Space of modular forms of level 620, weight 2, and trivial character

• Label: 620.2.a
• Level: $620$
• Weight: $2$
• Character orbit: 620.a
• Rep. character: $\chi_{620}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $5$
• Sturm bound: $192$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 620 = 2^{2} \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	620.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(192\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(620))\).

	Total	New	Old
Modular forms	102	10	92
Cusp forms	91	10	81
Eisenstein series	11	0	11

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

\(2\)	\(5\)	\(31\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(2\)
\(-\)	\(-\)	\(-\)	$-$	\(4\)
Plus space			\(+\)	\(3\)
Minus space			\(-\)	\(7\)

Trace form

\( 10 q + 4 q^{3} + 2 q^{5} + 14 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(620))\) 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	31
620.2.a.a	$620$	$2$	620.a	1.a	$1$	$1$	$1$	$4.951$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{5}-2q^{7}+6q^{9}+2q^{11}+\cdots\)
620.2.a.b	$620$	$2$	620.a	1.a	$1$	$1$	$1$	$4.951$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}-3q^{9}-4q^{11}-4q^{13}+\cdots\)
620.2.a.c	$620$	$2$	620.a	1.a	$1$	$1$	$1$	$4.951$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-4q^{7}-2q^{9}+2q^{13}+\cdots\)
620.2.a.d	$620$	$2$	620.a	1.a	$1$	$3$	$3$	$4.951$	3.3.756.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(-3\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-q^{5}-\beta _{2}q^{7}+(2-2\beta _{1}+\cdots)q^{9}+\cdots\)
620.2.a.e	$620$	$2$	620.a	1.a	$1$	$4$	$4$	$4.951$	4.4.25492.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(4\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+q^{5}+(2-\beta _{2}-\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(620))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(620)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(124))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(310))\)\(^{\oplus 2}\)