Space of modular forms of level 472, weight 2, and trivial character

• Label: 472.2.a
• Level: $472$
• Weight: $2$
• Character orbit: 472.a
• Rep. character: $\chi_{472}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $7$
• Sturm bound: $120$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 472 = 2^{3} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	472.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(120\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(472))\).

	Total	New	Old
Modular forms	64	15	49
Cusp forms	57	15	42
Eisenstein series	7	0	7

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

\(2\)	\(59\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(+\)	$-$	\(7\)
\(-\)	\(-\)	$+$	\(1\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(9\)

Trace form

\( 15 q - 2 q^{3} - 4 q^{7} + 21 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(472))\) 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	59
472.2.a.a	$472$	$2$	472.a	1.a	$1$	$1$	$1$	$3.769$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{5}+3q^{7}+6q^{9}-4q^{11}+\cdots\)
472.2.a.b	$472$	$2$	472.a	1.a	$1$	$1$	$1$	$3.769$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{7}-2q^{9}-2q^{13}+q^{15}+\cdots\)
472.2.a.c	$472$	$2$	472.a	1.a	$1$	$1$	$1$	$3.769$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{7}-2q^{9}+4q^{11}+2q^{13}+\cdots\)
472.2.a.d	$472$	$2$	472.a	1.a	$1$	$1$	$1$	$3.769$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(2\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{5}+q^{7}+q^{9}+q^{11}-q^{13}+\cdots\)
472.2.a.e	$472$	$2$	472.a	1.a	$1$	$1$	$1$	$3.769$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(-3\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-3q^{5}+3q^{7}+6q^{9}+6q^{11}+\cdots\)
472.2.a.f	$472$	$2$	472.a	1.a	$1$	$4$	$4$	$3.769$	4.4.6809.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-9\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{3}+(-1+\beta _{1}-\beta _{3})q^{5}+(-2+\cdots)q^{7}+\cdots\)
472.2.a.g	$472$	$2$	472.a	1.a	$1$	$6$	$6$	$3.769$	6.6.921465377.1	None	✓	$1$	$0$	\(0\)	\(-1\)	\(7\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{2})q^{5}+(-1-\beta _{4})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(472))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(472)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(118))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(236))\)\(^{\oplus 2}\)