ $$3.202$$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$-3$$ $$-5$$ $$-7$$ $$-20$$ $$+$$ $$q-\beta _{1}q^{2}+\beta _{9}q^{3}+(\beta _{2}+\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots$$
401.2.a.b $$21$$ $$3.202$$ None $$0$$ $$3$$ $$3$$ $$24$$ $$-$$