Space of modular forms of level 538, weight 2, and trivial character

• Label: 538.2.a
• Level: $538$
• Weight: $2$
• Character orbit: 538.a
• Rep. character: $\chi_{538}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $5$
• Sturm bound: $135$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 538 = 2 \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	538.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(135\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(538))\).

	Total	New	Old
Modular forms	69	22	47
Cusp forms	66	22	44
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.

\(2\)	\(269\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(13\)	\(7\)	\(6\)		\(13\)	\(7\)	\(6\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(21\)	\(4\)	\(17\)		\(20\)	\(4\)	\(16\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(16\)	\(9\)	\(7\)		\(15\)	\(9\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(19\)	\(2\)	\(17\)		\(18\)	\(2\)	\(16\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(32\)	\(9\)	\(23\)		\(31\)	\(9\)	\(22\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(37\)	\(13\)	\(24\)		\(35\)	\(13\)	\(22\)		\(2\)	\(0\)	\(2\)

Trace form

22 * q + 22 * q^4 + 2 * q^5 + 2 * q^6 - 4 * q^7 + 20 * q^9 + 4 * q^10 - 6 * q^11 + 6 * q^13 + 4 * q^14 + 4 * q^15 + 22 * q^16 - 4 * q^17 - 8 * q^19 + 2 * q^20 - 20 * q^21 + 4 * q^22 - 4 * q^23 + 2 * q^24 + 32 * q^25 - 8 * q^26 - 24 * q^27 - 4 * q^28 - 4 * q^29 + 4 * q^31 - 12 * q^33 + 4 * q^34 - 12 * q^35 + 20 * q^36 - 6 * q^37 + 2 * q^38 - 8 * q^39 + 4 * q^40 - 8 * q^41 - 26 * q^43 - 6 * q^44 - 2 * q^45 + 8 * q^46 - 24 * q^47 - 2 * q^49 + 16 * q^50 - 52 * q^51 + 6 * q^52 + 26 * q^53 + 8 * q^54 - 4 * q^55 + 4 * q^56 - 16 * q^57 + 10 * q^58 - 8 * q^59 + 4 * q^60 - 14 * q^61 + 12 * q^62 - 16 * q^63 + 22 * q^64 - 24 * q^65 + 24 * q^66 - 22 * q^67 - 4 * q^68 - 28 * q^69 + 32 * q^70 + 20 * q^71 - 20 * q^73 + 24 * q^74 - 28 * q^75 - 8 * q^76 + 4 * q^77 - 16 * q^78 - 16 * q^79 + 2 * q^80 + 46 * q^81 + 16 * q^82 - 20 * q^84 - 8 * q^85 - 16 * q^86 + 20 * q^87 + 4 * q^88 + 48 * q^89 + 16 * q^90 - 60 * q^91 - 4 * q^92 - 24 * q^93 - 4 * q^94 - 48 * q^95 + 2 * q^96 + 16 * q^97 + 16 * q^98 - 58 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(538))\) 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	269
538.2.a.a	$538$	$2$	538.a	1.a	$1$	$2$	$2$	$4.296$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	538.2.a.a	$1$	$1$	\(2\)	\(-1\)	\(-5\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta q^{3}+q^{4}+(-3+\beta )q^{5}-\beta q^{6}+\cdots\)
538.2.a.b	$538$	$2$	538.a	1.a	$1$	$2$	$2$	$4.296$	\(\Q(\sqrt{13}) \)	None	✓	✓			538.2.a.b	$1$	$0$	\(2\)	\(1\)	\(1\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+(1-\beta )q^{5}+\beta q^{6}+\cdots\)
538.2.a.c	$538$	$2$	538.a	1.a	$1$	$4$	$4$	$4.296$	4.4.4913.1	None	✓	✓	✓	✓	538.2.a.c	$1$	$0$	\(-4\)	\(3\)	\(5\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+(2-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
538.2.a.d	$538$	$2$	538.a	1.a	$1$	$7$	$7$	$4.296$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	538.2.a.d	$1$	$1$	\(-7\)	\(-4\)	\(-6\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta _{1}+\beta _{6})q^{3}+q^{4}+(\beta _{3}+\cdots)q^{5}+\cdots\)
538.2.a.e	$538$	$2$	538.a	1.a	$1$	$7$	$7$	$4.296$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		538.2.a.e	$1$	$0$	\(7\)	\(1\)	\(7\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{2}q^{3}+q^{4}+(1+\beta _{4})q^{5}+\beta _{2}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(538))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(538)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(269))\)\(^{\oplus 2}\)