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Properties

Label 13.3.f
Level $13$
Weight $3$
Character orbit 13.f
Rep. character $\chi_{13}(2,\cdot)$
Character field $\Q(\zeta_{12})$
Dimension $4$
Newform subspaces $1$
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: \(\Q(\zeta_{12})\)
Newform subspaces: \( 1 \)
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

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(13, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
13.3.f.a 13.f 13.f $4$ $0.354$ \(\Q(\zeta_{12})\) None \(-2\) \(-2\) \(-14\) \(16\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-1+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)