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Properties

Label 117.4.f
Level $117$
Weight $4$
Character orbit 117.f
Rep. character $\chi_{117}(61,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $80$
Newform subspaces $2$
Sturm bound $56$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.f (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 117 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(56\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(117, [\chi])\).

Total New Old
Modular forms 88 88 0
Cusp forms 80 80 0
Eisenstein series 8 8 0

Trace form

\( 80 q + q^{2} - q^{3} - 151 q^{4} - 2 q^{5} - 15 q^{6} + 22 q^{7} - 30 q^{8} + 15 q^{9} + O(q^{10}) \) \( 80 q + q^{2} - q^{3} - 151 q^{4} - 2 q^{5} - 15 q^{6} + 22 q^{7} - 30 q^{8} + 15 q^{9} + 6 q^{10} + 9 q^{11} + 39 q^{12} + 11 q^{13} - 130 q^{14} - 76 q^{15} - 535 q^{16} - 138 q^{17} + 238 q^{18} - 32 q^{19} + 286 q^{20} - 7 q^{21} + 33 q^{22} + 446 q^{23} + 420 q^{24} - 802 q^{25} + 414 q^{26} - 232 q^{27} - 314 q^{28} - 615 q^{29} - 173 q^{30} + 58 q^{31} + 121 q^{32} + 1067 q^{33} + 6 q^{34} - 573 q^{35} - q^{36} - 86 q^{37} - 172 q^{38} + 264 q^{39} - 204 q^{40} + 254 q^{41} - 948 q^{42} + 166 q^{43} + 776 q^{44} + 997 q^{45} - 18 q^{46} + 150 q^{47} + 986 q^{48} + 2262 q^{49} - 3104 q^{50} + 826 q^{51} + 218 q^{52} + 1096 q^{53} - 888 q^{54} + 123 q^{55} + 1288 q^{56} - 78 q^{57} + 627 q^{58} - 1195 q^{59} - 1083 q^{60} - 842 q^{61} - 1145 q^{62} - 1391 q^{63} + 3446 q^{64} + 139 q^{65} - 2412 q^{66} - 590 q^{67} + 5760 q^{68} + 887 q^{69} + 849 q^{70} + 302 q^{71} - 7803 q^{72} - 344 q^{73} + 3362 q^{74} + 3440 q^{75} + 1246 q^{76} - 1868 q^{77} + 551 q^{78} - 680 q^{79} - 4690 q^{80} + 1395 q^{81} + 1269 q^{82} + 1392 q^{83} - 2818 q^{84} + 594 q^{85} - 2064 q^{86} - 1751 q^{87} - 267 q^{88} + 1808 q^{89} + 6015 q^{90} + 775 q^{91} - 2728 q^{92} - 2706 q^{93} + 4386 q^{94} - 4450 q^{95} - 271 q^{96} + 2110 q^{97} - 1573 q^{98} - 5607 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(117, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
117.4.f.a 117.f 117.f $2$ $6.903$ \(\Q(\sqrt{-3}) \) None \(5\) \(-9\) \(19\) \(22\) $\mathrm{SU}(2)[C_{3}]$ \(q+5\zeta_{6}q^{2}+(-6+3\zeta_{6})q^{3}+(-17+\cdots)q^{4}+\cdots\)
117.4.f.b 117.f 117.f $78$ $6.903$ None \(-4\) \(8\) \(-21\) \(0\) $\mathrm{SU}(2)[C_{3}]$