Space of modular forms of level 58, weight 2, and trivial character

• Label: 58.2.a
• Level: $58$
• Weight: $2$
• Character orbit: 58.a
• Rep. character: $\chi_{58}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $15$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 58 = 2 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	58.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(15\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(58))\).

	Total	New	Old
Modular forms	9	2	7
Cusp forms	6	2	4
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.

\(2\)	\(29\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(1\)
\(-\)	\(+\)	\(-\)	\(1\)
Plus space		\(+\)	\(1\)
Minus space		\(-\)	\(1\)

Trace form

2 * q - 4 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 - 4 * q^7 + 4 * q^9 + 4 * q^10 - 4 * q^11 - 4 * q^12 + 2 * q^13 + 8 * q^15 + 2 * q^16 + 4 * q^17 - 8 * q^18 - 8 * q^19 - 2 * q^20 + 8 * q^21 - 2 * q^22 + 4 * q^23 + 2 * q^24 - 4 * q^26 - 4 * q^27 - 4 * q^28 - 2 * q^29 - 10 * q^30 + 6 * q^33 + 12 * q^34 + 4 * q^35 + 4 * q^36 + 8 * q^38 - 8 * q^39 + 4 * q^40 - 4 * q^42 - 4 * q^43 - 4 * q^44 - 20 * q^45 + 4 * q^46 + 24 * q^47 - 4 * q^48 - 6 * q^49 - 8 * q^50 + 4 * q^51 + 2 * q^52 - 10 * q^53 + 14 * q^54 + 24 * q^57 - 4 * q^59 + 8 * q^60 - 4 * q^61 - 6 * q^62 - 8 * q^63 + 2 * q^64 - 10 * q^65 - 16 * q^67 + 4 * q^68 - 4 * q^69 - 8 * q^70 - 8 * q^72 - 8 * q^73 + 16 * q^74 - 8 * q^75 - 8 * q^76 + 8 * q^77 + 10 * q^78 + 8 * q^79 - 2 * q^80 + 10 * q^81 + 4 * q^82 + 4 * q^83 + 8 * q^84 + 20 * q^85 - 18 * q^86 + 4 * q^87 - 2 * q^88 - 16 * q^89 + 16 * q^90 - 4 * q^91 + 4 * q^92 - 6 * q^93 + 2 * q^94 + 24 * q^95 + 2 * q^96 - 8 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(58))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	29
58.2.a.a	$58$	$2$	58.a	1.a	$1$	$1$	$1$	$0.463$	\(\Q\)	None	✓	✓	✓	✓	58.2.a.a	$1$	$1$	\(-1\)	\(-3\)	\(-3\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}+q^{4}-3q^{5}+3q^{6}-2q^{7}+\cdots\)
58.2.a.b	$58$	$2$	58.a	1.a	$1$	$1$	$1$	$0.463$	\(\Q\)	None	✓	✓	✓	✓	58.2.a.b	$1$	$0$	\(1\)	\(-1\)	\(1\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(58))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(58)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 2}\)