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Properties

Label 117.4.e
Level $117$
Weight $4$
Character orbit 117.e
Rep. character $\chi_{117}(40,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $72$
Newform subspaces $2$
Sturm bound $56$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(56\)
Trace bound: \(2\)
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 72 16
Cusp forms 80 72 8
Eisenstein series 8 0 8

Trace form

\( 72 q + 4 q^{2} + 2 q^{3} - 144 q^{4} + 28 q^{5} - 36 q^{6} - 96 q^{8} - 30 q^{9} + O(q^{10}) \) \( 72 q + 4 q^{2} + 2 q^{3} - 144 q^{4} + 28 q^{5} - 36 q^{6} - 96 q^{8} - 30 q^{9} + 138 q^{11} + 222 q^{12} + 38 q^{14} + 212 q^{15} - 576 q^{16} - 132 q^{17} - 362 q^{18} + 180 q^{19} + 214 q^{20} + 200 q^{21} - 36 q^{22} - 316 q^{23} + 342 q^{24} - 1044 q^{25} - 312 q^{26} + 902 q^{27} + 84 q^{29} - 134 q^{30} - 36 q^{31} - 992 q^{32} - 670 q^{33} + 576 q^{34} + 1092 q^{35} + 1496 q^{36} - 144 q^{37} - 1282 q^{38} + 180 q^{40} + 518 q^{41} + 1584 q^{42} + 342 q^{43} - 1420 q^{44} + 1900 q^{45} + 1080 q^{46} + 1716 q^{47} - 2668 q^{48} - 2160 q^{49} - 764 q^{50} + 40 q^{51} + 1384 q^{53} - 4044 q^{54} - 2016 q^{55} - 1160 q^{56} + 2346 q^{57} + 504 q^{58} + 926 q^{59} - 192 q^{60} + 36 q^{61} + 3868 q^{62} + 1024 q^{63} + 5976 q^{64} + 520 q^{65} - 2934 q^{66} + 1386 q^{67} + 1866 q^{68} + 56 q^{69} + 432 q^{70} - 4264 q^{71} - 408 q^{72} + 2484 q^{73} + 1418 q^{74} - 4222 q^{75} - 2196 q^{76} + 1336 q^{77} - 910 q^{78} - 252 q^{79} - 4456 q^{80} - 558 q^{81} - 2088 q^{82} + 1656 q^{83} - 3682 q^{84} + 3408 q^{86} - 1616 q^{87} + 612 q^{88} + 4016 q^{89} - 1110 q^{90} - 6880 q^{92} + 1752 q^{93} + 2052 q^{94} - 1156 q^{95} + 7148 q^{96} + 1206 q^{97} - 88 q^{98} - 4200 q^{99} + O(q^{100}) \)

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.e.a 117.e 9.c $36$ $6.903$ None \(-4\) \(1\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{3}]$
117.4.e.b 117.e 9.c $36$ $6.903$ None \(8\) \(1\) \(34\) \(0\) $\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{4}^{\mathrm{old}}(117, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(117, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 2}\)