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	Dim
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(5\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(7\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(9\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(8\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(6\)
Plus space				\(+\)	\(23\)
Minus space				\(-\)	\(27\)

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

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