Properties

Label 648.3.m
Level $648$
Weight $3$
Character orbit 648.m
Rep. character $\chi_{648}(377,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $48$
Newform subspaces $7$
Sturm bound $324$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 648.m (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 7 \)
Sturm bound: \(324\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(648, [\chi])\).

Total New Old
Modular forms 480 48 432
Cusp forms 384 48 336
Eisenstein series 96 0 96

Trace form

\( 48 q + O(q^{10}) \) \( 48 q - 24 q^{19} + 120 q^{25} - 120 q^{31} + 180 q^{43} - 216 q^{49} + 120 q^{55} + 96 q^{61} + 84 q^{67} + 72 q^{73} + 300 q^{79} + 24 q^{85} + 960 q^{91} + 132 q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(648, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
648.3.m.a 648.m 9.d $4$ $17.657$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-24\) $\mathrm{SU}(2)[C_{6}]$ \(q+5\beta _{1}q^{5}-12\beta _{2}q^{7}+(4\beta _{1}-4\beta _{3})q^{11}+\cdots\)
648.3.m.b 648.m 9.d $4$ $17.657$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{5}-3\beta _{2}q^{7}+(-\beta _{1}+\beta _{3})q^{11}+\cdots\)
648.3.m.c 648.m 9.d $4$ $17.657$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{5}+3\beta _{2}q^{7}+(-7\beta _{1}+7\beta _{3})q^{11}+\cdots\)
648.3.m.d 648.m 9.d $4$ $17.657$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{5}+6\beta _{2}q^{7}+(-\beta _{1}+\beta _{3})q^{11}+\cdots\)
648.3.m.e 648.m 9.d $8$ $17.657$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{24}^{6}q^{5}+(-3\zeta_{24}^{2}+\zeta_{24}^{3})q^{7}+\cdots\)
648.3.m.f 648.m 9.d $8$ $17.657$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{24}-\zeta_{24}^{3}-\zeta_{24}^{4})q^{5}+(3-3\zeta_{24}^{2}+\cdots)q^{7}+\cdots\)
648.3.m.g 648.m 9.d $16$ $17.657$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{11}+\beta _{12})q^{5}+(2+2\beta _{2}-\beta _{7})q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(648, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(648, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 2}\)