Properties

Label 324.4.a
Level $324$
Weight $4$
Character orbit 324.a
Rep. character $\chi_{324}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $5$
Sturm bound $216$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 324.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(216\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(324))\).

Total New Old
Modular forms 180 12 168
Cusp forms 144 12 132
Eisenstein series 36 0 36

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
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(7\)
Minus space\(-\)\(5\)

Trace form

\( 12 q - 12 q^{7} + O(q^{10}) \) \( 12 q - 12 q^{7} + 42 q^{13} + 150 q^{19} + 390 q^{25} + 24 q^{31} + 42 q^{37} + 78 q^{43} + 1512 q^{49} + 1080 q^{55} + 798 q^{61} + 258 q^{67} + 1176 q^{73} - 768 q^{79} - 1890 q^{85} - 3552 q^{91} + 2418 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(324))\) 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
324.4.a.a 324.a 1.a $1$ $19.117$ \(\Q\) None 324.4.a.a \(0\) \(0\) \(-3\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{5}-4q^{7}+24q^{11}-5^{2}q^{13}+\cdots\)
324.4.a.b 324.a 1.a $1$ $19.117$ \(\Q\) None 324.4.a.a \(0\) \(0\) \(3\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{5}-4q^{7}-24q^{11}-5^{2}q^{13}+\cdots\)
324.4.a.c 324.a 1.a $3$ $19.117$ 3.3.1509.1 None 36.4.e.a \(0\) \(0\) \(-6\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{2})q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
324.4.a.d 324.a 1.a $3$ $19.117$ 3.3.1509.1 None 36.4.e.a \(0\) \(0\) \(6\) \(6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(2+\beta _{2})q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+(17+\cdots)q^{11}+\cdots\)
324.4.a.e 324.a 1.a $4$ $19.117$ \(\Q(\sqrt{3}, \sqrt{7})\) None 324.4.a.e \(0\) \(0\) \(0\) \(-16\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{5}+(-4+\beta _{3})q^{7}+(-\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(324))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(324)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)