Properties

Label 1728.2.s
Level $1728$
Weight $2$
Character orbit 1728.s
Rep. character $\chi_{1728}(575,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $44$
Newform subspaces $7$
Sturm bound $576$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.s (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 36 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 7 \)
Sturm bound: \(576\)
Trace bound: \(7\)
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 648 52 596
Cusp forms 504 44 460
Eisenstein series 144 8 136

Trace form

\( 44 q - 6 q^{5} + O(q^{10}) \) \( 44 q - 6 q^{5} + 2 q^{13} + 12 q^{25} - 6 q^{29} + 8 q^{37} + 30 q^{41} + 8 q^{49} + 2 q^{61} + 6 q^{65} - 8 q^{73} - 6 q^{77} - 8 q^{85} - 2 q^{97} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1728.2.s.a 1728.s 36.h $2$ $13.798$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-6\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-4+2\zeta_{6})q^{5}+(-2-2\zeta_{6})q^{7}+\cdots\)
1728.2.s.b 1728.s 36.h $2$ $13.798$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-6\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-4+2\zeta_{6})q^{5}+(2+2\zeta_{6})q^{7}+(3+\cdots)q^{11}+\cdots\)
1728.2.s.c 1728.s 36.h $2$ $13.798$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(-3\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2-\zeta_{6})q^{5}+(-1-\zeta_{6})q^{7}+(3-3\zeta_{6})q^{11}+\cdots\)
1728.2.s.d 1728.s 36.h $2$ $13.798$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(3\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2-\zeta_{6})q^{5}+(1+\zeta_{6})q^{7}+(-3+3\zeta_{6})q^{11}+\cdots\)
1728.2.s.e 1728.s 36.h $4$ $13.798$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2-\zeta_{12}^{2})q^{5}-\zeta_{12}q^{7}+(-\zeta_{12}+\cdots)q^{11}+\cdots\)
1728.2.s.f 1728.s 36.h $8$ $13.798$ 8.0.170772624.1 None \(0\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\beta _{2})q^{5}+(\beta _{1}-\beta _{6})q^{7}+(-\beta _{3}+\cdots)q^{11}+\cdots\)
1728.2.s.g 1728.s 36.h $24$ $13.798$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

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

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