Space of modular forms of level 471, weight 2, and trivial character

• Label: 471.2.a
• Level: $471$
• Weight: $2$
• Character orbit: 471.a
• Rep. character: $\chi_{471}(1,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $5$
• Sturm bound: $105$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	471.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(105\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(471))\).

	Total	New	Old
Modular forms	54	27	27
Cusp forms	51	27	24
Eisenstein series	3	0	3

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

\(3\)	\(157\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(-\)	$-$	\(9\)
\(-\)	\(+\)	$-$	\(12\)
\(-\)	\(-\)	$+$	\(2\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(21\)

Trace form

\( 27 q - q^{2} + q^{3} + 25 q^{4} + 6 q^{5} - 3 q^{6} + 3 q^{8} + 27 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(471))\) 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	157
471.2.a.a	$471$	$2$	471.a	1.a	$1$	$1$	$1$	$3.761$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-2\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}+3q^{7}+\cdots\)
471.2.a.b	$471$	$2$	471.a	1.a	$1$	$2$	$2$	$3.761$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-1\)	\(2\)	\(-2\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-q^{5}-\beta q^{6}+\cdots\)
471.2.a.c	$471$	$2$	471.a	1.a	$1$	$3$	$3$	$3.761$	3.3.229.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
471.2.a.d	$471$	$2$	471.a	1.a	$1$	$9$	$9$	$3.761$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(-9\)	\(8\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{5}+\cdots)q^{5}+\cdots\)
471.2.a.e	$471$	$2$	471.a	1.a	$1$	$12$	$12$	$3.761$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(-1\)	\(12\)	\(4\)	\(8\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(471))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(471)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(157))\)\(^{\oplus 2}\)