Space of modular forms of level 224, weight 6, and trivial character

• Label: 224.6.a
• Level: $224$
• Weight: $6$
• Character orbit: 224.a
• Rep. character: $\chi_{224}(1,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $10$
• Sturm bound: $192$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 224 = 2^{5} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	224.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(192\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(224))\).

	Total	New	Old
Modular forms	168	30	138
Cusp forms	152	30	122
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(39\)	\(7\)	\(32\)		\(35\)	\(7\)	\(28\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(45\)	\(9\)	\(36\)		\(41\)	\(9\)	\(32\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(43\)	\(8\)	\(35\)		\(39\)	\(8\)	\(31\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(41\)	\(6\)	\(35\)		\(37\)	\(6\)	\(31\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(80\)	\(13\)	\(67\)		\(72\)	\(13\)	\(59\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(88\)	\(17\)	\(71\)		\(80\)	\(17\)	\(63\)		\(8\)	\(0\)	\(8\)

Trace form

30 * q - 76 * q^5 + 2342 * q^9 + 244 * q^13 - 3220 * q^17 + 28098 * q^25 + 8564 * q^29 - 7504 * q^33 + 18820 * q^37 + 42252 * q^41 - 102828 * q^45 + 72030 * q^49 + 117956 * q^53 + 198912 * q^57 - 91580 * q^61 - 118568 * q^65 + 285072 * q^69 + 122364 * q^73 + 115246 * q^81 + 131080 * q^85 + 426364 * q^89 + 364128 * q^93 - 470132 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(224))\) 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
224.6.a.a	$224$	$6$	224.a	1.a	$1$	$1$	$1$	$35.926$	\(\Q\)	None	✓	✓			224.6.a.a	$1$	$1$	\(0\)	\(-14\)	\(-64\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-14q^{3}-2^{6}q^{5}+7^{2}q^{7}-47q^{9}+\cdots\)
224.6.a.b	$224$	$6$	224.a	1.a	$1$	$1$	$1$	$35.926$	\(\Q\)	None	✓	✓			224.6.a.a	$1$	$0$	\(0\)	\(14\)	\(-64\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+14q^{3}-2^{6}q^{5}-7^{2}q^{7}-47q^{9}+\cdots\)
224.6.a.c	$224$	$6$	224.a	1.a	$1$	$2$	$2$	$35.926$	\(\Q(\sqrt{61}) \)	None	✓	✓			224.6.a.c	$1$	$0$	\(0\)	\(-14\)	\(34\)	\(-98\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{3}+(17-7\beta )q^{5}-7^{2}q^{7}+\cdots\)
224.6.a.d	$224$	$6$	224.a	1.a	$1$	$2$	$2$	$35.926$	\(\Q(\sqrt{61}) \)	None	✓	✓			224.6.a.c	$1$	$1$	\(0\)	\(14\)	\(34\)	\(98\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{3}+(17+7\beta )q^{5}+7^{2}q^{7}+\cdots\)
224.6.a.e	$224$	$6$	224.a	1.a	$1$	$3$	$3$	$35.926$	3.3.367637.1	None	✓	✓	✓		224.6.a.e	$1$	$1$	\(0\)	\(-8\)	\(-14\)	\(147\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{3}+(-5-\beta _{2})q^{5}+7^{2}q^{7}+\cdots\)
224.6.a.f	$224$	$6$	224.a	1.a	$1$	$3$	$3$	$35.926$	3.3.367637.1	None	✓	✓			224.6.a.e	$1$	$1$	\(0\)	\(8\)	\(-14\)	\(-147\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{3}+(-5-\beta _{2})q^{5}-7^{2}q^{7}+\cdots\)
224.6.a.g	$224$	$6$	224.a	1.a	$1$	$4$	$4$	$35.926$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		224.6.a.g	$1$	$1$	\(0\)	\(-18\)	\(-30\)	\(-196\)		$+$	$+$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{3}+(-7-\beta _{1}-\beta _{2})q^{5}+\cdots\)
224.6.a.h	$224$	$6$	224.a	1.a	$1$	$4$	$4$	$35.926$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			224.6.a.g	$1$	$0$	\(0\)	\(18\)	\(-30\)	\(196\)		$+$	$-$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1})q^{3}+(-7-\beta _{1}-\beta _{2})q^{5}+\cdots\)
224.6.a.i	$224$	$6$	224.a	1.a	$1$	$5$	$5$	$35.926$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		224.6.a.i	$1$	$0$	\(0\)	\(-10\)	\(36\)	\(245\)		$+$	$-$	$-$	$2^{12}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{3}+(7-\beta _{3})q^{5}+7^{2}q^{7}+\cdots\)
224.6.a.j	$224$	$6$	224.a	1.a	$1$	$5$	$5$	$35.926$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		224.6.a.i	$1$	$0$	\(0\)	\(10\)	\(36\)	\(-245\)		$-$	$+$	$-$	$2^{12}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(7-\beta _{3})q^{5}-7^{2}q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(224))\) into lower level spaces

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