Space of modular forms of level 3724, weight 2, and trivial character

• Label: 3724.2.a
• Level: $3724$
• Weight: $2$
• Character orbit: 3724.a
• Rep. character: $\chi_{3724}(1,\cdot)$
• Character field: $\Q$
• Dimension: $61$
• Newform subspaces: $16$
• Sturm bound: $1120$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 3724 = 2^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3724.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(1120\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3724))\).

	Total	New	Old
Modular forms	584	61	523
Cusp forms	537	61	476
Eisenstein series	47	0	47

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

\(2\)	\(7\)	\(19\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(68\)	\(0\)	\(68\)		\(61\)	\(0\)	\(61\)		\(7\)	\(0\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)		\(80\)	\(0\)	\(80\)		\(72\)	\(0\)	\(72\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)		\(77\)	\(0\)	\(77\)		\(69\)	\(0\)	\(69\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)		\(71\)	\(0\)	\(71\)		\(63\)	\(0\)	\(63\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)		\(72\)	\(15\)	\(57\)		\(68\)	\(15\)	\(53\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)		\(72\)	\(13\)	\(59\)		\(68\)	\(13\)	\(55\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)		\(72\)	\(15\)	\(57\)		\(68\)	\(15\)	\(53\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)		\(72\)	\(18\)	\(54\)		\(68\)	\(18\)	\(50\)		\(4\)	\(0\)	\(4\)
Plus space			\(+\)		\(283\)	\(28\)	\(255\)		\(260\)	\(28\)	\(232\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(301\)	\(33\)	\(268\)		\(277\)	\(33\)	\(244\)		\(24\)	\(0\)	\(24\)

Trace form

61 * q + 2 * q^3 + q^5 + 61 * q^9 - 5 * q^11 - 4 * q^13 - 2 * q^15 + 3 * q^17 + q^19 + 62 * q^25 + 8 * q^27 - 10 * q^29 + 4 * q^31 + 2 * q^33 - 2 * q^37 - 8 * q^39 - 14 * q^41 + 7 * q^43 - 11 * q^45 - 11 * q^47 + 14 * q^51 - 40 * q^53 + q^55 - 2 * q^57 - 10 * q^59 + 5 * q^61 - 44 * q^65 + 28 * q^67 + 4 * q^69 + 30 * q^71 + 3 * q^73 + 97 * q^81 - 20 * q^83 - 83 * q^85 + 48 * q^87 + 8 * q^89 + 16 * q^93 - q^95 + 3 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3724))\) 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	7	19
3724.2.a.a	$3724$	$2$	3724.a	1.a	$1$	$1$	$1$	$29.736$	\(\Q\)	None	✓				76.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+q^{9}+5q^{11}+4q^{13}+\cdots\)
3724.2.a.b	$3724$	$2$	3724.a	1.a	$1$	$1$	$1$	$29.736$	\(\Q\)	None	✓				532.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-3q^{9}+4q^{11}-4q^{13}-6q^{17}+\cdots\)
3724.2.a.c	$3724$	$2$	3724.a	1.a	$1$	$2$	$2$	$29.736$	\(\Q(\sqrt{3}) \)	None	✓				532.2.i.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+(1-2\beta )q^{9}+(-3+2\beta )q^{11}+\cdots\)
3724.2.a.d	$3724$	$2$	3724.a	1.a	$1$	$2$	$2$	$29.736$	\(\Q(\sqrt{5}) \)	None	✓				532.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(4\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1+2\beta )q^{5}+(-2+\beta )q^{9}+\cdots\)
3724.2.a.e	$3724$	$2$	3724.a	1.a	$1$	$2$	$2$	$29.736$	\(\Q(\sqrt{21}) \)	None	✓				532.2.a.c	$1$	$1$	\(0\)	\(1\)	\(-6\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-3q^{5}+(2+\beta )q^{9}+(-1-\beta )q^{11}+\cdots\)
3724.2.a.f	$3724$	$2$	3724.a	1.a	$1$	$2$	$2$	$29.736$	\(\Q(\sqrt{3}) \)	None	✓				532.2.i.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(1+2\beta )q^{9}+(-3-2\beta )q^{11}+\cdots\)
3724.2.a.g	$3724$	$2$	3724.a	1.a	$1$	$2$	$2$	$29.736$	\(\Q(\sqrt{5}) \)	None	✓				532.2.a.b	$1$	$0$	\(0\)	\(3\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+q^{5}+(-1+3\beta )q^{9}+(-2+\cdots)q^{11}+\cdots\)
3724.2.a.h	$3724$	$2$	3724.a	1.a	$1$	$3$	$3$	$29.736$	3.3.404.1	None	✓				532.2.i.b	$1$	$1$	\(0\)	\(-2\)	\(-3\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(-1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
3724.2.a.i	$3724$	$2$	3724.a	1.a	$1$	$3$	$3$	$29.736$	3.3.733.1	None	✓				532.2.a.e	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{1}+\beta _{2})q^{5}+(2+\beta _{2})q^{9}+\cdots\)
3724.2.a.j	$3724$	$2$	3724.a	1.a	$1$	$3$	$3$	$29.736$	3.3.404.1	None	✓				532.2.i.b	$1$	$1$	\(0\)	\(2\)	\(3\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(1+\beta _{1}-\beta _{2})q^{5}+(1+\cdots)q^{9}+\cdots\)
3724.2.a.k	$3724$	$2$	3724.a	1.a	$1$	$5$	$5$	$29.736$	5.5.11350832.1	None	✓	✓	✓		3724.2.a.k	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{2}q^{5}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
3724.2.a.l	$3724$	$2$	3724.a	1.a	$1$	$5$	$5$	$29.736$	5.5.11350832.1	None	✓	✓			3724.2.a.k	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{2}q^{5}+(1+\beta _{1}+\beta _{2})q^{9}+\cdots\)
3724.2.a.m	$3724$	$2$	3724.a	1.a	$1$	$7$	$7$	$29.736$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				532.2.i.c	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{6}q^{5}+(2-\beta _{4}+\beta _{5})q^{9}+\cdots\)
3724.2.a.n	$3724$	$2$	3724.a	1.a	$1$	$7$	$7$	$29.736$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		532.2.i.c	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{6}q^{5}+(2-\beta _{4}+\beta _{5})q^{9}+\cdots\)
3724.2.a.o	$3724$	$2$	3724.a	1.a	$1$	$8$	$8$	$29.736$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		3724.2.a.o	$1$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{2}q^{5}+(\beta _{1}+\beta _{2}+\beta _{3})q^{9}+\cdots\)
3724.2.a.p	$3724$	$2$	3724.a	1.a	$1$	$8$	$8$	$29.736$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		3724.2.a.o	$1$	$0$	\(0\)	\(4\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(\beta _{1}+\beta _{2}+\beta _{3})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3724))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3724)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(133))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(266))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(532))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(931))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1862))\)\(^{\oplus 2}\)