Properties

Label 384.2.a
Level $384$
Weight $2$
Character orbit 384.a
Rep. character $\chi_{384}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $8$
Sturm bound $128$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 384.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(128\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(384))\).

Total New Old
Modular forms 80 8 72
Cusp forms 49 8 41
Eisenstein series 31 0 31

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.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(6\)

Trace form

\( 8 q + 8 q^{9} + O(q^{10}) \) \( 8 q + 8 q^{9} + 16 q^{17} + 24 q^{25} + 16 q^{41} - 24 q^{49} - 32 q^{57} - 32 q^{65} - 80 q^{73} + 8 q^{81} - 48 q^{89} + 16 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(384))\) 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
384.2.a.a 384.a 1.a $1$ $3.066$ \(\Q\) None \(0\) \(-1\) \(-4\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{5}+2q^{7}+q^{9}+4q^{11}+\cdots\)
384.2.a.b 384.a 1.a $1$ $3.066$ \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
384.2.a.c 384.a 1.a $1$ $3.066$ \(\Q\) None \(0\) \(-1\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
384.2.a.d 384.a 1.a $1$ $3.066$ \(\Q\) None \(0\) \(-1\) \(4\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{5}-2q^{7}+q^{9}+4q^{11}+\cdots\)
384.2.a.e 384.a 1.a $1$ $3.066$ \(\Q\) None \(0\) \(1\) \(-4\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
384.2.a.f 384.a 1.a $1$ $3.066$ \(\Q\) None \(0\) \(1\) \(0\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
384.2.a.g 384.a 1.a $1$ $3.066$ \(\Q\) None \(0\) \(1\) \(0\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
384.2.a.h 384.a 1.a $1$ $3.066$ \(\Q\) None \(0\) \(1\) \(4\) \(2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{5}+2q^{7}+q^{9}-4q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(384))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(384)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)