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}$