Properties

Label 384.4.a
Level $384$
Weight $4$
Character orbit 384.a
Rep. character $\chi_{384}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $16$
Sturm bound $256$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 384.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(256\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(384))\).

Total New Old
Modular forms 208 24 184
Cusp forms 176 24 152
Eisenstein series 32 0 32

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.
\(+\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(14\)
Minus space\(-\)\(10\)

Trace form

\( 24 q + 216 q^{9} + O(q^{10}) \) \( 24 q + 216 q^{9} - 208 q^{17} + 776 q^{25} + 944 q^{41} + 2616 q^{49} + 672 q^{57} + 5024 q^{65} + 912 q^{73} + 1944 q^{81} - 528 q^{89} - 2512 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(384))\) 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
384.4.a.a 384.a 1.a $1$ $22.657$ \(\Q\) None \(0\) \(-3\) \(-8\) \(-10\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-8q^{5}-10q^{7}+9q^{9}+68q^{11}+\cdots\)
384.4.a.b 384.a 1.a $1$ $22.657$ \(\Q\) None \(0\) \(-3\) \(-4\) \(10\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-4q^{5}+10q^{7}+9q^{9}-4q^{11}+\cdots\)
384.4.a.c 384.a 1.a $1$ $22.657$ \(\Q\) None \(0\) \(-3\) \(4\) \(-10\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+4q^{5}-10q^{7}+9q^{9}-4q^{11}+\cdots\)
384.4.a.d 384.a 1.a $1$ $22.657$ \(\Q\) None \(0\) \(-3\) \(8\) \(10\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+8q^{5}+10q^{7}+9q^{9}+68q^{11}+\cdots\)
384.4.a.e 384.a 1.a $1$ $22.657$ \(\Q\) None \(0\) \(3\) \(-8\) \(10\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-8q^{5}+10q^{7}+9q^{9}-68q^{11}+\cdots\)
384.4.a.f 384.a 1.a $1$ $22.657$ \(\Q\) None \(0\) \(3\) \(-4\) \(-10\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-4q^{5}-10q^{7}+9q^{9}+4q^{11}+\cdots\)
384.4.a.g 384.a 1.a $1$ $22.657$ \(\Q\) None \(0\) \(3\) \(4\) \(10\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+4q^{5}+10q^{7}+9q^{9}+4q^{11}+\cdots\)
384.4.a.h 384.a 1.a $1$ $22.657$ \(\Q\) None \(0\) \(3\) \(8\) \(-10\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+8q^{5}-10q^{7}+9q^{9}-68q^{11}+\cdots\)
384.4.a.i 384.a 1.a $2$ $22.657$ \(\Q(\sqrt{7}) \) None \(0\) \(-6\) \(-16\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-8+\beta )q^{5}+(2+3\beta )q^{7}+\cdots\)
384.4.a.j 384.a 1.a $2$ $22.657$ \(\Q(\sqrt{15}) \) None \(0\) \(-6\) \(-8\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-4+\beta )q^{5}+(2-\beta )q^{7}+9q^{9}+\cdots\)
384.4.a.k 384.a 1.a $2$ $22.657$ \(\Q(\sqrt{15}) \) None \(0\) \(-6\) \(8\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(4+\beta )q^{5}+(-2-\beta )q^{7}+9q^{9}+\cdots\)
384.4.a.l 384.a 1.a $2$ $22.657$ \(\Q(\sqrt{7}) \) None \(0\) \(-6\) \(16\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(8+\beta )q^{5}+(-2+3\beta )q^{7}+\cdots\)
384.4.a.m 384.a 1.a $2$ $22.657$ \(\Q(\sqrt{7}) \) None \(0\) \(6\) \(-16\) \(-4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-8+\beta )q^{5}+(-2-3\beta )q^{7}+\cdots\)
384.4.a.n 384.a 1.a $2$ $22.657$ \(\Q(\sqrt{15}) \) None \(0\) \(6\) \(-8\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-4+\beta )q^{5}+(-2+\beta )q^{7}+\cdots\)
384.4.a.o 384.a 1.a $2$ $22.657$ \(\Q(\sqrt{15}) \) None \(0\) \(6\) \(8\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(4+\beta )q^{5}+(2+\beta )q^{7}+9q^{9}+\cdots\)
384.4.a.p 384.a 1.a $2$ $22.657$ \(\Q(\sqrt{7}) \) None \(0\) \(6\) \(16\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(8+\beta )q^{5}+(2-3\beta )q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(384))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(384)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 2}\)