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Properties

Label 84.6.i
Level $84$
Weight $6$
Character orbit 84.i
Rep. character $\chi_{84}(25,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $14$
Newform subspaces $3$
Sturm bound $96$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 84.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 3 \)
Sturm bound: \(96\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(84, [\chi])\).

Total New Old
Modular forms 172 14 158
Cusp forms 148 14 134
Eisenstein series 24 0 24

Trace form

\( 14 q + 9 q^{3} + 22 q^{5} + 29 q^{7} - 567 q^{9} + O(q^{10}) \) \( 14 q + 9 q^{3} + 22 q^{5} + 29 q^{7} - 567 q^{9} - 178 q^{11} - 314 q^{13} - 396 q^{15} + 320 q^{17} + 3217 q^{19} + 180 q^{21} - 568 q^{23} - 5485 q^{25} - 1458 q^{27} - 16760 q^{29} + 4705 q^{31} + 6714 q^{33} + 30890 q^{35} - 8441 q^{37} - 9423 q^{39} - 12252 q^{41} - 73802 q^{43} + 1782 q^{45} + 49746 q^{47} + 72845 q^{49} - 1224 q^{51} + 4572 q^{53} - 35400 q^{55} - 45018 q^{57} + 100540 q^{59} + 112738 q^{61} + 21303 q^{63} - 121722 q^{65} - 83189 q^{67} - 67104 q^{69} - 100612 q^{71} + 51745 q^{73} + 48807 q^{75} + 168112 q^{77} + 5719 q^{79} - 45927 q^{81} + 190628 q^{83} - 82704 q^{85} - 24156 q^{87} + 52416 q^{89} + 18709 q^{91} + 27405 q^{93} - 25498 q^{95} - 386156 q^{97} + 28836 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(84, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
84.6.i.a 84.i 7.c $2$ $13.472$ \(\Q(\sqrt{-3}) \) None \(0\) \(-9\) \(69\) \(245\) $\mathrm{SU}(2)[C_{3}]$ \(q-9\zeta_{6}q^{3}+(69-69\zeta_{6})q^{5}+(98+7^{2}\zeta_{6})q^{7}+\cdots\)
84.6.i.b 84.i 7.c $4$ $13.472$ \(\Q(\sqrt{-3}, \sqrt{7081})\) None \(0\) \(-18\) \(-47\) \(-174\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-9+9\beta _{2})q^{3}+(\beta _{1}-24\beta _{2})q^{5}+\cdots\)
84.6.i.c 84.i 7.c $8$ $13.472$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(36\) \(0\) \(-42\) $\mathrm{SU}(2)[C_{3}]$ \(q-9\beta _{1}q^{3}+(-\beta _{2}-\beta _{3})q^{5}+(10+30\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(84, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(84, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 2}\)