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Properties

Label 13.3.d
Level $13$
Weight $3$
Character orbit 13.d
Rep. character $\chi_{13}(5,\cdot)$
Character field $\Q(\zeta_{4})$
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.d (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(i)\)
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 8 8 0
Cusp forms 4 4 0
Eisenstein series 4 4 0

Trace form

\( 4 q - 4 q^{2} - 4 q^{3} + 8 q^{5} - 16 q^{6} - 12 q^{7} + 36 q^{8} + 8 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{2} - 4 q^{3} + 8 q^{5} - 16 q^{6} - 12 q^{7} + 36 q^{8} + 8 q^{9} - 4 q^{11} + 8 q^{13} + 4 q^{14} - 28 q^{15} - 84 q^{16} + 32 q^{18} + 16 q^{20} + 32 q^{21} + 88 q^{22} + 24 q^{24} - 88 q^{26} - 52 q^{27} + 4 q^{28} + 40 q^{29} + 40 q^{31} + 20 q^{32} - 76 q^{33} - 108 q^{34} - 68 q^{35} + 40 q^{37} + 92 q^{39} + 84 q^{40} + 32 q^{41} + 76 q^{42} - 172 q^{44} + 56 q^{45} + 132 q^{46} - 4 q^{47} - 76 q^{48} - 128 q^{50} + 80 q^{52} - 80 q^{53} + 152 q^{54} + 64 q^{55} - 40 q^{57} - 140 q^{58} + 56 q^{59} - 116 q^{60} - 296 q^{61} - 64 q^{63} + 56 q^{65} + 112 q^{66} - 84 q^{67} + 444 q^{68} - 32 q^{70} + 284 q^{71} - 48 q^{72} + 100 q^{74} + 80 q^{76} - 232 q^{78} + 64 q^{79} - 88 q^{80} - 220 q^{81} - 52 q^{83} - 184 q^{84} - 144 q^{85} - 24 q^{86} + 160 q^{87} + 200 q^{89} + 156 q^{91} - 456 q^{92} + 80 q^{93} - 452 q^{94} + 320 q^{96} - 68 q^{97} + 224 q^{98} + 152 q^{99} + O(q^{100}) \)

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.d.a 13.d 13.d $4$ $0.354$ \(\Q(i, \sqrt{10})\) None \(-4\) \(-4\) \(8\) \(-12\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{2}+(-1-\beta _{1}+\beta _{3})q^{3}+\cdots\)