Space of modular forms of level 2750, weight 2, and trivial character

• Label: 2750.2.a
• Level: $2750$
• Weight: $2$
• Character orbit: 2750.a
• Rep. character: $\chi_{2750}(1,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $20$
• Sturm bound: $900$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 2750 = 2 \cdot 5^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2750.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(900\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2750))\).

	Total	New	Old
Modular forms	470	80	390
Cusp forms	431	80	351
Eisenstein series	39	0	39

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

\(2\)	\(5\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(+\)	\(-\)	\(-\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)	\(14\)
\(+\)	\(-\)	\(-\)	\(+\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)	\(12\)
\(-\)	\(+\)	\(-\)	\(+\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)	\(8\)
\(-\)	\(-\)	\(-\)	\(-\)	\(14\)
Plus space			\(+\)	\(28\)
Minus space			\(-\)	\(52\)

Trace form

\( 80 q + 80 q^{4} - 4 q^{6} + 76 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2750))\) 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	11
2750.2.a.a	$2750$	$2$	2750.a	1.a	$1$	$2$	$2$	$21.959$	\(\Q(\sqrt{5}) \)	None	✓	✓			2750.2.a.a	$1$	$1$	\(-2\)	\(-3\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
2750.2.a.b	$2750$	$2$	2750.a	1.a	$1$	$2$	$2$	$21.959$	\(\Q(\sqrt{5}) \)	None	✓	✓			2750.2.a.b	$1$	$1$	\(-2\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+(1-2\beta )q^{7}+\cdots\)
2750.2.a.c	$2750$	$2$	2750.a	1.a	$1$	$2$	$2$	$21.959$	\(\Q(\sqrt{5}) \)	None	✓	✓			2750.2.a.c	$1$	$1$	\(-2\)	\(1\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+q^{7}-q^{8}+\cdots\)
2750.2.a.d	$2750$	$2$	2750.a	1.a	$1$	$2$	$2$	$21.959$	\(\Q(\sqrt{5}) \)	None	✓	✓			2750.2.a.c	$1$	$1$	\(2\)	\(-1\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta q^{3}+q^{4}-\beta q^{6}-q^{7}+q^{8}+\cdots\)
2750.2.a.e	$2750$	$2$	2750.a	1.a	$1$	$2$	$2$	$21.959$	\(\Q(\sqrt{5}) \)	None	✓	✓			2750.2.a.b	$1$	$1$	\(2\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta q^{3}+q^{4}-\beta q^{6}+(-1+2\beta )q^{7}+\cdots\)
2750.2.a.f	$2750$	$2$	2750.a	1.a	$1$	$2$	$2$	$21.959$	\(\Q(\sqrt{5}) \)	None	✓	✓			2750.2.a.a	$1$	$0$	\(2\)	\(3\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
2750.2.a.g	$2750$	$2$	2750.a	1.a	$1$	$4$	$4$	$21.959$	4.4.66025.1	None	✓	✓			2750.2.a.g	$1$	$0$	\(-4\)	\(-6\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-2+\beta _{2})q^{3}+q^{4}+(2-\beta _{2}+\cdots)q^{6}+\cdots\)
2750.2.a.h	$2750$	$2$	2750.a	1.a	$1$	$4$	$4$	$21.959$	4.4.4525.1	None	✓	✓	✓		2750.2.a.h	$1$	$1$	\(-4\)	\(-1\)	\(0\)	\(-6\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots\)
2750.2.a.i	$2750$	$2$	2750.a	1.a	$1$	$4$	$4$	$21.959$	4.4.2525.1	None	✓	✓	✓		2750.2.a.i	$1$	$1$	\(-4\)	\(3\)	\(0\)	\(-6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta _{2}-\beta _{3})q^{3}+q^{4}+(-1+\cdots)q^{6}+\cdots\)
2750.2.a.j	$2750$	$2$	2750.a	1.a	$1$	$4$	$4$	$21.959$	4.4.2525.1	None	✓	✓			2750.2.a.j	$1$	$0$	\(-4\)	\(3\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(\beta _{1}-\beta _{2}+\beta _{3})q^{3}+q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
2750.2.a.k	$2750$	$2$	2750.a	1.a	$1$	$4$	$4$	$21.959$	4.4.2525.1	None	✓	✓			2750.2.a.k	$1$	$0$	\(-4\)	\(3\)	\(0\)	\(10\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta _{2}-\beta _{3})q^{3}+q^{4}+(-1+\cdots)q^{6}+\cdots\)
2750.2.a.l	$2750$	$2$	2750.a	1.a	$1$	$4$	$4$	$21.959$	4.4.2525.1	None	✓	✓	✓		2750.2.a.k	$1$	$1$	\(4\)	\(-3\)	\(0\)	\(-10\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1-\beta _{3})q^{3}+q^{4}+(-1-\beta _{3})q^{6}+\cdots\)
2750.2.a.m	$2750$	$2$	2750.a	1.a	$1$	$4$	$4$	$21.959$	4.4.2525.1	None	✓	✓			2750.2.a.j	$1$	$0$	\(4\)	\(-3\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-\beta _{1}+\beta _{2}-\beta _{3})q^{3}+q^{4}+\cdots\)
2750.2.a.n	$2750$	$2$	2750.a	1.a	$1$	$4$	$4$	$21.959$	4.4.2525.1	None	✓	✓			2750.2.a.i	$1$	$0$	\(4\)	\(-3\)	\(0\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1-\beta _{3})q^{3}+q^{4}+(-1-\beta _{3})q^{6}+\cdots\)
2750.2.a.o	$2750$	$2$	2750.a	1.a	$1$	$4$	$4$	$21.959$	4.4.4525.1	None	✓	✓			2750.2.a.h	$1$	$0$	\(4\)	\(1\)	\(0\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1-\beta _{2}+\cdots)q^{7}+\cdots\)
2750.2.a.p	$2750$	$2$	2750.a	1.a	$1$	$4$	$4$	$21.959$	4.4.66025.1	None	✓	✓			2750.2.a.g	$1$	$0$	\(4\)	\(6\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(2-\beta _{2})q^{3}+q^{4}+(2-\beta _{2})q^{6}+\cdots\)
2750.2.a.q	$2750$	$2$	2750.a	1.a	$1$	$6$	$6$	$21.959$	6.6.11973625.1	None	✓	✓			2750.2.a.q	$1$	$0$	\(-6\)	\(4\)	\(0\)	\(9\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-\beta _{1})q^{3}+q^{4}+(-1+\beta _{1}+\cdots)q^{6}+\cdots\)
2750.2.a.r	$2750$	$2$	2750.a	1.a	$1$	$6$	$6$	$21.959$	6.6.11973625.1	None	✓	✓	✓		2750.2.a.q	$1$	$1$	\(6\)	\(-4\)	\(0\)	\(-9\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(-1+\beta _{1}+\cdots)q^{6}+\cdots\)
2750.2.a.s	$2750$	$2$	2750.a	1.a	$1$	$8$	$8$	$21.959$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		2750.2.a.s	$1$	$0$	\(-8\)	\(-3\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
2750.2.a.t	$2750$	$2$	2750.a	1.a	$1$	$8$	$8$	$21.959$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		2750.2.a.s	$1$	$0$	\(8\)	\(3\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(\beta _{1}-\beta _{6}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2750))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2750)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(550))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1375))\)\(^{\oplus 2}\)