Space of modular forms of level 1080, weight 6, and trivial character

• Label: 1080.6.a
• Level: $1080$
• Weight: $6$
• Character orbit: 1080.a
• Rep. character: $\chi_{1080}(1,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $20$
• Sturm bound: $1296$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1080.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(1296\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(1080))\).

	Total	New	Old
Modular forms	1104	80	1024
Cusp forms	1056	80	976
Eisenstein series	48	0	48

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.
\(+\)	\(+\)	\(+\)	\(+\)	\(9\)
\(+\)	\(+\)	\(-\)	\(-\)	\(10\)
\(+\)	\(-\)	\(+\)	\(-\)	\(11\)
\(+\)	\(-\)	\(-\)	\(+\)	\(10\)
\(-\)	\(+\)	\(+\)	\(-\)	\(10\)
\(-\)	\(+\)	\(-\)	\(+\)	\(9\)
\(-\)	\(-\)	\(+\)	\(+\)	\(10\)
\(-\)	\(-\)	\(-\)	\(-\)	\(11\)
Plus space			\(+\)	\(38\)
Minus space			\(-\)	\(42\)

Trace form

\( 80 q + 124 q^{7} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(1080))\) 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	5
1080.6.a.a	$1080$	$6$	1080.a	1.a	$1$	$1$	$1$	$173.215$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-25\)	\(-234\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}-234q^{7}-347q^{11}-33q^{13}+\cdots\)
1080.6.a.b	$1080$	$6$	1080.a	1.a	$1$	$1$	$1$	$173.215$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(25\)	\(-234\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}-234q^{7}+347q^{11}-33q^{13}+\cdots\)
1080.6.a.c	$1080$	$6$	1080.a	1.a	$1$	$2$	$2$	$173.215$	\(\Q(\sqrt{11}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-50\)	\(4\)		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(2+5\beta )q^{7}+(-124+19\beta )q^{11}+\cdots\)
1080.6.a.d	$1080$	$6$	1080.a	1.a	$1$	$2$	$2$	$173.215$	\(\Q(\sqrt{11}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(50\)	\(4\)		$+$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(2+5\beta )q^{7}+(124-19\beta )q^{11}+\cdots\)
1080.6.a.e	$1080$	$6$	1080.a	1.a	$1$	$3$	$3$	$173.215$	3.3.15881.1	None	✓	$1$	$0$	\(0\)	\(0\)	\(-75\)	\(-30\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3^{2}\cdot 11$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(-10+\beta _{2})q^{7}+(220+\beta _{1}+\cdots)q^{11}+\cdots\)
1080.6.a.f	$1080$	$6$	1080.a	1.a	$1$	$3$	$3$	$173.215$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-75\)	\(140\)		$+$	$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(47+\beta _{1})q^{7}+(54-\beta _{1}-2\beta _{2})q^{11}+\cdots\)
1080.6.a.g	$1080$	$6$	1080.a	1.a	$1$	$3$	$3$	$173.215$	3.3.15881.1	None	✓	$1$	$1$	\(0\)	\(0\)	\(75\)	\(-30\)		$+$	$-$	$-$	$+$	$2^{3}\cdot 3^{2}\cdot 11$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(-10+\beta _{2})q^{7}+(-220+\cdots)q^{11}+\cdots\)
1080.6.a.h	$1080$	$6$	1080.a	1.a	$1$	$3$	$3$	$173.215$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(75\)	\(140\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(47+\beta _{1})q^{7}+(-54+\beta _{1}+\cdots)q^{11}+\cdots\)
1080.6.a.i	$1080$	$6$	1080.a	1.a	$1$	$5$	$5$	$173.215$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-125\)	\(-35\)		$-$	$-$	$+$	$+$	$2^{6}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(-7-\beta _{3})q^{7}+(12-\beta _{2}+\cdots)q^{11}+\cdots\)
1080.6.a.j	$1080$	$6$	1080.a	1.a	$1$	$5$	$5$	$173.215$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-125\)	\(-10\)		$+$	$+$	$+$	$+$	$2^{9}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(-2-\beta _{1})q^{7}+(22-\beta _{2}+\cdots)q^{11}+\cdots\)
1080.6.a.k	$1080$	$6$	1080.a	1.a	$1$	$5$	$5$	$173.215$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-125\)	\(4\)		$-$	$+$	$+$	$-$	$2^{6}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(1+\beta _{1})q^{7}+(-39-\beta _{3}+\cdots)q^{11}+\cdots\)
1080.6.a.l	$1080$	$6$	1080.a	1.a	$1$	$5$	$5$	$173.215$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-125\)	\(72\)		$+$	$-$	$+$	$-$	$2^{9}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(14-\beta _{3})q^{7}+(34+\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
1080.6.a.m	$1080$	$6$	1080.a	1.a	$1$	$5$	$5$	$173.215$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-125\)	\(88\)		$-$	$-$	$+$	$+$	$2^{9}\cdot 3^{8}$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(18+\beta _{1})q^{7}+(-3^{3}-\beta _{3}+\cdots)q^{11}+\cdots\)
1080.6.a.n	$1080$	$6$	1080.a	1.a	$1$	$5$	$5$	$173.215$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(125\)	\(-35\)		$+$	$-$	$-$	$+$	$2^{6}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(-7-\beta _{3})q^{7}+(-12+\beta _{2}+\cdots)q^{11}+\cdots\)
1080.6.a.o	$1080$	$6$	1080.a	1.a	$1$	$5$	$5$	$173.215$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(125\)	\(-10\)		$-$	$-$	$-$	$-$	$2^{9}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(-2-\beta _{1})q^{7}+(-22+\beta _{2}+\cdots)q^{11}+\cdots\)
1080.6.a.p	$1080$	$6$	1080.a	1.a	$1$	$5$	$5$	$173.215$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(125\)	\(4\)		$+$	$+$	$-$	$-$	$2^{6}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(1+\beta _{1})q^{7}+(39+\beta _{3})q^{11}+\cdots\)
1080.6.a.q	$1080$	$6$	1080.a	1.a	$1$	$5$	$5$	$173.215$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(125\)	\(72\)		$-$	$+$	$-$	$+$	$2^{9}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(14-\beta _{3})q^{7}+(-34-\beta _{1}+\cdots)q^{11}+\cdots\)
1080.6.a.r	$1080$	$6$	1080.a	1.a	$1$	$5$	$5$	$173.215$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(125\)	\(88\)		$+$	$+$	$-$	$-$	$2^{9}\cdot 3^{8}$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(18+\beta _{1})q^{7}+(3^{3}+\beta _{3})q^{11}+\cdots\)
1080.6.a.s	$1080$	$6$	1080.a	1.a	$1$	$6$	$6$	$173.215$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-150\)	\(63\)		$+$	$-$	$+$	$-$	$2^{9}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q-5^{2}q^{5}+(11+\beta _{1})q^{7}+(7-\beta _{2})q^{11}+\cdots\)
1080.6.a.t	$1080$	$6$	1080.a	1.a	$1$	$6$	$6$	$173.215$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(150\)	\(63\)		$-$	$-$	$-$	$-$	$2^{9}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+5^{2}q^{5}+(11+\beta _{1})q^{7}+(-7+\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(1080))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(1080)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 24}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 16}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 16}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 18}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 16}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(270))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(360))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(540))\)\(^{\oplus 2}\)