Properties

Label 81.9.b
Level $81$
Weight $9$
Character orbit 81.b
Rep. character $\chi_{81}(80,\cdot)$
Character field $\Q$
Dimension $30$
Newform subspaces $2$
Sturm bound $81$
Trace bound $1$

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Defining parameters

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

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(81, [\chi])\).

Total New Old
Modular forms 78 34 44
Cusp forms 66 30 36
Eisenstein series 12 4 8

Trace form

\( 30 q - 3582 q^{4} + 1848 q^{7} + O(q^{10}) \) \( 30 q - 3582 q^{4} + 1848 q^{7} + 10236 q^{10} + 60492 q^{13} + 225678 q^{16} - 84522 q^{19} - 814602 q^{22} - 672066 q^{25} + 78576 q^{28} + 899400 q^{31} - 282330 q^{34} + 2682600 q^{37} - 4200420 q^{40} + 3869454 q^{43} - 3044508 q^{46} + 24255834 q^{49} - 14279520 q^{52} - 6345096 q^{55} - 27210576 q^{58} + 41549196 q^{61} - 2560830 q^{64} - 45779118 q^{67} - 44552388 q^{70} - 57995346 q^{73} + 155239794 q^{76} - 85501392 q^{79} + 195898278 q^{82} - 33976728 q^{85} + 317064414 q^{88} + 172166292 q^{91} - 471345096 q^{94} - 387977010 q^{97} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(81, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
81.9.b.a 81.b 3.b $14$ $32.998$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(0\) \(0\) \(-1844\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(-110+\beta _{2})q^{4}+\beta _{6}q^{5}+\cdots\)
81.9.b.b 81.b 3.b $16$ $32.998$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(3692\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{8}q^{2}+(-2^{7}+\beta _{1})q^{4}+(-2\beta _{8}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(81, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)