Space of modular forms of level 560, weight 6, and trivial character

• Label: 560.6.a
• Level: $560$
• Weight: $6$
• Character orbit: 560.a
• Rep. character: $\chi_{560}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $26$
• Sturm bound: $576$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(576\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(560))\).

	Total	New	Old
Modular forms	492	60	432
Cusp forms	468	60	408
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
\(+\)	\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(9\)
\(+\)	\(-\)	\(+\)	\(-\)	\(9\)
\(+\)	\(-\)	\(-\)	\(+\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)	\(7\)
\(-\)	\(+\)	\(-\)	\(+\)	\(8\)
\(-\)	\(-\)	\(+\)	\(+\)	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)	\(8\)
Plus space			\(+\)	\(27\)
Minus space			\(-\)	\(33\)

Trace form

\( 60 q + 98 q^{7} + 4772 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(560))\) 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	5	7
560.6.a.a	$560$	$6$	560.a	1.a	$1$	$1$	$1$	$89.815$	\(\Q\)	None	✓				70.6.a.d	$1$	$1$	\(0\)	\(-11\)	\(-25\)	\(49\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-11q^{3}-5^{2}q^{5}+7^{2}q^{7}-122q^{9}+\cdots\)
560.6.a.b	$560$	$6$	560.a	1.a	$1$	$1$	$1$	$89.815$	\(\Q\)	None	✓				280.6.a.b	$1$	$1$	\(0\)	\(-4\)	\(25\)	\(49\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+5^{2}q^{5}+7^{2}q^{7}-227q^{9}+\cdots\)
560.6.a.c	$560$	$6$	560.a	1.a	$1$	$1$	$1$	$89.815$	\(\Q\)	None	✓				35.6.a.a	$1$	$1$	\(0\)	\(-1\)	\(25\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+5^{2}q^{5}-7^{2}q^{7}-242q^{9}+453q^{11}+\cdots\)
560.6.a.d	$560$	$6$	560.a	1.a	$1$	$1$	$1$	$89.815$	\(\Q\)	None	✓				70.6.a.c	$1$	$0$	\(0\)	\(3\)	\(-25\)	\(-49\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5^{2}q^{5}-7^{2}q^{7}-234q^{9}+\cdots\)
560.6.a.e	$560$	$6$	560.a	1.a	$1$	$1$	$1$	$89.815$	\(\Q\)	None	✓				70.6.a.b	$1$	$0$	\(0\)	\(9\)	\(25\)	\(49\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+9q^{3}+5^{2}q^{5}+7^{2}q^{7}-162q^{9}+\cdots\)
560.6.a.f	$560$	$6$	560.a	1.a	$1$	$1$	$1$	$89.815$	\(\Q\)	None	✓				70.6.a.f	$1$	$0$	\(0\)	\(11\)	\(-25\)	\(-49\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+11q^{3}-5^{2}q^{5}-7^{2}q^{7}-122q^{9}+\cdots\)
560.6.a.g	$560$	$6$	560.a	1.a	$1$	$1$	$1$	$89.815$	\(\Q\)	None	✓				280.6.a.a	$1$	$1$	\(0\)	\(12\)	\(25\)	\(49\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+12q^{3}+5^{2}q^{5}+7^{2}q^{7}-99q^{9}+\cdots\)
560.6.a.h	$560$	$6$	560.a	1.a	$1$	$1$	$1$	$89.815$	\(\Q\)	None	✓				70.6.a.e	$1$	$0$	\(0\)	\(17\)	\(25\)	\(49\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+17q^{3}+5^{2}q^{5}+7^{2}q^{7}+46q^{9}+\cdots\)
560.6.a.i	$560$	$6$	560.a	1.a	$1$	$1$	$1$	$89.815$	\(\Q\)	None	✓				70.6.a.a	$1$	$1$	\(0\)	\(23\)	\(25\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+23q^{3}+5^{2}q^{5}-7^{2}q^{7}+286q^{9}+\cdots\)
560.6.a.j	$560$	$6$	560.a	1.a	$1$	$2$	$2$	$89.815$	\(\Q(\sqrt{37}) \)	None	✓				280.6.a.d	$1$	$1$	\(0\)	\(-26\)	\(50\)	\(98\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-13-\beta )q^{3}+5^{2}q^{5}+7^{2}q^{7}+(74+\cdots)q^{9}+\cdots\)
560.6.a.k	$560$	$6$	560.a	1.a	$1$	$2$	$2$	$89.815$	\(\Q(\sqrt{1129}) \)	None	✓				70.6.a.h	$1$	$1$	\(0\)	\(-5\)	\(50\)	\(-98\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{3}+5^{2}q^{5}-7^{2}q^{7}+(43+\cdots)q^{9}+\cdots\)
560.6.a.l	$560$	$6$	560.a	1.a	$1$	$2$	$2$	$89.815$	\(\Q(\sqrt{65}) \)	None	✓				35.6.a.b	$1$	$1$	\(0\)	\(-3\)	\(-50\)	\(98\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3\beta q^{3}-5^{2}q^{5}+7^{2}q^{7}+(-99+9\beta )q^{9}+\cdots\)
560.6.a.m	$560$	$6$	560.a	1.a	$1$	$2$	$2$	$89.815$	\(\Q(\sqrt{3369}) \)	None	✓				70.6.a.g	$1$	$1$	\(0\)	\(-3\)	\(-50\)	\(98\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}-5^{2}q^{5}+7^{2}q^{7}+(600+\cdots)q^{9}+\cdots\)
560.6.a.n	$560$	$6$	560.a	1.a	$1$	$2$	$2$	$89.815$	\(\Q(\sqrt{109}) \)	None	✓				280.6.a.c	$1$	$1$	\(0\)	\(-2\)	\(50\)	\(98\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+5^{2}q^{5}+7^{2}q^{7}+(194+\cdots)q^{9}+\cdots\)
560.6.a.o	$560$	$6$	560.a	1.a	$1$	$2$	$2$	$89.815$	\(\Q(\sqrt{1009}) \)	None	✓				140.6.a.b	$1$	$0$	\(0\)	\(17\)	\(50\)	\(98\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(9-\beta )q^{3}+5^{2}q^{5}+7^{2}q^{7}+(90-17\beta )q^{9}+\cdots\)
560.6.a.p	$560$	$6$	560.a	1.a	$1$	$2$	$2$	$89.815$	\(\Q(\sqrt{1009}) \)	None	✓				140.6.a.a	$1$	$0$	\(0\)	\(23\)	\(-50\)	\(-98\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(12-\beta )q^{3}-5^{2}q^{5}-7^{2}q^{7}+(153+\cdots)q^{9}+\cdots\)
560.6.a.q	$560$	$6$	560.a	1.a	$1$	$3$	$3$	$89.815$	3.3.577880.1	None	✓		✓		35.6.a.c	$1$	$0$	\(0\)	\(-26\)	\(-75\)	\(-147\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-9-\beta _{2})q^{3}-5^{2}q^{5}-7^{2}q^{7}+(166+\cdots)q^{9}+\cdots\)
560.6.a.r	$560$	$6$	560.a	1.a	$1$	$3$	$3$	$89.815$	3.3.996509.1	None	✓				280.6.a.f	$1$	$1$	\(0\)	\(-14\)	\(-75\)	\(-147\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-5-\beta _{2})q^{3}-5^{2}q^{5}-7^{2}q^{7}+(4+\cdots)q^{9}+\cdots\)
560.6.a.s	$560$	$6$	560.a	1.a	$1$	$3$	$3$	$89.815$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				280.6.a.e	$1$	$1$	\(0\)	\(-6\)	\(-75\)	\(-147\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}-5^{2}q^{5}-7^{2}q^{7}+(70+\cdots)q^{9}+\cdots\)
560.6.a.t	$560$	$6$	560.a	1.a	$1$	$3$	$3$	$89.815$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		140.6.a.d	$1$	$1$	\(0\)	\(-6\)	\(75\)	\(-147\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{3}+5^{2}q^{5}-7^{2}q^{7}+(94+\cdots)q^{9}+\cdots\)
560.6.a.u	$560$	$6$	560.a	1.a	$1$	$3$	$3$	$89.815$	3.3.3101016.1	None	✓		✓		140.6.a.c	$1$	$1$	\(0\)	\(10\)	\(-75\)	\(147\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}-5^{2}q^{5}+7^{2}q^{7}+(-23+\cdots)q^{9}+\cdots\)
560.6.a.v	$560$	$6$	560.a	1.a	$1$	$4$	$4$	$89.815$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		35.6.a.d	$1$	$0$	\(0\)	\(-14\)	\(100\)	\(196\)		$-$	$-$	$-$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{3}+5^{2}q^{5}+7^{2}q^{7}+(14^{2}+\cdots)q^{9}+\cdots\)
560.6.a.w	$560$	$6$	560.a	1.a	$1$	$4$	$4$	$89.815$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				280.6.a.h	$1$	$0$	\(0\)	\(-5\)	\(-100\)	\(196\)		$+$	$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}-5^{2}q^{5}+7^{2}q^{7}+(142+\cdots)q^{9}+\cdots\)
560.6.a.x	$560$	$6$	560.a	1.a	$1$	$4$	$4$	$89.815$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				280.6.a.g	$1$	$0$	\(0\)	\(13\)	\(100\)	\(-196\)		$+$	$-$	$+$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+5^{2}q^{5}-7^{2}q^{7}+(-7^{2}+\cdots)q^{9}+\cdots\)
560.6.a.y	$560$	$6$	560.a	1.a	$1$	$5$	$5$	$89.815$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		280.6.a.j	$1$	$0$	\(0\)	\(-15\)	\(125\)	\(-245\)		$+$	$-$	$+$	$-$	$2^{8}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{3}+5^{2}q^{5}-7^{2}q^{7}+(226+\cdots)q^{9}+\cdots\)
560.6.a.z	$560$	$6$	560.a	1.a	$1$	$5$	$5$	$89.815$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		280.6.a.i	$1$	$0$	\(0\)	\(3\)	\(-125\)	\(245\)		$+$	$+$	$-$	$-$	$2^{8}\cdot 3$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-5^{2}q^{5}+7^{2}q^{7}+(74+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(560))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(560)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 2}\)