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Properties

Label 39.4.j
Level $39$
Weight $4$
Character orbit 39.j
Rep. character $\chi_{39}(4,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $16$
Newform subspaces $3$
Sturm bound $18$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 39.j (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(18\)
Trace bound: \(1\)
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 16 16
Cusp forms 24 16 8
Eisenstein series 8 0 8

Trace form

\( 16 q - 6 q^{3} + 40 q^{4} - 18 q^{7} - 72 q^{9} + O(q^{10}) \) \( 16 q - 6 q^{3} + 40 q^{4} - 18 q^{7} - 72 q^{9} - 28 q^{10} + 24 q^{11} - 120 q^{12} + 144 q^{14} - 36 q^{15} - 204 q^{16} + 150 q^{17} + 360 q^{19} + 600 q^{20} - 256 q^{22} - 216 q^{23} - 644 q^{25} - 744 q^{26} + 108 q^{27} - 588 q^{28} - 402 q^{29} + 324 q^{30} + 1440 q^{32} - 288 q^{33} + 408 q^{35} + 360 q^{36} + 282 q^{37} - 768 q^{38} + 240 q^{39} + 1864 q^{40} + 1182 q^{41} + 396 q^{42} + 182 q^{43} - 378 q^{45} - 1104 q^{46} - 888 q^{48} + 886 q^{49} - 3096 q^{50} - 360 q^{51} + 2588 q^{52} - 2460 q^{53} - 436 q^{55} - 2568 q^{56} - 204 q^{58} + 1572 q^{59} - 816 q^{61} - 2208 q^{62} + 162 q^{63} + 408 q^{64} + 450 q^{65} + 1944 q^{66} - 126 q^{67} + 984 q^{68} + 288 q^{69} - 504 q^{71} + 3720 q^{74} + 1914 q^{75} - 2856 q^{76} - 264 q^{77} + 1728 q^{78} + 228 q^{79} + 8016 q^{80} - 648 q^{81} - 2356 q^{82} - 2664 q^{84} + 726 q^{85} - 1764 q^{87} + 3136 q^{88} + 4416 q^{89} + 504 q^{90} + 2166 q^{91} - 3936 q^{92} - 3402 q^{93} - 544 q^{94} - 1308 q^{95} - 6090 q^{97} - 6552 q^{98} + O(q^{100}) \)

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.j.a 39.j 13.e $2$ $2.301$ \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(0\) \(18\) $\mathrm{SU}(2)[C_{6}]$ \(q+(3-3\zeta_{6})q^{3}-8\zeta_{6}q^{4}+(3-6\zeta_{6})q^{5}+\cdots\)
39.4.j.b 39.j 13.e $4$ $2.301$ \(\Q(\sqrt{-3}, \sqrt{-17})\) None \(0\) \(6\) \(0\) \(-66\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+(3-3\beta _{2})q^{3}+9\beta _{2}q^{4}+(3+\cdots)q^{5}+\cdots\)
39.4.j.c 39.j 13.e $10$ $2.301$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(-15\) \(0\) \(30\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}-3\beta _{2}q^{3}+(6-6\beta _{2}-\beta _{4}+\cdots)q^{4}+\cdots\)

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

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