Space of modular forms of level 1224, weight 4, and trivial character

• Label: 1224.4.a
• Level: $1224$
• Weight: $4$
• Character orbit: 1224.a
• Rep. character: $\chi_{1224}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $17$
• Sturm bound: $864$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1224.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(864\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1224))\).

	Total	New	Old
Modular forms	664	60	604
Cusp forms	632	60	572
Eisenstein series	32	0	32

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

\(2\)	\(3\)	\(17\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(86\)	\(7\)	\(79\)		\(82\)	\(7\)	\(75\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(82\)	\(5\)	\(77\)		\(78\)	\(5\)	\(73\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(81\)	\(8\)	\(73\)		\(77\)	\(8\)	\(69\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(85\)	\(10\)	\(75\)		\(81\)	\(10\)	\(71\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(80\)	\(5\)	\(75\)		\(76\)	\(5\)	\(71\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(84\)	\(7\)	\(77\)		\(80\)	\(7\)	\(73\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(85\)	\(9\)	\(76\)		\(81\)	\(9\)	\(72\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(81\)	\(9\)	\(72\)		\(77\)	\(9\)	\(68\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(340\)	\(33\)	\(307\)		\(324\)	\(33\)	\(291\)		\(16\)	\(0\)	\(16\)
Minus space			\(-\)		\(324\)	\(27\)	\(297\)		\(308\)	\(27\)	\(281\)		\(16\)	\(0\)	\(16\)

Trace form

60 * q + 10 * q^5 + 48 * q^7 - 94 * q^11 - 36 * q^13 + 34 * q^17 + 248 * q^19 - 180 * q^23 + 1248 * q^25 + 258 * q^29 - 136 * q^31 + 136 * q^35 + 178 * q^37 + 260 * q^41 - 196 * q^43 + 1032 * q^47 + 3132 * q^49 - 524 * q^53 - 1592 * q^55 + 492 * q^59 + 570 * q^61 - 2132 * q^65 + 596 * q^67 + 1292 * q^71 + 1648 * q^73 - 2208 * q^77 - 996 * q^79 + 828 * q^83 - 170 * q^85 - 1392 * q^89 - 840 * q^91 + 872 * q^95 + 188 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1224))\) 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	17
1224.4.a.a	$1224$	$4$	1224.a	1.a	$1$	$1$	$1$	$72.218$	\(\Q\)	None	✓				408.4.a.a	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(-24\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{5}-24q^{7}-44q^{11}+6q^{13}+\cdots\)
1224.4.a.b	$1224$	$4$	1224.a	1.a	$1$	$1$	$1$	$72.218$	\(\Q\)	None	✓				408.4.a.b	$1$	$1$	\(0\)	\(0\)	\(7\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+7q^{5}+4q^{7}+21q^{11}-5^{2}q^{13}+\cdots\)
1224.4.a.c	$1224$	$4$	1224.a	1.a	$1$	$2$	$2$	$72.218$	\(\Q(\sqrt{241}) \)	None	✓				408.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-7\)	\(18\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{5}+(8+2\beta )q^{7}+(-9-3\beta )q^{11}+\cdots\)
1224.4.a.d	$1224$	$4$	1224.a	1.a	$1$	$2$	$2$	$72.218$	\(\Q(\sqrt{3}) \)	None	✓				136.4.a.a	$1$	$0$	\(0\)	\(0\)	\(12\)	\(-36\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(6+2\beta )q^{5}+(-18-3\beta )q^{7}+(10+\cdots)q^{11}+\cdots\)
1224.4.a.e	$1224$	$4$	1224.a	1.a	$1$	$3$	$3$	$72.218$	3.3.23321.1	None	✓				408.4.a.g	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(20\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{2})q^{5}+(7-\beta _{1})q^{7}+(-9+\cdots)q^{11}+\cdots\)
1224.4.a.f	$1224$	$4$	1224.a	1.a	$1$	$3$	$3$	$72.218$	3.3.1556.1	None	✓				136.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(12\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(4-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1224.4.a.g	$1224$	$4$	1224.a	1.a	$1$	$3$	$3$	$72.218$	3.3.12821.1	None	✓				408.4.a.e	$1$	$0$	\(0\)	\(0\)	\(4\)	\(28\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1}-2\beta _{2})q^{5}+(9+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1224.4.a.h	$1224$	$4$	1224.a	1.a	$1$	$3$	$3$	$72.218$	3.3.4481.1	None	✓				408.4.a.d	$1$	$0$	\(0\)	\(0\)	\(5\)	\(-4\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}-\beta _{2})q^{5}+(-2-2\beta _{1})q^{7}+\cdots\)
1224.4.a.i	$1224$	$4$	1224.a	1.a	$1$	$3$	$3$	$72.218$	3.3.8396.1	None	✓				136.4.a.b	$1$	$0$	\(0\)	\(0\)	\(8\)	\(-2\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1}+2\beta _{2})q^{5}+(-2-3\beta _{1}-\beta _{2})q^{7}+\cdots\)
1224.4.a.j	$1224$	$4$	1224.a	1.a	$1$	$3$	$3$	$72.218$	3.3.17717.1	None	✓				408.4.a.f	$1$	$1$	\(0\)	\(0\)	\(10\)	\(-22\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{5}+(-8+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1224.4.a.k	$1224$	$4$	1224.a	1.a	$1$	$4$	$4$	$72.218$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		408.4.a.h	$1$	$1$	\(0\)	\(0\)	\(-10\)	\(-4\)		$+$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{5}+(-1-\beta _{2}-\beta _{3})q^{7}+\cdots\)
1224.4.a.l	$1224$	$4$	1224.a	1.a	$1$	$4$	$4$	$72.218$	4.4.550476.1	None	✓		✓		136.4.a.d	$1$	$1$	\(0\)	\(0\)	\(-8\)	\(-22\)		$-$	$-$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{2}+2\beta _{3})q^{5}+(-4+\cdots)q^{7}+\cdots\)
1224.4.a.m	$1224$	$4$	1224.a	1.a	$1$	$4$	$4$	$72.218$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		408.4.a.i	$1$	$0$	\(0\)	\(0\)	\(2\)	\(32\)		$+$	$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(9+\beta _{1}-\beta _{2})q^{7}+(-15+\cdots)q^{11}+\cdots\)
1224.4.a.n	$1224$	$4$	1224.a	1.a	$1$	$5$	$5$	$72.218$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	1224.4.a.n	$1$	$1$	\(0\)	\(0\)	\(-13\)	\(-2\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{3})q^{5}+\beta _{1}q^{7}+(1+\beta _{2}-2\beta _{3}+\cdots)q^{11}+\cdots\)
1224.4.a.o	$1224$	$4$	1224.a	1.a	$1$	$5$	$5$	$72.218$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	1224.4.a.n	$1$	$1$	\(0\)	\(0\)	\(13\)	\(-2\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{3})q^{5}+\beta _{1}q^{7}+(-1-\beta _{2}+2\beta _{3}+\cdots)q^{11}+\cdots\)
1224.4.a.p	$1224$	$4$	1224.a	1.a	$1$	$7$	$7$	$72.218$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	1224.4.a.p	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(26\)		$+$	$+$	$+$	$+$	$2^{6}\cdot 13$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{5}+(4+\beta _{3})q^{7}+(-9-\beta _{2}+\beta _{4}+\cdots)q^{11}+\cdots\)
1224.4.a.q	$1224$	$4$	1224.a	1.a	$1$	$7$	$7$	$72.218$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	1224.4.a.p	$1$	$0$	\(0\)	\(0\)	\(3\)	\(26\)		$-$	$+$	$-$	$+$	$2^{6}\cdot 13$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{5}+(4+\beta _{3})q^{7}+(9+\beta _{2}-\beta _{4}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1224))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1224)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(204))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(306))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(408))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(612))\)\(^{\oplus 2}\)