Properties

Label 60.8.a
Level $60$
Weight $8$
Character orbit 60.a
Rep. character $\chi_{60}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $4$
Sturm bound $96$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 60.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(60))\).

Total New Old
Modular forms 90 4 86
Cusp forms 78 4 74
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
\(-\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(+\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(2\)

Trace form

\( 4 q - 1120 q^{7} + 2916 q^{9} + O(q^{10}) \) \( 4 q - 1120 q^{7} + 2916 q^{9} + 5664 q^{11} - 22000 q^{13} + 50640 q^{17} - 27040 q^{19} - 40824 q^{21} + 6720 q^{23} + 62500 q^{25} - 157296 q^{29} + 324176 q^{31} - 184680 q^{33} - 420000 q^{35} + 238400 q^{37} + 524880 q^{39} + 572040 q^{41} - 1576960 q^{43} + 408000 q^{47} + 445764 q^{49} + 1251936 q^{51} - 1276800 q^{53} - 930000 q^{55} - 1516320 q^{57} + 1729824 q^{59} + 2207528 q^{61} - 816480 q^{63} - 645000 q^{65} + 2594240 q^{67} - 3681936 q^{69} - 819360 q^{71} - 160840 q^{73} + 6568800 q^{77} - 5218672 q^{79} + 2125764 q^{81} - 8690880 q^{83} + 1485000 q^{85} + 5171040 q^{87} + 4408920 q^{89} + 1720480 q^{91} + 7698240 q^{93} + 4740000 q^{95} - 26632600 q^{97} + 4129056 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(60))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5
60.8.a.a 60.a 1.a $1$ $18.743$ \(\Q\) None \(0\) \(-27\) \(-125\) \(1028\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3^{3}q^{3}-5^{3}q^{5}+1028q^{7}+3^{6}q^{9}+\cdots\)
60.8.a.b 60.a 1.a $1$ $18.743$ \(\Q\) None \(0\) \(-27\) \(125\) \(-832\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3^{3}q^{3}+5^{3}q^{5}-832q^{7}+3^{6}q^{9}+\cdots\)
60.8.a.c 60.a 1.a $1$ $18.743$ \(\Q\) None \(0\) \(27\) \(-125\) \(92\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3^{3}q^{3}-5^{3}q^{5}+92q^{7}+3^{6}q^{9}+\cdots\)
60.8.a.d 60.a 1.a $1$ $18.743$ \(\Q\) None \(0\) \(27\) \(125\) \(-1408\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{3}q^{3}+5^{3}q^{5}-1408q^{7}+3^{6}q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(60))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(60)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)