Properties

Label 81.7.d
Level $81$
Weight $7$
Character orbit 81.d
Rep. character $\chi_{81}(26,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $46$
Newform subspaces $6$
Sturm bound $63$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 81.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 6 \)
Sturm bound: \(63\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(81, [\chi])\).

Total New Old
Modular forms 120 50 70
Cusp forms 96 46 50
Eisenstein series 24 4 20

Trace form

\( 46 q + 706 q^{4} + 602 q^{7} + O(q^{10}) \) \( 46 q + 706 q^{4} + 602 q^{7} + 252 q^{10} + 4202 q^{13} - 20606 q^{16} - 2284 q^{19} - 7326 q^{22} + 48721 q^{25} + 137468 q^{28} + 104642 q^{31} + 45144 q^{34} + 29900 q^{37} - 236934 q^{40} + 317582 q^{43} - 157752 q^{46} - 321621 q^{49} - 671824 q^{52} + 1431144 q^{55} + 1238724 q^{58} - 45190 q^{61} - 3195188 q^{64} + 237374 q^{67} + 389682 q^{70} - 1011004 q^{73} - 1972504 q^{76} - 2012194 q^{79} + 2717856 q^{82} + 1192536 q^{85} + 1547838 q^{88} + 3005464 q^{91} - 4810572 q^{94} + 1460006 q^{97} + O(q^{100}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(81, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
81.7.d.a 81.d 9.d $2$ $18.634$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(286\) $\mathrm{U}(1)[D_{6}]$ \(q-2^{6}\zeta_{6}q^{4}+(286-286\zeta_{6})q^{7}-506\zeta_{6}q^{13}+\cdots\)
81.7.d.b 81.d 9.d $4$ $18.634$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-598\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}-28\zeta_{12}^{2}q^{4}+(40\zeta_{12}-40\zeta_{12}^{3})q^{5}+\cdots\)
81.7.d.c 81.d 9.d $4$ $18.634$ \(\Q(\sqrt{-3}, \sqrt{-10})\) None \(0\) \(0\) \(0\) \(806\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+26\beta _{2}q^{4}+(-14\beta _{1}+14\beta _{3})q^{5}+\cdots\)
81.7.d.d 81.d 9.d $4$ $18.634$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-1048\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+98\beta _{2}q^{4}+(-5\beta _{1}+5\beta _{3})q^{5}+\cdots\)
81.7.d.e 81.d 9.d $8$ $18.634$ 8.0.\(\cdots\).3 None \(0\) \(0\) \(0\) \(676\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{3}q^{2}+(7^{2}+7^{2}\beta _{1}+\beta _{4}+\beta _{5})q^{4}+\cdots\)
81.7.d.f 81.d 9.d $24$ $18.634$ None \(0\) \(0\) \(0\) \(480\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{7}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(81, [\chi]) \cong \) \(S_{7}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 2}\)