↑

Properties

Label 81.3.d
Level $81$
Weight $3$
Character orbit 81.d
Rep. character $\chi_{81}(26,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $14$
Newform subspaces $3$
Sturm bound $27$
Trace bound $1$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 81.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(27\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(81, [\chi])\).

Total New Old
Modular forms 48 18 30
Cusp forms 24 14 10
Eisenstein series 24 4 20

Trace form

\( 14 q + 14 q^{4} + 7 q^{7} + O(q^{10}) \) \( 14 q + 14 q^{4} + 7 q^{7} + 12 q^{10} - 23 q^{13} - 22 q^{16} - 98 q^{19} - 30 q^{22} + 11 q^{25} + 28 q^{28} + 112 q^{31} + 72 q^{34} + 190 q^{37} - 174 q^{40} - 218 q^{43} - 456 q^{46} + 12 q^{49} + 208 q^{52} + 84 q^{55} + 204 q^{58} + 97 q^{61} + 860 q^{64} + 13 q^{67} - 318 q^{70} - 62 q^{73} + 64 q^{76} + 205 q^{79} - 168 q^{82} + 36 q^{85} + 102 q^{88} - 694 q^{91} - 372 q^{94} - 29 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(81, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
81.3.d.a 81.d 9.d $2$ $2.207$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(13\) $\mathrm{U}(1)[D_{6}]$ \(q-4\zeta_{6}q^{4}+(13-13\zeta_{6})q^{7}+\zeta_{6}q^{13}+\cdots\)
81.3.d.b 81.d 9.d $4$ $2.207$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-10\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+5\zeta_{12}^{2}q^{4}+(\zeta_{12}-\zeta_{12}^{3})q^{5}+\cdots\)
81.3.d.c 81.d 9.d $8$ $2.207$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{24}^{4}q^{2}+(2-2\zeta_{24}+\zeta_{24}^{2}-\zeta_{24}^{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(81, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)