Properties

Label 360.4.f
Level $360$
Weight $4$
Character orbit 360.f
Rep. character $\chi_{360}(289,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $6$
Sturm bound $288$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 360.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(288\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(360, [\chi])\).

Total New Old
Modular forms 232 22 210
Cusp forms 200 22 178
Eisenstein series 32 0 32

Trace form

\( 22 q - 4 q^{5} + O(q^{10}) \) \( 22 q - 4 q^{5} - 52 q^{11} - 80 q^{19} - 126 q^{25} + 16 q^{29} - 384 q^{31} - 292 q^{35} + 20 q^{41} - 1614 q^{49} + 88 q^{55} + 1284 q^{59} - 916 q^{61} + 356 q^{65} - 376 q^{71} - 2320 q^{79} + 108 q^{85} - 1452 q^{89} + 1304 q^{91} - 2704 q^{95} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(360, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
360.4.f.a 360.f 5.b $2$ $21.241$ \(\Q(\sqrt{-1}) \) None 120.4.f.c \(0\) \(0\) \(-10\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-5-5i)q^{5}+2iq^{7}+28q^{11}+\cdots\)
360.4.f.b 360.f 5.b $2$ $21.241$ \(\Q(\sqrt{-1}) \) None 120.4.f.b \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2-11i)q^{5}+10iq^{7}+14q^{11}+\cdots\)
360.4.f.c 360.f 5.b $2$ $21.241$ \(\Q(\sqrt{-1}) \) None 120.4.f.a \(0\) \(0\) \(20\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(10+5i)q^{5}+10iq^{7}+46q^{11}+\cdots\)
360.4.f.d 360.f 5.b $4$ $21.241$ \(\Q(i, \sqrt{129})\) None 120.4.f.d \(0\) \(0\) \(-22\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-5+\beta _{1}-\beta _{3})q^{5}+(-2\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots\)
360.4.f.e 360.f 5.b $4$ $21.241$ \(\Q(i, \sqrt{6})\) None 40.4.c.a \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta _{2}-\beta _{3})q^{5}+(\beta _{1}-2\beta _{2})q^{7}+\cdots\)
360.4.f.f 360.f 5.b $8$ $21.241$ 8.0.\(\cdots\).2 None 360.4.f.f \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{5}+\beta _{2}q^{7}+\beta _{6}q^{11}+\beta _{7}q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(360, [\chi])\) into lower level spaces

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