Properties

Label 39.4.f
Level $39$
Weight $4$
Character orbit 39.f
Rep. character $\chi_{39}(5,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $24$
Newform subspaces $2$
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.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 39 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
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 32 0
Cusp forms 24 24 0
Eisenstein series 8 8 0

Trace form

\( 24 q - 4 q^{3} + 44 q^{6} + 4 q^{7} - 4 q^{9} + O(q^{10}) \) \( 24 q - 4 q^{3} + 44 q^{6} + 4 q^{7} - 4 q^{9} - 76 q^{13} - 76 q^{15} - 272 q^{16} + 296 q^{18} + 148 q^{19} - 208 q^{21} - 224 q^{22} + 36 q^{24} - 592 q^{27} + 264 q^{28} - 84 q^{31} + 872 q^{33} + 816 q^{34} - 1440 q^{37} + 368 q^{39} + 3288 q^{40} + 124 q^{42} + 260 q^{45} - 1560 q^{46} - 1084 q^{48} - 2336 q^{52} - 232 q^{54} - 872 q^{55} + 1028 q^{57} - 1352 q^{58} - 1064 q^{60} + 1960 q^{61} - 652 q^{63} - 7664 q^{66} + 844 q^{67} + 1192 q^{70} + 6984 q^{72} - 416 q^{73} + 2712 q^{76} + 728 q^{78} + 6544 q^{79} + 3116 q^{81} + 5204 q^{84} - 8304 q^{85} + 3136 q^{87} + 4036 q^{91} - 4156 q^{93} - 6056 q^{94} - 5956 q^{96} - 5232 q^{97} + 1700 q^{99} + O(q^{100}) \)

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.f.a 39.f 39.f $4$ $2.301$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-40\) $\mathrm{U}(1)[D_{4}]$ \(q-\zeta_{12}^{3}q^{3}+8\zeta_{12}q^{4}+(-10+10\zeta_{12}+\cdots)q^{7}+\cdots\)
39.4.f.b 39.f 39.f $20$ $2.301$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(-4\) \(0\) \(44\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{8}q^{2}-\beta _{5}q^{3}+(-6\beta _{4}-\beta _{14})q^{4}+\cdots\)