Space of modular forms of level 224, weight 4, and trivial character

• Label: 224.4.a
• Level: $224$
• Weight: $4$
• Character orbit: 224.a
• Rep. character: $\chi_{224}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $8$
• Sturm bound: $128$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	224.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(128\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(224))\).

	Total	New	Old
Modular forms	104	18	86
Cusp forms	88	18	70
Eisenstein series	16	0	16

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	Dim
\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(6\)
Plus space		\(+\)	\(11\)
Minus space		\(-\)	\(7\)

Trace form

18 * q - 4 * q^5 + 122 * q^9 + 92 * q^13 + 308 * q^17 + 318 * q^25 + 284 * q^29 + 1232 * q^33 - 660 * q^37 - 588 * q^41 + 1788 * q^45 + 882 * q^49 + 1388 * q^53 - 768 * q^57 + 780 * q^61 - 1304 * q^65 - 1488 * q^69 - 1212 * q^73 - 2078 * q^81 + 1880 * q^85 + 388 * q^89 - 5856 * q^93 - 3628 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(224))\) 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
224.4.a.a	$224$	$4$	224.a	1.a	$1$	$1$	$1$	$13.216$	\(\Q\)	None	✓	✓			224.4.a.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+7q^{7}-23q^{9}+20q^{11}-20q^{13}+\cdots\)
224.4.a.b	$224$	$4$	224.a	1.a	$1$	$1$	$1$	$13.216$	\(\Q\)	None	✓	✓			224.4.a.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-7q^{7}-23q^{9}-20q^{11}-20q^{13}+\cdots\)
224.4.a.c	$224$	$4$	224.a	1.a	$1$	$2$	$2$	$13.216$	\(\Q(\sqrt{37}) \)	None	✓	✓	✓		224.4.a.c	$1$	$1$	\(0\)	\(-6\)	\(-6\)	\(14\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{3}+(-3+\beta )q^{5}+7q^{7}+\cdots\)
224.4.a.d	$224$	$4$	224.a	1.a	$1$	$2$	$2$	$13.216$	\(\Q(\sqrt{37}) \)	None	✓	✓			224.4.a.c	$1$	$0$	\(0\)	\(6\)	\(-6\)	\(-14\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}+(-3+\beta )q^{5}-7q^{7}+(19+\cdots)q^{9}+\cdots\)
224.4.a.e	$224$	$4$	224.a	1.a	$1$	$3$	$3$	$13.216$	3.3.2981.1	None	✓	✓	✓		224.4.a.e	$1$	$0$	\(0\)	\(-8\)	\(10\)	\(-21\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{3}+(3+\beta _{2})q^{5}-7q^{7}+\cdots\)
224.4.a.f	$224$	$4$	224.a	1.a	$1$	$3$	$3$	$13.216$	3.3.621.1	None	✓	✓	✓		224.4.a.f	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(-21\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}-7q^{7}+\cdots\)
224.4.a.g	$224$	$4$	224.a	1.a	$1$	$3$	$3$	$13.216$	3.3.621.1	None	✓	✓			224.4.a.f	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(21\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}+7q^{7}+\cdots\)
224.4.a.h	$224$	$4$	224.a	1.a	$1$	$3$	$3$	$13.216$	3.3.2981.1	None	✓	✓			224.4.a.e	$1$	$0$	\(0\)	\(8\)	\(10\)	\(21\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+(3+\beta _{2})q^{5}+7q^{7}+(11+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(224))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(224)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)