↑

Properties

Label 648.2.i
Level $648$
Weight $2$
Character orbit 648.i
Rep. character $\chi_{648}(217,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $24$
Newform subspaces $10$
Sturm bound $216$
Trace bound $7$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 10 \)
Sturm bound: \(216\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(648, [\chi])\).

Total New Old
Modular forms 264 24 240
Cusp forms 168 24 144
Eisenstein series 96 0 96

Trace form

\( 24 q + O(q^{10}) \) \( 24 q - 12 q^{19} - 12 q^{25} + 30 q^{31} + 36 q^{43} + 60 q^{55} + 12 q^{61} + 6 q^{67} + 36 q^{73} - 30 q^{79} - 6 q^{85} - 60 q^{91} - 24 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(648, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
648.2.i.a 648.i 9.c $2$ $5.174$ \(\Q(\sqrt{-3}) \) None 216.2.a.a \(0\) \(0\) \(-4\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q-4\zeta_{6}q^{5}+(3-3\zeta_{6})q^{7}+(-4+4\zeta_{6})q^{11}+\cdots\)
648.2.i.b 648.i 9.c $2$ $5.174$ \(\Q(\sqrt{-3}) \) None 24.2.a.a \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+2\zeta_{6}q^{13}+\cdots\)
648.2.i.c 648.i 9.c $2$ $5.174$ \(\Q(\sqrt{-3}) \) None 216.2.a.b \(0\) \(0\) \(-1\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(5-5\zeta_{6})q^{11}+\cdots\)
648.2.i.d 648.i 9.c $2$ $5.174$ \(\Q(\sqrt{-3}) \) None 648.2.a.b \(0\) \(0\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}+5\zeta_{6}q^{13}+\cdots\)
648.2.i.e 648.i 9.c $2$ $5.174$ \(\Q(\sqrt{-3}) \) None 216.2.a.b \(0\) \(0\) \(1\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(-5+5\zeta_{6})q^{11}+\cdots\)
648.2.i.f 648.i 9.c $2$ $5.174$ \(\Q(\sqrt{-3}) \) None 648.2.a.b \(0\) \(0\) \(1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+5\zeta_{6}q^{13}+\cdots\)
648.2.i.g 648.i 9.c $2$ $5.174$ \(\Q(\sqrt{-3}) \) None 24.2.a.a \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}+2\zeta_{6}q^{13}+\cdots\)
648.2.i.h 648.i 9.c $2$ $5.174$ \(\Q(\sqrt{-3}) \) None 216.2.a.a \(0\) \(0\) \(4\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+4\zeta_{6}q^{5}+(3-3\zeta_{6})q^{7}+(4-4\zeta_{6})q^{11}+\cdots\)
648.2.i.i 648.i 9.c $4$ $5.174$ \(\Q(\zeta_{12})\) None 648.2.a.e \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta_{2}-2\beta_1)q^{5}+(2\beta_{3}-2\beta_{2})q^{7}+\cdots\)
648.2.i.j 648.i 9.c $4$ $5.174$ \(\Q(\zeta_{12})\) None 648.2.a.e \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta_{3}-\beta_{2}-2\beta_1+2)q^{5}-2\beta_{2} q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(648, [\chi])\) into lower level spaces

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