Properties

Label 324.2.a
Level $324$
Weight $2$
Character orbit 324.a
Rep. character $\chi_{324}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $4$
Sturm bound $108$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(108\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(324))\).

Total New Old
Modular forms 72 4 68
Cusp forms 37 4 33
Eisenstein series 35 0 35

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)FrickeDim.
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(3\)

Trace form

\( 4 q + 2 q^{7} + O(q^{10}) \) \( 4 q + 2 q^{7} + 8 q^{13} - 4 q^{19} + 16 q^{25} + 2 q^{31} + 14 q^{37} - 22 q^{43} - 18 q^{49} - 18 q^{55} - 16 q^{61} - 10 q^{67} - 22 q^{73} + 2 q^{79} + 18 q^{85} + 22 q^{91} + 2 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(324))\) 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
324.2.a.a 324.a 1.a $1$ $2.587$ \(\Q\) None \(0\) \(0\) \(-3\) \(-1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{5}-q^{7}-3q^{11}-q^{13}-6q^{17}+\cdots\)
324.2.a.b 324.a 1.a $1$ $2.587$ \(\Q\) None \(0\) \(0\) \(-3\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{5}+2q^{7}+6q^{11}+5q^{13}+3q^{17}+\cdots\)
324.2.a.c 324.a 1.a $1$ $2.587$ \(\Q\) None \(0\) \(0\) \(3\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{5}-q^{7}+3q^{11}-q^{13}+6q^{17}+\cdots\)
324.2.a.d 324.a 1.a $1$ $2.587$ \(\Q\) None \(0\) \(0\) \(3\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{5}+2q^{7}-6q^{11}+5q^{13}-3q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(324))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(324)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)