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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(240))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5
240.2.a.a \(1\) \(1.916\) \(\Q\) None \(0\) \(-1\) \(-1\) \(-4\) \(+\) \(+\) \(+\) \(q-q^{3}-q^{5}-4q^{7}+q^{9}-6q^{13}+q^{15}+\cdots\)
240.2.a.b \(1\) \(1.916\) \(\Q\) None \(0\) \(-1\) \(-1\) \(4\) \(-\) \(+\) \(+\) \(q-q^{3}-q^{5}+4q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
240.2.a.c \(1\) \(1.916\) \(\Q\) None \(0\) \(-1\) \(1\) \(0\) \(+\) \(+\) \(-\) \(q-q^{3}+q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
240.2.a.d \(1\) \(1.916\) \(\Q\) None \(0\) \(1\) \(1\) \(0\) \(-\) \(-\) \(-\) \(q+q^{3}+q^{5}+q^{9}+4q^{11}-2q^{13}+\cdots\)

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ (\( 1 + T \))(\( 1 + T \))(\( 1 + T \))(\( 1 - T \))
$5$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 - T \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 7 T^{2} \))(\( 1 + 7 T^{2} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))(\( 1 - 4 T + 11 T^{2} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 - 2 T + 13 T^{2} \))(\( 1 - 6 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 - 6 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 - 4 T + 19 T^{2} \))(\( 1 + 4 T + 19 T^{2} \))
$23$ (\( 1 - 8 T + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))(\( 1 + 23 T^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))(\( 1 + 2 T + 29 T^{2} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 8 T + 31 T^{2} \))(\( 1 + 31 T^{2} \))
$37$ (\( 1 + 6 T + 37 T^{2} \))(\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( 1 + 10 T + 37 T^{2} \))
$41$ (\( 1 - 10 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 + 6 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 + 12 T + 43 T^{2} \))(\( 1 + 4 T + 43 T^{2} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))
$53$ (\( 1 - 10 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))(\( 1 + 10 T + 53 T^{2} \))
$59$ (\( 1 + 59 T^{2} \))(\( 1 + 59 T^{2} \))(\( 1 + 12 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))
$61$ (\( 1 - 6 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 - 14 T + 61 T^{2} \))(\( 1 + 2 T + 61 T^{2} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( 1 - 4 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 + 12 T + 67 T^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 8 T + 71 T^{2} \))(\( 1 - 8 T + 71 T^{2} \))
$73$ (\( 1 + 14 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 6 T + 73 T^{2} \))(\( 1 - 10 T + 73 T^{2} \))
$79$ (\( 1 + 16 T + 79 T^{2} \))(\( 1 + 8 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 79 T^{2} \))
$83$ (\( 1 + 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))
$89$ (\( 1 - 2 T + 89 T^{2} \))(\( 1 - 18 T + 89 T^{2} \))(\( 1 - 10 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))
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