Properties

Label 13.3.f
Level 13
Weight 3
Character orbit f
Rep. character \(\chi_{13}(2,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 4
Newforms 1
Sturm bound 3
Trace bound 0

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Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 13.f (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 13 \)
Character field: \(\Q(\zeta_{12})\)
Newforms: \( 1 \)
Sturm bound: \(3\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(13, [\chi])\).

Total New Old
Modular forms 12 12 0
Cusp forms 4 4 0
Eisenstein series 8 8 0

Trace form

\(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 14q^{5} \) \(\mathstrut +\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 16q^{7} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 14q^{5} \) \(\mathstrut +\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 16q^{7} \) \(\mathstrut -\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut 24q^{10} \) \(\mathstrut +\mathstrut 4q^{11} \) \(\mathstrut -\mathstrut 26q^{13} \) \(\mathstrut -\mathstrut 40q^{14} \) \(\mathstrut -\mathstrut 14q^{15} \) \(\mathstrut -\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut -\mathstrut 2q^{18} \) \(\mathstrut +\mathstrut 10q^{19} \) \(\mathstrut +\mathstrut 26q^{20} \) \(\mathstrut +\mathstrut 40q^{21} \) \(\mathstrut -\mathstrut 4q^{22} \) \(\mathstrut +\mathstrut 18q^{23} \) \(\mathstrut +\mathstrut 42q^{24} \) \(\mathstrut +\mathstrut 52q^{26} \) \(\mathstrut -\mathstrut 32q^{27} \) \(\mathstrut -\mathstrut 44q^{28} \) \(\mathstrut +\mathstrut 2q^{29} \) \(\mathstrut -\mathstrut 54q^{30} \) \(\mathstrut -\mathstrut 20q^{31} \) \(\mathstrut -\mathstrut 20q^{32} \) \(\mathstrut -\mathstrut 32q^{33} \) \(\mathstrut -\mathstrut 18q^{34} \) \(\mathstrut +\mathstrut 32q^{35} \) \(\mathstrut -\mathstrut 54q^{36} \) \(\mathstrut -\mathstrut 68q^{37} \) \(\mathstrut -\mathstrut 26q^{39} \) \(\mathstrut +\mathstrut 72q^{40} \) \(\mathstrut +\mathstrut 100q^{41} \) \(\mathstrut +\mathstrut 44q^{42} \) \(\mathstrut +\mathstrut 180q^{43} \) \(\mathstrut +\mathstrut 88q^{44} \) \(\mathstrut -\mathstrut 2q^{45} \) \(\mathstrut -\mathstrut 30q^{46} \) \(\mathstrut -\mathstrut 68q^{47} \) \(\mathstrut -\mathstrut 50q^{48} \) \(\mathstrut -\mathstrut 72q^{49} \) \(\mathstrut -\mathstrut 46q^{50} \) \(\mathstrut +\mathstrut 128q^{53} \) \(\mathstrut +\mathstrut 16q^{54} \) \(\mathstrut -\mathstrut 100q^{55} \) \(\mathstrut -\mathstrut 84q^{56} \) \(\mathstrut -\mathstrut 20q^{57} \) \(\mathstrut -\mathstrut 40q^{58} \) \(\mathstrut -\mathstrut 164q^{59} \) \(\mathstrut +\mathstrut 8q^{60} \) \(\mathstrut -\mathstrut 124q^{61} \) \(\mathstrut +\mathstrut 6q^{62} \) \(\mathstrut +\mathstrut 52q^{63} \) \(\mathstrut +\mathstrut 52q^{65} \) \(\mathstrut +\mathstrut 80q^{66} \) \(\mathstrut +\mathstrut 118q^{67} \) \(\mathstrut +\mathstrut 72q^{68} \) \(\mathstrut +\mathstrut 72q^{69} \) \(\mathstrut +\mathstrut 164q^{70} \) \(\mathstrut -\mathstrut 86q^{71} \) \(\mathstrut +\mathstrut 72q^{72} \) \(\mathstrut +\mathstrut 58q^{73} \) \(\mathstrut +\mathstrut 68q^{74} \) \(\mathstrut +\mathstrut 48q^{75} \) \(\mathstrut +\mathstrut 14q^{76} \) \(\mathstrut -\mathstrut 104q^{78} \) \(\mathstrut -\mathstrut 40q^{79} \) \(\mathstrut -\mathstrut 140q^{80} \) \(\mathstrut -\mathstrut 2q^{81} \) \(\mathstrut +\mathstrut 24q^{82} \) \(\mathstrut -\mathstrut 188q^{83} \) \(\mathstrut +\mathstrut 4q^{84} \) \(\mathstrut +\mathstrut 96q^{85} \) \(\mathstrut -\mathstrut 180q^{86} \) \(\mathstrut +\mathstrut 38q^{87} \) \(\mathstrut -\mathstrut 204q^{88} \) \(\mathstrut -\mathstrut 110q^{89} \) \(\mathstrut +\mathstrut 52q^{91} \) \(\mathstrut -\mathstrut 156q^{92} \) \(\mathstrut -\mathstrut 20q^{93} \) \(\mathstrut +\mathstrut 26q^{94} \) \(\mathstrut -\mathstrut 78q^{95} \) \(\mathstrut +\mathstrut 40q^{96} \) \(\mathstrut +\mathstrut 178q^{97} \) \(\mathstrut -\mathstrut 14q^{98} \) \(\mathstrut +\mathstrut 208q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(13, [\chi])\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
13.3.f.a \(4\) \(0.354\) \(\Q(\zeta_{12})\) None \(-2\) \(-2\) \(-14\) \(16\) \(q+(-1+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)