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Properties

Label 180.2.a
Level $180$
Weight $2$
Character orbit 180.a
Rep. character $\chi_{180}(1,\cdot)$
Character field $\Q$
Dimension $1$
Newform subspaces $1$
Sturm bound $72$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 180.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(72\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(180))\).

Total New Old
Modular forms 48 1 47
Cusp forms 25 1 24
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
\(-\)\(-\)\(-\)$-$\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(1\)

Trace form

\( q + q^{5} + 2 q^{7} + O(q^{10}) \) \( q + q^{5} + 2 q^{7} + 2 q^{13} + 6 q^{17} - 4 q^{19} - 6 q^{23} + q^{25} - 6 q^{29} - 4 q^{31} + 2 q^{35} + 2 q^{37} - 6 q^{41} - 10 q^{43} + 6 q^{47} - 3 q^{49} + 6 q^{53} - 12 q^{59} + 2 q^{61} + 2 q^{65} + 2 q^{67} + 12 q^{71} + 2 q^{73} + 8 q^{79} - 6 q^{83} + 6 q^{85} + 6 q^{89} + 4 q^{91} - 4 q^{95} + 2 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(180))\) 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
180.2.a.a 180.a 1.a $1$ $1.437$ \(\Q\) None \(0\) \(0\) \(1\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+2q^{7}+2q^{13}+6q^{17}-4q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(180))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(180)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)