Properties

Label 768.2.a
Level $768$
Weight $2$
Character orbit 768.a
Rep. character $\chi_{768}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $12$
Sturm bound $256$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 768.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(256\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(768))\).

Total New Old
Modular forms 152 16 136
Cusp forms 105 16 89
Eisenstein series 47 0 47

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.
\(+\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(4\)
\(-\)\(-\)\(+\)\(2\)
Plus space\(+\)\(6\)
Minus space\(-\)\(10\)

Trace form

\( 16q + 16q^{9} + O(q^{10}) \) \( 16q + 16q^{9} + 16q^{25} + 48q^{49} + 32q^{57} - 32q^{65} + 16q^{81} - 32q^{89} + 32q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(768))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
768.2.a.a \(1\) \(6.133\) \(\Q\) None \(0\) \(-1\) \(-2\) \(2\) \(+\) \(+\) \(q-q^{3}-2q^{5}+2q^{7}+q^{9}-4q^{13}+\cdots\)
768.2.a.b \(1\) \(6.133\) \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) \(+\) \(+\) \(q-q^{3}-4q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
768.2.a.c \(1\) \(6.133\) \(\Q\) None \(0\) \(-1\) \(0\) \(4\) \(-\) \(+\) \(q-q^{3}+4q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
768.2.a.d \(1\) \(6.133\) \(\Q\) None \(0\) \(-1\) \(2\) \(-2\) \(-\) \(+\) \(q-q^{3}+2q^{5}-2q^{7}+q^{9}+4q^{13}+\cdots\)
768.2.a.e \(1\) \(6.133\) \(\Q\) None \(0\) \(1\) \(-2\) \(-2\) \(-\) \(-\) \(q+q^{3}-2q^{5}-2q^{7}+q^{9}-4q^{13}+\cdots\)
768.2.a.f \(1\) \(6.133\) \(\Q\) None \(0\) \(1\) \(0\) \(-4\) \(-\) \(-\) \(q+q^{3}-4q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots\)
768.2.a.g \(1\) \(6.133\) \(\Q\) None \(0\) \(1\) \(0\) \(4\) \(+\) \(-\) \(q+q^{3}+4q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
768.2.a.h \(1\) \(6.133\) \(\Q\) None \(0\) \(1\) \(2\) \(2\) \(+\) \(-\) \(q+q^{3}+2q^{5}+2q^{7}+q^{9}+4q^{13}+\cdots\)
768.2.a.i \(2\) \(6.133\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(q-q^{3}+\beta q^{5}-\beta q^{7}+q^{9}-4q^{11}+\cdots\)
768.2.a.j \(2\) \(6.133\) \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(0\) \(-\) \(+\) \(q-q^{3}+\beta q^{5}+\beta q^{7}+q^{9}-\beta q^{15}+\cdots\)
768.2.a.k \(2\) \(6.133\) \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(0\) \(+\) \(-\) \(q+q^{3}+\beta q^{5}-\beta q^{7}+q^{9}+\beta q^{15}+\cdots\)
768.2.a.l \(2\) \(6.133\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) \(+\) \(-\) \(q+q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+4q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(768))\) into lower level spaces

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