Space of modular forms of level 1224, weight 2, and trivial character

• Label: 1224.2.a
• Level: $1224$
• Weight: $2$
• Character orbit: 1224.a
• Rep. character: $\chi_{1224}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $13$
• Sturm bound: $432$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1224.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(432\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1224))\).

	Total	New	Old
Modular forms	232	20	212
Cusp forms	201	20	181
Eisenstein series	31	0	31

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
\(+\)	\(+\)	\(+\)	\(+\)		\(26\)	\(1\)	\(25\)		\(23\)	\(1\)	\(22\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(30\)	\(3\)	\(27\)		\(26\)	\(3\)	\(23\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(31\)	\(4\)	\(27\)		\(27\)	\(4\)	\(23\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(27\)	\(2\)	\(25\)		\(23\)	\(2\)	\(21\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(32\)	\(3\)	\(29\)		\(28\)	\(3\)	\(25\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(28\)	\(1\)	\(27\)		\(24\)	\(1\)	\(23\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(27\)	\(3\)	\(24\)		\(23\)	\(3\)	\(20\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(31\)	\(3\)	\(28\)		\(27\)	\(3\)	\(24\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(108\)	\(7\)	\(101\)		\(93\)	\(7\)	\(86\)		\(15\)	\(0\)	\(15\)
Minus space			\(-\)		\(124\)	\(13\)	\(111\)		\(108\)	\(13\)	\(95\)		\(16\)	\(0\)	\(16\)

Trace form

20 * q - 2 * q^5 + 10 * q^11 + 4 * q^13 - 2 * q^17 + 8 * q^19 - 4 * q^23 + 32 * q^25 + 6 * q^29 - 8 * q^31 + 8 * q^35 + 6 * q^37 - 20 * q^41 - 4 * q^43 + 24 * q^47 + 36 * q^49 + 12 * q^53 + 24 * q^55 + 12 * q^59 + 14 * q^61 + 4 * q^65 + 20 * q^67 - 4 * q^71 + 16 * q^73 + 16 * q^77 + 28 * q^79 - 4 * q^83 + 2 * q^85 + 24 * q^91 - 24 * q^95 + 20 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1224))\) 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
1224.2.a.a	$1224$	$2$	1224.a	1.a	$1$	$1$	$1$	$9.774$	\(\Q\)	None	✓				408.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{5}+q^{11}+3q^{13}+q^{17}+q^{19}+\cdots\)
1224.2.a.b	$1224$	$2$	1224.a	1.a	$1$	$1$	$1$	$9.774$	\(\Q\)	None	✓				408.2.a.d	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-4q^{7}-4q^{11}+6q^{13}-q^{17}+\cdots\)
1224.2.a.c	$1224$	$2$	1224.a	1.a	$1$	$1$	$1$	$9.774$	\(\Q\)	None	✓	✓	✓	✓	1224.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}+3q^{11}-q^{13}-q^{17}+\cdots\)
1224.2.a.d	$1224$	$2$	1224.a	1.a	$1$	$1$	$1$	$9.774$	\(\Q\)	None	✓				136.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{11}-6q^{13}+q^{17}+4q^{19}-4q^{23}+\cdots\)
1224.2.a.e	$1224$	$2$	1224.a	1.a	$1$	$1$	$1$	$9.774$	\(\Q\)	None	✓				408.2.a.c	$1$	$0$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+2q^{13}+q^{17}+4q^{19}-2q^{23}+\cdots\)
1224.2.a.f	$1224$	$2$	1224.a	1.a	$1$	$1$	$1$	$9.774$	\(\Q\)	None	✓	✓	✓	✓	1224.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-2q^{7}-3q^{11}-q^{13}+q^{17}+\cdots\)
1224.2.a.g	$1224$	$2$	1224.a	1.a	$1$	$1$	$1$	$9.774$	\(\Q\)	None	✓				136.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{7}+6q^{11}+2q^{13}-q^{17}+\cdots\)
1224.2.a.h	$1224$	$2$	1224.a	1.a	$1$	$1$	$1$	$9.774$	\(\Q\)	None	✓				408.2.a.b	$1$	$1$	\(0\)	\(0\)	\(3\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-4q^{7}-q^{11}-5q^{13}-q^{17}+\cdots\)
1224.2.a.i	$1224$	$2$	1224.a	1.a	$1$	$2$	$2$	$9.774$	\(\Q(\sqrt{5}) \)	None	✓		✓		136.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(2\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{5}+(1-\beta )q^{7}+(-1+\beta )q^{11}+\cdots\)
1224.2.a.j	$1224$	$2$	1224.a	1.a	$1$	$2$	$2$	$9.774$	\(\Q(\sqrt{17}) \)	None	✓		✓		408.2.a.e	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(-2+2\beta )q^{7}+(4+\beta )q^{11}+\cdots\)
1224.2.a.k	$1224$	$2$	1224.a	1.a	$1$	$2$	$2$	$9.774$	\(\Q(\sqrt{57}) \)	None	✓		✓		408.2.a.f	$1$	$0$	\(0\)	\(0\)	\(1\)	\(8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+4q^{7}+(2-\beta )q^{11}+\beta q^{13}+\cdots\)
1224.2.a.l	$1224$	$2$	1224.a	1.a	$1$	$3$	$3$	$9.774$	3.3.1304.1	None	✓	✓	✓	✓	1224.2.a.l	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{2})q^{5}+(1+\beta _{2})q^{7}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
1224.2.a.m	$1224$	$2$	1224.a	1.a	$1$	$3$	$3$	$9.774$	3.3.1304.1	None	✓	✓	✓	✓	1224.2.a.l	$1$	$0$	\(0\)	\(0\)	\(1\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{2})q^{5}+(1+\beta _{2})q^{7}+(-1+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1224))\) into lower level spaces

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