Space of modular forms of level 3720, weight 2, and trivial character

• Label: 3720.2.a
• Level: $3720$
• Weight: $2$
• Character orbit: 3720.a
• Rep. character: $\chi_{3720}(1,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $23$
• Sturm bound: $1536$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(1536\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3720))\).

	Total	New	Old
Modular forms	784	60	724
Cusp forms	753	60	693
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.

\(2\)	\(3\)	\(5\)	\(31\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(44\)	\(3\)	\(41\)		\(43\)	\(3\)	\(40\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(50\)	\(4\)	\(46\)		\(48\)	\(4\)	\(44\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(50\)	\(5\)	\(45\)		\(48\)	\(5\)	\(43\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(50\)	\(3\)	\(47\)		\(48\)	\(3\)	\(45\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(54\)	\(5\)	\(49\)		\(52\)	\(5\)	\(47\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(48\)	\(2\)	\(46\)		\(46\)	\(2\)	\(44\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(48\)	\(2\)	\(46\)		\(46\)	\(2\)	\(44\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(48\)	\(6\)	\(42\)		\(46\)	\(6\)	\(40\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(52\)	\(4\)	\(48\)		\(50\)	\(4\)	\(46\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(46\)	\(4\)	\(42\)		\(44\)	\(4\)	\(40\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(50\)	\(4\)	\(46\)		\(48\)	\(4\)	\(44\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(50\)	\(3\)	\(47\)		\(48\)	\(3\)	\(45\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(46\)	\(3\)	\(43\)		\(44\)	\(3\)	\(41\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(52\)	\(5\)	\(47\)		\(50\)	\(5\)	\(45\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(48\)	\(4\)	\(44\)		\(46\)	\(4\)	\(42\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(48\)	\(3\)	\(45\)		\(46\)	\(3\)	\(43\)		\(2\)	\(0\)	\(2\)
Plus space				\(+\)		\(380\)	\(24\)	\(356\)		\(365\)	\(24\)	\(341\)		\(15\)	\(0\)	\(15\)
Minus space				\(-\)		\(404\)	\(36\)	\(368\)		\(388\)	\(36\)	\(352\)		\(16\)	\(0\)	\(16\)

Trace form

60 * q + 60 * q^9 + 8 * q^11 + 8 * q^19 + 60 * q^25 + 8 * q^33 + 8 * q^35 + 16 * q^37 - 24 * q^41 + 52 * q^49 + 8 * q^59 - 24 * q^65 - 24 * q^67 + 16 * q^69 - 32 * q^71 + 32 * q^79 + 60 * q^81 - 16 * q^83 + 16 * q^87 - 24 * q^89 + 48 * q^91 + 4 * q^93 + 16 * q^95 + 40 * q^97 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3720))\) 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	31
3720.2.a.a	$3720$	$2$	3720.a	1.a	$1$	$1$	$1$	$29.704$	\(\Q\)	None	✓	✓			3720.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{7}+q^{9}+4q^{13}+q^{15}+\cdots\)
3720.2.a.b	$3720$	$2$	3720.a	1.a	$1$	$1$	$1$	$29.704$	\(\Q\)	None	✓	✓			3720.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-q^{7}+q^{9}-3q^{11}-6q^{13}+\cdots\)
3720.2.a.c	$3720$	$2$	3720.a	1.a	$1$	$1$	$1$	$29.704$	\(\Q\)	None	✓	✓			3720.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(4\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+4q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
3720.2.a.d	$3720$	$2$	3720.a	1.a	$1$	$1$	$1$	$29.704$	\(\Q\)	None	✓	✓			3720.2.a.d	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-2\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-2q^{7}+q^{9}-q^{15}+2q^{17}+\cdots\)
3720.2.a.e	$3720$	$2$	3720.a	1.a	$1$	$1$	$1$	$29.704$	\(\Q\)	None	✓	✓			3720.2.a.e	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(0\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{9}-2q^{13}-q^{15}-2q^{17}+\cdots\)
3720.2.a.f	$3720$	$2$	3720.a	1.a	$1$	$1$	$1$	$29.704$	\(\Q\)	None	✓	✓			3720.2.a.f	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-3\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-3q^{7}+q^{9}+3q^{11}-2q^{13}+\cdots\)
3720.2.a.g	$3720$	$2$	3720.a	1.a	$1$	$1$	$1$	$29.704$	\(\Q\)	None	✓	✓			3720.2.a.g	$1$	$0$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{9}-2q^{13}+q^{15}+6q^{17}+\cdots\)
3720.2.a.h	$3720$	$2$	3720.a	1.a	$1$	$1$	$1$	$29.704$	\(\Q\)	None	✓	✓			3720.2.a.h	$1$	$0$	\(0\)	\(1\)	\(1\)	\(4\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+4q^{7}+q^{9}+6q^{11}-6q^{13}+\cdots\)
3720.2.a.i	$3720$	$2$	3720.a	1.a	$1$	$2$	$2$	$29.704$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		3720.2.a.i	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(2\)		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(1+\beta )q^{7}+q^{9}+(-1+\cdots)q^{13}+\cdots\)
3720.2.a.j	$3720$	$2$	3720.a	1.a	$1$	$2$	$2$	$29.704$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	3720.2.a.j	$1$	$1$	\(0\)	\(2\)	\(2\)	\(-4\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(-2+\beta )q^{7}+q^{9}-2\beta q^{11}+\cdots\)
3720.2.a.k	$3720$	$2$	3720.a	1.a	$1$	$2$	$2$	$29.704$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓		3720.2.a.k	$1$	$0$	\(0\)	\(2\)	\(2\)	\(6\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(3+\beta )q^{7}+q^{9}+(2-2\beta )q^{11}+\cdots\)
3720.2.a.l	$3720$	$2$	3720.a	1.a	$1$	$3$	$3$	$29.704$	3.3.404.1	None	✓	✓	✓		3720.2.a.l	$1$	$0$	\(0\)	\(-3\)	\(-3\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(-1-\beta _{1})q^{7}+q^{9}+4q^{11}+\cdots\)
3720.2.a.m	$3720$	$2$	3720.a	1.a	$1$	$3$	$3$	$29.704$	3.3.568.1	None	✓	✓	✓	✓	3720.2.a.m	$1$	$0$	\(0\)	\(-3\)	\(3\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+\beta _{1}q^{7}+q^{9}+(2+\beta _{2})q^{11}+\cdots\)
3720.2.a.n	$3720$	$2$	3720.a	1.a	$1$	$3$	$3$	$29.704$	3.3.316.1	None	✓	✓	✓	✓	3720.2.a.n	$1$	$1$	\(0\)	\(-3\)	\(3\)	\(2\)		$+$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(1+\beta _{2})q^{7}+q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
3720.2.a.o	$3720$	$2$	3720.a	1.a	$1$	$3$	$3$	$29.704$	3.3.148.1	None	✓	✓	✓	✓	3720.2.a.o	$1$	$1$	\(0\)	\(3\)	\(-3\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(-1+\beta _{1})q^{7}+q^{9}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)
3720.2.a.p	$3720$	$2$	3720.a	1.a	$1$	$3$	$3$	$29.704$	3.3.148.1	None	✓	✓	✓	✓	3720.2.a.p	$1$	$1$	\(0\)	\(3\)	\(3\)	\(-2\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+(-2\beta _{1}-\beta _{2})q^{7}+q^{9}+\cdots\)
3720.2.a.q	$3720$	$2$	3720.a	1.a	$1$	$4$	$4$	$29.704$	4.4.70164.1	None	✓	✓	✓	✓	3720.2.a.q	$1$	$1$	\(0\)	\(-4\)	\(-4\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-\beta _{3}q^{7}+q^{9}-2q^{11}+\cdots\)
3720.2.a.r	$3720$	$2$	3720.a	1.a	$1$	$4$	$4$	$29.704$	4.4.78292.1	None	✓	✓	✓	✓	3720.2.a.r	$1$	$0$	\(0\)	\(-4\)	\(-4\)	\(1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+\beta _{1}q^{7}+q^{9}+(\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
3720.2.a.s	$3720$	$2$	3720.a	1.a	$1$	$4$	$4$	$29.704$	4.4.17428.1	None	✓	✓	✓	✓	3720.2.a.s	$1$	$1$	\(0\)	\(-4\)	\(4\)	\(-4\)		$-$	$+$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(-1+\beta _{3})q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
3720.2.a.t	$3720$	$2$	3720.a	1.a	$1$	$4$	$4$	$29.704$	4.4.92692.1	None	✓	✓	✓		3720.2.a.t	$1$	$0$	\(0\)	\(-4\)	\(4\)	\(-1\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-\beta _{2}q^{7}+q^{9}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
3720.2.a.u	$3720$	$2$	3720.a	1.a	$1$	$5$	$5$	$29.704$	5.5.5547956.1	None	✓	✓	✓	✓	3720.2.a.u	$1$	$0$	\(0\)	\(5\)	\(-5\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(-\beta _{1}-\beta _{3})q^{7}+q^{9}+\cdots\)
3720.2.a.v	$3720$	$2$	3720.a	1.a	$1$	$5$	$5$	$29.704$	5.5.2294036.1	None	✓	✓	✓	✓	3720.2.a.v	$1$	$0$	\(0\)	\(5\)	\(-5\)	\(3\)		$-$	$-$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(1-\beta _{2})q^{7}+q^{9}+(1+\beta _{3}+\cdots)q^{11}+\cdots\)
3720.2.a.w	$3720$	$2$	3720.a	1.a	$1$	$5$	$5$	$29.704$	5.5.24504404.1	None	✓	✓	✓		3720.2.a.w	$1$	$0$	\(0\)	\(5\)	\(5\)	\(1\)		$+$	$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+\beta _{1}q^{7}+q^{9}-\beta _{4}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3720))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3720)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(93))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(124))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(186))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(248))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(310))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(372))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(465))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(620))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(744))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(930))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1860))\)\(^{\oplus 2}\)