Space of modular forms of level 5220, weight 2, and trivial character

• Label: 5220.2.a
• Level: $5220$
• Weight: $2$
• Character orbit: 5220.a
• Rep. character: $\chi_{5220}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $26$
• Sturm bound: $2160$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(2160\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5220))\).

	Total	New	Old
Modular forms	1104	48	1056
Cusp forms	1057	48	1009
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.

\(2\)	\(3\)	\(5\)	\(29\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(63\)	\(0\)	\(63\)		\(60\)	\(0\)	\(60\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(73\)	\(0\)	\(73\)		\(69\)	\(0\)	\(69\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(69\)	\(0\)	\(69\)		\(65\)	\(0\)	\(65\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(71\)	\(0\)	\(71\)		\(67\)	\(0\)	\(67\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(72\)	\(0\)	\(72\)		\(68\)	\(0\)	\(68\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(68\)	\(0\)	\(68\)		\(64\)	\(0\)	\(64\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(68\)	\(0\)	\(68\)		\(64\)	\(0\)	\(64\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(72\)	\(0\)	\(72\)		\(68\)	\(0\)	\(68\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(75\)	\(7\)	\(68\)		\(73\)	\(7\)	\(66\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(65\)	\(3\)	\(62\)		\(63\)	\(3\)	\(60\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(69\)	\(3\)	\(66\)		\(67\)	\(3\)	\(64\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(67\)	\(7\)	\(60\)		\(65\)	\(7\)	\(58\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(66\)	\(6\)	\(60\)		\(64\)	\(6\)	\(58\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(70\)	\(8\)	\(62\)		\(68\)	\(8\)	\(60\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(70\)	\(8\)	\(62\)		\(68\)	\(8\)	\(60\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(66\)	\(6\)	\(60\)		\(64\)	\(6\)	\(58\)		\(2\)	\(0\)	\(2\)
Plus space				\(+\)		\(536\)	\(18\)	\(518\)		\(513\)	\(18\)	\(495\)		\(23\)	\(0\)	\(23\)
Minus space				\(-\)		\(568\)	\(30\)	\(538\)		\(544\)	\(30\)	\(514\)		\(24\)	\(0\)	\(24\)

Trace form

48 * q - 4 * q^7 - 4 * q^11 - 8 * q^13 - 16 * q^17 + 12 * q^19 - 4 * q^23 + 48 * q^25 + 4 * q^31 + 4 * q^35 + 8 * q^37 + 8 * q^41 + 28 * q^43 + 4 * q^47 + 64 * q^49 - 8 * q^53 + 8 * q^55 + 16 * q^59 - 16 * q^61 + 4 * q^67 + 8 * q^71 + 12 * q^73 - 28 * q^77 + 12 * q^79 - 20 * q^83 + 4 * q^85 + 8 * q^89 + 24 * q^91 + 16 * q^95 + 12 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5220))\) 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	29
5220.2.a.a	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓				1740.2.a.f	$1$	$2$	\(0\)	\(0\)	\(-1\)	\(-5\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-5q^{7}-5q^{11}-5q^{13}-7q^{17}+\cdots\)
5220.2.a.b	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓				1740.2.a.g	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{7}+3q^{11}+q^{13}+3q^{17}+\cdots\)
5220.2.a.c	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓				1740.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-3q^{11}-2q^{13}+4q^{17}+\cdots\)
5220.2.a.d	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓	✓			5220.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+q^{11}-2q^{17}-2q^{19}+\cdots\)
5220.2.a.e	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓				580.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+4q^{11}-6q^{13}+4q^{17}+\cdots\)
5220.2.a.f	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓	✓			5220.2.a.f	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-q^{7}-3q^{11}-q^{13}+3q^{17}+\cdots\)
5220.2.a.g	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓				1740.2.a.h	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}-4q^{11}+2q^{13}+4q^{19}+\cdots\)
5220.2.a.h	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓				1740.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}+4q^{11}+2q^{13}+8q^{17}+\cdots\)
5220.2.a.i	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓				1740.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+3q^{7}-3q^{11}+3q^{13}-q^{17}+\cdots\)
5220.2.a.j	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓	✓			5220.2.a.j	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(3\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+3q^{7}+q^{11}-5q^{13}+3q^{17}+\cdots\)
5220.2.a.k	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓	✓			5220.2.a.d	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}-q^{11}+2q^{17}-2q^{19}+\cdots\)
5220.2.a.l	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓				1740.2.a.e	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{7}+3q^{11}-q^{13}-3q^{17}+\cdots\)
5220.2.a.m	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓	✓			5220.2.a.f	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-q^{7}+3q^{11}-q^{13}-3q^{17}+\cdots\)
5220.2.a.n	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓				580.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2q^{11}-2q^{13}-2q^{19}+8q^{23}+\cdots\)
5220.2.a.o	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓				1740.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+6q^{11}+6q^{13}-4q^{17}+2q^{19}+\cdots\)
5220.2.a.p	$5220$	$2$	5220.a	1.a	$1$	$1$	$1$	$41.682$	\(\Q\)	None	✓	✓			5220.2.a.j	$1$	$1$	\(0\)	\(0\)	\(1\)	\(3\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+3q^{7}-q^{11}-5q^{13}-3q^{17}+\cdots\)
5220.2.a.q	$5220$	$2$	5220.a	1.a	$1$	$2$	$2$	$41.682$	\(\Q(\sqrt{17}) \)	None	✓				1740.2.a.k	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-5\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+(-2-\beta )q^{7}+\beta q^{11}+(-2+\cdots)q^{13}+\cdots\)
5220.2.a.r	$5220$	$2$	5220.a	1.a	$1$	$2$	$2$	$41.682$	\(\Q(\sqrt{57}) \)	None	✓				1740.2.a.n	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(4\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}+(2-\beta )q^{11}+2q^{13}+\cdots\)
5220.2.a.s	$5220$	$2$	5220.a	1.a	$1$	$2$	$2$	$41.682$	\(\Q(\sqrt{33}) \)	None	✓				1740.2.a.i	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-\beta q^{7}+(-2-\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
5220.2.a.t	$5220$	$2$	5220.a	1.a	$1$	$2$	$2$	$41.682$	\(\Q(\sqrt{33}) \)	None	✓				1740.2.a.l	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+\beta q^{7}+(-2+\beta )q^{11}+(-2+\cdots)q^{13}+\cdots\)
5220.2.a.u	$5220$	$2$	5220.a	1.a	$1$	$2$	$2$	$41.682$	\(\Q(\sqrt{17}) \)	None	✓				1740.2.a.m	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+2\beta q^{7}+(2-\beta )q^{11}+6q^{13}+\cdots\)
5220.2.a.v	$5220$	$2$	5220.a	1.a	$1$	$2$	$2$	$41.682$	\(\Q(\sqrt{17}) \)	None	✓				1740.2.a.j	$1$	$1$	\(0\)	\(0\)	\(2\)	\(3\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(1+\beta )q^{7}+(1-2\beta )q^{11}+(-3+\cdots)q^{13}+\cdots\)
5220.2.a.w	$5220$	$2$	5220.a	1.a	$1$	$3$	$3$	$41.682$	3.3.148.1	None	✓		✓		580.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(2\)		$-$	$-$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{5}+(1-\beta _{2})q^{7}+(-1+\beta _{2})q^{11}+\cdots\)
5220.2.a.x	$5220$	$2$	5220.a	1.a	$1$	$3$	$3$	$41.682$	3.3.564.1	None	✓		✓		580.2.a.c	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-4\)		$-$	$-$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{5}+(-1+\beta _{2})q^{7}+(-3-\beta _{2})q^{11}+\cdots\)
5220.2.a.y	$5220$	$2$	5220.a	1.a	$1$	$7$	$7$	$41.682$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	5220.2.a.y	$1$	$0$	\(0\)	\(0\)	\(-7\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-q^{5}+\beta _{2}q^{7}-\beta _{5}q^{11}-\beta _{6}q^{13}+\cdots\)
5220.2.a.z	$5220$	$2$	5220.a	1.a	$1$	$7$	$7$	$41.682$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	5220.2.a.y	$1$	$0$	\(0\)	\(0\)	\(7\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+q^{5}+\beta _{2}q^{7}+\beta _{5}q^{11}-\beta _{6}q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5220))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5220)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(290))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(348))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(435))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(522))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(580))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(870))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1044))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1305))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1740))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2610))\)\(^{\oplus 2}\)