Properties

Label 1728.3.b
Level $1728$
Weight $3$
Character orbit 1728.b
Rep. character $\chi_{1728}(1567,\cdot)$
Character field $\Q$
Dimension $64$
Newform subspaces $10$
Sturm bound $864$
Trace bound $49$

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Defining parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(864\)
Trace bound: \(49\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\), \(17\)

Dimensions

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

Total New Old
Modular forms 612 64 548
Cusp forms 540 64 476
Eisenstein series 72 0 72

Trace form

\( 64 q + O(q^{10}) \) \( 64 q - 320 q^{25} - 480 q^{49} + 160 q^{73} + 320 q^{97} + O(q^{100}) \)

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.b.a \(4\) \(47.085\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+6\zeta_{12}q^{5}-5\zeta_{12}q^{7}-6q^{11}-\zeta_{12}^{2}q^{13}+\cdots\)
1728.3.b.b \(4\) \(47.085\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q-2\zeta_{12}^{2}q^{5}+\zeta_{12}q^{7}-10\zeta_{12}^{3}q^{11}+\cdots\)
1728.3.b.c \(4\) \(47.085\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+13\zeta_{12}q^{7}-5\zeta_{12}^{2}q^{13}-7\zeta_{12}^{3}q^{19}+\cdots\)
1728.3.b.d \(4\) \(47.085\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+11\zeta_{12}q^{7}-7\zeta_{12}^{2}q^{13}+5\zeta_{12}^{3}q^{19}+\cdots\)
1728.3.b.e \(4\) \(47.085\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q-2\zeta_{12}^{2}q^{5}-\zeta_{12}q^{7}+10\zeta_{12}^{3}q^{11}+\cdots\)
1728.3.b.f \(4\) \(47.085\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+6\zeta_{12}q^{5}+5\zeta_{12}q^{7}+6q^{11}+\zeta_{12}^{2}q^{13}+\cdots\)
1728.3.b.g \(8\) \(47.085\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{3}q^{5}-7\zeta_{24}q^{7}-\zeta_{24}^{6}q^{11}+\cdots\)
1728.3.b.h \(8\) \(47.085\) \(\Q(\zeta_{24})\) \(\Q(\sqrt{-6}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{3}q^{5}+(2\zeta_{24}-\zeta_{24}^{4})q^{7}+(2\zeta_{24}^{2}+\cdots)q^{11}+\cdots\)
1728.3.b.i \(12\) \(47.085\) 12.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{4}q^{5}+\beta _{2}q^{7}+(-2\beta _{1}+\beta _{9})q^{11}+\cdots\)
1728.3.b.j \(12\) \(47.085\) 12.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{4}q^{5}-\beta _{2}q^{7}+(2\beta _{1}-\beta _{9})q^{11}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)