Properties

Label 720.2.x
Level $720$
Weight $2$
Character orbit 720.x
Rep. character $\chi_{720}(127,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $30$
Newform subspaces $7$
Sturm bound $288$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.x (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 20 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 7 \)
Sturm bound: \(288\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(720, [\chi])\).

Total New Old
Modular forms 336 30 306
Cusp forms 240 30 210
Eisenstein series 96 0 96

Trace form

\( 30 q + O(q^{10}) \) \( 30 q - 6 q^{13} - 6 q^{17} - 6 q^{25} + 6 q^{37} + 24 q^{41} + 30 q^{53} + 54 q^{65} + 30 q^{73} + 24 q^{77} + 18 q^{85} + 30 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(720, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
720.2.x.a 720.x 20.e $2$ $5.749$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) 80.2.n.a \(0\) \(0\) \(-4\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-2-i)q^{5}+(-5+5i)q^{13}+(5+\cdots)q^{17}+\cdots\)
720.2.x.b 720.x 20.e $2$ $5.749$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) 720.2.x.b \(0\) \(0\) \(-2\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-1-2i)q^{5}+(1-i)q^{13}+(-5+\cdots)q^{17}+\cdots\)
720.2.x.c 720.x 20.e $2$ $5.749$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) 720.2.x.b \(0\) \(0\) \(2\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(1+2i)q^{5}+(1-i)q^{13}+(5+5i)q^{17}+\cdots\)
720.2.x.d 720.x 20.e $4$ $5.749$ \(\Q(\zeta_{12})\) None 80.2.n.b \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+2\zeta_{12})q^{5}-\zeta_{12}^{3}q^{7}+(\zeta_{12}^{2}+\cdots)q^{11}+\cdots\)
720.2.x.e 720.x 20.e $4$ $5.749$ \(\Q(\zeta_{8})\) None 240.2.w.a \(0\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2+\zeta_{8}^{2})q^{5}+\zeta_{8}q^{7}+(-\zeta_{8}-\zeta_{8}^{3})q^{11}+\cdots\)
720.2.x.f 720.x 20.e $8$ $5.749$ \(\Q(\zeta_{24})\) None 240.2.w.b \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\zeta_{24}^{2}-\zeta_{24}^{3})q^{5}+(\zeta_{24}^{4}+\cdots)q^{7}+\cdots\)
720.2.x.g 720.x 20.e $8$ $5.749$ 8.0.3317760000.2 None 720.2.x.g \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{3}q^{5}+\beta _{6}q^{7}-\beta _{7}q^{11}+(-2+2\beta _{2}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(720, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(720, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 2}\)