Properties

Label 1536.2.a
Level $1536$
Weight $2$
Character orbit 1536.a
Rep. character $\chi_{1536}(1,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $14$
Sturm bound $512$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(512\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1536))\).

Total New Old
Modular forms 288 32 256
Cusp forms 225 32 193
Eisenstein series 63 0 63

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.
\(+\)\(+\)\(+\)\(6\)
\(+\)\(-\)\(-\)\(10\)
\(-\)\(+\)\(-\)\(10\)
\(-\)\(-\)\(+\)\(6\)
Plus space\(+\)\(12\)
Minus space\(-\)\(20\)

Trace form

\( 32q + 32q^{9} + O(q^{10}) \) \( 32q + 32q^{9} + 32q^{25} + 32q^{49} + 64q^{65} + 64q^{73} + 32q^{81} + 64q^{89} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1536))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3
1536.2.a.a \(2\) \(12.265\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-4\) \(0\) \(+\) \(+\) \(q-q^{3}+(-2+\beta )q^{5}+\beta q^{7}+q^{9}+(2+\cdots)q^{11}+\cdots\)
1536.2.a.b \(2\) \(12.265\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-4\) \(4\) \(-\) \(+\) \(q-q^{3}+(-2+\beta )q^{5}+(2+\beta )q^{7}+q^{9}+\cdots\)
1536.2.a.c \(2\) \(12.265\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) \(+\) \(+\) \(q-q^{3}+\beta q^{5}-\beta q^{7}+q^{9}-2q^{11}+\cdots\)
1536.2.a.d \(2\) \(12.265\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(0\) \(0\) \(-\) \(+\) \(q-q^{3}+\beta q^{5}+3\beta q^{7}+q^{9}+6q^{11}+\cdots\)
1536.2.a.e \(2\) \(12.265\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(4\) \(-4\) \(+\) \(+\) \(q-q^{3}+(2+\beta )q^{5}+(-2+\beta )q^{7}+q^{9}+\cdots\)
1536.2.a.f \(2\) \(12.265\) \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(4\) \(0\) \(-\) \(+\) \(q-q^{3}+(2+\beta )q^{5}+\beta q^{7}+q^{9}+(2+2\beta )q^{11}+\cdots\)
1536.2.a.g \(2\) \(12.265\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(-4\) \(-4\) \(-\) \(-\) \(q+q^{3}+(-2+\beta )q^{5}+(-2-\beta )q^{7}+\cdots\)
1536.2.a.h \(2\) \(12.265\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(-4\) \(0\) \(-\) \(-\) \(q+q^{3}+(-2+\beta )q^{5}-\beta q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
1536.2.a.i \(2\) \(12.265\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) \(-\) \(-\) \(q+q^{3}+\beta q^{5}-3\beta q^{7}+q^{9}-6q^{11}+\cdots\)
1536.2.a.j \(2\) \(12.265\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(0\) \(0\) \(+\) \(-\) \(q+q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+2q^{11}+\cdots\)
1536.2.a.k \(2\) \(12.265\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(4\) \(0\) \(+\) \(-\) \(q+q^{3}+(2+\beta )q^{5}-\beta q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
1536.2.a.l \(2\) \(12.265\) \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(4\) \(4\) \(+\) \(-\) \(q+q^{3}+(2+\beta )q^{5}+(2-\beta )q^{7}+q^{9}+\cdots\)
1536.2.a.m \(4\) \(12.265\) 4.4.4352.1 None \(0\) \(-4\) \(0\) \(0\) \(-\) \(+\) \(q-q^{3}+\beta _{1}q^{5}-\beta _{3}q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
1536.2.a.n \(4\) \(12.265\) 4.4.4352.1 None \(0\) \(4\) \(0\) \(0\) \(+\) \(-\) \(q+q^{3}+\beta _{1}q^{5}+\beta _{3}q^{7}+q^{9}+(2+\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1536))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1536)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(192))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(384))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(512))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(768))\)\(^{\oplus 2}\)