Space of modular forms of level 3700, weight 2, and trivial character

• Label: 3700.2.a
• Level: $3700$
• Weight: $2$
• Character orbit: 3700.a
• Rep. character: $\chi_{3700}(1,\cdot)$
• Character field: $\Q$
• Dimension: $57$
• Newform subspaces: $16$
• Sturm bound: $1140$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3700.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(1140\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3700))\).

	Total	New	Old
Modular forms	588	57	531
Cusp forms	553	57	496
Eisenstein series	35	0	35

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

\(2\)	\(5\)	\(37\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(72\)	\(0\)	\(72\)		\(67\)	\(0\)	\(67\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(75\)	\(0\)	\(75\)		\(69\)	\(0\)	\(69\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(76\)	\(0\)	\(76\)		\(70\)	\(0\)	\(70\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(73\)	\(0\)	\(73\)		\(67\)	\(0\)	\(67\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(75\)	\(15\)	\(60\)		\(72\)	\(15\)	\(57\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(72\)	\(12\)	\(60\)		\(69\)	\(12\)	\(57\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(71\)	\(14\)	\(57\)		\(68\)	\(14\)	\(54\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(74\)	\(16\)	\(58\)		\(71\)	\(16\)	\(55\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(288\)	\(26\)	\(262\)		\(271\)	\(26\)	\(245\)		\(17\)	\(0\)	\(17\)
Minus space			\(-\)		\(300\)	\(31\)	\(269\)		\(282\)	\(31\)	\(251\)		\(18\)	\(0\)	\(18\)

Trace form

57 * q - 2 * q^3 + 2 * q^7 + 55 * q^9 - 6 * q^11 - 4 * q^13 - 16 * q^17 - 8 * q^19 - 10 * q^21 + 6 * q^23 - 14 * q^27 - 2 * q^29 + 2 * q^31 - 10 * q^33 - q^37 + 2 * q^39 + 4 * q^41 - 32 * q^43 - 14 * q^47 + 63 * q^49 - 30 * q^51 - 12 * q^53 + 2 * q^57 - 14 * q^59 + 2 * q^61 + 20 * q^63 + 16 * q^67 + 4 * q^69 + 6 * q^71 - 4 * q^73 + 14 * q^77 - 12 * q^79 + 57 * q^81 - 6 * q^83 + 8 * q^87 - 4 * q^89 - 22 * q^91 + 12 * q^93 - 2 * q^97 - 28 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3700))\) 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	5	37
3700.2.a.a	$3700$	$2$	3700.a	1.a	$1$	$1$	$1$	$29.545$	\(\Q\)	None	✓				740.2.a.c	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+3q^{7}+6q^{9}+5q^{11}-2q^{13}+\cdots\)
3700.2.a.b	$3700$	$2$	3700.a	1.a	$1$	$1$	$1$	$29.545$	\(\Q\)	None	✓	✓			3700.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}+6q^{11}-4q^{13}+\cdots\)
3700.2.a.c	$3700$	$2$	3700.a	1.a	$1$	$1$	$1$	$29.545$	\(\Q\)	None	✓				740.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{7}-2q^{9}-3q^{11}+4q^{13}+\cdots\)
3700.2.a.d	$3700$	$2$	3700.a	1.a	$1$	$1$	$1$	$29.545$	\(\Q\)	None	✓				740.2.a.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{7}-2q^{9}-3q^{11}+6q^{13}+\cdots\)
3700.2.a.e	$3700$	$2$	3700.a	1.a	$1$	$1$	$1$	$29.545$	\(\Q\)	None	✓				148.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{7}-2q^{9}+5q^{11}+6q^{17}+\cdots\)
3700.2.a.f	$3700$	$2$	3700.a	1.a	$1$	$1$	$1$	$29.545$	\(\Q\)	None	✓	✓			3700.2.a.b	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}+6q^{11}+4q^{13}+\cdots\)
3700.2.a.g	$3700$	$2$	3700.a	1.a	$1$	$2$	$2$	$29.545$	\(\Q(\sqrt{17}) \)	None	✓				148.2.a.b	$1$	$1$	\(0\)	\(1\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{7}+(1+\beta )q^{9}+\beta q^{11}+\cdots\)
3700.2.a.h	$3700$	$2$	3700.a	1.a	$1$	$2$	$2$	$29.545$	\(\Q(\sqrt{3}) \)	None	✓				740.2.a.d	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-3+\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
3700.2.a.i	$3700$	$2$	3700.a	1.a	$1$	$3$	$3$	$29.545$	3.3.148.1	None	✓				740.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+(-2\beta _{1}+\beta _{2})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
3700.2.a.j	$3700$	$2$	3700.a	1.a	$1$	$4$	$4$	$29.545$	4.4.286164.1	None	✓				740.2.a.f	$1$	$0$	\(0\)	\(-3\)	\(0\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+(1+\beta _{1})q^{7}+(3+\beta _{2}+\cdots)q^{9}+\cdots\)
3700.2.a.k	$3700$	$2$	3700.a	1.a	$1$	$5$	$5$	$29.545$	5.5.454057.1	None	✓	✓			3700.2.a.k	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(\beta _{1}+\beta _{2})q^{7}+(\beta _{1}+\beta _{2})q^{9}+\cdots\)
3700.2.a.l	$3700$	$2$	3700.a	1.a	$1$	$5$	$5$	$29.545$	5.5.454057.1	None	✓	✓	✓		3700.2.a.k	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
3700.2.a.m	$3700$	$2$	3700.a	1.a	$1$	$6$	$6$	$29.545$	6.6.17268201.1	None	✓	✓	✓		3700.2.a.m	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+\beta _{3}q^{7}+(1+\beta _{1}+\beta _{3}+\beta _{5})q^{9}+\cdots\)
3700.2.a.n	$3700$	$2$	3700.a	1.a	$1$	$6$	$6$	$29.545$	6.6.17268201.1	None	✓	✓			3700.2.a.m	$1$	$0$	\(0\)	\(1\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-\beta _{3}q^{7}+(1+\beta _{1}+\beta _{3}+\beta _{5})q^{9}+\cdots\)
3700.2.a.o	$3700$	$2$	3700.a	1.a	$1$	$9$	$9$	$29.545$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		740.2.d.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{5}q^{3}+\beta _{4}q^{7}+(\beta _{1}+\beta _{3}-\beta _{4})q^{9}+\cdots\)
3700.2.a.p	$3700$	$2$	3700.a	1.a	$1$	$9$	$9$	$29.545$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		740.2.d.a	$1$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{5}q^{3}-\beta _{4}q^{7}+(\beta _{1}+\beta _{3}-\beta _{4})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3700))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3700)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(100))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(148))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(740))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(925))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1850))\)\(^{\oplus 2}\)