↑

Properties

Label 720.2.a
Level $720$
Weight $2$
Character orbit 720.a
Rep. character $\chi_{720}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $10$
Sturm bound $288$
Trace bound $13$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(288\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(720))\).

Total New Old
Modular forms 168 10 158
Cusp forms 121 10 111
Eisenstein series 47 0 47

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
\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)$+$\(1\)
\(-\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(+\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(6\)

Trace form

\( 10 q - 6 q^{7} + O(q^{10}) \) \( 10 q - 6 q^{7} - 4 q^{11} + 4 q^{17} + 2 q^{23} + 10 q^{25} + 12 q^{29} + 12 q^{31} - 6 q^{35} + 8 q^{37} - 8 q^{41} + 10 q^{43} + 22 q^{47} - 2 q^{49} - 4 q^{55} + 16 q^{59} - 16 q^{61} - 4 q^{65} - 2 q^{67} - 12 q^{71} - 20 q^{73} + 16 q^{77} + 8 q^{79} + 14 q^{83} - 8 q^{85} - 12 q^{89} + 20 q^{91} + 8 q^{95} - 28 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(720))\) 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
720.2.a.a 720.a 1.a $1$ $5.749$ \(\Q\) None \(0\) \(0\) \(-1\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}-2q^{11}+4q^{13}-2q^{17}+\cdots\)
720.2.a.b 720.a 1.a $1$ $5.749$ \(\Q\) None \(0\) \(0\) \(-1\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}+6q^{11}-4q^{13}+6q^{17}+\cdots\)
720.2.a.c 720.a 1.a $1$ $5.749$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-4q^{11}-2q^{13}-2q^{17}-4q^{19}+\cdots\)
720.2.a.d 720.a 1.a $1$ $5.749$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-4q^{11}+6q^{13}+6q^{17}+4q^{19}+\cdots\)
720.2.a.e 720.a 1.a $1$ $5.749$ \(\Q\) None \(0\) \(0\) \(-1\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}+4q^{7}+4q^{11}-2q^{13}-2q^{17}+\cdots\)
720.2.a.f 720.a 1.a $1$ $5.749$ \(\Q\) None \(0\) \(0\) \(1\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-4q^{7}-6q^{13}+2q^{17}-4q^{19}+\cdots\)
720.2.a.g 720.a 1.a $1$ $5.749$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}-6q^{11}-4q^{13}-6q^{17}+\cdots\)
720.2.a.h 720.a 1.a $1$ $5.749$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}+2q^{13}+6q^{17}+4q^{19}+\cdots\)
720.2.a.i 720.a 1.a $1$ $5.749$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}+2q^{11}+4q^{13}+2q^{17}+\cdots\)
720.2.a.j 720.a 1.a $1$ $5.749$ \(\Q\) None \(0\) \(0\) \(1\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+4q^{7}+2q^{13}-6q^{17}+4q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(720))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(720)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)