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.

\(2\)\(3\)\(5\)FrickeTotalCuspEisenstein
AllNewOldAllNewOldAllNewOld
\(+\)\(+\)\(+\)\(+\)\(10\)\(0\)\(10\)\(7\)\(0\)\(7\)\(3\)\(0\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(12\)\(1\)\(11\)\(8\)\(1\)\(7\)\(4\)\(0\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(12\)\(1\)\(11\)\(8\)\(1\)\(7\)\(4\)\(0\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(10\)\(0\)\(10\)\(6\)\(0\)\(6\)\(4\)\(0\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(12\)\(1\)\(11\)\(8\)\(1\)\(7\)\(4\)\(0\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(10\)\(0\)\(10\)\(6\)\(0\)\(6\)\(4\)\(0\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(10\)\(1\)\(9\)\(6\)\(1\)\(5\)\(4\)\(0\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(12\)\(1\)\(11\)\(8\)\(1\)\(7\)\(4\)\(0\)\(4\)
Plus space\(+\)\(40\)\(1\)\(39\)\(25\)\(1\)\(24\)\(15\)\(0\)\(15\)
Minus space\(-\)\(48\)\(4\)\(44\)\(32\)\(4\)\(28\)\(16\)\(0\)\(16\)

Trace form

\( 5 q - q^{5} + 4 q^{7} + 6 q^{13} + 6 q^{17} + 12 q^{19} + 4 q^{23} + 5 q^{25} + 10 q^{29} - 8 q^{31} + 8 q^{35} - 2 q^{37} + 2 q^{41} - 16 q^{43} - 20 q^{47} + 5 q^{49} - 22 q^{53} - 4 q^{55} - 8 q^{59}+ \cdots - 38 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(360))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5
360.2.a.a 360.a 1.a $1$ $2.875$ \(\Q\) None 40.2.a.a \(0\) \(0\) \(-1\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-4q^{7}-4q^{11}-2q^{13}-2q^{17}+\cdots\)
360.2.a.b 360.a 1.a $1$ $2.875$ \(\Q\) None 120.2.a.b \(0\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{11}+6q^{13}+6q^{17}-4q^{19}+\cdots\)
360.2.a.c 360.a 1.a $1$ $2.875$ \(\Q\) None 360.2.a.c \(0\) \(0\) \(-1\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}+2q^{11}+4q^{13}-2q^{17}+\cdots\)
360.2.a.d 360.a 1.a $1$ $2.875$ \(\Q\) None 360.2.a.c \(0\) \(0\) \(1\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+2q^{7}-2q^{11}+4q^{13}+2q^{17}+\cdots\)
360.2.a.e 360.a 1.a $1$ $2.875$ \(\Q\) None 120.2.a.a \(0\) \(0\) \(1\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+4q^{7}-6q^{13}+2q^{17}+4q^{19}+\cdots\)

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

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