Space of modular forms of level 6288, weight 2, and trivial character

• Label: 6288.2.a
• Level: $6288$
• Weight: $2$
• Character orbit: 6288.a
• Rep. character: $\chi_{6288}(1,\cdot)$
• Character field: $\Q$
• Dimension: $130$
• Newform subspaces: $41$
• Sturm bound: $2112$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 6288 = 2^{4} \cdot 3 \cdot 131 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6288.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 41 \)
Sturm bound:			\(2112\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(7\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6288))\).

	Total	New	Old
Modular forms	1068	130	938
Cusp forms	1045	130	915
Eisenstein series	23	0	23

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

\(2\)	\(3\)	\(131\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(15\)
\(+\)	\(+\)	\(-\)	$-$	\(18\)
\(+\)	\(-\)	\(+\)	$-$	\(17\)
\(+\)	\(-\)	\(-\)	$+$	\(14\)
\(-\)	\(+\)	\(+\)	$-$	\(18\)
\(-\)	\(+\)	\(-\)	$+$	\(15\)
\(-\)	\(-\)	\(+\)	$+$	\(15\)
\(-\)	\(-\)	\(-\)	$-$	\(18\)
Plus space			\(+\)	\(59\)
Minus space			\(-\)	\(71\)

Trace form

\( 130 q - 2 q^{3} + 4 q^{5} + 130 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6288))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	131
6288.2.a.a	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}-5q^{7}+q^{9}-3q^{11}+\cdots\)
6288.2.a.b	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}+3q^{7}+q^{9}+5q^{11}+\cdots\)
6288.2.a.c	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+2q^{7}+q^{9}-3q^{11}+\cdots\)
6288.2.a.d	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-3q^{7}+q^{9}+3q^{11}+4q^{13}+\cdots\)
6288.2.a.e	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+5q^{7}+q^{9}+3q^{11}+4q^{13}+\cdots\)
6288.2.a.f	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+2q^{7}+q^{9}+q^{11}-5q^{13}+\cdots\)
6288.2.a.g	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+q^{9}+3q^{11}+q^{13}+\cdots\)
6288.2.a.h	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}-3q^{7}+q^{9}+2q^{11}+\cdots\)
6288.2.a.i	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-2q^{7}+q^{9}+3q^{11}+\cdots\)
6288.2.a.j	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{7}+q^{9}+q^{11}+4q^{13}+\cdots\)
6288.2.a.k	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+3q^{7}+q^{9}-q^{11}-2q^{13}+\cdots\)
6288.2.a.l	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+3q^{7}+q^{9}-q^{11}-2q^{13}+\cdots\)
6288.2.a.m	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{7}+q^{9}-2q^{13}-6q^{17}+\cdots\)
6288.2.a.n	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}-4q^{11}-2q^{13}+\cdots\)
6288.2.a.o	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+4q^{7}+q^{9}+4q^{11}+\cdots\)
6288.2.a.p	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(3\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}-3q^{7}+q^{9}+q^{11}-4q^{13}+\cdots\)
6288.2.a.q	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(4\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+q^{7}+q^{9}+6q^{11}-6q^{13}+\cdots\)
6288.2.a.r	$6288$	$2$	6288.a	1.a	$1$	$1$	$1$	$50.210$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(4\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+4q^{5}+4q^{7}+q^{9}+6q^{13}+\cdots\)
6288.2.a.s	$6288$	$2$	6288.a	1.a	$1$	$2$	$2$	$50.210$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2\beta q^{7}+q^{9}+(1-2\beta )q^{11}+\cdots\)
6288.2.a.t	$6288$	$2$	6288.a	1.a	$1$	$2$	$2$	$50.210$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(-3\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta )q^{5}+(1+\beta )q^{7}+q^{9}+\cdots\)
6288.2.a.u	$6288$	$2$	6288.a	1.a	$1$	$2$	$2$	$50.210$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2\beta q^{5}-4q^{7}+q^{9}-q^{11}+\cdots\)
6288.2.a.v	$6288$	$2$	6288.a	1.a	$1$	$3$	$3$	$50.210$	3.3.316.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-4\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{1}+\beta _{2})q^{5}+2\beta _{1}q^{7}+\cdots\)
6288.2.a.w	$6288$	$2$	6288.a	1.a	$1$	$3$	$3$	$50.210$	3.3.316.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(2\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+(-1-\beta _{2})q^{7}+\cdots\)
6288.2.a.x	$6288$	$2$	6288.a	1.a	$1$	$3$	$3$	$50.210$	3.3.961.1	None	✓	$1$	$0$	\(0\)	\(3\)	\(1\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{5}-\beta _{1}q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
6288.2.a.y	$6288$	$2$	6288.a	1.a	$1$	$4$	$4$	$50.210$	4.4.725.1	None	✓	$1$	$0$	\(0\)	\(-4\)	\(-8\)	\(8\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-2+\beta _{2}-\beta _{3})q^{5}+(2-2\beta _{1}+\cdots)q^{7}+\cdots\)
6288.2.a.z	$6288$	$2$	6288.a	1.a	$1$	$4$	$4$	$50.210$	4.4.1957.1	None	✓	$1$	$1$	\(0\)	\(-4\)	\(-4\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{1}-\beta _{2}-\beta _{3})q^{5}+(-2\beta _{1}+\cdots)q^{7}+\cdots\)
6288.2.a.ba	$6288$	$2$	6288.a	1.a	$1$	$4$	$4$	$50.210$	4.4.64268.1	None	✓	$1$	$0$	\(0\)	\(-4\)	\(0\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta _{3}q^{5}+(-\beta _{1}-\beta _{3})q^{7}+q^{9}+\cdots\)
6288.2.a.bb	$6288$	$2$	6288.a	1.a	$1$	$4$	$4$	$50.210$	4.4.9909.1	None	✓	$1$	$1$	\(0\)	\(4\)	\(-4\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1-\beta _{1})q^{5}-\beta _{2}q^{7}+q^{9}+\cdots\)
6288.2.a.bc	$6288$	$2$	6288.a	1.a	$1$	$4$	$4$	$50.210$	4.4.1957.1	None	✓	$1$	$0$	\(0\)	\(4\)	\(0\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3})q^{5}+(2-\beta _{2}+\cdots)q^{7}+\cdots\)
6288.2.a.bd	$6288$	$2$	6288.a	1.a	$1$	$5$	$5$	$50.210$	5.5.81589.1	None	✓	$1$	$1$	\(0\)	\(-5\)	\(-5\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(-1-\beta _{3})q^{5}+(1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
6288.2.a.be	$6288$	$2$	6288.a	1.a	$1$	$5$	$5$	$50.210$	5.5.324301.1	None	✓	$1$	$0$	\(0\)	\(-5\)	\(-1\)	\(9\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(\beta _{2}+\beta _{3})q^{5}+(2+\beta _{2})q^{7}+q^{9}+\cdots\)
6288.2.a.bf	$6288$	$2$	6288.a	1.a	$1$	$5$	$5$	$50.210$	5.5.301117.1	None	✓	$1$	$1$	\(0\)	\(5\)	\(-2\)	\(-5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-\beta _{1}+\beta _{4})q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots\)
6288.2.a.bg	$6288$	$2$	6288.a	1.a	$1$	$5$	$5$	$50.210$	5.5.535221.1	None	✓	$1$	$1$	\(0\)	\(5\)	\(-2\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(-1+\beta _{3}-\beta _{4})q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots\)
6288.2.a.bh	$6288$	$2$	6288.a	1.a	$1$	$6$	$6$	$50.210$	6.6.254064397.1	None	✓	$1$	$0$	\(0\)	\(-6\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-q^{3}+\beta _{3}q^{5}+\beta _{2}q^{7}+q^{9}+(-2+\cdots)q^{11}+\cdots\)
6288.2.a.bi	$6288$	$2$	6288.a	1.a	$1$	$6$	$6$	$50.210$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(3\)	\(-8\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{2})q^{5}+(-1-\beta _{1})q^{7}+\cdots\)
6288.2.a.bj	$6288$	$2$	6288.a	1.a	$1$	$6$	$6$	$50.210$	6.6.12062776.1	None	✓	$1$	$1$	\(0\)	\(-6\)	\(8\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{4})q^{5}+(1-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
6288.2.a.bk	$6288$	$2$	6288.a	1.a	$1$	$6$	$6$	$50.210$	6.6.45984149.1	None	✓	$1$	$1$	\(0\)	\(6\)	\(-2\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta _{5}q^{5}+(-\beta _{2}-\beta _{3})q^{7}+q^{9}+\cdots\)
6288.2.a.bl	$6288$	$2$	6288.a	1.a	$1$	$6$	$6$	$50.210$	6.6.72098748.1	None	✓	$1$	$0$	\(0\)	\(6\)	\(4\)	\(-4\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta _{1})q^{5}+(-1-\beta _{2}+\beta _{5})q^{7}+\cdots\)
6288.2.a.bm	$6288$	$2$	6288.a	1.a	$1$	$8$	$8$	$50.210$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(8\)	\(1\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+\beta _{1}q^{5}+\beta _{4}q^{7}+q^{9}+(2+\beta _{3}+\cdots)q^{11}+\cdots\)
6288.2.a.bn	$6288$	$2$	6288.a	1.a	$1$	$8$	$8$	$50.210$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(8\)	\(5\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1-\beta _{1})q^{5}-\beta _{5}q^{7}+q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
6288.2.a.bo	$6288$	$2$	6288.a	1.a	$1$	$11$	$11$	$50.210$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-11\)	\(7\)	\(-10\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{3}+(1-\beta _{1})q^{5}+(-1-\beta _{7})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6288))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6288)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(131))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(262))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(393))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(524))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(786))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1048))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1572))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2096))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3144))\)\(^{\oplus 2}\)