Properties

Label 13.4.c
Level 13
Weight 4
Character orbit c
Rep. character \(\chi_{13}(3,\cdot)\)
Character field \(\Q(\zeta_{3})\)
Dimension 6
Newforms 2
Sturm bound 4
Trace bound 1

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Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 13.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newforms: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(13, [\chi])\).

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

Trace form

\(6q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut 7q^{3} \) \(\mathstrut -\mathstrut 13q^{4} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 30q^{6} \) \(\mathstrut -\mathstrut 35q^{7} \) \(\mathstrut -\mathstrut 30q^{8} \) \(\mathstrut -\mathstrut 12q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(6q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut 7q^{3} \) \(\mathstrut -\mathstrut 13q^{4} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut +\mathstrut 30q^{6} \) \(\mathstrut -\mathstrut 35q^{7} \) \(\mathstrut -\mathstrut 30q^{8} \) \(\mathstrut -\mathstrut 12q^{9} \) \(\mathstrut -\mathstrut 63q^{10} \) \(\mathstrut +\mathstrut 15q^{11} \) \(\mathstrut +\mathstrut 312q^{12} \) \(\mathstrut +\mathstrut 34q^{13} \) \(\mathstrut +\mathstrut 68q^{14} \) \(\mathstrut -\mathstrut 124q^{15} \) \(\mathstrut +\mathstrut 7q^{16} \) \(\mathstrut -\mathstrut 57q^{17} \) \(\mathstrut -\mathstrut 614q^{18} \) \(\mathstrut +\mathstrut 111q^{19} \) \(\mathstrut -\mathstrut 311q^{20} \) \(\mathstrut +\mathstrut 206q^{21} \) \(\mathstrut +\mathstrut 298q^{22} \) \(\mathstrut -\mathstrut 223q^{23} \) \(\mathstrut +\mathstrut 216q^{24} \) \(\mathstrut +\mathstrut 478q^{25} \) \(\mathstrut +\mathstrut 237q^{26} \) \(\mathstrut +\mathstrut 470q^{27} \) \(\mathstrut -\mathstrut 240q^{28} \) \(\mathstrut -\mathstrut 231q^{29} \) \(\mathstrut +\mathstrut 4q^{30} \) \(\mathstrut -\mathstrut 428q^{31} \) \(\mathstrut +\mathstrut 151q^{32} \) \(\mathstrut -\mathstrut 361q^{33} \) \(\mathstrut -\mathstrut 182q^{34} \) \(\mathstrut -\mathstrut 270q^{35} \) \(\mathstrut -\mathstrut 541q^{36} \) \(\mathstrut +\mathstrut 417q^{37} \) \(\mathstrut +\mathstrut 860q^{38} \) \(\mathstrut -\mathstrut 51q^{39} \) \(\mathstrut -\mathstrut 370q^{40} \) \(\mathstrut -\mathstrut 373q^{41} \) \(\mathstrut +\mathstrut 210q^{42} \) \(\mathstrut -\mathstrut 299q^{43} \) \(\mathstrut +\mathstrut 848q^{44} \) \(\mathstrut +\mathstrut 16q^{45} \) \(\mathstrut -\mathstrut 230q^{46} \) \(\mathstrut -\mathstrut 204q^{47} \) \(\mathstrut +\mathstrut 368q^{48} \) \(\mathstrut +\mathstrut 508q^{49} \) \(\mathstrut -\mathstrut 1106q^{50} \) \(\mathstrut -\mathstrut 518q^{51} \) \(\mathstrut -\mathstrut 102q^{52} \) \(\mathstrut +\mathstrut 1276q^{53} \) \(\mathstrut +\mathstrut 1314q^{54} \) \(\mathstrut +\mathstrut 34q^{55} \) \(\mathstrut +\mathstrut 172q^{56} \) \(\mathstrut -\mathstrut 330q^{57} \) \(\mathstrut -\mathstrut 193q^{58} \) \(\mathstrut +\mathstrut 1673q^{59} \) \(\mathstrut +\mathstrut 144q^{60} \) \(\mathstrut -\mathstrut 647q^{61} \) \(\mathstrut +\mathstrut 796q^{62} \) \(\mathstrut +\mathstrut 70q^{63} \) \(\mathstrut -\mathstrut 3566q^{64} \) \(\mathstrut -\mathstrut 2102q^{65} \) \(\mathstrut -\mathstrut 3708q^{66} \) \(\mathstrut -\mathstrut 387q^{67} \) \(\mathstrut +\mathstrut 609q^{68} \) \(\mathstrut +\mathstrut 323q^{69} \) \(\mathstrut +\mathstrut 2560q^{70} \) \(\mathstrut -\mathstrut 781q^{71} \) \(\mathstrut +\mathstrut 1155q^{72} \) \(\mathstrut +\mathstrut 1600q^{73} \) \(\mathstrut +\mathstrut 59q^{74} \) \(\mathstrut +\mathstrut 1397q^{75} \) \(\mathstrut -\mathstrut 100q^{76} \) \(\mathstrut -\mathstrut 770q^{77} \) \(\mathstrut +\mathstrut 3278q^{78} \) \(\mathstrut +\mathstrut 328q^{79} \) \(\mathstrut +\mathstrut 2153q^{80} \) \(\mathstrut -\mathstrut 543q^{81} \) \(\mathstrut +\mathstrut 2175q^{82} \) \(\mathstrut +\mathstrut 1776q^{83} \) \(\mathstrut -\mathstrut 1540q^{84} \) \(\mathstrut +\mathstrut 1426q^{85} \) \(\mathstrut -\mathstrut 5172q^{86} \) \(\mathstrut -\mathstrut 2009q^{87} \) \(\mathstrut +\mathstrut 1020q^{88} \) \(\mathstrut -\mathstrut 655q^{89} \) \(\mathstrut -\mathstrut 4578q^{90} \) \(\mathstrut -\mathstrut 767q^{91} \) \(\mathstrut -\mathstrut 832q^{92} \) \(\mathstrut -\mathstrut 2052q^{93} \) \(\mathstrut -\mathstrut 392q^{94} \) \(\mathstrut -\mathstrut 1780q^{95} \) \(\mathstrut +\mathstrut 2816q^{96} \) \(\mathstrut +\mathstrut 177q^{97} \) \(\mathstrut -\mathstrut 1513q^{98} \) \(\mathstrut +\mathstrut 5892q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(13, [\chi])\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
13.4.c.a \(2\) \(0.767\) \(\Q(\sqrt{-3}) \) None \(-4\) \(-2\) \(34\) \(-20\) \(q+(-4+4\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}+\cdots\)
13.4.c.b \(4\) \(0.767\) \(\Q(\sqrt{-3}, \sqrt{17})\) None \(5\) \(-5\) \(-30\) \(-15\) \(q+(2-\beta _{1}-2\beta _{2}-\beta _{3})q^{2}+(-1+3\beta _{1}+\cdots)q^{3}+\cdots\)