Properties

Label 81.4.c
Level $81$
Weight $4$
Character orbit 81.c
Rep. character $\chi_{81}(28,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $22$
Newform subspaces $7$
Sturm bound $36$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 81.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 7 \)
Sturm bound: \(36\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(81, [\chi])\).

Total New Old
Modular forms 66 26 40
Cusp forms 42 22 20
Eisenstein series 24 4 20

Trace form

\( 22 q - 38 q^{4} + 32 q^{7} + O(q^{10}) \) \( 22 q - 38 q^{4} + 32 q^{7} - 36 q^{10} - 58 q^{13} - 110 q^{16} + 476 q^{19} - 18 q^{22} - 317 q^{25} - 1252 q^{28} + 266 q^{31} + 1188 q^{34} + 692 q^{37} + 846 q^{40} - 76 q^{43} + 1152 q^{46} - 1239 q^{49} - 1732 q^{52} - 4356 q^{55} + 684 q^{58} - 22 q^{61} + 4180 q^{64} + 1346 q^{67} + 1494 q^{70} + 2528 q^{73} - 6124 q^{76} - 2614 q^{79} - 4464 q^{82} - 54 q^{85} + 5958 q^{88} + 6172 q^{91} + 6732 q^{94} + 86 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(81, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
81.4.c.a 81.c 9.c $2$ $4.779$ \(\Q(\sqrt{-3}) \) None \(-3\) \(0\) \(-15\) \(25\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{2}-\zeta_{6}q^{4}-15\zeta_{6}q^{5}+\cdots\)
81.4.c.b 81.c 9.c $2$ $4.779$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-20\) $\mathrm{U}(1)[D_{3}]$ \(q+8\zeta_{6}q^{4}+(-20+20\zeta_{6})q^{7}+70\zeta_{6}q^{13}+\cdots\)
81.4.c.c 81.c 9.c $2$ $4.779$ \(\Q(\sqrt{-3}) \) None \(3\) \(0\) \(15\) \(25\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{2}-\zeta_{6}q^{4}+15\zeta_{6}q^{5}+\cdots\)
81.4.c.d 81.c 9.c $4$ $4.779$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(-3\) \(0\) \(12\) \(-10\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{1}-\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
81.4.c.e 81.c 9.c $4$ $4.779$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-22\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+10\beta _{2}q^{4}+(4\beta _{1}+4\beta _{3})q^{5}+\cdots\)
81.4.c.f 81.c 9.c $4$ $4.779$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(44\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{12}^{2}q^{2}+(5-5\zeta_{12})q^{4}+(-7\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
81.4.c.g 81.c 9.c $4$ $4.779$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(3\) \(0\) \(-12\) \(-10\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+3\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(81, [\chi])\) into lower level spaces

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