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Properties

Label 86.6.a
Level $86$
Weight $6$
Character orbit 86.a
Rep. character $\chi_{86}(1,\cdot)$
Character field $\Q$
Dimension $17$
Newform subspaces $4$
Sturm bound $66$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 86 = 2 \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 86.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(66\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(86))\).

Total New Old
Modular forms 57 17 40
Cusp forms 53 17 36
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.

\(2\)\(43\)FrickeDim
\(+\)\(+\)$+$\(3\)
\(+\)\(-\)$-$\(5\)
\(-\)\(+\)$-$\(6\)
\(-\)\(-\)$+$\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(11\)

Trace form

\( 17 q + 4 q^{2} - 4 q^{3} + 272 q^{4} + 94 q^{5} - 72 q^{6} + 76 q^{7} + 64 q^{8} + 651 q^{9} + O(q^{10}) \) \( 17 q + 4 q^{2} - 4 q^{3} + 272 q^{4} + 94 q^{5} - 72 q^{6} + 76 q^{7} + 64 q^{8} + 651 q^{9} + 320 q^{11} - 64 q^{12} + 798 q^{13} - 80 q^{14} - 792 q^{15} + 4352 q^{16} - 460 q^{17} + 2900 q^{18} + 1864 q^{19} + 1504 q^{20} + 4244 q^{21} - 928 q^{22} + 2586 q^{23} - 1152 q^{24} + 5125 q^{25} + 2792 q^{26} + 10796 q^{27} + 1216 q^{28} + 1434 q^{29} + 14464 q^{30} + 2246 q^{31} + 1024 q^{32} + 24384 q^{33} + 1320 q^{34} - 12624 q^{35} + 10416 q^{36} + 21146 q^{37} - 8840 q^{38} - 4944 q^{39} + 19828 q^{41} - 31200 q^{42} - 1849 q^{43} + 5120 q^{44} - 23718 q^{45} + 1552 q^{46} + 51562 q^{47} - 1024 q^{48} - 6439 q^{49} - 3684 q^{50} - 62492 q^{51} + 12768 q^{52} - 67090 q^{53} - 10464 q^{54} - 39740 q^{55} - 1280 q^{56} - 3736 q^{57} - 65520 q^{58} - 96672 q^{59} - 12672 q^{60} + 79446 q^{61} - 9040 q^{62} - 40024 q^{63} + 69632 q^{64} - 124176 q^{65} + 32352 q^{66} - 60436 q^{67} - 7360 q^{68} + 31068 q^{69} - 41584 q^{70} - 98376 q^{71} + 46400 q^{72} - 111118 q^{73} + 34752 q^{74} - 175272 q^{75} + 29824 q^{76} - 3336 q^{77} - 82192 q^{78} + 45274 q^{79} + 24064 q^{80} - 99479 q^{81} - 12408 q^{82} + 178924 q^{83} + 67904 q^{84} - 45464 q^{85} - 36980 q^{86} - 323682 q^{87} - 14848 q^{88} + 190290 q^{89} + 121016 q^{90} + 87440 q^{91} + 41376 q^{92} + 352320 q^{93} - 96304 q^{94} + 229230 q^{95} - 18432 q^{96} + 135248 q^{97} - 135708 q^{98} + 187348 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(86))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 43
86.6.a.a 86.a 1.a $3$ $13.793$ 3.3.159992.1 None \(-12\) \(8\) \(-14\) \(-74\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(3+\beta _{2})q^{3}+2^{4}q^{4}+(-5+\cdots)q^{5}+\cdots\)
86.6.a.b 86.a 1.a $3$ $13.793$ 3.3.146508.1 None \(12\) \(-28\) \(-14\) \(-182\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-9+\beta _{2})q^{3}+2^{4}q^{4}+(-5+\cdots)q^{5}+\cdots\)
86.6.a.c 86.a 1.a $5$ $13.793$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-20\) \(-1\) \(61\) \(122\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+(12-2\beta _{1}+\cdots)q^{5}+\cdots\)
86.6.a.d 86.a 1.a $6$ $13.793$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(24\) \(17\) \(61\) \(210\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(3-\beta _{1})q^{3}+2^{4}q^{4}+(10-\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(86))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(86)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 2}\)