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Properties

Label 13.5.f
Level $13$
Weight $5$
Character orbit 13.f
Rep. character $\chi_{13}(2,\cdot)$
Character field $\Q(\zeta_{12})$
Dimension $16$
Newform subspaces $1$
Sturm bound $5$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 13 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(5\)
Trace bound: \(0\)

Dimensions

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

Total New Old
Modular forms 24 24 0
Cusp forms 16 16 0
Eisenstein series 8 8 0

Trace form

\( 16 q - 4 q^{2} - 2 q^{3} - 6 q^{4} + 8 q^{5} - 38 q^{6} + 56 q^{7} + 90 q^{8} - 164 q^{9} + O(q^{10}) \) \( 16 q - 4 q^{2} - 2 q^{3} - 6 q^{4} + 8 q^{5} - 38 q^{6} + 56 q^{7} + 90 q^{8} - 164 q^{9} - 486 q^{10} - 100 q^{11} + 294 q^{13} + 808 q^{14} + 346 q^{15} + 230 q^{16} + 984 q^{17} + 2434 q^{18} - 1498 q^{19} - 3962 q^{20} - 1076 q^{21} - 1524 q^{22} - 1014 q^{23} - 2142 q^{24} + 614 q^{26} + 3352 q^{27} + 5764 q^{28} + 814 q^{29} + 9162 q^{30} + 4060 q^{31} - 4996 q^{32} - 5636 q^{33} - 2502 q^{34} - 4892 q^{35} - 15750 q^{36} - 1790 q^{37} + 6982 q^{39} + 18816 q^{40} + 4280 q^{41} - 1204 q^{42} - 1368 q^{43} + 10736 q^{44} - 6806 q^{45} - 15246 q^{46} + 1484 q^{47} - 3002 q^{48} - 11820 q^{49} - 13574 q^{50} + 1432 q^{52} + 7204 q^{53} + 13240 q^{54} + 6936 q^{55} + 8124 q^{56} + 12736 q^{57} + 3030 q^{58} - 2380 q^{59} - 6472 q^{60} - 162 q^{61} + 19614 q^{62} + 12004 q^{63} - 5248 q^{65} - 23872 q^{66} - 14854 q^{67} - 6444 q^{68} + 2412 q^{69} - 34524 q^{70} - 8050 q^{71} + 15420 q^{72} - 15448 q^{73} - 2882 q^{74} + 8280 q^{75} + 10622 q^{76} - 11672 q^{78} - 17064 q^{79} + 2564 q^{80} + 2128 q^{81} - 5346 q^{82} + 12788 q^{83} + 25948 q^{84} + 35382 q^{85} + 67260 q^{86} + 29342 q^{87} + 40836 q^{88} + 20492 q^{89} + 8996 q^{91} - 49884 q^{92} - 78920 q^{93} - 30606 q^{94} - 98574 q^{95} - 94664 q^{96} + 50944 q^{97} + 61484 q^{98} - 21632 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{5}^{\mathrm{new}}(13, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
13.5.f.a 13.f 13.f $16$ $1.344$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(-4\) \(-2\) \(8\) \(56\) $\mathrm{SU}(2)[C_{12}]$ \(q-\beta _{12}q^{2}+(-1-\beta _{8}+\beta _{9}+\beta _{12}+\cdots)q^{3}+\cdots\)