Space of modular forms of level 57, weight 10, and trivial character

• Label: 57.10.a
• Level: $57$
• Weight: $10$
• Character orbit: 57.a
• Rep. character: $\chi_{57}(1,\cdot)$
• Character field: $\Q$
• Dimension: $26$
• Newform subspaces: $4$
• Sturm bound: $66$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	57.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(66\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(57))\).

	Total	New	Old
Modular forms	62	26	36
Cusp forms	58	26	32
Eisenstein series	4	0	4

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

\(3\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(-\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(8\)
\(-\)	\(-\)	\(+\)	\(5\)
Plus space		\(+\)	\(12\)
Minus space		\(-\)	\(14\)

Trace form

\( 26 q + 36 q^{2} + 5804 q^{4} + 5120 q^{5} - 2592 q^{6} - 3660 q^{7} + 43356 q^{8} + 170586 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(57))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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	19
57.10.a.a	$57$	$10$	57.a	1.a	$1$	$5$	$5$	$29.357$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	57.10.a.a	$1$	$1$	\(-15\)	\(405\)	\(-1104\)	\(-11318\)		$-$	$-$	$+$	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+3^{4}q^{3}+(157+\beta _{3}+\cdots)q^{4}+\cdots\)
57.10.a.b	$57$	$10$	57.a	1.a	$1$	$6$	$6$	$29.357$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	57.10.a.b	$1$	$0$	\(33\)	\(-486\)	\(1158\)	\(7890\)		$+$	$-$	$-$	$2^{3}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+(6-\beta _{1})q^{2}-3^{4}q^{3}+(12^{2}-8\beta _{1}+\cdots)q^{4}+\cdots\)
57.10.a.c	$57$	$10$	57.a	1.a	$1$	$7$	$7$	$29.357$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	57.10.a.c	$1$	$1$	\(1\)	\(-567\)	\(1164\)	\(-9720\)		$+$	$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3^{4}q^{3}+(193-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
57.10.a.d	$57$	$10$	57.a	1.a	$1$	$8$	$8$	$29.357$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	57.10.a.d	$1$	$0$	\(17\)	\(648\)	\(3902\)	\(9488\)		$-$	$+$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+3^{4}q^{3}+(354+4\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(57))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(57)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)