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Properties

Label 720.2.q
Level $720$
Weight $2$
Character orbit 720.q
Rep. character $\chi_{720}(241,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $48$
Newform subspaces $12$
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.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(288\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

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

Total New Old
Modular forms 312 48 264
Cusp forms 264 48 216
Eisenstein series 48 0 48

Trace form

\( 48 q + 4 q^{9} + O(q^{10}) \) \( 48 q + 4 q^{9} + 8 q^{17} + 4 q^{21} + 12 q^{23} - 24 q^{25} + 12 q^{27} - 4 q^{29} - 12 q^{31} + 12 q^{33} + 24 q^{35} + 36 q^{39} + 4 q^{41} - 12 q^{43} - 8 q^{45} + 20 q^{47} - 24 q^{49} + 8 q^{51} - 12 q^{57} - 12 q^{59} - 40 q^{63} - 28 q^{69} - 24 q^{71} + 24 q^{73} - 16 q^{77} - 8 q^{81} - 56 q^{83} - 60 q^{87} - 24 q^{89} + 24 q^{91} - 8 q^{93} + 16 q^{95} - 12 q^{97} + 76 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
720.2.q.a 720.q 9.c $2$ $5.749$ \(\Q(\sqrt{-3}) \) None 180.2.i.a \(0\) \(-3\) \(-1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+\cdots\)
720.2.q.b 720.q 9.c $2$ $5.749$ \(\Q(\sqrt{-3}) \) None 90.2.e.a \(0\) \(-3\) \(1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+\cdots\)
720.2.q.c 720.q 9.c $2$ $5.749$ \(\Q(\sqrt{-3}) \) None 90.2.e.b \(0\) \(3\) \(-1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{7}+\cdots\)
720.2.q.d 720.q 9.c $2$ $5.749$ \(\Q(\sqrt{-3}) \) None 45.2.e.a \(0\) \(3\) \(1\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+\cdots\)
720.2.q.e 720.q 9.c $2$ $5.749$ \(\Q(\sqrt{-3}) \) None 360.2.q.a \(0\) \(3\) \(1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(-5+\cdots)q^{11}+\cdots\)
720.2.q.f 720.q 9.c $4$ $5.749$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None 90.2.e.c \(0\) \(-2\) \(2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}+\beta _{3})q^{3}+(1-\beta _{2})q^{5}+(-1+\cdots)q^{7}+\cdots\)
720.2.q.g 720.q 9.c $4$ $5.749$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 360.2.q.c \(0\) \(-2\) \(2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{2}-\beta _{3})q^{3}+\beta _{2}q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
720.2.q.h 720.q 9.c $4$ $5.749$ \(\Q(\zeta_{12})\) None 360.2.q.b \(0\) \(0\) \(2\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta_{2} q^{3}+\beta_1 q^{5}+(-\beta_{3}+\beta_{2}-2\beta_1+2)q^{7}+\cdots\)
720.2.q.i 720.q 9.c $6$ $5.749$ 6.0.954288.1 None 45.2.e.b \(0\) \(-1\) \(-3\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{4}q^{3}+\beta _{3}q^{5}+(2-\beta _{1}+\beta _{2}+2\beta _{3}+\cdots)q^{7}+\cdots\)
720.2.q.j 720.q 9.c $6$ $5.749$ 6.0.954288.1 None 360.2.q.d \(0\) \(1\) \(-3\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}-\beta _{5})q^{3}+\beta _{3}q^{5}+(-2-2\beta _{3}+\cdots)q^{7}+\cdots\)
720.2.q.k 720.q 9.c $6$ $5.749$ 6.0.954288.1 None 180.2.i.b \(0\) \(1\) \(3\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{4}q^{3}+(1+\beta _{2})q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
720.2.q.l 720.q 9.c $8$ $5.749$ 8.0.856615824.2 None 360.2.q.e \(0\) \(0\) \(-4\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{4}q^{3}-\beta _{1}q^{5}+(\beta _{2}-\beta _{4})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(720, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 2}\)