Properties

Label 396.2.a
Level $396$
Weight $2$
Character orbit 396.a
Rep. character $\chi_{396}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
Sturm bound $144$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 396.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(144\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(396))\).

Total New Old
Modular forms 84 3 81
Cusp forms 61 3 58
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.

\(2\)\(3\)\(11\)FrickeDim.
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(2\)

Trace form

\( 3 q - q^{5} + 2 q^{7} + O(q^{10}) \) \( 3 q - q^{5} + 2 q^{7} + q^{11} - 6 q^{17} + 11 q^{23} + 2 q^{25} + 8 q^{29} - 3 q^{31} + 6 q^{35} + 3 q^{37} - 8 q^{41} - 2 q^{43} + 8 q^{47} - 9 q^{49} - 6 q^{53} + 3 q^{55} - 3 q^{59} - 8 q^{61} - 20 q^{65} + 15 q^{67} - 23 q^{71} + 8 q^{73} + 6 q^{77} + 2 q^{79} - 38 q^{83} - 18 q^{85} + 37 q^{89} + 8 q^{91} + 40 q^{95} - 11 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(396))\) 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 11
396.2.a.a 396.a 1.a $1$ $3.162$ \(\Q\) None \(0\) \(0\) \(-2\) \(-2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}-2q^{7}-q^{11}-2q^{13}-4q^{17}+\cdots\)
396.2.a.b 396.a 1.a $1$ $3.162$ \(\Q\) None \(0\) \(0\) \(-2\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}+2q^{7}+q^{11}+6q^{13}+4q^{17}+\cdots\)
396.2.a.c 396.a 1.a $1$ $3.162$ \(\Q\) None \(0\) \(0\) \(3\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{5}+2q^{7}+q^{11}-4q^{13}-6q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(396))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(396)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(66))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(99))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(132))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(198))\)\(^{\oplus 2}\)