Space of modular forms of level 6750, weight 2, and trivial character

• Label: 6750.2.a
• Level: $6750$
• Weight: $2$
• Character orbit: 6750.a
• Rep. character: $\chi_{6750}(1,\cdot)$
• Character field: $\Q$
• Dimension: $128$
• Newform subspaces: $38$
• Sturm bound: $2700$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 6750 = 2 \cdot 3^{3} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6750.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 38 \)
Sturm bound:			\(2700\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6750))\).

	Total	New	Old
Modular forms	1410	128	1282
Cusp forms	1291	128	1163
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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(14\)
\(+\)	\(+\)	\(-\)	\(-\)	\(18\)
\(+\)	\(-\)	\(+\)	\(-\)	\(18\)
\(+\)	\(-\)	\(-\)	\(+\)	\(14\)
\(-\)	\(+\)	\(+\)	\(-\)	\(18\)
\(-\)	\(+\)	\(-\)	\(+\)	\(14\)
\(-\)	\(-\)	\(+\)	\(+\)	\(14\)
\(-\)	\(-\)	\(-\)	\(-\)	\(18\)
Plus space			\(+\)	\(56\)
Minus space			\(-\)	\(72\)

Trace form

\( 128 q + 128 q^{4} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6750))\) 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
6750.2.a.a	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-3-\beta )q^{7}-q^{8}+(-1+\cdots)q^{11}+\cdots\)
6750.2.a.b	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.b	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1-2\beta )q^{7}-q^{8}+(-5+\cdots)q^{11}+\cdots\)
6750.2.a.c	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.c	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}+(-1+2\beta )q^{11}+\cdots\)
6750.2.a.d	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.d	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-\beta q^{7}-q^{8}+(-4+2\beta )q^{11}+\cdots\)
6750.2.a.e	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.d	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta q^{7}-q^{8}+(4-2\beta )q^{11}+\cdots\)
6750.2.a.f	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.c	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}+(-1+2\beta )q^{11}+\cdots\)
6750.2.a.g	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.b	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+2\beta )q^{7}-q^{8}+(5-\beta )q^{11}+\cdots\)
6750.2.a.h	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.a	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(4-\beta )q^{7}-q^{8}+(-1+\cdots)q^{11}+\cdots\)
6750.2.a.i	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.a	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-3-\beta )q^{7}+q^{8}+(1+\cdots)q^{11}+\cdots\)
6750.2.a.j	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.b	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1-2\beta )q^{7}+q^{8}+(5+\cdots)q^{11}+\cdots\)
6750.2.a.k	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.c	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}+(1-2\beta )q^{11}+\cdots\)
6750.2.a.l	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.d	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-\beta q^{7}+q^{8}+(4-2\beta )q^{11}+\cdots\)
6750.2.a.m	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.d	$1$	$1$	\(2\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{7}+q^{8}+(-4+2\beta )q^{11}+\cdots\)
6750.2.a.n	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.c	$1$	$1$	\(2\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}+(-1+2\beta )q^{11}+\cdots\)
6750.2.a.o	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.b	$1$	$1$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+2\beta )q^{7}+q^{8}+(-5+\cdots)q^{11}+\cdots\)
6750.2.a.p	$6750$	$2$	6750.a	1.a	$1$	$2$	$2$	$53.899$	\(\Q(\sqrt{5}) \)	None	✓	✓			6750.2.a.a	$1$	$0$	\(2\)	\(0\)	\(0\)	\(7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(4-\beta )q^{7}+q^{8}+(1-2\beta )q^{11}+\cdots\)
6750.2.a.q	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.4525.1	None	✓	✓			6750.2.a.q	$1$	$0$	\(-4\)	\(0\)	\(0\)	\(-5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1-\beta _{1})q^{7}-q^{8}+(2+\cdots)q^{11}+\cdots\)
6750.2.a.r	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.4525.1	None	✓	✓			6750.2.a.r	$1$	$0$	\(-4\)	\(0\)	\(0\)	\(-5\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1+\beta _{1}+\beta _{3})q^{7}-q^{8}+\cdots\)
6750.2.a.s	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	\(\Q(\sqrt{5}, \sqrt{29})\)	None	✓	✓			6750.2.a.s	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{2}q^{7}-q^{8}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
6750.2.a.t	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.12400.1	None	✓	✓			6750.2.a.t	$2$	$1$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{1}q^{7}-q^{8}-\beta _{3}q^{11}+\cdots\)
6750.2.a.u	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.24400.1	None	✓	✓			6750.2.a.u	$2$	$1$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-\beta _{3}q^{7}-q^{8}+(\beta _{1}+\beta _{3})q^{11}+\cdots\)
6750.2.a.v	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.12400.1	None	✓	✓			6750.2.a.v	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{1}q^{7}-q^{8}+(\beta _{1}+\beta _{3})q^{11}+\cdots\)
6750.2.a.w	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	\(\Q(\sqrt{5}, \sqrt{29})\)	None	✓	✓			6750.2.a.s	$1$	$0$	\(-4\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-\beta _{2}q^{7}-q^{8}+(-2+\beta _{1}+\cdots)q^{11}+\cdots\)
6750.2.a.x	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.4525.1	None	✓	✓			6750.2.a.q	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+\beta _{1})q^{7}-q^{8}+(-2+\cdots)q^{11}+\cdots\)
6750.2.a.y	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.4525.1	None	✓	✓			6750.2.a.r	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(1+\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots\)
6750.2.a.z	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.4525.1	None	✓	✓			6750.2.a.r	$1$	$1$	\(4\)	\(0\)	\(0\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1+\beta _{1}+\beta _{3})q^{7}+q^{8}+\cdots\)
6750.2.a.ba	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.4525.1	None	✓	✓	✓		6750.2.a.q	$1$	$1$	\(4\)	\(0\)	\(0\)	\(-5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-1-\beta _{1})q^{7}+q^{8}+(-2+\cdots)q^{11}+\cdots\)
6750.2.a.bb	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	\(\Q(\sqrt{5}, \sqrt{29})\)	None	✓	✓			6750.2.a.s	$1$	$1$	\(4\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta _{2}q^{7}+q^{8}+(-1-\beta _{1}+\cdots)q^{11}+\cdots\)
6750.2.a.bc	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.12400.1	None	✓	✓			6750.2.a.v	$2$	$1$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta _{1}q^{7}+q^{8}+(-\beta _{1}-\beta _{3})q^{11}+\cdots\)
6750.2.a.bd	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.24400.1	None	✓	✓			6750.2.a.u	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-\beta _{3}q^{7}+q^{8}+(-\beta _{1}-\beta _{3})q^{11}+\cdots\)
6750.2.a.be	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.12400.1	None	✓	✓			6750.2.a.t	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta _{1}q^{7}+q^{8}+\beta _{3}q^{11}+\cdots\)
6750.2.a.bf	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	\(\Q(\sqrt{5}, \sqrt{29})\)	None	✓	✓			6750.2.a.s	$1$	$0$	\(4\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+\beta _{2})q^{7}+q^{8}+(1-\beta _{1}+\cdots)q^{11}+\cdots\)
6750.2.a.bg	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.4525.1	None	✓	✓			6750.2.a.r	$1$	$0$	\(4\)	\(0\)	\(0\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots\)
6750.2.a.bh	$6750$	$2$	6750.a	1.a	$1$	$4$	$4$	$53.899$	4.4.4525.1	None	✓	✓			6750.2.a.q	$1$	$0$	\(4\)	\(0\)	\(0\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+\beta _{1})q^{7}+q^{8}+(2-\beta _{1}+\cdots)q^{11}+\cdots\)
6750.2.a.bi	$6750$	$2$	6750.a	1.a	$1$	$6$	$6$	$53.899$	6.6.2589817625.1	None	✓	✓	✓		6750.2.a.bi	$1$	$0$	\(-6\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-\beta _{1}q^{7}-q^{8}+(-1-\beta _{2}+\cdots)q^{11}+\cdots\)
6750.2.a.bj	$6750$	$2$	6750.a	1.a	$1$	$6$	$6$	$53.899$	6.6.2589817625.1	None	✓	✓	✓		6750.2.a.bi	$1$	$0$	\(-6\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{1}q^{7}-q^{8}+(1+\beta _{2}+\cdots)q^{11}+\cdots\)
6750.2.a.bk	$6750$	$2$	6750.a	1.a	$1$	$6$	$6$	$53.899$	6.6.2589817625.1	None	✓	✓	✓		6750.2.a.bi	$1$	$0$	\(6\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-\beta _{1}q^{7}+q^{8}+(1+\beta _{2}+\cdots)q^{11}+\cdots\)
6750.2.a.bl	$6750$	$2$	6750.a	1.a	$1$	$6$	$6$	$53.899$	6.6.2589817625.1	None	✓	✓	✓		6750.2.a.bi	$1$	$0$	\(6\)	\(0\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta _{1}q^{7}+q^{8}+(-1-\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6750))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6750)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 18}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(375))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(450))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(675))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(750))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1125))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1350))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2250))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3375))\)\(^{\oplus 2}\)