Space of modular forms of level 224, weight 2, and trivial character

• Label: 224.2.a
• Level: $224$
• Weight: $2$
• Character orbit: 224.a
• Rep. character: $\chi_{224}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $4$
• Sturm bound: $64$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	224.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(64\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(224))\).

	Total	New	Old
Modular forms	40	6	34
Cusp forms	25	6	19
Eisenstein series	15	0	15

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

\(2\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(7\)	\(1\)	\(6\)		\(4\)	\(1\)	\(3\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(13\)	\(3\)	\(10\)		\(9\)	\(3\)	\(6\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(11\)	\(2\)	\(9\)		\(7\)	\(2\)	\(5\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(9\)	\(0\)	\(9\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(16\)	\(1\)	\(15\)		\(9\)	\(1\)	\(8\)		\(7\)	\(0\)	\(7\)
Minus space		\(-\)		\(24\)	\(5\)	\(19\)		\(16\)	\(5\)	\(11\)		\(8\)	\(0\)	\(8\)

Trace form

6 * q + 4 * q^5 + 14 * q^9 + 4 * q^13 - 4 * q^17 - 6 * q^25 + 4 * q^29 - 16 * q^33 + 20 * q^37 - 36 * q^41 - 28 * q^45 + 6 * q^49 - 44 * q^53 + 20 * q^61 - 8 * q^65 - 48 * q^69 + 12 * q^73 + 22 * q^81 + 40 * q^85 + 12 * q^89 - 32 * q^93 + 28 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(224))\) 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	7
224.2.a.a	$224$	$2$	224.a	1.a	$1$	$1$	$1$	$1.789$	\(\Q\)	None	✓	✓	✓	✓	224.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots\)
224.2.a.b	$224$	$2$	224.a	1.a	$1$	$1$	$1$	$1.789$	\(\Q\)	None	✓	✓			224.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
224.2.a.c	$224$	$2$	224.a	1.a	$1$	$2$	$2$	$1.789$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		224.2.a.c	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(1-\beta )q^{5}+q^{7}+(3+\cdots)q^{9}+\cdots\)
224.2.a.d	$224$	$2$	224.a	1.a	$1$	$2$	$2$	$1.789$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	224.2.a.c	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1-\beta )q^{5}-q^{7}+(3+2\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(224))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(224)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)