Properties

Label 731.2.f
Level 731
Weight 2
Character orbit f
Rep. character \(\chi_{731}(259,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 128
Newforms 4
Sturm bound 132
Trace bound 3

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Defining parameters

Level: \( N \) = \( 731 = 17 \cdot 43 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 731.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 17 \)
Character field: \(\Q(i)\)
Newforms: \( 4 \)
Sturm bound: \(132\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(731, [\chi])\).

Total New Old
Modular forms 136 128 8
Cusp forms 128 128 0
Eisenstein series 8 0 8

Trace form

\( 128q - 132q^{4} - 8q^{5} + 8q^{6} + O(q^{10}) \) \( 128q - 132q^{4} - 8q^{5} + 8q^{6} + 4q^{10} - 10q^{11} - 20q^{12} - 4q^{13} - 12q^{14} + 124q^{16} - 4q^{17} - 4q^{18} + 20q^{20} - 8q^{21} + 16q^{22} + 10q^{23} - 12q^{24} - 24q^{27} - 28q^{28} + 4q^{29} + 64q^{30} - 18q^{31} - 8q^{33} + 32q^{34} + 32q^{35} - 12q^{37} + 16q^{38} - 48q^{39} - 20q^{40} + 34q^{41} + 20q^{44} + 52q^{45} + 4q^{46} - 48q^{47} - 24q^{48} - 20q^{50} + 32q^{51} - 40q^{52} + 52q^{54} + 32q^{55} + 36q^{56} - 52q^{57} - 28q^{58} - 16q^{61} - 24q^{62} + 32q^{63} - 172q^{64} - 4q^{65} - 8q^{67} + 40q^{68} - 16q^{69} - 28q^{71} + 12q^{72} + 36q^{73} - 60q^{74} + 60q^{75} - 28q^{79} - 60q^{80} - 144q^{81} + 24q^{82} + 88q^{84} + 16q^{85} + 20q^{86} + 32q^{88} + 32q^{89} + 128q^{90} - 16q^{91} - 116q^{92} - 28q^{95} + 52q^{96} - 30q^{97} - 76q^{98} - 42q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(731, [\chi])\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
731.2.f.a \(2\) \(5.837\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(-4\) \(q-iq^{2}+q^{4}+(-1-i)q^{5}+(-2+2i)q^{7}+\cdots\)
731.2.f.b \(2\) \(5.837\) \(\Q(\sqrt{-1}) \) None \(0\) \(4\) \(0\) \(0\) \(q+iq^{2}+(2+2i)q^{3}+q^{4}+(-2+2i)q^{6}+\cdots\)
731.2.f.c \(56\) \(5.837\) None \(0\) \(-4\) \(-2\) \(4\)
731.2.f.d \(68\) \(5.837\) None \(0\) \(0\) \(-4\) \(0\)