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	Dim
$-$	$+$	$+$	$-$	$15$
$-$	$+$	$-$	$+$	$13$
$-$	$-$	$+$	$+$	$15$
$-$	$-$	$-$	$-$	$18$
Plus space			$+$	$28$
Minus space			$-$	$33$

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}$