Properties

Label 1728.3.q
Level $1728$
Weight $3$
Character orbit 1728.q
Rep. character $\chi_{1728}(449,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $92$
Newform subspaces $12$
Sturm bound $864$
Trace bound $41$

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Defining parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.q (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 12 \)
Sturm bound: \(864\)
Trace bound: \(41\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1728, [\chi])\).

Total New Old
Modular forms 1224 100 1124
Cusp forms 1080 92 988
Eisenstein series 144 8 136

Trace form

\( 92 q - 6 q^{5} + O(q^{10}) \) \( 92 q - 6 q^{5} + 2 q^{13} + 188 q^{25} - 6 q^{29} + 8 q^{37} - 138 q^{41} - 240 q^{49} + 2 q^{61} + 6 q^{65} - 8 q^{73} - 6 q^{77} - 48 q^{85} - 2 q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1728, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1728.3.q.a 1728.q 9.d $2$ $47.085$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(6\) \(-2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(4-2\zeta_{6})q^{5}+(-2+2\zeta_{6})q^{7}+(-1+\cdots)q^{11}+\cdots\)
1728.3.q.b 1728.q 9.d $2$ $47.085$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(6\) \(2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(4-2\zeta_{6})q^{5}+(2-2\zeta_{6})q^{7}+(1+\zeta_{6})q^{11}+\cdots\)
1728.3.q.c 1728.q 9.d $4$ $47.085$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(-18\) \(-2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-3-3\beta _{2})q^{5}+(\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots\)
1728.3.q.d 1728.q 9.d $4$ $47.085$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(-18\) \(2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-3-3\beta _{2})q^{5}+(-\beta _{1}+\beta _{2}-\beta _{3})q^{7}+\cdots\)
1728.3.q.e 1728.q 9.d $4$ $47.085$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(6\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\beta _{1}+\beta _{2})q^{5}+(-\beta _{1}-3\beta _{2}-\beta _{3})q^{7}+\cdots\)
1728.3.q.f 1728.q 9.d $4$ $47.085$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(6\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\beta _{1}+\beta _{2})q^{5}+(\beta _{1}+3\beta _{2}+\beta _{3})q^{7}+\cdots\)
1728.3.q.g 1728.q 9.d $4$ $47.085$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(0\) \(9\) \(-1\) $\mathrm{SU}(2)[C_{6}]$ \(q+(3-\beta _{1}+\beta _{2})q^{5}+(-1+2\beta _{1}-\beta _{3})q^{7}+\cdots\)
1728.3.q.h 1728.q 9.d $4$ $47.085$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(0\) \(9\) \(1\) $\mathrm{SU}(2)[C_{6}]$ \(q+(3-\beta _{1}+\beta _{2})q^{5}+(1-2\beta _{1}+\beta _{3})q^{7}+\cdots\)
1728.3.q.i 1728.q 9.d $8$ $47.085$ 8.0.\(\cdots\).9 None \(0\) \(0\) \(-6\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1+\beta _{1}+\beta _{2}-\beta _{4})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
1728.3.q.j 1728.q 9.d $8$ $47.085$ 8.0.\(\cdots\).9 None \(0\) \(0\) \(-6\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1+\beta _{1}+\beta _{2}-\beta _{4})q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
1728.3.q.k 1728.q 9.d $24$ $47.085$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
1728.3.q.l 1728.q 9.d $24$ $47.085$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{3}^{\mathrm{old}}(1728, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 14}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)