Space of modular forms of level 560, weight 4, and trivial character

• Label: 560.4.a
• Level: $560$
• Weight: $4$
• Character orbit: 560.a
• Rep. character: $\chi_{560}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $24$
• Sturm bound: $384$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(384\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(560))\).

	Total	New	Old
Modular forms	300	36	264
Cusp forms	276	36	240
Eisenstein series	24	0	24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(6\)
\(+\)	\(+\)	\(-\)	$-$	\(3\)
\(+\)	\(-\)	\(+\)	$-$	\(3\)
\(+\)	\(-\)	\(-\)	$+$	\(6\)
\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(-\)	\(+\)	\(-\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	$+$	\(5\)
\(-\)	\(-\)	\(-\)	$-$	\(4\)
Plus space			\(+\)	\(21\)
Minus space			\(-\)	\(15\)

Trace form

\( 36 q - 14 q^{7} + 284 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(560))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	5	7
560.4.a.a	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-9\)	\(5\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-9q^{3}+5q^{5}+7q^{7}+54q^{9}-55q^{11}+\cdots\)
560.4.a.b	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-8\)	\(-5\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-5q^{5}-7q^{7}+37q^{9}-28q^{11}+\cdots\)
560.4.a.c	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-7\)	\(-5\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}-5q^{5}-7q^{7}+22q^{9}+33q^{11}+\cdots\)
560.4.a.d	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-7\)	\(5\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-7q^{3}+5q^{5}-7q^{7}+22q^{9}-9q^{11}+\cdots\)
560.4.a.e	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-5\)	\(-5\)	\(7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}-5q^{5}+7q^{7}-2q^{9}+39q^{11}+\cdots\)
560.4.a.f	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-5\)	\(5\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}+5q^{5}+7q^{7}-2q^{9}+q^{11}+\cdots\)
560.4.a.g	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-4\)	\(5\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+5q^{5}-7q^{7}-11q^{9}-60q^{11}+\cdots\)
560.4.a.h	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-5\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-5q^{5}+7q^{7}-26q^{9}+7q^{11}+\cdots\)
560.4.a.i	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-5\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-5q^{5}-7q^{7}-26q^{9}+65q^{11}+\cdots\)
560.4.a.j	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(5\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+5q^{5}-7q^{7}-26q^{9}+39q^{11}+\cdots\)
560.4.a.k	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(5\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+7q^{7}-18q^{9}+17q^{11}+\cdots\)
560.4.a.l	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(4\)	\(5\)	\(-7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+5q^{5}-7q^{7}-11q^{9}-20q^{11}+\cdots\)
560.4.a.m	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(4\)	\(5\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+5q^{5}+7q^{7}-11q^{9}-68q^{11}+\cdots\)
560.4.a.n	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(5\)	\(-5\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}-5q^{5}-7q^{7}-2q^{9}-15q^{11}+\cdots\)
560.4.a.o	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(5\)	\(5\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{3}+5q^{5}-7q^{7}-2q^{9}+15q^{11}+\cdots\)
560.4.a.p	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(8\)	\(-5\)	\(-7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-5q^{5}-7q^{7}+37q^{9}-12q^{11}+\cdots\)
560.4.a.q	$560$	$4$	560.a	1.a	$1$	$1$	$1$	$33.041$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(8\)	\(-5\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-5q^{5}+7q^{7}+37q^{9}-68q^{11}+\cdots\)
560.4.a.r	$560$	$4$	560.a	1.a	$1$	$2$	$2$	$33.041$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-10\)	\(14\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-5q^{5}+7q^{7}+(6-2\beta )q^{9}+\cdots\)
560.4.a.s	$560$	$4$	560.a	1.a	$1$	$2$	$2$	$33.041$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$1$	\(0\)	\(3\)	\(-10\)	\(14\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-5q^{5}+7q^{7}+(-8+3\beta )q^{9}+\cdots\)
560.4.a.t	$560$	$4$	560.a	1.a	$1$	$3$	$3$	$33.041$	3.3.6053.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(-15\)	\(-21\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-5q^{5}-7q^{7}+(13+\cdots)q^{9}+\cdots\)
560.4.a.u	$560$	$4$	560.a	1.a	$1$	$3$	$3$	$33.041$	3.3.14360.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(15\)	\(-21\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}+5q^{5}-7q^{7}+(28+\cdots)q^{9}+\cdots\)
560.4.a.v	$560$	$4$	560.a	1.a	$1$	$3$	$3$	$33.041$	3.3.78693.1	None	✓	$1$	$0$	\(0\)	\(-2\)	\(15\)	\(21\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+5q^{5}+7q^{7}+(20+\cdots)q^{9}+\cdots\)
560.4.a.w	$560$	$4$	560.a	1.a	$1$	$3$	$3$	$33.041$	3.3.11045.1	None	✓	$1$	$0$	\(0\)	\(6\)	\(-15\)	\(-21\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}-5q^{5}-7q^{7}+(8-7\beta _{1}+\cdots)q^{9}+\cdots\)
560.4.a.x	$560$	$4$	560.a	1.a	$1$	$3$	$3$	$33.041$	3.3.11853.1	None	✓	$1$	$0$	\(0\)	\(6\)	\(15\)	\(21\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+5q^{5}+7q^{7}+(3-6\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(560))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(560)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 2}\)