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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(4\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(3\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(6\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(4\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(5\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(3\)
Plus space				\(+\)	\(24\)
Minus space				\(-\)	\(36\)

Trace form

\( 60 q + 60 q^{9} + O(q^{10}) \)

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

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

\( S_{2}^{\mathrm{old}}(\Gamma_0(3720)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(93))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(124))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(186))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(248))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(310))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(372))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(465))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(620))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(744))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(930))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1240))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1860))\)\(^{\oplus 2}\)