Properties

Label 768.4.a
Level $768$
Weight $4$
Character orbit 768.a
Rep. character $\chi_{768}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $22$
Sturm bound $512$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(512\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(768))\).

Total New Old
Modular forms 408 48 360
Cusp forms 360 48 312
Eisenstein series 48 0 48

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
\(+\)\(+\)\(+\)\(12\)
\(+\)\(-\)\(-\)\(10\)
\(-\)\(+\)\(-\)\(12\)
\(-\)\(-\)\(+\)\(14\)
Plus space\(+\)\(26\)
Minus space\(-\)\(22\)

Trace form

\( 48 q + 432 q^{9} + O(q^{10}) \) \( 48 q + 432 q^{9} + 1200 q^{25} + 912 q^{49} - 672 q^{57} + 1952 q^{65} + 2048 q^{73} + 3888 q^{81} - 352 q^{89} + 3168 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(768))\) 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
768.4.a.a 768.a 1.a $1$ $45.313$ \(\Q\) None 384.4.d.a \(0\) \(-3\) \(-8\) \(12\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-8q^{5}+12q^{7}+9q^{9}+12q^{11}+\cdots\)
768.4.a.b 768.a 1.a $1$ $45.313$ \(\Q\) None 384.4.d.a \(0\) \(-3\) \(8\) \(-12\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+8q^{5}-12q^{7}+9q^{9}+12q^{11}+\cdots\)
768.4.a.c 768.a 1.a $1$ $45.313$ \(\Q\) None 384.4.d.a \(0\) \(3\) \(-8\) \(-12\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-8q^{5}-12q^{7}+9q^{9}-12q^{11}+\cdots\)
768.4.a.d 768.a 1.a $1$ $45.313$ \(\Q\) None 384.4.d.a \(0\) \(3\) \(8\) \(12\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+8q^{5}+12q^{7}+9q^{9}-12q^{11}+\cdots\)
768.4.a.e 768.a 1.a $2$ $45.313$ \(\Q(\sqrt{13}) \) None 384.4.d.c \(0\) \(-6\) \(-8\) \(-16\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-4-\beta )q^{5}+(-8-\beta )q^{7}+\cdots\)
768.4.a.f 768.a 1.a $2$ $45.313$ \(\Q(\sqrt{3}) \) None 192.4.d.c \(0\) \(-6\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+\beta q^{5}+7\beta q^{7}+9q^{9}-48q^{11}+\cdots\)
768.4.a.g 768.a 1.a $2$ $45.313$ \(\Q(\sqrt{3}) \) None 192.4.d.a \(0\) \(-6\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+3\beta q^{5}+\beta q^{7}+9q^{9}-2^{4}\beta q^{13}+\cdots\)
768.4.a.h 768.a 1.a $2$ $45.313$ \(\Q(\sqrt{2}) \) None 384.4.d.d \(0\) \(-6\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+\beta q^{5}+5\beta q^{7}+9q^{9}+20q^{11}+\cdots\)
768.4.a.i 768.a 1.a $2$ $45.313$ \(\Q(\sqrt{11}) \) None 192.4.d.b \(0\) \(-6\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+\beta q^{5}-\beta q^{7}+9q^{9}+48q^{11}+\cdots\)
768.4.a.j 768.a 1.a $2$ $45.313$ \(\Q(\sqrt{13}) \) None 384.4.d.c \(0\) \(-6\) \(8\) \(16\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(4+\beta )q^{5}+(8+\beta )q^{7}+9q^{9}+\cdots\)
768.4.a.k 768.a 1.a $2$ $45.313$ \(\Q(\sqrt{13}) \) None 384.4.d.c \(0\) \(6\) \(-8\) \(16\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-4-\beta )q^{5}+(8+\beta )q^{7}+9q^{9}+\cdots\)
768.4.a.l 768.a 1.a $2$ $45.313$ \(\Q(\sqrt{11}) \) None 192.4.d.b \(0\) \(6\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+\beta q^{5}+\beta q^{7}+9q^{9}-48q^{11}+\cdots\)
768.4.a.m 768.a 1.a $2$ $45.313$ \(\Q(\sqrt{2}) \) None 384.4.d.d \(0\) \(6\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+\beta q^{5}-5\beta q^{7}+9q^{9}-20q^{11}+\cdots\)
768.4.a.n 768.a 1.a $2$ $45.313$ \(\Q(\sqrt{3}) \) None 192.4.d.a \(0\) \(6\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+3\beta q^{5}-\beta q^{7}+9q^{9}-2^{4}\beta q^{13}+\cdots\)
768.4.a.o 768.a 1.a $2$ $45.313$ \(\Q(\sqrt{3}) \) None 192.4.d.c \(0\) \(6\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+\beta q^{5}-7\beta q^{7}+9q^{9}+48q^{11}+\cdots\)
768.4.a.p 768.a 1.a $2$ $45.313$ \(\Q(\sqrt{13}) \) None 384.4.d.c \(0\) \(6\) \(8\) \(-16\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(4+\beta )q^{5}+(-8-\beta )q^{7}+9q^{9}+\cdots\)
768.4.a.q 768.a 1.a $3$ $45.313$ 3.3.1436.1 None 24.4.d.a \(0\) \(-9\) \(-10\) \(14\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-3-\beta _{2})q^{5}+(5-\beta _{1})q^{7}+\cdots\)
768.4.a.r 768.a 1.a $3$ $45.313$ 3.3.1436.1 None 24.4.d.a \(0\) \(-9\) \(10\) \(-14\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(3+\beta _{2})q^{5}+(-5+\beta _{1})q^{7}+\cdots\)
768.4.a.s 768.a 1.a $3$ $45.313$ 3.3.1436.1 None 24.4.d.a \(0\) \(9\) \(-10\) \(-14\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(-3-\beta _{2})q^{5}+(-5+\beta _{1}+\cdots)q^{7}+\cdots\)
768.4.a.t 768.a 1.a $3$ $45.313$ 3.3.1436.1 None 24.4.d.a \(0\) \(9\) \(10\) \(14\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(3+\beta _{2})q^{5}+(5-\beta _{1})q^{7}+9q^{9}+\cdots\)
768.4.a.u 768.a 1.a $4$ $45.313$ 4.4.9792.1 None 384.4.d.f \(0\) \(-12\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+\beta _{1}q^{5}+(-\beta _{1}+\beta _{3})q^{7}+9q^{9}+\cdots\)
768.4.a.v 768.a 1.a $4$ $45.313$ 4.4.9792.1 None 384.4.d.f \(0\) \(12\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+\beta _{1}q^{5}+(\beta _{1}-\beta _{3})q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(768))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(768)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 2}\)