Properties

Label 512.2.a
Level $512$
Weight $2$
Character orbit 512.a
Rep. character $\chi_{512}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $7$
Sturm bound $128$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 512 = 2^{9} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 512.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(128\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(512))\).

Total New Old
Modular forms 80 16 64
Cusp forms 49 16 33
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.

\(2\)Dim.
\(+\)\(6\)
\(-\)\(10\)

Trace form

\( 16q + 16q^{9} + O(q^{10}) \) \( 16q + 16q^{9} + 16q^{25} + 16q^{49} - 32q^{65} - 32q^{73} + 16q^{81} - 32q^{89} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(512))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
512.2.a.a \(2\) \(4.088\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(0\) \(-4\) \(0\) \(0\) \(+\) \(q+(-2+\beta )q^{3}+(3-4\beta )q^{9}+(-2-3\beta )q^{11}+\cdots\)
512.2.a.b \(2\) \(4.088\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-4\) \(0\) \(+\) \(q+\beta q^{3}-2q^{5}-2\beta q^{7}-q^{9}-3\beta q^{11}+\cdots\)
512.2.a.c \(2\) \(4.088\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-8\) \(+\) \(q+\beta q^{3}-2\beta q^{5}-4q^{7}-q^{9}+\beta q^{11}+\cdots\)
512.2.a.d \(2\) \(4.088\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(8\) \(-\) \(q+\beta q^{3}+2\beta q^{5}+4q^{7}-q^{9}+\beta q^{11}+\cdots\)
512.2.a.e \(2\) \(4.088\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(4\) \(0\) \(-\) \(q+\beta q^{3}+2q^{5}+2\beta q^{7}-q^{9}-3\beta q^{11}+\cdots\)
512.2.a.f \(2\) \(4.088\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(0\) \(4\) \(0\) \(0\) \(-\) \(q+(2+\beta )q^{3}+(3+4\beta )q^{9}+(2-3\beta )q^{11}+\cdots\)
512.2.a.g \(4\) \(4.088\) \(\Q(\zeta_{24})^+\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(q+\beta _{1}q^{3}-\beta _{2}q^{5}-\beta _{3}q^{7}+3q^{9}+\beta _{1}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(512))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(512)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 2}\)