Space of modular forms of level 7220, weight 2, and trivial character

• Label: 7220.2.a
• Level: $7220$
• Weight: $2$
• Character orbit: 7220.a
• Rep. character: $\chi_{7220}(1,\cdot)$
• Character field: $\Q$
• Dimension: $113$
• Newform subspaces: $26$
• Sturm bound: $2280$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 7220 = 2^{2} \cdot 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7220.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(2280\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7220))\).

	Total	New	Old
Modular forms	1200	113	1087
Cusp forms	1081	113	968
Eisenstein series	119	0	119

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

\(2\)	\(5\)	\(19\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(33\)
\(-\)	\(+\)	\(-\)	$+$	\(24\)
\(-\)	\(-\)	\(+\)	$+$	\(23\)
\(-\)	\(-\)	\(-\)	$-$	\(33\)
Plus space			\(+\)	\(47\)
Minus space			\(-\)	\(66\)

Trace form

\( 113 q + 2 q^{3} - q^{5} - 2 q^{7} + 113 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7220))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	5	19
7220.2.a.a	$7220$	$2$	7220.a	1.a	$1$	$1$	$1$	$57.652$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}-4q^{7}+q^{9}-3q^{11}+\cdots\)
7220.2.a.b	$7220$	$2$	7220.a	1.a	$1$	$1$	$1$	$57.652$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+2q^{7}+q^{9}-6q^{13}+\cdots\)
7220.2.a.c	$7220$	$2$	7220.a	1.a	$1$	$1$	$1$	$57.652$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
7220.2.a.d	$7220$	$2$	7220.a	1.a	$1$	$1$	$1$	$57.652$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-3q^{9}-4q^{11}+4q^{13}+\cdots\)
7220.2.a.e	$7220$	$2$	7220.a	1.a	$1$	$1$	$1$	$57.652$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}-4q^{7}+q^{9}-3q^{11}+\cdots\)
7220.2.a.f	$7220$	$2$	7220.a	1.a	$1$	$1$	$1$	$57.652$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(-1\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}+2q^{7}+q^{9}-2q^{13}+\cdots\)
7220.2.a.g	$7220$	$2$	7220.a	1.a	$1$	$1$	$1$	$57.652$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+q^{9}-4q^{11}+2q^{13}+\cdots\)
7220.2.a.h	$7220$	$2$	7220.a	1.a	$1$	$2$	$2$	$57.652$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+q^{5}+2q^{7}+(1-2\beta )q^{9}+\cdots\)
7220.2.a.i	$7220$	$2$	7220.a	1.a	$1$	$2$	$2$	$57.652$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(-6\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-q^{5}-3q^{7}+(-2+\beta )q^{9}+\cdots\)
7220.2.a.j	$7220$	$2$	7220.a	1.a	$1$	$2$	$2$	$57.652$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{3}-q^{5}+(1-\beta )q^{7}+2q^{9}+\cdots\)
7220.2.a.k	$7220$	$2$	7220.a	1.a	$1$	$2$	$2$	$57.652$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{3}-q^{5}+\beta q^{7}+2q^{9}+(-2+\cdots)q^{11}+\cdots\)
7220.2.a.l	$7220$	$2$	7220.a	1.a	$1$	$2$	$2$	$57.652$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(-6\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}-3q^{7}+(-2+\beta )q^{9}+\cdots\)
7220.2.a.m	$7220$	$2$	7220.a	1.a	$1$	$2$	$2$	$57.652$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(4\)	\(2\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}+q^{5}+(-2+2\beta )q^{7}+(3+\cdots)q^{9}+\cdots\)
7220.2.a.n	$7220$	$2$	7220.a	1.a	$1$	$3$	$3$	$57.652$	3.3.257.1	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(2\)		$-$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-q^{5}+(1+\beta _{2})q^{7}+(2-\beta _{1}+\cdots)q^{9}+\cdots\)
7220.2.a.o	$7220$	$2$	7220.a	1.a	$1$	$3$	$3$	$57.652$	3.3.257.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(2\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-q^{5}+(1+\beta _{2})q^{7}+(2-\beta _{1}+\cdots)q^{9}+\cdots\)
7220.2.a.p	$7220$	$2$	7220.a	1.a	$1$	$4$	$4$	$57.652$	4.4.133593.1	None	✓	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}-\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
7220.2.a.q	$7220$	$2$	7220.a	1.a	$1$	$4$	$4$	$57.652$	4.4.7168.1	None	✓	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}-\beta _{3}q^{7}+(3+\beta _{3})q^{9}+\cdots\)
7220.2.a.r	$7220$	$2$	7220.a	1.a	$1$	$4$	$4$	$57.652$	4.4.133593.1	None	✓	$1$	$1$	\(0\)	\(1\)	\(4\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}-\beta _{3}q^{7}+(1+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
7220.2.a.s	$7220$	$2$	7220.a	1.a	$1$	$8$	$8$	$57.652$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(8\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+(1+\beta _{6})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
7220.2.a.t	$7220$	$2$	7220.a	1.a	$1$	$8$	$8$	$57.652$	8.8.\(\cdots\).2	None	✓	$2$	$1$	\(0\)	\(0\)	\(8\)	\(-10\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}+(-1-\beta _{2}+\beta _{4})q^{7}+\cdots\)
7220.2.a.u	$7220$	$2$	7220.a	1.a	$1$	$8$	$8$	$57.652$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(8\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}+(1+\beta _{6})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
7220.2.a.v	$7220$	$2$	7220.a	1.a	$1$	$9$	$9$	$57.652$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-3\)	\(-9\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+(-\beta _{4}+\beta _{7})q^{7}+(1+\cdots)q^{9}+\cdots\)
7220.2.a.w	$7220$	$2$	7220.a	1.a	$1$	$9$	$9$	$57.652$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(9\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+(-\beta _{2}-\beta _{5}+\beta _{8})q^{7}+\cdots\)
7220.2.a.x	$7220$	$2$	7220.a	1.a	$1$	$9$	$9$	$57.652$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(-9\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(-\beta _{4}+\beta _{7})q^{7}+(1+\cdots)q^{9}+\cdots\)
7220.2.a.y	$7220$	$2$	7220.a	1.a	$1$	$9$	$9$	$57.652$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(9\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}+(-\beta _{2}-\beta _{5}+\beta _{8})q^{7}+\cdots\)
7220.2.a.z	$7220$	$2$	7220.a	1.a	$1$	$16$	$16$	$57.652$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(-16\)	\(10\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(1-\beta _{2})q^{7}+(1-\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7220))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7220)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(361))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(722))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1444))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1805))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3610))\)\(^{\oplus 2}\)