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Properties

Label 39.4.e
Level $39$
Weight $4$
Character orbit 39.e
Rep. character $\chi_{39}(16,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $12$
Newform subspaces $3$
Sturm bound $18$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 39.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 3 \)
Sturm bound: \(18\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(39, [\chi])\).

Total New Old
Modular forms 32 12 20
Cusp forms 24 12 12
Eisenstein series 8 0 8

Trace form

\( 12 q - 4 q^{2} + 6 q^{3} - 16 q^{4} - 16 q^{5} + 22 q^{7} + 96 q^{8} - 54 q^{9} + O(q^{10}) \) \( 12 q - 4 q^{2} + 6 q^{3} - 16 q^{4} - 16 q^{5} + 22 q^{7} + 96 q^{8} - 54 q^{9} + 96 q^{10} - 48 q^{11} - 168 q^{12} - 86 q^{13} + 112 q^{14} - 12 q^{15} - 92 q^{16} - 24 q^{17} + 72 q^{18} - 108 q^{19} + 524 q^{20} + 36 q^{21} - 332 q^{22} + 64 q^{23} + 180 q^{24} - 356 q^{25} - 324 q^{26} - 108 q^{27} + 72 q^{28} + 24 q^{29} - 84 q^{30} + 244 q^{31} - 448 q^{32} + 96 q^{33} + 1384 q^{34} - 144 q^{36} - 132 q^{37} + 328 q^{38} + 264 q^{39} - 1408 q^{40} + 1000 q^{41} + 72 q^{42} + 310 q^{43} + 280 q^{44} + 72 q^{45} + 28 q^{46} - 1440 q^{47} + 456 q^{48} - 488 q^{49} - 940 q^{50} - 1032 q^{51} + 2124 q^{52} + 728 q^{53} + 220 q^{55} - 1972 q^{56} - 840 q^{57} - 520 q^{58} - 1484 q^{59} + 2448 q^{60} + 1330 q^{61} - 2284 q^{62} + 198 q^{63} + 2584 q^{64} + 1124 q^{65} - 648 q^{66} + 6 q^{67} - 1416 q^{68} + 816 q^{69} - 4616 q^{70} + 952 q^{71} - 432 q^{72} - 212 q^{73} - 1196 q^{74} + 186 q^{75} + 1964 q^{76} + 5576 q^{77} + 636 q^{78} - 2372 q^{79} + 1996 q^{80} - 486 q^{81} - 756 q^{82} + 1848 q^{83} - 648 q^{84} + 136 q^{85} + 3744 q^{86} + 1236 q^{87} - 2100 q^{88} - 2312 q^{89} - 1728 q^{90} + 3070 q^{91} + 2920 q^{92} - 210 q^{93} + 796 q^{94} - 2828 q^{95} - 5160 q^{96} - 2322 q^{97} + 2680 q^{98} + 864 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(39, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
39.4.e.a 39.e 13.c $2$ $2.301$ \(\Q(\sqrt{-3}) \) None \(-3\) \(-3\) \(-18\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}+\cdots\)
39.4.e.b 39.e 13.c $2$ $2.301$ \(\Q(\sqrt{-3}) \) None \(1\) \(-3\) \(14\) \(10\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}+7\zeta_{6}q^{4}+\cdots\)
39.4.e.c 39.e 13.c $8$ $2.301$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-2\) \(12\) \(-12\) \(14\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(3+3\beta _{2})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(39, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(39, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 2}\)