Space of modular forms of level 555, weight 4, and trivial character

• Label: 555.4.a
• Level: $555$
• Weight: $4$
• Character orbit: 555.a
• Rep. character: $\chi_{555}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $10$
• Sturm bound: $304$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 555 = 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	555.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(304\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(555))\).

	Total	New	Old
Modular forms	232	72	160
Cusp forms	224	72	152
Eisenstein series	8	0	8

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

\(3\)	\(5\)	\(37\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(10\)
\(+\)	\(+\)	\(-\)	$-$	\(7\)
\(+\)	\(-\)	\(+\)	$-$	\(9\)
\(+\)	\(-\)	\(-\)	$+$	\(10\)
\(-\)	\(+\)	\(+\)	$-$	\(9\)
\(-\)	\(+\)	\(-\)	$+$	\(10\)
\(-\)	\(-\)	\(+\)	$+$	\(10\)
\(-\)	\(-\)	\(-\)	$-$	\(7\)
Plus space			\(+\)	\(40\)
Minus space			\(-\)	\(32\)

Trace form

\( 72 q + 276 q^{4} + 12 q^{6} + 648 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(555))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	37
555.4.a.a	$555$	$4$	555.a	1.a	$1$	$1$	$1$	$32.746$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-3\)	\(5\)	\(25\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}-7q^{4}+5q^{5}-3q^{6}+\cdots\)
555.4.a.b	$555$	$4$	555.a	1.a	$1$	$1$	$1$	$32.746$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(3\)	\(5\)	\(25\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}-7q^{4}+5q^{5}+3q^{6}+\cdots\)
555.4.a.c	$555$	$4$	555.a	1.a	$1$	$7$	$7$	$32.746$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-6\)	\(21\)	\(35\)	\(-42\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+3q^{3}+(3-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
555.4.a.d	$555$	$4$	555.a	1.a	$1$	$7$	$7$	$32.746$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(2\)	\(-21\)	\(-35\)	\(-14\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
555.4.a.e	$555$	$4$	555.a	1.a	$1$	$8$	$8$	$32.746$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(-24\)	\(40\)	\(-53\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(5+\beta _{2})q^{4}+5q^{5}+\cdots\)
555.4.a.f	$555$	$4$	555.a	1.a	$1$	$9$	$9$	$32.746$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(27\)	\(-45\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(3+\beta _{1}+\beta _{2})q^{4}+\cdots\)
555.4.a.g	$555$	$4$	555.a	1.a	$1$	$9$	$9$	$32.746$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(9\)	\(27\)	\(45\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+3q^{3}+(7-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
555.4.a.h	$555$	$4$	555.a	1.a	$1$	$10$	$10$	$32.746$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(-6\)	\(-30\)	\(-50\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-3q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
555.4.a.i	$555$	$4$	555.a	1.a	$1$	$10$	$10$	$32.746$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(30\)	\(-50\)	\(14\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}-5q^{5}+\cdots\)
555.4.a.j	$555$	$4$	555.a	1.a	$1$	$10$	$10$	$32.746$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(-30\)	\(50\)	\(42\)		$+$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(555))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(555)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 2}\)