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 Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1728.3.q.a \(2\) \(47.085\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(6\) \(-2\) \(q+(4-2\zeta_{6})q^{5}+(-2+2\zeta_{6})q^{7}+(-1+\cdots)q^{11}+\cdots\)
1728.3.q.b \(2\) \(47.085\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(6\) \(2\) \(q+(4-2\zeta_{6})q^{5}+(2-2\zeta_{6})q^{7}+(1+\zeta_{6})q^{11}+\cdots\)
1728.3.q.c \(4\) \(47.085\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(-18\) \(-2\) \(q+(-3-3\beta _{2})q^{5}+(\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots\)
1728.3.q.d \(4\) \(47.085\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(-18\) \(2\) \(q+(-3-3\beta _{2})q^{5}+(-\beta _{1}+\beta _{2}-\beta _{3})q^{7}+\cdots\)
1728.3.q.e \(4\) \(47.085\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(6\) \(-6\) \(q+(1-\beta _{1}+\beta _{2})q^{5}+(-\beta _{1}-3\beta _{2}-\beta _{3})q^{7}+\cdots\)
1728.3.q.f \(4\) \(47.085\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(6\) \(6\) \(q+(1-\beta _{1}+\beta _{2})q^{5}+(\beta _{1}+3\beta _{2}+\beta _{3})q^{7}+\cdots\)
1728.3.q.g \(4\) \(47.085\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(0\) \(9\) \(-1\) \(q+(3-\beta _{1}+\beta _{2})q^{5}+(-1+2\beta _{1}-\beta _{3})q^{7}+\cdots\)
1728.3.q.h \(4\) \(47.085\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(0\) \(9\) \(1\) \(q+(3-\beta _{1}+\beta _{2})q^{5}+(1-2\beta _{1}+\beta _{3})q^{7}+\cdots\)
1728.3.q.i \(8\) \(47.085\) 8.0.\(\cdots\).9 None \(0\) \(0\) \(-6\) \(-6\) \(q+(-1+\beta _{1}+\beta _{2}-\beta _{4})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
1728.3.q.j \(8\) \(47.085\) 8.0.\(\cdots\).9 None \(0\) \(0\) \(-6\) \(6\) \(q+(-1+\beta _{1}+\beta _{2}-\beta _{4})q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
1728.3.q.k \(24\) \(47.085\) None \(0\) \(0\) \(0\) \(0\)
1728.3.q.l \(24\) \(47.085\) None \(0\) \(0\) \(0\) \(0\)

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

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