Properties

Label 39.4.a
Level $39$
Weight $4$
Character orbit 39.a
Rep. character $\chi_{39}(1,\cdot)$
Character field $\Q$
Dimension $6$
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.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(18\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(39))\).

Total New Old
Modular forms 16 6 10
Cusp forms 12 6 6
Eisenstein series 4 0 4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(5\)
Minus space\(-\)\(1\)

Trace form

\( 6 q + 4 q^{2} + 16 q^{4} + 16 q^{5} + 32 q^{7} + 48 q^{8} + 54 q^{9} + O(q^{10}) \) \( 6 q + 4 q^{2} + 16 q^{4} + 16 q^{5} + 32 q^{7} + 48 q^{8} + 54 q^{9} - 36 q^{10} - 96 q^{11} + 12 q^{12} + 26 q^{13} - 232 q^{14} - 24 q^{15} - 76 q^{16} - 60 q^{17} + 36 q^{18} + 216 q^{19} - 92 q^{20} + 84 q^{21} - 436 q^{22} - 136 q^{23} - 180 q^{24} + 314 q^{25} - 48 q^{28} + 420 q^{29} + 84 q^{30} + 128 q^{31} + 28 q^{32} + 192 q^{33} + 440 q^{34} + 168 q^{35} + 144 q^{36} - 12 q^{37} + 416 q^{38} + 156 q^{39} + 76 q^{40} + 296 q^{41} - 360 q^{42} - 352 q^{43} + 20 q^{44} + 144 q^{45} + 248 q^{46} - 48 q^{47} - 432 q^{48} - 466 q^{49} - 380 q^{50} - 696 q^{51} - 156 q^{52} - 284 q^{53} - 976 q^{55} + 40 q^{56} - 84 q^{57} - 896 q^{58} - 1072 q^{59} - 1188 q^{60} + 932 q^{61} + 1264 q^{62} + 288 q^{63} - 1456 q^{64} - 416 q^{65} + 972 q^{66} - 360 q^{67} + 1536 q^{68} + 120 q^{69} + 656 q^{70} - 976 q^{71} + 432 q^{72} - 1804 q^{73} + 2408 q^{74} - 72 q^{75} + 2248 q^{76} + 712 q^{77} + 156 q^{78} + 248 q^{79} - 700 q^{80} + 486 q^{81} + 3396 q^{82} - 3072 q^{83} + 624 q^{84} + 2432 q^{85} + 2640 q^{86} - 1272 q^{87} + 84 q^{88} + 1376 q^{89} - 324 q^{90} + 416 q^{91} + 2216 q^{92} + 1428 q^{93} - 1348 q^{94} - 4456 q^{95} + 840 q^{96} + 2820 q^{97} - 5644 q^{98} - 864 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(39))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 13
39.4.a.a 39.a 1.a $1$ $2.301$ \(\Q\) None 39.4.a.a \(0\) \(-3\) \(-12\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-8q^{4}-12q^{5}+2q^{7}+9q^{9}+\cdots\)
39.4.a.b 39.a 1.a $2$ $2.301$ \(\Q(\sqrt{14}) \) None 39.4.a.b \(2\) \(-6\) \(24\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-3q^{3}+(7+2\beta )q^{4}+(12+\cdots)q^{5}+\cdots\)
39.4.a.c 39.a 1.a $3$ $2.301$ 3.3.3144.1 None 39.4.a.c \(2\) \(9\) \(4\) \(30\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(3+\beta _{2})q^{4}+(2+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(39))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(39)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)