Space of modular forms of level 4440, weight 2, and trivial character

• Label: 4440.2.a
• Level: $4440$
• Weight: $2$
• Character orbit: 4440.a
• Rep. character: $\chi_{4440}(1,\cdot)$
• Character field: $\Q$
• Dimension: $72$
• Newform subspaces: $27$
• Sturm bound: $1824$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 4440 = 2^{3} \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4440.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 27 \)
Sturm bound:			\(1824\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4440))\).

	Total	New	Old
Modular forms	928	72	856
Cusp forms	897	72	825
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.

\(2\)	\(3\)	\(5\)	\(37\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(50\)	\(6\)	\(44\)		\(49\)	\(6\)	\(43\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(65\)	\(3\)	\(62\)		\(63\)	\(3\)	\(60\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(53\)	\(4\)	\(49\)		\(51\)	\(4\)	\(47\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(64\)	\(5\)	\(59\)		\(62\)	\(5\)	\(57\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(61\)	\(4\)	\(57\)		\(59\)	\(4\)	\(55\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(56\)	\(5\)	\(51\)		\(54\)	\(5\)	\(49\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(58\)	\(3\)	\(55\)		\(56\)	\(3\)	\(53\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(57\)	\(6\)	\(51\)		\(55\)	\(6\)	\(49\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(58\)	\(4\)	\(54\)		\(56\)	\(4\)	\(52\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(59\)	\(5\)	\(54\)		\(57\)	\(5\)	\(52\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(55\)	\(3\)	\(52\)		\(53\)	\(3\)	\(50\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(60\)	\(6\)	\(54\)		\(58\)	\(6\)	\(52\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(63\)	\(3\)	\(60\)		\(61\)	\(3\)	\(58\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(52\)	\(6\)	\(46\)		\(50\)	\(6\)	\(44\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(66\)	\(7\)	\(59\)		\(64\)	\(7\)	\(57\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(51\)	\(2\)	\(49\)		\(49\)	\(2\)	\(47\)		\(2\)	\(0\)	\(2\)
Plus space				\(+\)		\(456\)	\(32\)	\(424\)		\(441\)	\(32\)	\(409\)		\(15\)	\(0\)	\(15\)
Minus space				\(-\)		\(472\)	\(40\)	\(432\)		\(456\)	\(40\)	\(416\)		\(16\)	\(0\)	\(16\)

Trace form

72 * q + 72 * q^9 + 8 * q^13 + 16 * q^17 + 16 * q^23 + 72 * q^25 + 16 * q^29 + 16 * q^31 + 4 * q^37 + 8 * q^41 - 8 * q^43 - 32 * q^47 + 88 * q^49 + 16 * q^51 - 40 * q^53 + 8 * q^57 - 48 * q^59 - 32 * q^61 + 16 * q^67 + 16 * q^71 + 24 * q^73 - 32 * q^77 - 48 * q^79 + 72 * q^81 + 32 * q^83 - 48 * q^89 + 48 * q^91 + 24 * q^93 - 16 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4440))\) 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	5	37
4440.2.a.a	$4440$	$2$	4440.a	1.a	$1$	$1$	$1$	$35.454$	\(\Q\)	None	✓	✓			4440.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-4\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-4q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
4440.2.a.b	$4440$	$2$	4440.a	1.a	$1$	$1$	$1$	$35.454$	\(\Q\)	None	✓	✓			4440.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}-2q^{13}+q^{15}-2q^{17}+\cdots\)
4440.2.a.c	$4440$	$2$	4440.a	1.a	$1$	$1$	$1$	$35.454$	\(\Q\)	None	✓	✓			4440.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}+2q^{11}+6q^{13}+\cdots\)
4440.2.a.d	$4440$	$2$	4440.a	1.a	$1$	$1$	$1$	$35.454$	\(\Q\)	None	✓	✓			4440.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(3\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+3q^{7}+q^{9}-3q^{11}-3q^{13}+\cdots\)
4440.2.a.e	$4440$	$2$	4440.a	1.a	$1$	$1$	$1$	$35.454$	\(\Q\)	None	✓	✓			4440.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}+6q^{13}-q^{15}+2q^{17}+\cdots\)
4440.2.a.f	$4440$	$2$	4440.a	1.a	$1$	$1$	$1$	$35.454$	\(\Q\)	None	✓	✓			4440.2.a.f	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
4440.2.a.g	$4440$	$2$	4440.a	1.a	$1$	$1$	$1$	$35.454$	\(\Q\)	None	✓	✓			4440.2.a.g	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-2\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{7}+q^{9}-3q^{13}+q^{15}+\cdots\)
4440.2.a.h	$4440$	$2$	4440.a	1.a	$1$	$1$	$1$	$35.454$	\(\Q\)	None	✓	✓			4440.2.a.h	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-1\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}+q^{9}+3q^{11}+6q^{13}+\cdots\)
4440.2.a.i	$4440$	$2$	4440.a	1.a	$1$	$1$	$1$	$35.454$	\(\Q\)	None	✓	✓			4440.2.a.i	$1$	$0$	\(0\)	\(1\)	\(1\)	\(3\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+3q^{7}+q^{9}-q^{11}+3q^{13}+\cdots\)
4440.2.a.j	$4440$	$2$	4440.a	1.a	$1$	$2$	$2$	$35.454$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		4440.2.a.j	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(0\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(1-2\beta )q^{7}+q^{9}+(3-\beta )q^{11}+\cdots\)
4440.2.a.k	$4440$	$2$	4440.a	1.a	$1$	$2$	$2$	$35.454$	\(\Q(\sqrt{17}) \)	None	✓	✓			4440.2.a.k	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-2\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2\beta q^{7}+q^{9}-4q^{11}+\cdots\)
4440.2.a.l	$4440$	$2$	4440.a	1.a	$1$	$2$	$2$	$35.454$	\(\Q(\sqrt{17}) \)	None	✓	✓			4440.2.a.l	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(-3\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(-1-\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
4440.2.a.m	$4440$	$2$	4440.a	1.a	$1$	$2$	$2$	$35.454$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		4440.2.a.m	$1$	$1$	\(0\)	\(2\)	\(2\)	\(-4\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(-1-2\beta )q^{7}+q^{9}+(-4+\cdots)q^{11}+\cdots\)
4440.2.a.n	$4440$	$2$	4440.a	1.a	$1$	$2$	$2$	$35.454$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	4440.2.a.n	$1$	$1$	\(0\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(1-2\beta )q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
4440.2.a.o	$4440$	$2$	4440.a	1.a	$1$	$3$	$3$	$35.454$	3.3.2429.1	None	✓	✓	✓		4440.2.a.o	$1$	$0$	\(0\)	\(-3\)	\(-3\)	\(2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(1-\beta _{1})q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
4440.2.a.p	$4440$	$2$	4440.a	1.a	$1$	$3$	$3$	$35.454$	3.3.2677.1	None	✓	✓	✓		4440.2.a.p	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(-3\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-q^{7}+q^{9}-\beta _{1}q^{11}+(-1+\cdots)q^{13}+\cdots\)
4440.2.a.q	$4440$	$2$	4440.a	1.a	$1$	$3$	$3$	$35.454$	3.3.229.1	None	✓	✓	✓	✓	4440.2.a.q	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(3\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{7}+q^{9}+(-\beta _{1}+2\beta _{2})q^{11}+\cdots\)
4440.2.a.r	$4440$	$2$	4440.a	1.a	$1$	$3$	$3$	$35.454$	3.3.229.1	None	✓	✓	✓	✓	4440.2.a.r	$1$	$1$	\(0\)	\(3\)	\(-3\)	\(-3\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-q^{7}+q^{9}+(-1+2\beta _{1}+\cdots)q^{11}+\cdots\)
4440.2.a.s	$4440$	$2$	4440.a	1.a	$1$	$3$	$3$	$35.454$	3.3.469.1	None	✓	✓	✓		4440.2.a.s	$1$	$1$	\(0\)	\(3\)	\(-3\)	\(-2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(-\beta _{1}+\beta _{2})q^{7}+q^{9}+\cdots\)
4440.2.a.t	$4440$	$2$	4440.a	1.a	$1$	$4$	$4$	$35.454$	4.4.119893.1	None	✓	✓	✓		4440.2.a.t	$1$	$1$	\(0\)	\(-4\)	\(-4\)	\(-1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-\beta _{3}q^{7}+q^{9}+(2+\beta _{1}+\cdots)q^{11}+\cdots\)
4440.2.a.u	$4440$	$2$	4440.a	1.a	$1$	$4$	$4$	$35.454$	4.4.48389.1	None	✓	✓	✓	✓	4440.2.a.u	$1$	$0$	\(0\)	\(-4\)	\(4\)	\(1\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(\beta _{1}+\beta _{3})q^{7}+q^{9}+(3+\cdots)q^{11}+\cdots\)
4440.2.a.v	$4440$	$2$	4440.a	1.a	$1$	$4$	$4$	$35.454$	\(\Q(\sqrt{50 +2 \sqrt{33}})\)	None	✓	✓	✓	✓	4440.2.a.v	$1$	$0$	\(0\)	\(4\)	\(-4\)	\(3\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(\beta _{1}-\beta _{2}-\beta _{3})q^{7}+q^{9}+\cdots\)
4440.2.a.w	$4440$	$2$	4440.a	1.a	$1$	$5$	$5$	$35.454$	5.5.20193189.1	None	✓	✓	✓		4440.2.a.w	$1$	$1$	\(0\)	\(-5\)	\(-5\)	\(2\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-\beta _{4}q^{7}+q^{9}+(\beta _{1}+\beta _{4})q^{11}+\cdots\)
4440.2.a.x	$4440$	$2$	4440.a	1.a	$1$	$5$	$5$	$35.454$	5.5.15147037.1	None	✓	✓	✓		4440.2.a.x	$1$	$0$	\(0\)	\(-5\)	\(5\)	\(-1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+\beta _{2}q^{7}+q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
4440.2.a.y	$4440$	$2$	4440.a	1.a	$1$	$5$	$5$	$35.454$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		4440.2.a.y	$1$	$0$	\(0\)	\(5\)	\(-5\)	\(3\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(1-\beta _{1})q^{7}+q^{9}+\beta _{3}q^{11}+\cdots\)
4440.2.a.z	$4440$	$2$	4440.a	1.a	$1$	$5$	$5$	$35.454$	5.5.13350941.1	None	✓	✓	✓		4440.2.a.z	$1$	$0$	\(0\)	\(5\)	\(5\)	\(7\)		$+$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(1-\beta _{2})q^{7}+q^{9}-\beta _{4}q^{11}+\cdots\)
4440.2.a.ba	$4440$	$2$	4440.a	1.a	$1$	$6$	$6$	$35.454$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		4440.2.a.ba	$1$	$0$	\(0\)	\(6\)	\(6\)	\(-1\)		$-$	$-$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-\beta _{1}q^{7}+q^{9}+(2-\beta _{4}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4440))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4440)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(222))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(296))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(444))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(555))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(740))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(888))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1110))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1480))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2220))\)\(^{\oplus 2}\)