Space of modular forms of level 112, weight 2, and trivial character

• Label: 112.2.a
• Level: $112$
• Weight: $2$
• Character orbit: 112.a
• Rep. character: $\chi_{112}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $32$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(32\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(112))\).

	Total	New	Old
Modular forms	22	3	19
Cusp forms	11	3	8
Eisenstein series	11	0	11

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(4\)	\(1\)	\(3\)		\(2\)	\(1\)	\(1\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(7\)	\(1\)	\(6\)		\(4\)	\(1\)	\(3\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(2\)	\(1\)	\(1\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(6\)	\(0\)	\(6\)		\(3\)	\(0\)	\(3\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(10\)	\(1\)	\(9\)		\(5\)	\(1\)	\(4\)		\(5\)	\(0\)	\(5\)
Minus space		\(-\)		\(12\)	\(2\)	\(10\)		\(6\)	\(2\)	\(4\)		\(6\)	\(0\)	\(6\)

Trace form

3 * q - 2 * q^5 - q^7 - q^9 + 4 * q^11 - 2 * q^13 + 8 * q^15 - 2 * q^17 - 8 * q^19 - 8 * q^23 + 5 * q^25 + 2 * q^29 - 8 * q^31 + 6 * q^35 - 6 * q^37 - 8 * q^39 + 6 * q^41 - 12 * q^43 - 10 * q^45 + 24 * q^47 + 3 * q^49 + 16 * q^51 + 2 * q^53 + 8 * q^55 - 8 * q^57 + 6 * q^61 - 5 * q^63 + 4 * q^65 + 20 * q^67 + 16 * q^69 + 8 * q^71 - 2 * q^73 - 32 * q^75 + 4 * q^77 - 16 * q^79 - 13 * q^81 - 8 * q^83 - 4 * q^85 - 16 * q^87 - 2 * q^89 + 6 * q^91 + 16 * q^93 - 24 * q^95 - 18 * q^97 - 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(112))\) 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	7
112.2.a.a	$112$	$2$	112.a	1.a	$1$	$1$	$1$	$0.894$	\(\Q\)	None	✓		✓	✓	56.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{5}-q^{7}+q^{9}+8q^{15}+\cdots\)
112.2.a.b	$112$	$2$	112.a	1.a	$1$	$1$	$1$	$0.894$	\(\Q\)	None	✓		✓	✓	56.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+q^{7}-3q^{9}+4q^{11}+2q^{13}+\cdots\)
112.2.a.c	$112$	$2$	112.a	1.a	$1$	$1$	$1$	$0.894$	\(\Q\)	None	✓		✓	✓	14.2.a.a	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{7}+q^{9}-4q^{13}+6q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(112))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(112)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)