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Properties

Label 360.2.a
Level $360$
Weight $2$
Character orbit 360.a
Rep. character $\chi_{360}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $5$
Sturm bound $144$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(144\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(360))\).

Total New Old
Modular forms 88 5 83
Cusp forms 57 5 52
Eisenstein series 31 0 31

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.
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(4\)

Trace form

\( 5 q - q^{5} + 4 q^{7} + O(q^{10}) \) \( 5 q - q^{5} + 4 q^{7} + 6 q^{13} + 6 q^{17} + 12 q^{19} + 4 q^{23} + 5 q^{25} + 10 q^{29} - 8 q^{31} + 8 q^{35} - 2 q^{37} + 2 q^{41} - 16 q^{43} - 20 q^{47} + 5 q^{49} - 22 q^{53} - 4 q^{55} - 8 q^{59} - 10 q^{61} - 10 q^{65} - 8 q^{71} - 14 q^{73} + 16 q^{77} - 16 q^{79} + 16 q^{83} + 2 q^{85} - 6 q^{89} + 4 q^{95} - 38 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(360))\) 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
360.2.a.a $1$ $2.875$ \(\Q\) None \(0\) \(0\) \(-1\) \(-4\) $-$ $-$ $+$ \(q-q^{5}-4q^{7}-4q^{11}-2q^{13}-2q^{17}+\cdots\)
360.2.a.b $1$ $2.875$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ \(q-q^{5}+4q^{11}+6q^{13}+6q^{17}-4q^{19}+\cdots\)
360.2.a.c $1$ $2.875$ \(\Q\) None \(0\) \(0\) \(-1\) \(2\) $-$ $+$ $+$ \(q-q^{5}+2q^{7}+2q^{11}+4q^{13}-2q^{17}+\cdots\)
360.2.a.d $1$ $2.875$ \(\Q\) None \(0\) \(0\) \(1\) \(2\) $+$ $+$ $-$ \(q+q^{5}+2q^{7}-2q^{11}+4q^{13}+2q^{17}+\cdots\)
360.2.a.e $1$ $2.875$ \(\Q\) None \(0\) \(0\) \(1\) \(4\) $-$ $-$ $-$ \(q+q^{5}+4q^{7}-6q^{13}+2q^{17}+4q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(360))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(360)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)