Space of modular forms of level 1024, weight 4, and trivial character

• Label: 1024.4.a
• Level: $1024$
• Weight: $4$
• Character orbit: 1024.a
• Rep. character: $\chi_{1024}(1,\cdot)$
• Character field: $\Q$
• Dimension: $92$
• Newform subspaces: $16$
• Sturm bound: $512$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1024 = 2^{10} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1024.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(512\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1024))\).

	Total	New	Old
Modular forms	408	100	308
Cusp forms	360	92	268
Eisenstein series	48	8	40

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
\(+\)		\(208\)	\(52\)	\(156\)		\(184\)	\(48\)	\(136\)		\(24\)	\(4\)	\(20\)
\(-\)		\(200\)	\(48\)	\(152\)		\(176\)	\(44\)	\(132\)		\(24\)	\(4\)	\(20\)

Trace form

\( 92 q + 756 q^{9} + 8 q^{17} + 1900 q^{25} - 8 q^{33} + 3340 q^{49} + 216 q^{57} - 8 q^{65} + 4868 q^{81} - 8 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1024))\) 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
1024.4.a.a	$1024$	$4$	1024.a	1.a	$1$	$2$	$2$	$60.418$	\(\Q(\sqrt{2}) \)	None	✓				512.4.e.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-16\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4\beta q^{3}+\beta q^{5}-8q^{7}+5q^{9}+4\beta q^{11}+\cdots\)
1024.4.a.b	$1024$	$4$	1024.a	1.a	$1$	$2$	$2$	$60.418$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-1}) \)	✓				512.4.e.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+13\beta q^{5}-3^{3}q^{9}+55\beta q^{13}-104q^{17}+\cdots\)
1024.4.a.c	$1024$	$4$	1024.a	1.a	$1$	$2$	$2$	$60.418$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-1}) \)	✓				512.4.e.d	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-9\beta q^{5}-3^{3}q^{9}+37\beta q^{13}+104q^{17}+\cdots\)
1024.4.a.d	$1024$	$4$	1024.a	1.a	$1$	$2$	$2$	$60.418$	\(\Q(\sqrt{2}) \)	None	✓				512.4.e.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(16\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4\beta q^{3}-\beta q^{5}+8q^{7}+5q^{9}+4\beta q^{11}+\cdots\)
1024.4.a.e	$1024$	$4$	1024.a	1.a	$1$	$4$	$4$	$60.418$	\(\Q(\sqrt{2}, \sqrt{43})\)	None	✓				256.4.e.a	$2$	$1$	\(0\)	\(-12\)	\(0\)	\(0\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{3})q^{3}+(-4\beta _{1}+\beta _{2})q^{5}+\cdots\)
1024.4.a.f	$1024$	$4$	1024.a	1.a	$1$	$4$	$4$	$60.418$	\(\Q(\sqrt{2}, \sqrt{17})\)	None	✓				512.4.e.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-104\)		$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3\beta _{1}+\beta _{2})q^{3}+(3\beta _{1}-2\beta _{2})q^{5}+\cdots\)
1024.4.a.g	$1024$	$4$	1024.a	1.a	$1$	$4$	$4$	$60.418$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓				512.4.e.k	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-11\beta _{1}q^{5}+3\beta _{3}q^{7}+13q^{9}+\cdots\)
1024.4.a.h	$1024$	$4$	1024.a	1.a	$1$	$4$	$4$	$60.418$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓				512.4.e.l	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-7\beta _{1}q^{5}-\beta _{3}q^{7}+13q^{9}+\cdots\)
1024.4.a.i	$1024$	$4$	1024.a	1.a	$1$	$4$	$4$	$60.418$	\(\Q(\sqrt{2}, \sqrt{17})\)	None	✓				512.4.e.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(104\)		$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3\beta _{1}+\beta _{2})q^{3}+(-3\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)
1024.4.a.j	$1024$	$4$	1024.a	1.a	$1$	$4$	$4$	$60.418$	\(\Q(\sqrt{2}, \sqrt{43})\)	None	✓				256.4.e.a	$2$	$0$	\(0\)	\(12\)	\(0\)	\(0\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(3-\beta _{3})q^{3}+(-4\beta _{1}+\beta _{2})q^{5}+(5\beta _{1}+\cdots)q^{7}+\cdots\)
1024.4.a.k	$1024$	$4$	1024.a	1.a	$1$	$8$	$8$	$60.418$	8.8.\(\cdots\).3	None	✓				256.4.e.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2^{12}$	$\mathrm{SU}(2)$	\(q+\beta _{5}q^{3}+(\beta _{1}-\beta _{3}+\beta _{6})q^{5}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
1024.4.a.l	$1024$	$4$	1024.a	1.a	$1$	$8$	$8$	$60.418$	8.8.\(\cdots\).3	None	✓				256.4.e.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2^{12}$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{3}+(\beta _{1}-\beta _{3}+\beta _{6})q^{5}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
1024.4.a.m	$1024$	$4$	1024.a	1.a	$1$	$10$	$10$	$60.418$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				16.4.e.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-28\)		$-$	$-$	$2^{22}$	$\mathrm{SU}(2)$	\(q+\beta _{7}q^{3}+\beta _{4}q^{5}+(-3-\beta _{2})q^{7}+(6+\cdots)q^{9}+\cdots\)
1024.4.a.n	$1024$	$4$	1024.a	1.a	$1$	$10$	$10$	$60.418$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				16.4.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(28\)		$+$	$+$	$2^{22}$	$\mathrm{SU}(2)$	\(q+\beta _{7}q^{3}-\beta _{4}q^{5}+(3+\beta _{2})q^{7}+(6+\beta _{2}+\cdots)q^{9}+\cdots\)
1024.4.a.o	$1024$	$4$	1024.a	1.a	$1$	$12$	$12$	$60.418$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		512.4.e.q	$2$	$1$	\(0\)	\(0\)	\(-40\)	\(0\)		$-$	$-$	$2^{33}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(-3+\beta _{7})q^{5}+(\beta _{3}-\beta _{9}+\cdots)q^{7}+\cdots\)
1024.4.a.p	$1024$	$4$	1024.a	1.a	$1$	$12$	$12$	$60.418$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		512.4.e.q	$2$	$0$	\(0\)	\(0\)	\(40\)	\(0\)		$+$	$+$	$2^{33}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(3-\beta _{7})q^{5}+(-\beta _{3}+\beta _{9}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1024))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1024)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(256))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(512))\)\(^{\oplus 2}\)