Properties

Label 1728.2.c
Level $1728$
Weight $2$
Character orbit 1728.c
Rep. character $\chi_{1728}(1727,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $7$
Sturm bound $576$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 12 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(576\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1728, [\chi])\).

Total New Old
Modular forms 324 32 292
Cusp forms 252 32 220
Eisenstein series 72 0 72

Trace form

\( 32q + O(q^{10}) \) \( 32q + 8q^{13} - 32q^{25} - 8q^{37} - 32q^{49} - 24q^{61} + 16q^{85} + 16q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1728, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1728.2.c.a \(2\) \(13.798\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{6}q^{7}-7q^{13}+5\zeta_{6}q^{19}+5q^{25}+\cdots\)
1728.2.c.b \(2\) \(13.798\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{6}q^{7}+5q^{13}+\zeta_{6}q^{19}+5q^{25}+\cdots\)
1728.2.c.c \(4\) \(13.798\) \(\Q(\sqrt{3}, \sqrt{-5})\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{5}-\beta _{2}q^{7}-\beta _{3}q^{11}-2q^{13}+\cdots\)
1728.2.c.d \(4\) \(13.798\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{5}+\beta _{2}q^{7}+\beta _{3}q^{11}+q^{13}+\cdots\)
1728.2.c.e \(4\) \(13.798\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{5}-\zeta_{12}^{2}q^{7}+\zeta_{12}^{3}q^{11}+\cdots\)
1728.2.c.f \(8\) \(13.798\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{24}^{2}q^{5}+(\zeta_{24}+\zeta_{24}^{3})q^{7}-\zeta_{24}^{6}q^{11}+\cdots\)
1728.2.c.g \(8\) \(13.798\) \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{3}q^{5}-\zeta_{24}^{7}q^{7}+(\zeta_{24}+\zeta_{24}^{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1728, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)