Properties

Label 120.4.a
Level $120$
Weight $4$
Character orbit 120.a
Rep. character $\chi_{120}(1,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $6$
Sturm bound $96$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 120.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(96\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(120))\).

Total New Old
Modular forms 80 6 74
Cusp forms 64 6 58
Eisenstein series 16 0 16

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
\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(-\)\(-\)$+$\(1\)
\(-\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(+\)$+$\(1\)
Plus space\(+\)\(4\)
Minus space\(-\)\(2\)

Trace form

\( 6 q - 6 q^{3} + 36 q^{7} + 54 q^{9} + O(q^{10}) \) \( 6 q - 6 q^{3} + 36 q^{7} + 54 q^{9} + 28 q^{11} + 16 q^{13} + 8 q^{17} + 112 q^{19} + 60 q^{21} + 152 q^{23} + 150 q^{25} - 54 q^{27} - 280 q^{29} + 8 q^{31} + 132 q^{33} - 260 q^{35} + 568 q^{37} + 432 q^{39} - 244 q^{41} - 120 q^{43} - 1168 q^{47} - 922 q^{49} + 120 q^{51} - 1264 q^{53} - 180 q^{55} + 144 q^{57} + 12 q^{59} + 300 q^{61} + 324 q^{63} + 420 q^{65} - 64 q^{67} - 600 q^{69} - 1784 q^{71} + 196 q^{73} - 150 q^{75} + 96 q^{77} + 1784 q^{79} + 486 q^{81} + 672 q^{83} + 340 q^{85} + 408 q^{87} + 228 q^{89} + 8 q^{91} + 888 q^{93} - 40 q^{95} - 916 q^{97} + 252 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(120))\) 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
120.4.a.a 120.a 1.a $1$ $7.080$ \(\Q\) None \(0\) \(-3\) \(-5\) \(4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}+4q^{7}+9q^{9}+72q^{11}+\cdots\)
120.4.a.b 120.a 1.a $1$ $7.080$ \(\Q\) None \(0\) \(-3\) \(-5\) \(20\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-5q^{5}+20q^{7}+9q^{9}-56q^{11}+\cdots\)
120.4.a.c 120.a 1.a $1$ $7.080$ \(\Q\) None \(0\) \(-3\) \(5\) \(-16\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}-2^{4}q^{7}+9q^{9}-28q^{11}+\cdots\)
120.4.a.d 120.a 1.a $1$ $7.080$ \(\Q\) None \(0\) \(-3\) \(5\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}+5q^{5}+9q^{9}+4q^{11}+54q^{13}+\cdots\)
120.4.a.e 120.a 1.a $1$ $7.080$ \(\Q\) None \(0\) \(3\) \(-5\) \(20\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-5q^{5}+20q^{7}+9q^{9}+2^{4}q^{11}+\cdots\)
120.4.a.f 120.a 1.a $1$ $7.080$ \(\Q\) None \(0\) \(3\) \(5\) \(8\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+5q^{5}+8q^{7}+9q^{9}+20q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(120))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(120)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)