Properties

Label 720.4.f
Level $720$
Weight $4$
Character orbit 720.f
Rep. character $\chi_{720}(289,\cdot)$
Character field $\Q$
Dimension $44$
Newform subspaces $14$
Sturm bound $576$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 720.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(576\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(720, [\chi])\).

Total New Old
Modular forms 456 46 410
Cusp forms 408 44 364
Eisenstein series 48 2 46

Trace form

\( 44 q + 2 q^{5} + O(q^{10}) \) \( 44 q + 2 q^{5} + 64 q^{11} + 104 q^{19} - 96 q^{25} + 172 q^{29} + 56 q^{31} + 184 q^{35} - 28 q^{41} - 2220 q^{49} + 264 q^{55} - 720 q^{59} - 536 q^{61} - 328 q^{65} - 176 q^{71} + 1432 q^{79} - 652 q^{85} - 324 q^{89} - 2000 q^{91} + 1216 q^{95} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(720, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
720.4.f.a 720.f 5.b $2$ $42.481$ \(\Q(\sqrt{-19}) \) None \(0\) \(0\) \(-14\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-7-\beta )q^{5}-\beta q^{7}+20q^{11}+6\beta q^{13}+\cdots\)
720.4.f.b 720.f 5.b $2$ $42.481$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-10\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-5+5i)q^{5}+2iq^{7}-28q^{11}+\cdots\)
720.4.f.c 720.f 5.b $2$ $42.481$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-2-11i)q^{5}+2iq^{7}+70q^{11}+\cdots\)
720.4.f.d 720.f 5.b $2$ $42.481$ \(\Q(\sqrt{-5}) \) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-5\beta q^{5}+62\beta q^{17}-164q^{19}-44\beta q^{23}+\cdots\)
720.4.f.e 720.f 5.b $2$ $42.481$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2+11i)q^{5}+10iq^{7}-14q^{11}+\cdots\)
720.4.f.f 720.f 5.b $2$ $42.481$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(10\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(5-5i)q^{5}+13iq^{7}-28q^{11}+6iq^{13}+\cdots\)
720.4.f.g 720.f 5.b $2$ $42.481$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(20\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(10-5i)q^{5}+10iq^{7}-46q^{11}+\cdots\)
720.4.f.h 720.f 5.b $2$ $42.481$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(20\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(10+5i)q^{5}+22iq^{7}-14q^{11}+\cdots\)
720.4.f.i 720.f 5.b $4$ $42.481$ \(\Q(i, \sqrt{129})\) None \(0\) \(0\) \(-22\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-5-\beta _{1}-\beta _{2})q^{5}+(-2\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots\)
720.4.f.j 720.f 5.b $4$ $42.481$ \(\Q(i, \sqrt{41})\) None \(0\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta _{2}-\beta _{3})q^{5}+(2\beta _{1}-\beta _{2})q^{7}+\cdots\)
720.4.f.k 720.f 5.b $4$ $42.481$ \(\Q(\sqrt{10}, \sqrt{-34})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{5}+\beta _{1}q^{7}+(3\beta _{2}-\beta _{3})q^{11}+\cdots\)
720.4.f.l 720.f 5.b $4$ $42.481$ \(\Q(i, \sqrt{31})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{1}-\beta _{2})q^{5}-\beta _{3}q^{7}+(2\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
720.4.f.m 720.f 5.b $4$ $42.481$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta _{1}-\beta _{3})q^{5}+(\beta _{1}-2\beta _{2})q^{7}+\cdots\)
720.4.f.n 720.f 5.b $8$ $42.481$ 8.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{5}+\beta _{2}q^{7}-\beta _{6}q^{11}-\beta _{7}q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(720, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(720, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(120, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(240, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(360, [\chi])\)\(^{\oplus 2}\)