Space of modular forms of level 3750, weight 2, and trivial character

• Label: 3750.2.a
• Level: $3750$
• Weight: $2$
• Character orbit: 3750.a
• Rep. character: $\chi_{3750}(1,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $22$
• Sturm bound: $1500$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3750.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(1500\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3750))\).

	Total	New	Old
Modular forms	810	80	730
Cusp forms	691	80	611
Eisenstein series	119	0	119

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

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(91\)	\(8\)	\(83\)		\(77\)	\(8\)	\(69\)		\(14\)	\(0\)	\(14\)
\(+\)	\(+\)	\(-\)	\(-\)		\(111\)	\(12\)	\(99\)		\(96\)	\(12\)	\(84\)		\(15\)	\(0\)	\(15\)
\(+\)	\(-\)	\(+\)	\(-\)		\(104\)	\(10\)	\(94\)		\(89\)	\(10\)	\(79\)		\(15\)	\(0\)	\(15\)
\(+\)	\(-\)	\(-\)	\(+\)		\(99\)	\(10\)	\(89\)		\(84\)	\(10\)	\(74\)		\(15\)	\(0\)	\(15\)
\(-\)	\(+\)	\(+\)	\(-\)		\(104\)	\(12\)	\(92\)		\(89\)	\(12\)	\(77\)		\(15\)	\(0\)	\(15\)
\(-\)	\(+\)	\(-\)	\(+\)		\(99\)	\(8\)	\(91\)		\(84\)	\(8\)	\(76\)		\(15\)	\(0\)	\(15\)
\(-\)	\(-\)	\(+\)	\(+\)		\(106\)	\(6\)	\(100\)		\(91\)	\(6\)	\(85\)		\(15\)	\(0\)	\(15\)
\(-\)	\(-\)	\(-\)	\(-\)		\(96\)	\(14\)	\(82\)		\(81\)	\(14\)	\(67\)		\(15\)	\(0\)	\(15\)
Plus space			\(+\)		\(395\)	\(32\)	\(363\)		\(336\)	\(32\)	\(304\)		\(59\)	\(0\)	\(59\)
Minus space			\(-\)		\(415\)	\(48\)	\(367\)		\(355\)	\(48\)	\(307\)		\(60\)	\(0\)	\(60\)

Trace form

80 * q + 80 * q^4 + 80 * q^9 + 80 * q^16 - 20 * q^19 - 20 * q^21 + 20 * q^26 + 20 * q^29 - 20 * q^31 + 20 * q^34 + 80 * q^36 - 20 * q^39 + 20 * q^41 + 60 * q^49 + 80 * q^64 + 20 * q^74 - 20 * q^76 - 20 * q^79 + 80 * q^81 - 20 * q^84 + 20 * q^89 - 40 * q^91

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3750))\) 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
3750.2.a.a	$3750$	$2$	3750.a	1.a	$1$	$2$	$2$	$29.944$	\(\Q(\sqrt{5}) \)	None	✓	✓			3750.2.a.a	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}-\beta q^{7}-q^{8}+\cdots\)
3750.2.a.b	$3750$	$2$	3750.a	1.a	$1$	$2$	$2$	$29.944$	\(\Q(\sqrt{5}) \)	None	✓				150.2.g.b	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}+2q^{7}-q^{8}+\cdots\)
3750.2.a.c	$3750$	$2$	3750.a	1.a	$1$	$2$	$2$	$29.944$	\(\Q(\sqrt{5}) \)	None	✓	✓			3750.2.a.c	$1$	$1$	\(-2\)	\(2\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}-3\beta q^{7}-q^{8}+\cdots\)
3750.2.a.d	$3750$	$2$	3750.a	1.a	$1$	$2$	$2$	$29.944$	\(\Q(\sqrt{5}) \)	None	✓				150.2.g.a	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}+(-1-\beta )q^{7}+\cdots\)
3750.2.a.e	$3750$	$2$	3750.a	1.a	$1$	$2$	$2$	$29.944$	\(\Q(\sqrt{5}) \)	None	✓	✓			3750.2.a.c	$1$	$0$	\(2\)	\(-2\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+3\beta q^{7}+q^{8}+\cdots\)
3750.2.a.f	$3750$	$2$	3750.a	1.a	$1$	$2$	$2$	$29.944$	\(\Q(\sqrt{5}) \)	None	✓				150.2.g.a	$1$	$0$	\(2\)	\(-2\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+(1+\beta )q^{7}+\cdots\)
3750.2.a.g	$3750$	$2$	3750.a	1.a	$1$	$2$	$2$	$29.944$	\(\Q(\sqrt{5}) \)	None	✓				150.2.g.b	$1$	$1$	\(2\)	\(2\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}-2q^{7}+q^{8}+\cdots\)
3750.2.a.h	$3750$	$2$	3750.a	1.a	$1$	$2$	$2$	$29.944$	\(\Q(\sqrt{5}) \)	None	✓	✓			3750.2.a.a	$1$	$0$	\(2\)	\(2\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}+\beta q^{7}+q^{8}+\cdots\)
3750.2.a.i	$3750$	$2$	3750.a	1.a	$1$	$4$	$4$	$29.944$	\(\Q(\zeta_{15})^+\)	None	✓	✓	✓		3750.2.a.i	$1$	$1$	\(-4\)	\(-4\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)
3750.2.a.j	$3750$	$2$	3750.a	1.a	$1$	$4$	$4$	$29.944$	\(\Q(\zeta_{15})^+\)	None	✓	✓			3750.2.a.j	$1$	$0$	\(-4\)	\(-4\)	\(0\)	\(8\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}+(2+\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
3750.2.a.k	$3750$	$2$	3750.a	1.a	$1$	$4$	$4$	$29.944$	\(\Q(\zeta_{15})^+\)	None	✓	✓			3750.2.a.k	$1$	$1$	\(-4\)	\(4\)	\(0\)	\(-6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)
3750.2.a.l	$3750$	$2$	3750.a	1.a	$1$	$4$	$4$	$29.944$	\(\Q(\sqrt{150 +18 \sqrt{5}})\)	None	✓				150.2.g.c	$1$	$0$	\(-4\)	\(4\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}-\beta _{1}q^{7}-q^{8}+\cdots\)
3750.2.a.m	$3750$	$2$	3750.a	1.a	$1$	$4$	$4$	$29.944$	\(\Q(\zeta_{20})^+\)	None	✓				150.2.h.a	$1$	$1$	\(-4\)	\(4\)	\(0\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}+(1-\beta _{1}+\beta _{3})q^{7}+\cdots\)
3750.2.a.n	$3750$	$2$	3750.a	1.a	$1$	$4$	$4$	$29.944$	\(\Q(\zeta_{15})^+\)	None	✓	✓			3750.2.a.n	$1$	$0$	\(-4\)	\(4\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
3750.2.a.o	$3750$	$2$	3750.a	1.a	$1$	$4$	$4$	$29.944$	\(\Q(\zeta_{20})^+\)	None	✓				150.2.h.a	$1$	$1$	\(4\)	\(-4\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
3750.2.a.p	$3750$	$2$	3750.a	1.a	$1$	$4$	$4$	$29.944$	\(\Q(\zeta_{15})^+\)	None	✓	✓			3750.2.a.n	$1$	$1$	\(4\)	\(-4\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
3750.2.a.q	$3750$	$2$	3750.a	1.a	$1$	$4$	$4$	$29.944$	\(\Q(\sqrt{150 +18 \sqrt{5}})\)	None	✓				150.2.g.c	$1$	$0$	\(4\)	\(-4\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+\beta _{1}q^{7}+q^{8}+\cdots\)
3750.2.a.r	$3750$	$2$	3750.a	1.a	$1$	$4$	$4$	$29.944$	\(\Q(\zeta_{15})^+\)	None	✓	✓			3750.2.a.k	$1$	$0$	\(4\)	\(-4\)	\(0\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
3750.2.a.s	$3750$	$2$	3750.a	1.a	$1$	$4$	$4$	$29.944$	\(\Q(\zeta_{15})^+\)	None	✓	✓	✓		3750.2.a.j	$1$	$1$	\(4\)	\(4\)	\(0\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}+(-2+\beta _{1}+\cdots)q^{7}+\cdots\)
3750.2.a.t	$3750$	$2$	3750.a	1.a	$1$	$4$	$4$	$29.944$	\(\Q(\zeta_{15})^+\)	None	✓	✓			3750.2.a.i	$1$	$0$	\(4\)	\(4\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}+(2-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
3750.2.a.u	$3750$	$2$	3750.a	1.a	$1$	$8$	$8$	$29.944$	8.8.\(\cdots\).1	None	✓		✓		150.2.h.b	$1$	$0$	\(-8\)	\(-8\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$5^{2}$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}+(-1+\beta _{6}+\cdots)q^{7}+\cdots\)
3750.2.a.v	$3750$	$2$	3750.a	1.a	$1$	$8$	$8$	$29.944$	8.8.\(\cdots\).1	None	✓		✓		150.2.h.b	$1$	$0$	\(8\)	\(8\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$5^{2}$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}+(1-\beta _{6})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3750))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3750)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(375))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(625))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(750))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1250))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\)\(^{\oplus 2}\)