Properties

Label 60.3.f
Level $60$
Weight $3$
Character orbit 60.f
Rep. character $\chi_{60}(19,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $2$
Sturm bound $36$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 60.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 20 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(36\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(60, [\chi])\).

Total New Old
Modular forms 28 12 16
Cusp forms 20 12 8
Eisenstein series 8 0 8

Trace form

\( 12 q - 2 q^{4} + 4 q^{5} + 6 q^{6} + 36 q^{9} + O(q^{10}) \) \( 12 q - 2 q^{4} + 4 q^{5} + 6 q^{6} + 36 q^{9} - 22 q^{10} - 52 q^{14} - 78 q^{16} + 52 q^{20} - 18 q^{24} - 68 q^{25} + 156 q^{26} - 40 q^{29} - 60 q^{30} - 28 q^{34} - 6 q^{36} + 154 q^{40} - 184 q^{41} + 204 q^{44} + 12 q^{45} + 160 q^{46} + 212 q^{49} + 72 q^{50} + 18 q^{54} - 244 q^{56} + 126 q^{60} + 8 q^{61} - 266 q^{64} - 192 q^{65} - 84 q^{66} - 96 q^{69} - 104 q^{70} - 468 q^{74} + 168 q^{76} - 308 q^{80} + 108 q^{81} - 348 q^{84} + 224 q^{85} - 136 q^{86} + 632 q^{89} - 66 q^{90} + 376 q^{94} + 234 q^{96} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(60, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
60.3.f.a 60.f 20.d $4$ $1.635$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{12}+\zeta_{12}^{3})q^{2}+\zeta_{12}^{3}q^{3}+(2+2\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
60.3.f.b 60.f 20.d $8$ $1.635$ 8.0.\(\cdots\).4 None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(-1+\beta _{4})q^{4}+(1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(60, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(60, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 2}\)