Space of modular forms of level 5580, weight 2, and trivial character

• Label: 5580.2.a
• Level: $5580$
• Weight: $2$
• Character orbit: 5580.a
• Rep. character: $\chi_{5580}(1,\cdot)$
• Character field: $\Q$
• Dimension: $50$
• Newform subspaces: $18$
• Sturm bound: $2304$
• Trace bound: $23$

Defining parameters

Level:	$N$	$=$	$5580 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 31$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	5580.a (trivial)
Character field:			$\Q$
Newform subspaces:			$18$
Sturm bound:			$2304$
Trace bound:			$23$
Distinguishing $T_p$:			$7$, $11$, $23$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(5580))$.

	Total	New	Old
Modular forms	1176	50	1126
Cusp forms	1129	50	1079
Eisenstein series	47	0	47

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

$2$	$3$	$5$	$31$	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$	$+$	$+$		$62$	$0$	$62$		$59$	$0$	$59$		$3$	$0$	$3$
$+$	$+$	$+$	$-$	$-$		$86$	$0$	$86$		$82$	$0$	$82$		$4$	$0$	$4$
$+$	$+$	$-$	$+$	$-$		$74$	$0$	$74$		$70$	$0$	$70$		$4$	$0$	$4$
$+$	$+$	$-$	$-$	$+$		$74$	$0$	$74$		$70$	$0$	$70$		$4$	$0$	$4$
$+$	$-$	$+$	$+$	$-$		$80$	$0$	$80$		$76$	$0$	$76$		$4$	$0$	$4$
$+$	$-$	$+$	$-$	$+$		$68$	$0$	$68$		$64$	$0$	$64$		$4$	$0$	$4$
$+$	$-$	$-$	$+$	$+$		$80$	$0$	$80$		$76$	$0$	$76$		$4$	$0$	$4$
$+$	$-$	$-$	$-$	$-$		$68$	$0$	$68$		$64$	$0$	$64$		$4$	$0$	$4$
$-$	$+$	$+$	$+$	$-$		$70$	$5$	$65$		$68$	$5$	$63$		$2$	$0$	$2$
$-$	$+$	$+$	$-$	$+$		$76$	$5$	$71$		$74$	$5$	$69$		$2$	$0$	$2$
$-$	$+$	$-$	$+$	$+$		$70$	$5$	$65$		$68$	$5$	$63$		$2$	$0$	$2$
$-$	$+$	$-$	$-$	$-$		$76$	$5$	$71$		$74$	$5$	$69$		$2$	$0$	$2$
$-$	$-$	$+$	$+$	$+$		$76$	$7$	$69$		$74$	$7$	$67$		$2$	$0$	$2$
$-$	$-$	$+$	$-$	$-$		$70$	$9$	$61$		$68$	$9$	$59$		$2$	$0$	$2$
$-$	$-$	$-$	$+$	$-$		$76$	$8$	$68$		$74$	$8$	$66$		$2$	$0$	$2$
$-$	$-$	$-$	$-$	$+$		$70$	$6$	$64$		$68$	$6$	$62$		$2$	$0$	$2$
Plus space				$+$		$576$	$23$	$553$		$553$	$23$	$530$		$23$	$0$	$23$
Minus space				$-$		$600$	$27$	$573$		$576$	$27$	$549$		$24$	$0$	$24$

Trace form

50 * q - 2 * q^5 + 4 * q^11 - 8 * q^17 + 12 * q^19 - 4 * q^23 + 50 * q^25 + 8 * q^29 - 20 * q^37 - 4 * q^41 + 8 * q^43 - 28 * q^47 + 50 * q^49 - 12 * q^53 + 8 * q^55 + 8 * q^59 + 12 * q^61 + 4 * q^65 + 16 * q^67 - 28 * q^71 + 4 * q^73 - 12 * q^77 - 20 * q^79 + 4 * q^83 - 12 * q^85 - 40 * q^89 + 4 * q^95 - 12 * q^97

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(5580))$ 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	3	5	31
5580.2.a.a	$5580$	$2$	5580.a	1.a	$1$	$1$	$1$	$44.557$	$\Q$	None	✓				620.2.a.a	$1$	$1$	$0$	$0$	$-1$	$-2$		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{5}-2q^{7}-2q^{11}+2q^{13}+3q^{17}+\cdots$
5580.2.a.b	$5580$	$2$	5580.a	1.a	$1$	$1$	$1$	$44.557$	$\Q$	None	✓				1860.2.a.b	$1$	$0$	$0$	$0$	$-1$	$-2$		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{5}-2q^{7}+4q^{11}-4q^{13}+q^{25}+\cdots$
5580.2.a.c	$5580$	$2$	5580.a	1.a	$1$	$1$	$1$	$44.557$	$\Q$	None	✓				620.2.a.b	$1$	$1$	$0$	$0$	$-1$	$-2$		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{5}-2q^{7}+4q^{11}-4q^{13}+4q^{23}+\cdots$
5580.2.a.d	$5580$	$2$	5580.a	1.a	$1$	$1$	$1$	$44.557$	$\Q$	None	✓				1860.2.a.a	$1$	$1$	$0$	$0$	$-1$	$0$		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{5}-2q^{13}+4q^{17}-4q^{19}-4q^{23}+\cdots$
5580.2.a.e	$5580$	$2$	5580.a	1.a	$1$	$1$	$1$	$44.557$	$\Q$	None	✓				1860.2.a.c	$1$	$1$	$0$	$0$	$-1$	$2$		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{5}+2q^{7}+4q^{11}+4q^{13}-4q^{17}+\cdots$
5580.2.a.f	$5580$	$2$	5580.a	1.a	$1$	$1$	$1$	$44.557$	$\Q$	None	✓				620.2.a.c	$1$	$1$	$0$	$0$	$1$	$-4$		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{5}-4q^{7}+2q^{13}+3q^{17}-7q^{19}+\cdots$
5580.2.a.g	$5580$	$2$	5580.a	1.a	$1$	$2$	$2$	$44.557$	$\Q(\sqrt{6})$	None	✓				1860.2.a.d	$1$	$1$	$0$	$0$	$2$	$0$		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{5}+\beta q^{7}+(-2-\beta )q^{13}-2q^{17}+\cdots$
5580.2.a.h	$5580$	$2$	5580.a	1.a	$1$	$2$	$2$	$44.557$	$\Q(\sqrt{3})$	None	✓				1860.2.a.e	$1$	$0$	$0$	$0$	$2$	$2$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{5}+(1+\beta )q^{7}+4q^{11}+(-1-3\beta )q^{13}+\cdots$
5580.2.a.i	$5580$	$2$	5580.a	1.a	$1$	$3$	$3$	$44.557$	3.3.7636.1	None	✓		✓		1860.2.a.h	$1$	$1$	$0$	$0$	$-3$	$0$		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{5}-\beta _{1}q^{7}-2q^{11}+\beta _{1}q^{13}+(-1+\cdots)q^{17}+\cdots$
5580.2.a.j	$5580$	$2$	5580.a	1.a	$1$	$3$	$3$	$44.557$	3.3.404.1	None	✓		✓		1860.2.a.g	$1$	$1$	$0$	$0$	$3$	$-2$		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{5}+(-1+\beta _{2})q^{7}-2\beta _{2}q^{11}+(1+\cdots)q^{13}+\cdots$
5580.2.a.k	$5580$	$2$	5580.a	1.a	$1$	$3$	$3$	$44.557$	3.3.756.1	None	✓				620.2.a.d	$1$	$0$	$0$	$0$	$3$	$0$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{5}-\beta _{2}q^{7}+(-2-\beta _{2})q^{11}-\beta _{2}q^{13}+\cdots$
5580.2.a.l	$5580$	$2$	5580.a	1.a	$1$	$3$	$3$	$44.557$	3.3.564.1	None	✓				1860.2.a.f	$1$	$0$	$0$	$0$	$3$	$4$		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+2\beta _{1}q^{11}+\cdots$
5580.2.a.m	$5580$	$2$	5580.a	1.a	$1$	$4$	$4$	$44.557$	4.4.224148.1	None	✓				1860.2.a.i	$1$	$0$	$0$	$0$	$-4$	$-4$		$-$	$-$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	$q-q^{5}+(-1+\beta _{1})q^{7}+(-\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots$
5580.2.a.n	$5580$	$2$	5580.a	1.a	$1$	$4$	$4$	$44.557$	4.4.25492.1	None	✓				620.2.a.e	$1$	$0$	$0$	$0$	$-4$	$8$		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{5}+(2-\beta _{2}-\beta _{3})q^{7}+(-\beta _{2}-\beta _{3})q^{11}+\cdots$
5580.2.a.o	$5580$	$2$	5580.a	1.a	$1$	$5$	$5$	$44.557$	5.5.10758096.1	None	✓	✓	✓	✓	5580.2.a.o	$1$	$0$	$0$	$0$	$-5$	$-4$		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{5}+(-1+\beta _{1})q^{7}+(-\beta _{3}-\beta _{4})q^{11}+\cdots$
5580.2.a.p	$5580$	$2$	5580.a	1.a	$1$	$5$	$5$	$44.557$	5.5.223952.1	None	✓	✓	✓	✓	5580.2.a.p	$1$	$1$	$0$	$0$	$-5$	$4$		$-$	$+$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	$q-q^{5}+(1-\beta _{2}+\beta _{3})q^{7}+(-2+\beta _{1}+\cdots)q^{11}+\cdots$
5580.2.a.q	$5580$	$2$	5580.a	1.a	$1$	$5$	$5$	$44.557$	5.5.10758096.1	None	✓	✓	✓	✓	5580.2.a.o	$1$	$1$	$0$	$0$	$5$	$-4$		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{5}+(-1+\beta _{1})q^{7}+(\beta _{3}+\beta _{4})q^{11}+\cdots$
5580.2.a.r	$5580$	$2$	5580.a	1.a	$1$	$5$	$5$	$44.557$	5.5.223952.1	None	✓	✓	✓	✓	5580.2.a.p	$1$	$0$	$0$	$0$	$5$	$4$		$-$	$+$	$-$	$-$	$-$	$3$	$\mathrm{SU}(2)$	$q+q^{5}+(1-\beta _{2}+\beta _{3})q^{7}+(2-\beta _{1}+\beta _{4})q^{11}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(5580))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(5580)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(15))$$^{\oplus 12}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(20))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(30))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(31))$$^{\oplus 18}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(36))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(45))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(62))$$^{\oplus 12}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(90))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(93))$$^{\oplus 12}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(124))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(155))$$^{\oplus 9}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(180))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(186))$$^{\oplus 8}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(279))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(310))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(372))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(465))$$^{\oplus 6}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(558))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(620))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(930))$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(1116))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(1395))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(1860))$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(2790))$$^{\oplus 2}$