Space of modular forms of level 1560, weight 4, and trivial character

• Label: 1560.4.a
• Level: $1560$
• Weight: $4$
• Character orbit: 1560.a
• Rep. character: $\chi_{1560}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $21$
• Sturm bound: $1344$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1560.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(1344\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1560))\).

	Total	New	Old
Modular forms	1024	72	952
Cusp forms	992	72	920
Eisenstein series	32	0	32

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

\(2\)	\(3\)	\(5\)	\(13\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)	\(5\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)	\(4\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)	\(5\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)	\(4\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(6\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(6\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(6\)
Plus space				\(+\)	\(40\)
Minus space				\(-\)	\(32\)

Trace form

\( 72 q + 648 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1560))\) 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	3	5	13
1560.4.a.a	$1560$	$4$	1560.a	1.a	$1$	$1$	$1$	$92.043$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(-5\)	\(-18\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}-18q^{7}+9q^{9}+28q^{11}+\cdots\)
1560.4.a.b	$1560$	$4$	1560.a	1.a	$1$	$1$	$1$	$92.043$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(-32\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}-2^{5}q^{7}+9q^{9}+60q^{11}+\cdots\)
1560.4.a.c	$1560$	$4$	1560.a	1.a	$1$	$1$	$1$	$92.043$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(5\)	\(10\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+10q^{7}+9q^{9}+24q^{11}+\cdots\)
1560.4.a.d	$1560$	$4$	1560.a	1.a	$1$	$1$	$1$	$92.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(2\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+2q^{7}+9q^{9}-52q^{11}+\cdots\)
1560.4.a.e	$1560$	$4$	1560.a	1.a	$1$	$1$	$1$	$92.043$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(24\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+24q^{7}+9q^{9}-8q^{11}+\cdots\)
1560.4.a.f	$1560$	$4$	1560.a	1.a	$1$	$2$	$2$	$92.043$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(-6\)	\(10\)	\(21\)		$+$	$+$	$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+(11-\beta )q^{7}+9q^{9}+\cdots\)
1560.4.a.g	$1560$	$4$	1560.a	1.a	$1$	$3$	$3$	$92.043$	3.3.18257.1	None	✓	$1$	$0$	\(0\)	\(-9\)	\(-15\)	\(29\)		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+(10+2\beta _{1}-\beta _{2})q^{7}+\cdots\)
1560.4.a.h	$1560$	$4$	1560.a	1.a	$1$	$3$	$3$	$92.043$	3.3.9153.1	None	✓	$1$	$1$	\(0\)	\(-9\)	\(15\)	\(15\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+(5-\beta _{1}+2\beta _{2})q^{7}+\cdots\)
1560.4.a.i	$1560$	$4$	1560.a	1.a	$1$	$3$	$3$	$92.043$	3.3.37133.1	None	✓	$1$	$1$	\(0\)	\(9\)	\(-15\)	\(-20\)		$+$	$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+(-7+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1560.4.a.j	$1560$	$4$	1560.a	1.a	$1$	$3$	$3$	$92.043$	3.3.2700.1	None	✓	$1$	$1$	\(0\)	\(9\)	\(-15\)	\(-15\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+(-5-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1560.4.a.k	$1560$	$4$	1560.a	1.a	$1$	$3$	$3$	$92.043$	3.3.18257.1	None	✓	$1$	$1$	\(0\)	\(9\)	\(15\)	\(-13\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+(-5+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1560.4.a.l	$1560$	$4$	1560.a	1.a	$1$	$4$	$4$	$92.043$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-12\)	\(-20\)	\(-11\)		$-$	$+$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+(-3-\beta _{3})q^{7}+9q^{9}+\cdots\)
1560.4.a.m	$1560$	$4$	1560.a	1.a	$1$	$4$	$4$	$92.043$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(12\)	\(-20\)	\(15\)		$+$	$-$	$+$	$-$	$+$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+(4+\beta _{1})q^{7}+9q^{9}+\cdots\)
1560.4.a.n	$1560$	$4$	1560.a	1.a	$1$	$4$	$4$	$92.043$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(12\)	\(20\)	\(-34\)		$+$	$-$	$-$	$-$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+(-8-\beta _{3})q^{7}+9q^{9}+\cdots\)
1560.4.a.o	$1560$	$4$	1560.a	1.a	$1$	$5$	$5$	$92.043$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-15\)	\(-25\)	\(-8\)		$+$	$+$	$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+(-2+\beta _{1})q^{7}+9q^{9}+\cdots\)
1560.4.a.p	$1560$	$4$	1560.a	1.a	$1$	$5$	$5$	$92.043$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-15\)	\(-25\)	\(8\)		$-$	$+$	$+$	$-$	$+$	$2^{5}\cdot 5$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+(2+\beta _{1})q^{7}+9q^{9}+\cdots\)
1560.4.a.q	$1560$	$4$	1560.a	1.a	$1$	$5$	$5$	$92.043$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-15\)	\(25\)	\(4\)		$+$	$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+(1-\beta _{1})q^{7}+9q^{9}+\cdots\)
1560.4.a.r	$1560$	$4$	1560.a	1.a	$1$	$5$	$5$	$92.043$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(15\)	\(25\)	\(27\)		$+$	$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+(5+\beta _{1})q^{7}+9q^{9}+\cdots\)
1560.4.a.s	$1560$	$4$	1560.a	1.a	$1$	$6$	$6$	$92.043$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-18\)	\(30\)	\(-18\)		$-$	$+$	$-$	$+$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q-3q^{3}+5q^{5}+(-3+\beta _{2})q^{7}+9q^{9}+\cdots\)
1560.4.a.t	$1560$	$4$	1560.a	1.a	$1$	$6$	$6$	$92.043$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(18\)	\(-30\)	\(-6\)		$-$	$-$	$+$	$+$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+3q^{3}-5q^{5}+(-1+\beta _{1})q^{7}+9q^{9}+\cdots\)
1560.4.a.u	$1560$	$4$	1560.a	1.a	$1$	$6$	$6$	$92.043$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(18\)	\(30\)	\(20\)		$-$	$-$	$-$	$-$	$+$	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}+(3+\beta _{2})q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1560))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1560)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(390))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(520))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(780))\)\(^{\oplus 2}\)