Space of modular forms of level 501, weight 2, and trivial character

• Label: 501.2.a
• Level: $501$
• Weight: $2$
• Character orbit: 501.a
• Rep. character: $\chi_{501}(1,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $5$
• Sturm bound: $112$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 501 = 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	501.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(112\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(501))\).

	Total	New	Old
Modular forms	58	27	31
Cusp forms	55	27	28
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.

\(3\)	\(167\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(7\)	\(5\)	\(2\)		\(7\)	\(5\)	\(2\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(22\)	\(9\)	\(13\)		\(21\)	\(9\)	\(12\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(11\)	\(8\)	\(3\)		\(10\)	\(8\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(18\)	\(5\)	\(13\)		\(17\)	\(5\)	\(12\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(25\)	\(10\)	\(15\)		\(24\)	\(10\)	\(14\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(33\)	\(17\)	\(16\)		\(31\)	\(17\)	\(14\)		\(2\)	\(0\)	\(2\)

Trace form

27 * q - 3 * q^2 - q^3 + 25 * q^4 - 6 * q^5 + q^6 - 8 * q^7 - 15 * q^8 + 27 * q^9 - 2 * q^10 + q^12 - 2 * q^13 + 8 * q^14 + 2 * q^15 + 33 * q^16 - 2 * q^17 - 3 * q^18 + 4 * q^19 - 6 * q^20 - 8 * q^21 - 8 * q^23 - 3 * q^24 + 9 * q^25 + 22 * q^26 - q^27 - 20 * q^28 - 10 * q^29 + 10 * q^30 - 8 * q^31 - 23 * q^32 - 4 * q^33 + 2 * q^34 - 4 * q^35 + 25 * q^36 - 14 * q^37 + 20 * q^38 + 2 * q^39 + 42 * q^40 + 10 * q^41 - 4 * q^42 + 16 * q^43 - 6 * q^45 + 8 * q^46 + 12 * q^47 + q^48 + 11 * q^49 - 13 * q^50 + 2 * q^51 - 26 * q^52 + 2 * q^53 + q^54 + 24 * q^55 - 4 * q^56 - 12 * q^57 - 46 * q^58 + 32 * q^59 + 6 * q^60 + 2 * q^61 - 20 * q^62 - 8 * q^63 + q^64 - 16 * q^65 - 20 * q^66 - 8 * q^67 + 46 * q^68 + 4 * q^69 - 8 * q^70 + 36 * q^71 - 15 * q^72 - 34 * q^73 - 6 * q^74 - 7 * q^75 - 20 * q^76 - 2 * q^78 + 24 * q^79 - 42 * q^80 + 27 * q^81 + 38 * q^82 + 36 * q^83 - 16 * q^84 - 24 * q^85 - 76 * q^86 + 10 * q^87 - 4 * q^88 - 6 * q^89 - 2 * q^90 - 8 * q^91 - 60 * q^92 - 32 * q^93 - 8 * q^94 + 32 * q^95 + 25 * q^96 - 46 * q^97 - 59 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(501))\) 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	167
501.2.a.a	$501$	$2$	501.a	1.a	$1$	$1$	$1$	$4.001$	\(\Q\)	None	✓	✓			501.2.a.a	$1$	$0$	\(1\)	\(-1\)	\(-4\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-4q^{5}-q^{6}+4q^{7}+\cdots\)
501.2.a.b	$501$	$2$	501.a	1.a	$1$	$5$	$5$	$4.001$	5.5.36497.1	None	✓	✓	✓	✓	501.2.a.b	$1$	$1$	\(-4\)	\(5\)	\(-9\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3})q^{2}+q^{3}+(1+\beta _{1}-2\beta _{3}+\cdots)q^{4}+\cdots\)
501.2.a.c	$501$	$2$	501.a	1.a	$1$	$5$	$5$	$4.001$	5.5.38569.1	None	✓	✓	✓	✓	501.2.a.c	$1$	$1$	\(0\)	\(-5\)	\(-1\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(\beta _{2}+\beta _{3})q^{4}+(\beta _{1}+\beta _{4})q^{5}+\cdots\)
501.2.a.d	$501$	$2$	501.a	1.a	$1$	$8$	$8$	$4.001$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		501.2.a.d	$1$	$0$	\(-3\)	\(-8\)	\(1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}-q^{3}+(1-\beta _{2}+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots\)
501.2.a.e	$501$	$2$	501.a	1.a	$1$	$8$	$8$	$4.001$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	501.2.a.e	$1$	$0$	\(3\)	\(8\)	\(7\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{3}+\beta _{5}+\beta _{7})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(501))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(501)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(167))\)\(^{\oplus 2}\)