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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$		$196$	$36$	$160$		$176$	$36$	$140$		$20$	$0$	$20$
$+$	$-$	$-$		$217$	$36$	$181$		$196$	$36$	$160$		$21$	$0$	$21$
$-$	$+$	$-$		$203$	$39$	$164$		$182$	$39$	$143$		$21$	$0$	$21$
$-$	$-$	$+$		$210$	$33$	$177$		$189$	$33$	$156$		$21$	$0$	$21$
Plus space		$+$		$406$	$69$	$337$		$365$	$69$	$296$		$41$	$0$	$41$
Minus space		$-$		$420$	$75$	$345$		$378$	$75$	$303$		$42$	$0$	$42$

Trace form

144 * q + 144 * q^9 + 2 * q^11 + 2 * q^23 + 144 * q^25 - 4 * q^29 - 4 * q^37 - 12 * q^39 + 2 * q^43 - 12 * q^51 - 4 * q^53 + 8 * q^57 + 8 * q^65 + 10 * q^67 - 10 * q^71 + 10 * q^79 + 152 * q^81 + 8 * q^85 + 8 * q^93 - 12 * q^95 - 2 * q^99

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$$S_{2}^{\mathrm{new}}(\Gamma_0(49))$$^{\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}$