Space of modular forms of level 1875, weight 2, and trivial character

• Label: 1875.2.a
• Level: $1875$
• Weight: $2$
• Character orbit: 1875.a
• Rep. character: $\chi_{1875}(1,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $16$
• Sturm bound: $500$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(500\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1875))\).

	Total	New	Old
Modular forms	280	80	200
Cusp forms	221	80	141
Eisenstein series	59	0	59

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

\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(65\)	\(18\)	\(47\)		\(51\)	\(18\)	\(33\)		\(14\)	\(0\)	\(14\)
\(+\)	\(-\)	\(-\)		\(75\)	\(22\)	\(53\)		\(60\)	\(22\)	\(38\)		\(15\)	\(0\)	\(15\)
\(-\)	\(+\)	\(-\)		\(75\)	\(26\)	\(49\)		\(60\)	\(26\)	\(34\)		\(15\)	\(0\)	\(15\)
\(-\)	\(-\)	\(+\)		\(65\)	\(14\)	\(51\)		\(50\)	\(14\)	\(36\)		\(15\)	\(0\)	\(15\)
Plus space		\(+\)		\(130\)	\(32\)	\(98\)		\(101\)	\(32\)	\(69\)		\(29\)	\(0\)	\(29\)
Minus space		\(-\)		\(150\)	\(48\)	\(102\)		\(120\)	\(48\)	\(72\)		\(30\)	\(0\)	\(30\)

Trace form

80 * q + 80 * q^4 + 80 * q^9 + 80 * q^16 + 10 * q^19 + 10 * q^21 - 20 * q^26 - 20 * q^29 + 10 * q^31 - 20 * q^34 + 80 * q^36 + 10 * q^39 - 20 * q^41 + 60 * q^44 + 60 * q^46 + 90 * q^49 + 60 * q^56 - 10 * q^61 + 140 * q^64 + 40 * q^74 + 40 * q^76 + 40 * q^79 + 80 * q^81 + 40 * q^84 + 60 * q^86 - 20 * q^89 + 50 * q^91 - 40 * q^94 - 40 * q^96

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5
1875.2.a.a	$1875$	$2$	1875.a	1.a	$1$	$2$	$2$	$14.972$	\(\Q(\sqrt{5}) \)	None	✓				75.2.g.a	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}+(2-4\beta )q^{7}+\cdots\)
1875.2.a.b	$1875$	$2$	1875.a	1.a	$1$	$2$	$2$	$14.972$	\(\Q(\sqrt{5}) \)	None	✓	✓			1875.2.a.b	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
1875.2.a.c	$1875$	$2$	1875.a	1.a	$1$	$2$	$2$	$14.972$	\(\Q(\sqrt{5}) \)	None	✓	✓			1875.2.a.b	$1$	$1$	\(1\)	\(2\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
1875.2.a.d	$1875$	$2$	1875.a	1.a	$1$	$2$	$2$	$14.972$	\(\Q(\sqrt{5}) \)	None	✓				75.2.g.a	$1$	$0$	\(2\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{6}+(-2+4\beta )q^{7}+\cdots\)
1875.2.a.e	$1875$	$2$	1875.a	1.a	$1$	$4$	$4$	$14.972$	\(\Q(\sqrt{15 +2 \sqrt{5}})\)	None	✓				75.2.g.b	$1$	$1$	\(-2\)	\(-4\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
1875.2.a.f	$1875$	$2$	1875.a	1.a	$1$	$4$	$4$	$14.972$	\(\Q(\zeta_{15})^+\)	None	✓	✓			1875.2.a.f	$1$	$1$	\(-1\)	\(4\)	\(0\)	\(-5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{2}+\beta _{3})q^{2}+q^{3}+\beta _{1}q^{4}+(\beta _{2}+\beta _{3})q^{6}+\cdots\)
1875.2.a.g	$1875$	$2$	1875.a	1.a	$1$	$4$	$4$	$14.972$	\(\Q(\zeta_{15})^+\)	None	✓	✓			1875.2.a.f	$1$	$1$	\(1\)	\(-4\)	\(0\)	\(5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{2}-\beta _{3})q^{2}-q^{3}+\beta _{1}q^{4}+(\beta _{2}+\cdots)q^{6}+\cdots\)
1875.2.a.h	$1875$	$2$	1875.a	1.a	$1$	$4$	$4$	$14.972$	\(\Q(\sqrt{15 +2 \sqrt{5}})\)	None	✓				75.2.g.b	$1$	$0$	\(2\)	\(4\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
1875.2.a.i	$1875$	$2$	1875.a	1.a	$1$	$6$	$6$	$14.972$	6.6.46840000.1	None	✓	✓			1875.2.a.i	$1$	$0$	\(-1\)	\(6\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
1875.2.a.j	$1875$	$2$	1875.a	1.a	$1$	$6$	$6$	$14.972$	6.6.44400625.1	None	✓		✓		75.2.g.c	$1$	$1$	\(0\)	\(-6\)	\(0\)	\(-6\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1-\beta _{3}+\beta _{4})q^{4}+\beta _{1}q^{6}+\cdots\)
1875.2.a.k	$1875$	$2$	1875.a	1.a	$1$	$6$	$6$	$14.972$	6.6.44400625.1	None	✓				75.2.g.c	$1$	$0$	\(0\)	\(6\)	\(0\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(1-\beta _{3}+\beta _{4})q^{4}+\beta _{1}q^{6}+\cdots\)
1875.2.a.l	$1875$	$2$	1875.a	1.a	$1$	$6$	$6$	$14.972$	6.6.46840000.1	None	✓	✓			1875.2.a.i	$1$	$0$	\(1\)	\(-6\)	\(0\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
1875.2.a.m	$1875$	$2$	1875.a	1.a	$1$	$8$	$8$	$14.972$	8.8.5444000000.1	None	✓		✓		75.2.i.a	$1$	$1$	\(-4\)	\(8\)	\(0\)	\(-8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1875.2.a.n	$1875$	$2$	1875.a	1.a	$1$	$8$	$8$	$14.972$	8.8.\(\cdots\).1	None	✓	✓			1875.2.a.n	$1$	$0$	\(-1\)	\(-8\)	\(0\)	\(-12\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{4}+\beta _{5})q^{4}+\beta _{1}q^{6}+\cdots\)
1875.2.a.o	$1875$	$2$	1875.a	1.a	$1$	$8$	$8$	$14.972$	8.8.\(\cdots\).1	None	✓	✓	✓		1875.2.a.n	$1$	$0$	\(1\)	\(8\)	\(0\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{4}+\beta _{5})q^{4}+\beta _{1}q^{6}+\cdots\)
1875.2.a.p	$1875$	$2$	1875.a	1.a	$1$	$8$	$8$	$14.972$	8.8.5444000000.1	None	✓				75.2.i.a	$1$	$0$	\(4\)	\(-8\)	\(0\)	\(8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1875))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1875)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(375))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(625))\)\(^{\oplus 2}\)