Space of modular forms of level 112, weight 6, and trivial character

• Label: 112.6.a
• Level: $112$
• Weight: $6$
• Character orbit: 112.a
• Rep. character: $\chi_{112}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $11$
• Sturm bound: $96$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(96\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(112))\).

	Total	New	Old
Modular forms	86	15	71
Cusp forms	74	15	59
Eisenstein series	12	0	12

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

\(2\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(-\)	$-$	\(4\)
\(-\)	\(+\)	$-$	\(4\)
\(-\)	\(-\)	$+$	\(3\)
Plus space		\(+\)	\(7\)
Minus space		\(-\)	\(8\)

Trace form

\( 15 q + 38 q^{5} - 49 q^{7} + 1259 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(112))\) 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	7
112.6.a.a	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-30\)	\(32\)	\(-49\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-30q^{3}+2^{5}q^{5}-7^{2}q^{7}+657q^{9}+\cdots\)
112.6.a.b	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-26\)	\(16\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-26q^{3}+2^{4}q^{5}+7^{2}q^{7}+433q^{9}+\cdots\)
112.6.a.c	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-10\)	\(84\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{3}+84q^{5}-7^{2}q^{7}-143q^{9}+\cdots\)
112.6.a.d	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-8\)	\(10\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+10q^{5}+7^{2}q^{7}-179q^{9}+\cdots\)
112.6.a.e	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-96\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-96q^{5}-7^{2}q^{7}-239q^{9}+\cdots\)
112.6.a.f	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(6\)	\(4\)	\(-49\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}+4q^{5}-7^{2}q^{7}-207q^{9}+240q^{11}+\cdots\)
112.6.a.g	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(14\)	\(-56\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+14q^{3}-56q^{5}+7^{2}q^{7}-47q^{9}+\cdots\)
112.6.a.h	$112$	$6$	112.a	1.a	$1$	$2$	$2$	$17.963$	\(\Q(\sqrt{57}) \)	None	✓	$1$	$0$	\(0\)	\(6\)	\(-18\)	\(-98\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+3\beta )q^{3}+(-9+5\beta )q^{5}-7^{2}q^{7}+\cdots\)
112.6.a.i	$112$	$6$	112.a	1.a	$1$	$2$	$2$	$17.963$	\(\Q(\sqrt{345}) \)	None	✓	$1$	$0$	\(0\)	\(6\)	\(82\)	\(98\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}+(41-3\beta )q^{5}+7^{2}q^{7}+\cdots\)
112.6.a.j	$112$	$6$	112.a	1.a	$1$	$2$	$2$	$17.963$	\(\Q(\sqrt{193}) \)	None	✓	$1$	$0$	\(0\)	\(14\)	\(42\)	\(98\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{3}+(21-5\beta )q^{5}+7^{2}q^{7}+\cdots\)
112.6.a.k	$112$	$6$	112.a	1.a	$1$	$2$	$2$	$17.963$	\(\Q(\sqrt{177}) \)	None	✓	$1$	$1$	\(0\)	\(26\)	\(-62\)	\(-98\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(13-\beta )q^{3}+(-31+5\beta )q^{5}-7^{2}q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(112))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(112)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)