Properties

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

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Defining parameters

Level: \( N \) = \( 13 \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 13.e (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newforms: \( 2 \)
Sturm bound: \(4\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

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

Trace form

\(4q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut -\mathstrut 48q^{6} \) \(\mathstrut +\mathstrut 15q^{7} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 3q^{2} \) \(\mathstrut +\mathstrut 5q^{3} \) \(\mathstrut -\mathstrut q^{4} \) \(\mathstrut -\mathstrut 48q^{6} \) \(\mathstrut +\mathstrut 15q^{7} \) \(\mathstrut +\mathstrut q^{9} \) \(\mathstrut +\mathstrut 51q^{10} \) \(\mathstrut -\mathstrut 15q^{11} \) \(\mathstrut +\mathstrut 76q^{12} \) \(\mathstrut +\mathstrut 65q^{13} \) \(\mathstrut -\mathstrut 204q^{14} \) \(\mathstrut +\mathstrut 174q^{15} \) \(\mathstrut +\mathstrut 79q^{16} \) \(\mathstrut -\mathstrut 144q^{17} \) \(\mathstrut -\mathstrut 351q^{19} \) \(\mathstrut -\mathstrut 111q^{20} \) \(\mathstrut +\mathstrut 102q^{22} \) \(\mathstrut +\mathstrut 21q^{23} \) \(\mathstrut +\mathstrut 246q^{24} \) \(\mathstrut +\mathstrut 110q^{25} \) \(\mathstrut +\mathstrut 351q^{26} \) \(\mathstrut -\mathstrut 130q^{27} \) \(\mathstrut +\mathstrut 276q^{28} \) \(\mathstrut +\mathstrut 210q^{29} \) \(\mathstrut -\mathstrut 330q^{30} \) \(\mathstrut -\mathstrut 603q^{32} \) \(\mathstrut -\mathstrut 321q^{33} \) \(\mathstrut +\mathstrut 336q^{35} \) \(\mathstrut +\mathstrut 203q^{36} \) \(\mathstrut -\mathstrut 318q^{37} \) \(\mathstrut +\mathstrut 216q^{38} \) \(\mathstrut +\mathstrut 325q^{39} \) \(\mathstrut -\mathstrut 462q^{40} \) \(\mathstrut -\mathstrut 210q^{41} \) \(\mathstrut -\mathstrut 498q^{42} \) \(\mathstrut -\mathstrut 19q^{43} \) \(\mathstrut +\mathstrut 597q^{45} \) \(\mathstrut +\mathstrut 576q^{46} \) \(\mathstrut -\mathstrut 562q^{48} \) \(\mathstrut +\mathstrut 13q^{49} \) \(\mathstrut +\mathstrut 768q^{50} \) \(\mathstrut +\mathstrut 90q^{51} \) \(\mathstrut -\mathstrut 494q^{52} \) \(\mathstrut +\mathstrut 1038q^{53} \) \(\mathstrut -\mathstrut 510q^{54} \) \(\mathstrut +\mathstrut 336q^{55} \) \(\mathstrut +\mathstrut 624q^{56} \) \(\mathstrut +\mathstrut 9q^{58} \) \(\mathstrut -\mathstrut 525q^{59} \) \(\mathstrut -\mathstrut 128q^{61} \) \(\mathstrut -\mathstrut 522q^{62} \) \(\mathstrut -\mathstrut 1410q^{63} \) \(\mathstrut -\mathstrut 742q^{64} \) \(\mathstrut -\mathstrut 1209q^{65} \) \(\mathstrut +\mathstrut 996q^{66} \) \(\mathstrut -\mathstrut 1077q^{67} \) \(\mathstrut -\mathstrut 477q^{68} \) \(\mathstrut +\mathstrut 555q^{69} \) \(\mathstrut +\mathstrut 2841q^{71} \) \(\mathstrut +\mathstrut 1425q^{72} \) \(\mathstrut -\mathstrut 111q^{74} \) \(\mathstrut -\mathstrut 713q^{75} \) \(\mathstrut +\mathstrut 378q^{76} \) \(\mathstrut -\mathstrut 1398q^{77} \) \(\mathstrut -\mathstrut 858q^{78} \) \(\mathstrut +\mathstrut 64q^{79} \) \(\mathstrut +\mathstrut 1923q^{80} \) \(\mathstrut +\mathstrut 418q^{81} \) \(\mathstrut +\mathstrut 1833q^{82} \) \(\mathstrut +\mathstrut 852q^{84} \) \(\mathstrut +\mathstrut 297q^{85} \) \(\mathstrut -\mathstrut 201q^{87} \) \(\mathstrut +\mathstrut 624q^{88} \) \(\mathstrut -\mathstrut 1161q^{89} \) \(\mathstrut -\mathstrut 1974q^{90} \) \(\mathstrut -\mathstrut 741q^{91} \) \(\mathstrut -\mathstrut 1236q^{92} \) \(\mathstrut -\mathstrut 1422q^{93} \) \(\mathstrut -\mathstrut 1710q^{94} \) \(\mathstrut -\mathstrut 1026q^{95} \) \(\mathstrut +\mathstrut 2487q^{97} \) \(\mathstrut -\mathstrut 1437q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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.e.a \(2\) \(0.767\) \(\Q(\sqrt{-3}) \) None \(-6\) \(7\) \(0\) \(39\) \(q+(-2-2\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}+4\zeta_{6}q^{4}+\cdots\)
13.4.e.b \(2\) \(0.767\) \(\Q(\sqrt{-3}) \) None \(3\) \(-2\) \(0\) \(-24\) \(q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-5\zeta_{6}q^{4}+\cdots\)