Properties

Label 13.5.d
Level 13
Weight 5
Character orbit d
Rep. character \(\chi_{13}(5,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 6
Newforms 1
Sturm bound 5
Trace bound 0

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Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 13.d (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 13 \)
Character field: \(\Q(i)\)
Newforms: \( 1 \)
Sturm bound: \(5\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(13, [\chi])\).

Total New Old
Modular forms 10 10 0
Cusp forms 6 6 0
Eisenstein series 4 4 0

Trace form

\(6q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 14q^{5} \) \(\mathstrut +\mathstrut 32q^{6} \) \(\mathstrut +\mathstrut 48q^{7} \) \(\mathstrut -\mathstrut 96q^{8} \) \(\mathstrut -\mathstrut 58q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut -\mathstrut 14q^{5} \) \(\mathstrut +\mathstrut 32q^{6} \) \(\mathstrut +\mathstrut 48q^{7} \) \(\mathstrut -\mathstrut 96q^{8} \) \(\mathstrut -\mathstrut 58q^{9} \) \(\mathstrut -\mathstrut 32q^{11} \) \(\mathstrut -\mathstrut 244q^{14} \) \(\mathstrut +\mathstrut 404q^{15} \) \(\mathstrut +\mathstrut 1044q^{16} \) \(\mathstrut -\mathstrut 802q^{18} \) \(\mathstrut +\mathstrut 732q^{19} \) \(\mathstrut +\mathstrut 428q^{20} \) \(\mathstrut -\mathstrut 2128q^{21} \) \(\mathstrut -\mathstrut 1632q^{22} \) \(\mathstrut -\mathstrut 24q^{24} \) \(\mathstrut +\mathstrut 910q^{26} \) \(\mathstrut +\mathstrut 236q^{27} \) \(\mathstrut +\mathstrut 1884q^{28} \) \(\mathstrut +\mathstrut 4184q^{29} \) \(\mathstrut -\mathstrut 3468q^{31} \) \(\mathstrut +\mathstrut 2092q^{32} \) \(\mathstrut +\mathstrut 2324q^{33} \) \(\mathstrut -\mathstrut 5304q^{34} \) \(\mathstrut -\mathstrut 4204q^{35} \) \(\mathstrut -\mathstrut 1758q^{37} \) \(\mathstrut +\mathstrut 1196q^{39} \) \(\mathstrut -\mathstrut 708q^{40} \) \(\mathstrut +\mathstrut 4750q^{41} \) \(\mathstrut +\mathstrut 9532q^{42} \) \(\mathstrut -\mathstrut 3956q^{44} \) \(\mathstrut +\mathstrut 830q^{45} \) \(\mathstrut +\mathstrut 516q^{46} \) \(\mathstrut -\mathstrut 6872q^{47} \) \(\mathstrut -\mathstrut 9436q^{48} \) \(\mathstrut -\mathstrut 322q^{50} \) \(\mathstrut +\mathstrut 3900q^{52} \) \(\mathstrut +\mathstrut 2108q^{53} \) \(\mathstrut -\mathstrut 184q^{54} \) \(\mathstrut +\mathstrut 6408q^{55} \) \(\mathstrut -\mathstrut 5800q^{57} \) \(\mathstrut +\mathstrut 6516q^{58} \) \(\mathstrut +\mathstrut 4372q^{59} \) \(\mathstrut +\mathstrut 1324q^{60} \) \(\mathstrut +\mathstrut 5988q^{61} \) \(\mathstrut -\mathstrut 652q^{63} \) \(\mathstrut -\mathstrut 5018q^{65} \) \(\mathstrut -\mathstrut 4592q^{66} \) \(\mathstrut +\mathstrut 72q^{67} \) \(\mathstrut -\mathstrut 10572q^{68} \) \(\mathstrut +\mathstrut 7368q^{70} \) \(\mathstrut -\mathstrut 14672q^{71} \) \(\mathstrut -\mathstrut 7980q^{72} \) \(\mathstrut +\mathstrut 5874q^{73} \) \(\mathstrut +\mathstrut 1544q^{74} \) \(\mathstrut +\mathstrut 3576q^{76} \) \(\mathstrut +\mathstrut 5720q^{78} \) \(\mathstrut +\mathstrut 2616q^{79} \) \(\mathstrut -\mathstrut 12080q^{80} \) \(\mathstrut -\mathstrut 19450q^{81} \) \(\mathstrut +\mathstrut 19264q^{83} \) \(\mathstrut +\mathstrut 6296q^{84} \) \(\mathstrut +\mathstrut 4164q^{85} \) \(\mathstrut +\mathstrut 29376q^{86} \) \(\mathstrut +\mathstrut 35584q^{87} \) \(\mathstrut -\mathstrut 986q^{89} \) \(\mathstrut -\mathstrut 30888q^{91} \) \(\mathstrut +\mathstrut 5304q^{92} \) \(\mathstrut -\mathstrut 9520q^{93} \) \(\mathstrut -\mathstrut 36156q^{94} \) \(\mathstrut +\mathstrut 20720q^{96} \) \(\mathstrut -\mathstrut 23154q^{97} \) \(\mathstrut -\mathstrut 41426q^{98} \) \(\mathstrut +\mathstrut 17492q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(13, [\chi])\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
13.5.d.a \(6\) \(1.344\) 6.0.\(\cdots\).1 None \(-2\) \(-4\) \(-14\) \(48\) \(q-\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)