Space of modular forms of level 1044, weight 4, and trivial character

• Label: 1044.4.a
• Level: $1044$
• Weight: $4$
• Character orbit: 1044.a
• Rep. character: $\chi_{1044}(1,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $10$
• Sturm bound: $720$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1044 = 2^{2} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1044.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(720\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1044))\).

	Total	New	Old
Modular forms	552	35	517
Cusp forms	528	35	493
Eisenstein series	24	0	24

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

\(2\)	\(3\)	\(29\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(73\)	\(0\)	\(73\)		\(69\)	\(0\)	\(69\)		\(4\)	\(0\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)		\(67\)	\(0\)	\(67\)		\(63\)	\(0\)	\(63\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)		\(69\)	\(0\)	\(69\)		\(65\)	\(0\)	\(65\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)		\(69\)	\(0\)	\(69\)		\(65\)	\(0\)	\(65\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)		\(65\)	\(7\)	\(58\)		\(63\)	\(7\)	\(56\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(71\)	\(7\)	\(64\)		\(69\)	\(7\)	\(62\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(69\)	\(12\)	\(57\)		\(67\)	\(12\)	\(55\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(69\)	\(9\)	\(60\)		\(67\)	\(9\)	\(58\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(282\)	\(19\)	\(263\)		\(270\)	\(19\)	\(251\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(270\)	\(16\)	\(254\)		\(258\)	\(16\)	\(242\)		\(12\)	\(0\)	\(12\)

Trace form

35 * q - 16 * q^5 - 4 * q^7 + 64 * q^11 - 28 * q^13 + 50 * q^17 + 244 * q^19 + 68 * q^23 + 771 * q^25 - 87 * q^29 - 40 * q^31 + 68 * q^35 + 726 * q^37 + 246 * q^41 - 464 * q^43 + 1056 * q^47 + 1867 * q^49 + 424 * q^53 + 144 * q^55 - 1724 * q^59 - 142 * q^61 + 706 * q^65 - 1564 * q^67 + 528 * q^71 - 630 * q^73 + 808 * q^77 - 1392 * q^79 - 244 * q^83 + 600 * q^85 - 2018 * q^89 - 1860 * q^91 + 288 * q^95 + 1506 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1044))\) 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	29
1044.4.a.a	$1044$	$4$	1044.a	1.a	$1$	$1$	$1$	$61.598$	\(\Q\)	None	✓				348.4.a.a	$1$	$0$	\(0\)	\(0\)	\(9\)	\(21\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+9q^{5}+21q^{7}+66q^{11}-72q^{13}+\cdots\)
1044.4.a.b	$1044$	$4$	1044.a	1.a	$1$	$2$	$2$	$61.598$	\(\Q(\sqrt{22}) \)	None	✓				116.4.a.b	$1$	$0$	\(0\)	\(0\)	\(-30\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-15q^{5}-2\beta q^{7}+(23+5\beta )q^{11}+(-5+\cdots)q^{13}+\cdots\)
1044.4.a.c	$1044$	$4$	1044.a	1.a	$1$	$2$	$2$	$61.598$	\(\Q(\sqrt{13}) \)	None	✓				348.4.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-40\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(-20+3\beta )q^{7}+(-4+\cdots)q^{11}+\cdots\)
1044.4.a.d	$1044$	$4$	1044.a	1.a	$1$	$2$	$2$	$61.598$	\(\Q(\sqrt{13}) \)	None	✓				116.4.a.a	$1$	$1$	\(0\)	\(0\)	\(10\)	\(-20\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(5+2\beta )q^{5}+(-10-4\beta )q^{7}+(2^{4}+\cdots)q^{11}+\cdots\)
1044.4.a.e	$1044$	$4$	1044.a	1.a	$1$	$3$	$3$	$61.598$	3.3.189021.1	None	✓				348.4.a.c	$1$	$1$	\(0\)	\(0\)	\(-19\)	\(9\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-6+\beta _{2})q^{5}+(3+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1044.4.a.f	$1044$	$4$	1044.a	1.a	$1$	$3$	$3$	$61.598$	3.3.148344.1	None	✓				116.4.a.c	$1$	$0$	\(0\)	\(0\)	\(20\)	\(8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(7-\beta _{2})q^{5}+(4-2\beta _{1}-2\beta _{2})q^{7}+\cdots\)
1044.4.a.g	$1044$	$4$	1044.a	1.a	$1$	$4$	$4$	$61.598$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		348.4.a.e	$1$	$1$	\(0\)	\(0\)	\(-9\)	\(-11\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{2})q^{5}+(-3+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1044.4.a.h	$1044$	$4$	1044.a	1.a	$1$	$4$	$4$	$61.598$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		348.4.a.d	$1$	$0$	\(0\)	\(0\)	\(1\)	\(17\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}+\beta _{2})q^{5}+(4+\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots\)
1044.4.a.i	$1044$	$4$	1044.a	1.a	$1$	$7$	$7$	$61.598$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	1044.4.a.i	$1$	$1$	\(0\)	\(0\)	\(-20\)	\(6\)		$-$	$+$	$+$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{5}+(1+\beta _{2})q^{7}+(5+\beta _{1}+\cdots)q^{11}+\cdots\)
1044.4.a.j	$1044$	$4$	1044.a	1.a	$1$	$7$	$7$	$61.598$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	1044.4.a.i	$1$	$0$	\(0\)	\(0\)	\(20\)	\(6\)		$-$	$+$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{5}+(1+\beta _{2})q^{7}+(-5-\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1044))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1044)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(348))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(522))\)\(^{\oplus 2}\)