Space of modular forms of level 1110, weight 6, and trivial character

• Label: 1110.6.a
• Level: $1110$
• Weight: $6$
• Character orbit: 1110.a
• Rep. character: $\chi_{1110}(1,\cdot)$
• Character field: $\Q$
• Dimension: $120$
• Newform subspaces: $17$
• Sturm bound: $1368$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1110.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(1368\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(1110))\).

	Total	New	Old
Modular forms	1148	120	1028
Cusp forms	1132	120	1012
Eisenstein series	16	0	16

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

\(2\)	\(3\)	\(5\)	\(37\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	$+$	\(9\)
\(+\)	\(+\)	\(+\)	\(-\)	$-$	\(6\)
\(+\)	\(+\)	\(-\)	\(+\)	$-$	\(8\)
\(+\)	\(+\)	\(-\)	\(-\)	$+$	\(7\)
\(+\)	\(-\)	\(+\)	\(+\)	$-$	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)	$+$	\(7\)
\(+\)	\(-\)	\(-\)	\(+\)	$+$	\(6\)
\(+\)	\(-\)	\(-\)	\(-\)	$-$	\(9\)
\(-\)	\(+\)	\(+\)	\(+\)	$-$	\(7\)
\(-\)	\(+\)	\(+\)	\(-\)	$+$	\(8\)
\(-\)	\(+\)	\(-\)	\(+\)	$+$	\(7\)
\(-\)	\(+\)	\(-\)	\(-\)	$-$	\(8\)
\(-\)	\(-\)	\(+\)	\(+\)	$+$	\(7\)
\(-\)	\(-\)	\(+\)	\(-\)	$-$	\(8\)
\(-\)	\(-\)	\(-\)	\(+\)	$-$	\(10\)
\(-\)	\(-\)	\(-\)	\(-\)	$+$	\(5\)
Plus space				\(+\)	\(56\)
Minus space				\(-\)	\(64\)

Trace form

\( 120 q + 1920 q^{4} + 9720 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(1110))\) 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	37
1110.6.a.a	$1110$	$6$	1110.a	1.a	$1$	$1$	$1$	$178.026$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(-9\)	\(25\)	\(33\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+5^{2}q^{5}-6^{2}q^{6}+\cdots\)
1110.6.a.b	$1110$	$6$	1110.a	1.a	$1$	$5$	$5$	$178.026$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(20\)	\(45\)	\(125\)	\(-354\)		$-$	$-$	$-$	$-$	$+$	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}+5^{2}q^{5}+6^{2}q^{6}+\cdots\)
1110.6.a.c	$1110$	$6$	1110.a	1.a	$1$	$6$	$6$	$178.026$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-24\)	\(-54\)	\(-150\)	\(-38\)		$+$	$+$	$+$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}-5^{2}q^{5}+6^{2}q^{6}+\cdots\)
1110.6.a.d	$1110$	$6$	1110.a	1.a	$1$	$6$	$6$	$178.026$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(-24\)	\(54\)	\(150\)	\(-38\)		$+$	$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+5^{2}q^{5}-6^{2}q^{6}+\cdots\)
1110.6.a.e	$1110$	$6$	1110.a	1.a	$1$	$6$	$6$	$178.026$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(24\)	\(-54\)	\(150\)	\(-142\)		$-$	$+$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+5^{2}q^{5}-6^{2}q^{6}+\cdots\)
1110.6.a.f	$1110$	$6$	1110.a	1.a	$1$	$7$	$7$	$178.026$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-28\)	\(-63\)	\(175\)	\(109\)		$+$	$+$	$-$	$-$	$+$	$2^{2}\cdot 3\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+5^{2}q^{5}+6^{2}q^{6}+\cdots\)
1110.6.a.g	$1110$	$6$	1110.a	1.a	$1$	$7$	$7$	$178.026$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-28\)	\(63\)	\(-175\)	\(-87\)		$+$	$-$	$+$	$-$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}-5^{2}q^{5}-6^{2}q^{6}+\cdots\)
1110.6.a.h	$1110$	$6$	1110.a	1.a	$1$	$7$	$7$	$178.026$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(28\)	\(-63\)	\(-175\)	\(38\)		$-$	$+$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}-5^{2}q^{5}-6^{2}q^{6}+\cdots\)
1110.6.a.i	$1110$	$6$	1110.a	1.a	$1$	$7$	$7$	$178.026$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(28\)	\(63\)	\(-175\)	\(-207\)		$-$	$-$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}-5^{2}q^{5}+6^{2}q^{6}+\cdots\)
1110.6.a.j	$1110$	$6$	1110.a	1.a	$1$	$8$	$8$	$178.026$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(-32\)	\(-72\)	\(200\)	\(-87\)		$+$	$+$	$-$	$+$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+5^{2}q^{5}+6^{2}q^{6}+\cdots\)
1110.6.a.k	$1110$	$6$	1110.a	1.a	$1$	$8$	$8$	$178.026$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(-32\)	\(72\)	\(-200\)	\(207\)		$+$	$-$	$+$	$+$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}-5^{2}q^{5}-6^{2}q^{6}+\cdots\)
1110.6.a.l	$1110$	$6$	1110.a	1.a	$1$	$8$	$8$	$178.026$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(32\)	\(-72\)	\(-200\)	\(-60\)		$-$	$+$	$+$	$-$	$+$	$2^{2}\cdot 13$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}-5^{2}q^{5}-6^{2}q^{6}+\cdots\)
1110.6.a.m	$1110$	$6$	1110.a	1.a	$1$	$8$	$8$	$178.026$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(32\)	\(-72\)	\(200\)	\(185\)		$-$	$+$	$-$	$-$	$-$	$2^{4}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+4q^{2}-9q^{3}+2^{4}q^{4}+5^{2}q^{5}-6^{2}q^{6}+\cdots\)
1110.6.a.n	$1110$	$6$	1110.a	1.a	$1$	$8$	$8$	$178.026$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(32\)	\(72\)	\(-200\)	\(185\)		$-$	$-$	$+$	$-$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}-5^{2}q^{5}+6^{2}q^{6}+\cdots\)
1110.6.a.o	$1110$	$6$	1110.a	1.a	$1$	$9$	$9$	$178.026$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(-36\)	\(-81\)	\(-225\)	\(-38\)		$+$	$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}-5^{2}q^{5}+6^{2}q^{6}+\cdots\)
1110.6.a.p	$1110$	$6$	1110.a	1.a	$1$	$9$	$9$	$178.026$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(-36\)	\(81\)	\(225\)	\(60\)		$+$	$-$	$-$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q-4q^{2}+9q^{3}+2^{4}q^{4}+5^{2}q^{5}-6^{2}q^{6}+\cdots\)
1110.6.a.q	$1110$	$6$	1110.a	1.a	$1$	$10$	$10$	$178.026$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(40\)	\(90\)	\(250\)	\(234\)		$-$	$-$	$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+4q^{2}+9q^{3}+2^{4}q^{4}+5^{2}q^{5}+6^{2}q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(1110))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(1110)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(222))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(555))\)\(^{\oplus 2}\)