Space of modular forms of level 51, weight 2, and trivial character

• Label: 51.2.a
• Level: $51$
• Weight: $2$
• Character orbit: 51.a
• Rep. character: $\chi_{51}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $12$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 51 = 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	51.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(12\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(51))\).

	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.

\(3\)	\(17\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(-\)	\(-\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(2\)	\(1\)	\(1\)		\(1\)	\(1\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(6\)	\(3\)	\(3\)		\(4\)	\(3\)	\(1\)		\(2\)	\(0\)	\(2\)

Trace form

3 * q - q^2 - q^3 + 3 * q^4 + 6 * q^5 + q^6 - 4 * q^7 - 9 * q^8 + 3 * q^9 - 10 * q^10 - 4 * q^11 - 7 * q^12 + 4 * q^13 + 7 * q^16 + q^17 - q^18 + 2 * q^19 + 10 * q^20 - 4 * q^21 - 8 * q^22 + 9 * q^24 + 7 * q^25 + 6 * q^26 - q^27 + 8 * q^28 + 6 * q^29 + 10 * q^30 - 9 * q^32 - 2 * q^33 - q^34 - 12 * q^35 + 3 * q^36 - 6 * q^37 + 24 * q^38 - 6 * q^39 - 22 * q^40 - 6 * q^41 - 10 * q^43 + 12 * q^44 + 6 * q^45 - 4 * q^46 - 20 * q^47 + q^48 - 5 * q^49 - 27 * q^50 - 3 * q^51 + 6 * q^52 + 2 * q^53 + q^54 - 2 * q^55 - 4 * q^57 + 34 * q^58 + 12 * q^59 - 22 * q^60 + 18 * q^61 - 16 * q^62 - 4 * q^63 - q^64 - 4 * q^65 + 8 * q^66 + 4 * q^67 + 7 * q^68 + 18 * q^69 + 16 * q^71 - 9 * q^72 - 6 * q^73 + 18 * q^74 + q^75 - 16 * q^76 + 12 * q^77 - 6 * q^78 - 4 * q^79 + 42 * q^80 + 3 * q^81 + 10 * q^82 - 16 * q^83 + 8 * q^84 - 24 * q^86 + 6 * q^87 - 4 * q^88 + 6 * q^89 - 10 * q^90 + 4 * q^91 - 32 * q^92 + 4 * q^93 + 24 * q^94 - 24 * q^95 + 9 * q^96 - 30 * q^97 + 7 * q^98 - 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(51))\) 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	17
51.2.a.a	$51$	$2$	51.a	1.a	$1$	$1$	$1$	$0.407$	\(\Q\)	None	✓	✓	✓	✓	51.2.a.a	$1$	$0$	\(0\)	\(1\)	\(3\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+3q^{5}-4q^{7}+q^{9}-3q^{11}+\cdots\)
51.2.a.b	$51$	$2$	51.a	1.a	$1$	$2$	$2$	$0.407$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	51.2.a.b	$1$	$0$	\(-1\)	\(-2\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(2+\beta )q^{4}+(1+\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(51))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(51)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)