Space of modular forms of level 57, weight 2, and trivial character

• Label: 57.2.a
• Level: $57$
• Weight: $2$
• Character orbit: 57.a
• Rep. character: $\chi_{57}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $13$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 57 = 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	57.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(13\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(57))\).

	Total	New	Old
Modular forms	8	3	5
Cusp forms	5	3	2
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.

\(3\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(1\)
\(-\)	\(+\)	\(-\)	\(2\)
Plus space		\(+\)	\(1\)
Minus space		\(-\)	\(2\)

Trace form

3 * q - 3 * q^2 + q^3 + 3 * q^4 - 4 * q^5 + q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9 + 2 * q^10 - 2 * q^11 - q^12 + 2 * q^13 + 4 * q^14 + 2 * q^15 - 9 * q^16 - 4 * q^17 - 3 * q^18 - 3 * q^19 - 2 * q^20 + 8 * q^21 + 4 * q^22 + 4 * q^23 - 3 * q^24 - q^25 + 14 * q^26 + q^27 - 4 * q^28 - 10 * q^29 - 10 * q^30 + 4 * q^31 + 21 * q^32 - 4 * q^33 - 10 * q^34 + 18 * q^35 + 3 * q^36 - 2 * q^37 + 3 * q^38 - 2 * q^39 + 6 * q^40 - 10 * q^41 - 16 * q^42 - 6 * q^43 - 4 * q^44 - 4 * q^45 + 4 * q^46 + 6 * q^47 - q^48 + 13 * q^49 - q^50 - 2 * q^51 - 14 * q^52 - 2 * q^53 + q^54 - 6 * q^55 - q^57 + 26 * q^58 - 20 * q^59 + 10 * q^60 + 4 * q^61 + 16 * q^62 - 2 * q^63 - 9 * q^64 - 24 * q^65 + 8 * q^66 + 12 * q^67 + 10 * q^68 + 12 * q^69 - 36 * q^70 - 3 * q^72 - 12 * q^73 - 26 * q^74 - 9 * q^75 - 3 * q^76 - 14 * q^77 + 22 * q^78 + 16 * q^79 + 10 * q^80 + 3 * q^81 + 14 * q^82 + 32 * q^83 + 16 * q^84 + 18 * q^85 - 6 * q^87 + 2 * q^89 + 2 * q^90 - 28 * q^91 - 4 * q^92 + 16 * q^93 + 24 * q^94 + 4 * q^95 + 5 * q^96 - 2 * q^97 - 47 * q^98 - 2 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(57))\) 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
57.2.a.a	$57$	$2$	57.a	1.a	$1$	$1$	$1$	$0.455$	\(\Q\)	None	✓	✓	✓	✓	57.2.a.a	$1$	$1$	\(-2\)	\(-1\)	\(-3\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+2q^{4}-3q^{5}+2q^{6}+\cdots\)
57.2.a.b	$57$	$2$	57.a	1.a	$1$	$1$	$1$	$0.455$	\(\Q\)	None	✓	✓			57.2.a.b	$1$	$0$	\(-2\)	\(1\)	\(1\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}+q^{5}-2q^{6}+3q^{7}+\cdots\)
57.2.a.c	$57$	$2$	57.a	1.a	$1$	$1$	$1$	$0.455$	\(\Q\)	None	✓	✓			57.2.a.c	$1$	$0$	\(1\)	\(1\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}-2q^{5}+q^{6}-3q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(57))\) into lower level spaces

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