Space of modular forms of level 5488, weight 2, and trivial character

• Label: 5488.2.a
• Level: $5488$
• Weight: $2$
• Character orbit: 5488.a
• Rep. character: $\chi_{5488}(1,\cdot)$
• Character field: $\Q$
• Dimension: $144$
• Newform subspaces: $23$
• Sturm bound: $1568$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 5488 = 2^{4} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5488.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(1568\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5488))\).

	Total	New	Old
Modular forms	826	144	682
Cusp forms	743	144	599
Eisenstein series	83	0	83

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

\(2\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(36\)
\(+\)	\(-\)	\(-\)	\(36\)
\(-\)	\(+\)	\(-\)	\(39\)
\(-\)	\(-\)	\(+\)	\(33\)
Plus space		\(+\)	\(69\)
Minus space		\(-\)	\(75\)

Trace form

\( 144 q + 144 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5488))\) 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	7
5488.2.a.a	$5488$	$2$	5488.a	1.a	$1$	$3$	$3$	$43.822$	\(\Q(\zeta_{14})^+\)	None	✓				686.2.a.c	$1$	$1$	\(0\)	\(-5\)	\(4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{2})q^{3}+(1-\beta _{2})q^{5}+(2+\beta _{1}+\cdots)q^{9}+\cdots\)
5488.2.a.b	$5488$	$2$	5488.a	1.a	$1$	$3$	$3$	$43.822$	\(\Q(\zeta_{14})^+\)	None	✓				686.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{1}+2\beta _{2})q^{5}+(-1+\cdots)q^{9}+\cdots\)
5488.2.a.c	$5488$	$2$	5488.a	1.a	$1$	$3$	$3$	$43.822$	\(\Q(\zeta_{14})^+\)	\(\Q(\sqrt{-7}) \)	✓				343.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}+(1-5\beta _{1}+3\beta _{2})q^{11}+(-4+\cdots)q^{23}+\cdots\)
5488.2.a.d	$5488$	$2$	5488.a	1.a	$1$	$3$	$3$	$43.822$	\(\Q(\zeta_{14})^+\)	\(\Q(\sqrt{-7}) \)	✓				343.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-3q^{9}+(3+\beta _{1}+\beta _{2})q^{11}+(4-2\beta _{1}+\cdots)q^{23}+\cdots\)
5488.2.a.e	$5488$	$2$	5488.a	1.a	$1$	$3$	$3$	$43.822$	\(\Q(\zeta_{14})^+\)	None	✓				686.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{1}-2\beta _{2})q^{5}+(-1+\cdots)q^{9}+\cdots\)
5488.2.a.f	$5488$	$2$	5488.a	1.a	$1$	$3$	$3$	$43.822$	\(\Q(\zeta_{14})^+\)	None	✓				686.2.a.c	$1$	$0$	\(0\)	\(5\)	\(-4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(-1-\beta _{1})q^{5}+(3-4\beta _{1}+\cdots)q^{9}+\cdots\)
5488.2.a.g	$5488$	$2$	5488.a	1.a	$1$	$6$	$6$	$43.822$	6.6.2624293.1	None	✓				1372.2.a.a	$1$	$1$	\(0\)	\(-7\)	\(7\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{4})q^{3}+(1+\beta _{5})q^{5}+\cdots\)
5488.2.a.h	$5488$	$2$	5488.a	1.a	$1$	$6$	$6$	$43.822$	6.6.1279733.1	None	✓				343.2.a.c	$1$	$0$	\(0\)	\(-5\)	\(11\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{5})q^{3}+(1+\beta _{2}+\beta _{4}-\beta _{5})q^{5}+\cdots\)
5488.2.a.i	$5488$	$2$	5488.a	1.a	$1$	$6$	$6$	$43.822$	6.6.1229312.1	None	✓				2744.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{3}+(\beta _{1}+\beta _{3}-\beta _{5})q^{5}-q^{9}+\cdots\)
5488.2.a.j	$5488$	$2$	5488.a	1.a	$1$	$6$	$6$	$43.822$	6.6.1229312.1	None	✓				1372.2.a.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3}+2\beta _{5})q^{5}+(1+\cdots)q^{9}+\cdots\)
5488.2.a.k	$5488$	$2$	5488.a	1.a	$1$	$6$	$6$	$43.822$	6.6.1229312.1	None	✓				2744.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3}-\beta _{5})q^{5}+(1+2\beta _{2}+\cdots)q^{9}+\cdots\)
5488.2.a.l	$5488$	$2$	5488.a	1.a	$1$	$6$	$6$	$43.822$	6.6.1229312.1	None	✓				1372.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{5})q^{5}+(1+2\beta _{2})q^{9}+\cdots\)
5488.2.a.m	$5488$	$2$	5488.a	1.a	$1$	$6$	$6$	$43.822$	6.6.10910144.1	None	✓				686.2.a.f	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{5}q^{5}+(1+\beta _{2})q^{9}+(-4+\cdots)q^{11}+\cdots\)
5488.2.a.n	$5488$	$2$	5488.a	1.a	$1$	$6$	$6$	$43.822$	6.6.4456256.1	None	✓				686.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{3}-\beta _{5})q^{3}+(\beta _{1}+\beta _{3}-\beta _{5})q^{5}+\cdots\)
5488.2.a.o	$5488$	$2$	5488.a	1.a	$1$	$6$	$6$	$43.822$	6.6.35650048.1	None	✓				343.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{5}q^{5}+(3+2\beta _{2})q^{9}-\beta _{2}q^{11}+\cdots\)
5488.2.a.p	$5488$	$2$	5488.a	1.a	$1$	$6$	$6$	$43.822$	6.6.1279733.1	None	✓				343.2.a.c	$1$	$1$	\(0\)	\(5\)	\(-11\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{5})q^{3}+(-1-\beta _{2}-\beta _{4}+\beta _{5})q^{5}+\cdots\)
5488.2.a.q	$5488$	$2$	5488.a	1.a	$1$	$6$	$6$	$43.822$	6.6.2624293.1	None	✓				1372.2.a.a	$1$	$0$	\(0\)	\(7\)	\(-7\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}-\beta _{4})q^{3}+(-1-\beta _{5})q^{5}+\cdots\)
5488.2.a.r	$5488$	$2$	5488.a	1.a	$1$	$9$	$9$	$43.822$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				2744.2.a.c	$1$	$1$	\(0\)	\(-6\)	\(-5\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1+\beta _{8})q^{5}+(1+\cdots)q^{9}+\cdots\)
5488.2.a.s	$5488$	$2$	5488.a	1.a	$1$	$9$	$9$	$43.822$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				2744.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-13\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{4})q^{5}+(1+\beta _{4}+\beta _{5}+\cdots)q^{9}+\cdots\)
5488.2.a.t	$5488$	$2$	5488.a	1.a	$1$	$9$	$9$	$43.822$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				2744.2.a.d	$1$	$0$	\(0\)	\(0\)	\(13\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{4})q^{5}+(1+\beta _{4}+\beta _{5}+\cdots)q^{9}+\cdots\)
5488.2.a.u	$5488$	$2$	5488.a	1.a	$1$	$9$	$9$	$43.822$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓				2744.2.a.c	$1$	$0$	\(0\)	\(6\)	\(5\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1-\beta _{8})q^{5}+(1-2\beta _{1}+\cdots)q^{9}+\cdots\)
5488.2.a.v	$5488$	$2$	5488.a	1.a	$1$	$12$	$12$	$43.822$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		2744.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{4}-\beta _{7})q^{5}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
5488.2.a.w	$5488$	$2$	5488.a	1.a	$1$	$12$	$12$	$43.822$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		2744.2.a.h	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{10}q^{5}+(2+\beta _{2}+\beta _{3})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5488))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5488)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(343))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(686))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(784))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1372))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2744))\)\(^{\oplus 2}\)