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Properties

Label 39.8.a
Level $39$
Weight $8$
Character orbit 39.a
Rep. character $\chi_{39}(1,\cdot)$
Character field $\Q$
Dimension $14$
Newform subspaces $4$
Sturm bound $37$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 39.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(37\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(39))\).

Total New Old
Modular forms 34 14 20
Cusp forms 30 14 16
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
\(+\)\(+\)$+$\(4\)
\(+\)\(-\)$-$\(3\)
\(-\)\(+\)$-$\(2\)
\(-\)\(-\)$+$\(5\)
Plus space\(+\)\(9\)
Minus space\(-\)\(5\)

Trace form

\( 14 q - 28 q^{2} + 1208 q^{4} - 4 q^{5} + 324 q^{6} - 764 q^{7} - 432 q^{8} + 10206 q^{9} + O(q^{10}) \) \( 14 q - 28 q^{2} + 1208 q^{4} - 4 q^{5} + 324 q^{6} - 764 q^{7} - 432 q^{8} + 10206 q^{9} - 4236 q^{10} + 5316 q^{11} - 7668 q^{12} + 4394 q^{13} - 3824 q^{14} + 34776 q^{15} + 121172 q^{16} + 17916 q^{17} - 20412 q^{18} - 28388 q^{19} + 153188 q^{20} + 37044 q^{21} - 46596 q^{22} - 76256 q^{23} + 28188 q^{24} + 85034 q^{25} + 369376 q^{28} + 315636 q^{29} - 150876 q^{30} + 652780 q^{31} - 613924 q^{32} - 282420 q^{33} - 467400 q^{34} - 782472 q^{35} + 880632 q^{36} - 712148 q^{37} + 1412296 q^{38} + 237276 q^{39} - 1441524 q^{40} - 522308 q^{41} - 977184 q^{42} - 1910864 q^{43} - 283628 q^{44} - 2916 q^{45} + 4392 q^{46} - 170700 q^{47} - 436752 q^{48} + 3483054 q^{49} - 9840220 q^{50} + 1585872 q^{51} + 834860 q^{52} + 1249604 q^{53} + 236196 q^{54} - 667296 q^{55} - 1691320 q^{56} + 3412260 q^{57} + 1581648 q^{58} + 3164260 q^{59} + 2424492 q^{60} - 5720996 q^{61} - 5378872 q^{62} - 556956 q^{63} + 9162032 q^{64} + 1783964 q^{65} - 1410372 q^{66} + 5678284 q^{67} + 11239632 q^{68} - 2345328 q^{69} + 19855296 q^{70} - 824180 q^{71} - 314928 q^{72} + 15930220 q^{73} - 20781416 q^{74} + 933768 q^{75} - 35244008 q^{76} + 18610496 q^{77} + 949104 q^{78} - 9196232 q^{79} + 16936420 q^{80} + 7440174 q^{81} + 5039772 q^{82} + 13441788 q^{83} + 13319424 q^{84} + 2234232 q^{85} + 24412032 q^{86} - 9953712 q^{87} - 27425100 q^{88} - 17827940 q^{89} - 3088044 q^{90} + 8128900 q^{91} - 76862072 q^{92} - 16104852 q^{93} - 2877348 q^{94} + 24544504 q^{95} + 15582888 q^{96} + 39048556 q^{97} - 59078732 q^{98} + 3875364 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(39))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 13
39.8.a.a 39.a 1.a $2$ $12.183$ \(\Q(\sqrt{29}) \) None \(-8\) \(54\) \(-132\) \(-1116\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{2}+3^{3}q^{3}+(4+8\beta )q^{4}+\cdots\)
39.8.a.b 39.a 1.a $3$ $12.183$ 3.3.1035048.1 None \(-14\) \(-81\) \(-370\) \(48\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-5-\beta _{1})q^{2}-3^{3}q^{3}+(114+2\beta _{1}+\cdots)q^{4}+\cdots\)
39.8.a.c 39.a 1.a $4$ $12.183$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-6\) \(-108\) \(-276\) \(-1116\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{2}-3^{3}q^{3}+(102-\beta _{1}+\cdots)q^{4}+\cdots\)
39.8.a.d 39.a 1.a $5$ $12.183$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(135\) \(774\) \(1420\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3^{3}q^{3}+(91+\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(39))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(39)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)