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Properties

Label 216.6.i
Level $216$
Weight $6$
Character orbit 216.i
Rep. character $\chi_{216}(73,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $30$
Newform subspaces $2$
Sturm bound $216$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 216.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(216\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(216, [\chi])\).

Total New Old
Modular forms 384 30 354
Cusp forms 336 30 306
Eisenstein series 48 0 48

Trace form

\( 30 q - 50 q^{5} + O(q^{10}) \) \( 30 q - 50 q^{5} + 955 q^{11} + 806 q^{17} + 894 q^{19} - 2636 q^{23} - 9375 q^{25} + 924 q^{29} + 1626 q^{31} - 14292 q^{35} + 315 q^{41} + 17019 q^{43} - 25272 q^{47} - 37845 q^{49} + 9088 q^{53} - 28572 q^{55} + 51493 q^{59} + 23370 q^{61} - 34526 q^{65} - 10527 q^{67} - 96184 q^{71} - 80442 q^{73} - 87132 q^{77} - 44754 q^{79} + 53834 q^{83} + 40812 q^{85} - 13812 q^{89} + 47148 q^{91} + 30344 q^{95} - 148647 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(216, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
216.6.i.a 216.i 9.c $14$ $34.643$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(0\) \(-25\) \(93\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-4+4\beta _{1}-\beta _{3}+\beta _{6})q^{5}+(13\beta _{1}+\cdots)q^{7}+\cdots\)
216.6.i.b 216.i 9.c $16$ $34.643$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(-25\) \(-93\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3\beta _{8}-\beta _{10})q^{5}+(-12+\beta _{2}+12\beta _{8}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(216, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(216, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)