Properties

Label 1024.2.a
Level $1024$
Weight $2$
Character orbit 1024.a
Rep. character $\chi_{1024}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $10$
Sturm bound $256$
Trace bound $17$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 1024 = 2^{10} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1024.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(256\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1024))\).

Total New Old
Modular forms 152 36 116
Cusp forms 105 28 77
Eisenstein series 47 8 39

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.
\(+\)\(12\)
\(-\)\(16\)

Trace form

\( 28 q + 20 q^{9} + O(q^{10}) \) \( 28 q + 20 q^{9} + 8 q^{17} + 12 q^{25} - 8 q^{33} + 12 q^{49} + 24 q^{57} - 8 q^{65} + 4 q^{81} - 8 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1024))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
1024.2.a.a \(2\) \(8.177\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-8\) \(-\) \(q+2\beta q^{3}+\beta q^{5}-4q^{7}+5q^{9}+2\beta q^{11}+\cdots\)
1024.2.a.b \(2\) \(8.177\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(-4\) \(+\) \(q+\beta q^{3}-\beta q^{5}-2q^{7}-q^{9}-\beta q^{11}+\cdots\)
1024.2.a.c \(2\) \(8.177\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(+\) \(q+\beta q^{5}-3q^{9}-5\beta q^{13}-8q^{17}-3q^{25}+\cdots\)
1024.2.a.d \(2\) \(8.177\) \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(-\) \(q+3\beta q^{5}-3q^{9}+\beta q^{13}+8q^{17}+13q^{25}+\cdots\)
1024.2.a.e \(2\) \(8.177\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(4\) \(-\) \(q+\beta q^{3}+\beta q^{5}+2q^{7}-q^{9}-\beta q^{11}+\cdots\)
1024.2.a.f \(2\) \(8.177\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(8\) \(-\) \(q+2\beta q^{3}-\beta q^{5}+4q^{7}+5q^{9}+2\beta q^{11}+\cdots\)
1024.2.a.g \(4\) \(8.177\) \(\Q(\zeta_{24})^+\) None \(0\) \(-4\) \(0\) \(0\) \(+\) \(q+(-1+\beta _{1})q^{3}-\beta _{3}q^{5}+(\beta _{2}+\beta _{3})q^{7}+\cdots\)
1024.2.a.h \(4\) \(8.177\) \(\Q(\zeta_{16})^+\) None \(0\) \(0\) \(-8\) \(0\) \(+\) \(q+\beta _{1}q^{3}+(-2-\beta _{2})q^{5}+(-\beta _{1}+\beta _{3})q^{7}+\cdots\)
1024.2.a.i \(4\) \(8.177\) \(\Q(\zeta_{16})^+\) None \(0\) \(0\) \(8\) \(0\) \(-\) \(q+\beta _{1}q^{3}+(2+\beta _{2})q^{5}+(\beta _{1}-\beta _{3})q^{7}+\cdots\)
1024.2.a.j \(4\) \(8.177\) \(\Q(\zeta_{24})^+\) None \(0\) \(4\) \(0\) \(0\) \(-\) \(q+(1-\beta _{1})q^{3}+\beta _{3}q^{5}+(\beta _{2}+\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1024))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1024)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(512))\)\(^{\oplus 2}\)