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Properties

Label 60.4.a
Level $60$
Weight $4$
Character orbit 60.a
Rep. character $\chi_{60}(1,\cdot)$
Character field $\Q$
Dimension $2$
Newform subspaces $2$
Sturm bound $48$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 60.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(48\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(60))\).

Total New Old
Modular forms 42 2 40
Cusp forms 30 2 28
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(5\)FrickeDim
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(1\)

Trace form

\( 2 q - 6 q^{3} + 4 q^{7} + 18 q^{9} + O(q^{10}) \) \( 2 q - 6 q^{3} + 4 q^{7} + 18 q^{9} + 12 q^{11} - 80 q^{13} + 24 q^{17} + 160 q^{19} - 12 q^{21} - 264 q^{23} + 50 q^{25} - 54 q^{27} + 312 q^{29} - 152 q^{31} - 36 q^{33} + 300 q^{35} - 248 q^{37} + 240 q^{39} - 540 q^{41} - 344 q^{43} + 336 q^{47} + 1122 q^{49} - 72 q^{51} - 528 q^{53} + 300 q^{55} - 480 q^{57} - 228 q^{59} - 476 q^{61} + 36 q^{63} + 300 q^{65} - 704 q^{67} + 792 q^{69} + 360 q^{71} - 212 q^{73} - 150 q^{75} + 1824 q^{77} + 184 q^{79} + 162 q^{81} - 1344 q^{83} - 900 q^{85} - 936 q^{87} - 1620 q^{89} + 1640 q^{91} + 456 q^{93} + 600 q^{95} - 92 q^{97} + 108 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(60))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5
60.4.a.a 60.a 1.a $1$ $3.540$ \(\Q\) None 60.4.a.a \(0\) \(-3\) \(-5\) \(-28\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}-28q^{7}+9q^{9}-24q^{11}+\cdots\)
60.4.a.b 60.a 1.a $1$ $3.540$ \(\Q\) None 60.4.a.b \(0\) \(-3\) \(5\) \(32\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}+2^{5}q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(60))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(60)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)