Space of modular forms of level 1053, weight 2, and trivial character

• Label: 1053.2.a
• Level: $1053$
• Weight: $2$
• Character orbit: 1053.a
• Rep. character: $\chi_{1053}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $14$
• Sturm bound: $252$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1053 = 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1053.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(252\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1053))\).

	Total	New	Old
Modular forms	138	48	90
Cusp forms	115	48	67
Eisenstein series	23	0	23

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

\(3\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(30\)	\(10\)	\(20\)		\(25\)	\(10\)	\(15\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(36\)	\(14\)	\(22\)		\(30\)	\(14\)	\(16\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(39\)	\(14\)	\(25\)		\(33\)	\(14\)	\(19\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(33\)	\(10\)	\(23\)		\(27\)	\(10\)	\(17\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(63\)	\(20\)	\(43\)		\(52\)	\(20\)	\(32\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(75\)	\(28\)	\(47\)		\(63\)	\(28\)	\(35\)		\(12\)	\(0\)	\(12\)

Trace form

48 * q + 48 * q^4 + 72 * q^16 - 12 * q^19 + 12 * q^22 + 36 * q^25 + 24 * q^28 - 12 * q^31 - 36 * q^34 - 12 * q^37 - 24 * q^40 + 12 * q^43 - 24 * q^46 + 48 * q^49 - 12 * q^55 - 24 * q^61 + 72 * q^64 + 24 * q^67 + 24 * q^70 - 12 * q^73 + 12 * q^76 + 24 * q^79 + 60 * q^82 + 24 * q^85 + 12 * q^88 - 48 * q^94 - 48 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1053))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	13
1053.2.a.a	$1053$	$2$	1053.a	1.a	$1$	$1$	$1$	$8.408$	\(\Q\)	None	✓				117.2.e.a	$1$	$1$	\(-2\)	\(0\)	\(-4\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-4q^{5}+2q^{7}+8q^{10}+\cdots\)
1053.2.a.b	$1053$	$2$	1053.a	1.a	$1$	$1$	$1$	$8.408$	\(\Q\)	None	✓	✓			1053.2.a.b	$1$	$0$	\(-1\)	\(0\)	\(-2\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-2q^{5}+5q^{7}+3q^{8}+2q^{10}+\cdots\)
1053.2.a.c	$1053$	$2$	1053.a	1.a	$1$	$1$	$1$	$8.408$	\(\Q\)	None	✓	✓			1053.2.a.b	$1$	$0$	\(1\)	\(0\)	\(2\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}+2q^{5}+5q^{7}-3q^{8}+2q^{10}+\cdots\)
1053.2.a.d	$1053$	$2$	1053.a	1.a	$1$	$1$	$1$	$8.408$	\(\Q\)	None	✓				117.2.e.a	$1$	$0$	\(2\)	\(0\)	\(4\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+4q^{5}+2q^{7}+8q^{10}+\cdots\)
1053.2.a.e	$1053$	$2$	1053.a	1.a	$1$	$2$	$2$	$8.408$	\(\Q(\sqrt{5}) \)	None	✓	✓			1053.2.a.e	$1$	$1$	\(-1\)	\(0\)	\(3\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(2-\beta )q^{5}+(-1+\cdots)q^{7}+\cdots\)
1053.2.a.f	$1053$	$2$	1053.a	1.a	$1$	$2$	$2$	$8.408$	\(\Q(\sqrt{5}) \)	None	✓	✓			1053.2.a.e	$1$	$1$	\(1\)	\(0\)	\(-3\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
1053.2.a.g	$1053$	$2$	1053.a	1.a	$1$	$3$	$3$	$8.408$	3.3.621.1	None	✓	✓			1053.2.a.g	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{1}+\cdots)q^{5}+\cdots\)
1053.2.a.h	$1053$	$2$	1053.a	1.a	$1$	$3$	$3$	$8.408$	3.3.621.1	None	✓	✓			1053.2.a.g	$1$	$0$	\(0\)	\(0\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
1053.2.a.i	$1053$	$2$	1053.a	1.a	$1$	$4$	$4$	$8.408$	\(\Q(\sqrt{3}, \sqrt{7})\)	None	✓	✓			1053.2.a.i	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}-\beta _{2}q^{4}-\beta _{3}q^{5}+(-2+\beta _{2}+\cdots)q^{7}+\cdots\)
1053.2.a.j	$1053$	$2$	1053.a	1.a	$1$	$5$	$5$	$8.408$	5.5.403137.1	None	✓		✓		117.2.e.b	$1$	$1$	\(-2\)	\(0\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}-\beta _{3}q^{5}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
1053.2.a.k	$1053$	$2$	1053.a	1.a	$1$	$5$	$5$	$8.408$	5.5.403137.1	None	✓		✓		117.2.e.b	$1$	$0$	\(2\)	\(0\)	\(1\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+\beta _{3}q^{5}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
1053.2.a.l	$1053$	$2$	1053.a	1.a	$1$	$6$	$6$	$8.408$	6.6.22931361.1	None	✓		✓		117.2.e.c	$1$	$1$	\(-2\)	\(0\)	\(-3\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{1}-\beta _{4}+\beta _{5})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1053.2.a.m	$1053$	$2$	1053.a	1.a	$1$	$6$	$6$	$8.408$	6.6.22931361.1	None	✓				117.2.e.c	$1$	$0$	\(2\)	\(0\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{1}-\beta _{4}+\beta _{5})q^{4}+(1+\cdots)q^{5}+\cdots\)
1053.2.a.n	$1053$	$2$	1053.a	1.a	$1$	$8$	$8$	$8.408$	8.8.\(\cdots\).1	None	✓	✓	✓		1053.2.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(6\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{1}-\beta _{7})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1053))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1053)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(351))\)\(^{\oplus 2}\)