Space of modular forms of level 228, weight 2, and trivial character

• Label: 228.2.a
• Level: $228$
• Weight: $2$
• Character orbit: 228.a
• Rep. character: $\chi_{228}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $3$
• Sturm bound: $80$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 228 = 2^{2} \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	228.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(80\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(228))\).

	Total	New	Old
Modular forms	46	4	42
Cusp forms	35	4	31
Eisenstein series	11	0	11

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(19\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(6\)	\(2\)	\(4\)		\(5\)	\(2\)	\(3\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(21\)	\(1\)	\(20\)		\(16\)	\(1\)	\(15\)		\(5\)	\(0\)	\(5\)
Minus space			\(-\)		\(25\)	\(3\)	\(22\)		\(19\)	\(3\)	\(16\)		\(6\)	\(0\)	\(6\)

Trace form

4 * q + 2 * q^5 + 2 * q^7 + 4 * q^9 - 6 * q^11 + 4 * q^15 - 2 * q^17 + 2 * q^19 + 14 * q^25 + 4 * q^29 - 4 * q^31 - 18 * q^35 - 12 * q^37 + 8 * q^39 - 16 * q^41 + 2 * q^43 + 2 * q^45 - 18 * q^47 - 10 * q^49 - 4 * q^51 + 4 * q^53 - 2 * q^55 + 2 * q^57 - 8 * q^59 + 2 * q^61 + 2 * q^63 + 28 * q^65 + 16 * q^67 - 12 * q^69 - 32 * q^71 - 10 * q^73 + 8 * q^75 + 10 * q^77 - 8 * q^79 + 4 * q^81 + 8 * q^83 + 6 * q^85 - 16 * q^87 + 28 * q^89 - 4 * q^91 - 2 * q^95 + 16 * q^97 - 6 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(228))\) 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	3	19
228.2.a.a	$228$	$2$	228.a	1.a	$1$	$1$	$1$	$1.821$	\(\Q\)	None	✓	✓	✓	✓	228.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}+q^{7}+q^{9}-5q^{11}-6q^{13}+\cdots\)
228.2.a.b	$228$	$2$	228.a	1.a	$1$	$1$	$1$	$1.821$	\(\Q\)	None	✓	✓	✓	✓	228.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+q^{9}+2q^{11}+2q^{13}+\cdots\)
228.2.a.c	$228$	$2$	228.a	1.a	$1$	$2$	$2$	$1.821$	\(\Q(\sqrt{33}) \)	None	✓	✓	✓	✓	228.2.a.c	$1$	$0$	\(0\)	\(2\)	\(3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta )q^{5}+(1-\beta )q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(228))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(228)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 2}\)