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Properties

Label 720.2.bm
Level $720$
Weight $2$
Character orbit 720.bm
Rep. character $\chi_{720}(109,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $116$
Newform subspaces $8$
Sturm bound $288$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.bm (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 80 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 8 \)
Sturm bound: \(288\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

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

Total New Old
Modular forms 304 124 180
Cusp forms 272 116 156
Eisenstein series 32 8 24

Trace form

\( 116 q - 4 q^{4} + 2 q^{5} + O(q^{10}) \) \( 116 q - 4 q^{4} + 2 q^{5} - 8 q^{10} + 4 q^{11} + 20 q^{14} - 8 q^{16} + 4 q^{19} + 16 q^{20} + 8 q^{26} + 4 q^{29} + 16 q^{31} - 24 q^{34} - 48 q^{40} + 24 q^{44} - 4 q^{46} + 84 q^{49} - 4 q^{50} + 8 q^{56} + 20 q^{59} + 12 q^{61} - 64 q^{64} - 4 q^{65} - 52 q^{70} - 16 q^{74} - 40 q^{76} + 16 q^{79} - 64 q^{80} + 8 q^{85} + 36 q^{86} - 16 q^{91} - 84 q^{94} + 28 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
720.2.bm.a 720.bm 80.q $2$ $5.749$ \(\Q(\sqrt{-1}) \) None \(-2\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+2iq^{4}+(-2-i)q^{5}+\cdots\)
720.2.bm.b 720.bm 80.q $2$ $5.749$ \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{2}+2iq^{4}+(-1-2i)q^{5}+\cdots\)
720.2.bm.c 720.bm 80.q $4$ $5.749$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\zeta_{8}+\zeta_{8}^{3})q^{2}-2q^{4}+(2\zeta_{8}+\zeta_{8}^{3})q^{5}+\cdots\)
720.2.bm.d 720.bm 80.q $4$ $5.749$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\zeta_{8}+\zeta_{8}^{3})q^{2}-2q^{4}+(-2\zeta_{8}+\zeta_{8}^{3})q^{5}+\cdots\)
720.2.bm.e 720.bm 80.q $8$ $5.749$ 8.0.3317760000.5 \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{4}-\beta _{6})q^{5}+(2\beta _{4}+\cdots)q^{8}+\cdots\)
720.2.bm.f 720.bm 80.q $16$ $5.749$ 16.0.\(\cdots\).1 None \(0\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{12}q^{2}+(-1+\beta _{4}-\beta _{7}+\beta _{8})q^{4}+\cdots\)
720.2.bm.g 720.bm 80.q $32$ $5.749$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
720.2.bm.h 720.bm 80.q $48$ $5.749$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$

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

\( S_{2}^{\mathrm{old}}(720, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)