Space of modular forms of level 1024, weight 2, and trivial character

• 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: $17$

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.

\(2\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(72\)	\(16\)	\(56\)		\(49\)	\(12\)	\(37\)		\(23\)	\(4\)	\(19\)
\(-\)		\(80\)	\(20\)	\(60\)		\(56\)	\(16\)	\(40\)		\(24\)	\(4\)	\(20\)

Trace form

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

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

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2
1024.2.a.a	$1024$	$2$	1024.a	1.a	$1$	$2$	$2$	$8.177$	\(\Q(\sqrt{2}) \)	None	✓				512.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}+\beta q^{5}-4q^{7}+5q^{9}+2\beta q^{11}+\cdots\)
1024.2.a.b	$1024$	$2$	1024.a	1.a	$1$	$2$	$2$	$8.177$	\(\Q(\sqrt{2}) \)	None	✓				16.2.e.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{5}-2q^{7}-q^{9}-\beta q^{11}+\cdots\)
1024.2.a.c	$1024$	$2$	1024.a	1.a	$1$	$2$	$2$	$8.177$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-1}) \)	✓				512.2.e.d	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+\beta q^{5}-3q^{9}-5\beta q^{13}-8q^{17}-3q^{25}+\cdots\)
1024.2.a.d	$1024$	$2$	1024.a	1.a	$1$	$2$	$2$	$8.177$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-1}) \)	✓				512.2.e.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+3\beta q^{5}-3q^{9}+\beta q^{13}+8q^{17}+13q^{25}+\cdots\)
1024.2.a.e	$1024$	$2$	1024.a	1.a	$1$	$2$	$2$	$8.177$	\(\Q(\sqrt{2}) \)	None	✓				16.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{5}+2q^{7}-q^{9}-\beta q^{11}+\cdots\)
1024.2.a.f	$1024$	$2$	1024.a	1.a	$1$	$2$	$2$	$8.177$	\(\Q(\sqrt{2}) \)	None	✓				512.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}-\beta q^{5}+4q^{7}+5q^{9}+2\beta q^{11}+\cdots\)
1024.2.a.g	$1024$	$2$	1024.a	1.a	$1$	$4$	$4$	$8.177$	\(\Q(\zeta_{24})^+\)	None	✓				256.2.e.a	$2$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-\beta _{3}q^{5}+(\beta _{2}+\beta _{3})q^{7}+\cdots\)
1024.2.a.h	$1024$	$2$	1024.a	1.a	$1$	$4$	$4$	$8.177$	\(\Q(\zeta_{16})^+\)	None	✓				512.2.e.i	$2$	$1$	\(0\)	\(0\)	\(-8\)	\(0\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-2-\beta _{2})q^{5}+(-\beta _{1}+\beta _{3})q^{7}+\cdots\)
1024.2.a.i	$1024$	$2$	1024.a	1.a	$1$	$4$	$4$	$8.177$	\(\Q(\zeta_{16})^+\)	None	✓				512.2.e.i	$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(2+\beta _{2})q^{5}+(\beta _{1}-\beta _{3})q^{7}+\cdots\)
1024.2.a.j	$1024$	$2$	1024.a	1.a	$1$	$4$	$4$	$8.177$	\(\Q(\zeta_{24})^+\)	None	✓				256.2.e.a	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+\beta _{3}q^{5}+(\beta _{2}+\beta _{3})q^{7}+\cdots\)

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

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