Properties

Label 240.2.a
Level $240$
Weight $2$
Character orbit 240.a
Rep. character $\chi_{240}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $4$
Sturm bound $96$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 240.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(96\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(240))\).

Total New Old
Modular forms 60 4 56
Cusp forms 37 4 33
Eisenstein series 23 0 23

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\(+\)\(1\)
Minus space\(-\)\(3\)

Trace form

\( 4 q - 2 q^{3} + 4 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{3} + 4 q^{9} + 8 q^{11} + 2 q^{15} + 8 q^{23} + 4 q^{25} - 2 q^{27} - 16 q^{29} - 16 q^{37} - 4 q^{39} + 8 q^{41} - 8 q^{43} - 24 q^{47} + 4 q^{49} + 4 q^{51} + 8 q^{55} - 8 q^{57} - 8 q^{59} + 8 q^{61} + 8 q^{65} - 8 q^{67} - 8 q^{69} - 8 q^{73} - 2 q^{75} - 16 q^{79} + 4 q^{81} - 24 q^{83} - 8 q^{85} + 12 q^{87} + 24 q^{89} + 32 q^{91} + 8 q^{97} + 8 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(240))\) 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
240.2.a.a 240.a 1.a $1$ $1.916$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-4q^{7}+q^{9}-6q^{13}+q^{15}+\cdots\)
240.2.a.b 240.a 1.a $1$ $1.916$ \(\Q\) None \(0\) \(-1\) \(-1\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+4q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
240.2.a.c 240.a 1.a $1$ $1.916$ \(\Q\) None \(0\) \(-1\) \(1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
240.2.a.d 240.a 1.a $1$ $1.916$ \(\Q\) None \(0\) \(1\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+q^{9}+4q^{11}-2q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(240))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(240)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)