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Properties

Label 401.2.a
Level $401$
Weight $2$
Character orbit 401.a
Rep. character $\chi_{401}(1,\cdot)$
Character field $\Q$
Dimension $33$
Newform subspaces $2$
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: \(\Q\)
Newform subspaces: \( 2 \)
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 and the Fricke involution.

\(401\)Dim
\(+\)\(12\)
\(-\)\(21\)

Trace form

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(401))\) into newform subspaces

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