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

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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

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

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 \(0\) \(3\) \(3\) \(24\) \(-\)