Space of modular forms of level 112, weight 4, and trivial character

• Label: 112.4.a
• Level: $112$
• Weight: $4$
• Character orbit: 112.a
• Rep. character: $\chi_{112}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $8$
• Sturm bound: $64$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(64\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(112))\).

	Total	New	Old
Modular forms	54	9	45
Cusp forms	42	9	33
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(15\)	\(2\)	\(13\)		\(12\)	\(2\)	\(10\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(12\)	\(2\)	\(10\)		\(9\)	\(2\)	\(7\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(14\)	\(2\)	\(12\)		\(11\)	\(2\)	\(9\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(13\)	\(3\)	\(10\)		\(10\)	\(3\)	\(7\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(28\)	\(5\)	\(23\)		\(22\)	\(5\)	\(17\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(26\)	\(4\)	\(22\)		\(20\)	\(4\)	\(16\)		\(6\)	\(0\)	\(6\)

Trace form

9 * q + 2 * q^5 + 7 * q^7 + 101 * q^9 - 76 * q^11 - 46 * q^13 + 72 * q^15 - 102 * q^17 + 24 * q^19 + 56 * q^23 + 247 * q^25 + 576 * q^27 - 242 * q^29 - 264 * q^31 - 384 * q^33 - 210 * q^35 + 326 * q^37 - 264 * q^39 + 98 * q^41 - 28 * q^43 + 90 * q^45 + 792 * q^47 + 441 * q^49 - 1200 * q^51 - 98 * q^53 - 1080 * q^55 + 40 * q^57 - 960 * q^59 - 1302 * q^61 + 315 * q^63 - 116 * q^65 + 36 * q^67 - 528 * q^69 + 840 * q^71 + 458 * q^73 + 2720 * q^75 + 476 * q^77 + 304 * q^79 + 1577 * q^81 + 1112 * q^83 - 700 * q^85 - 16 * q^87 - 150 * q^89 - 546 * q^91 - 1680 * q^93 + 1192 * q^95 + 906 * q^97 + 2308 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(112))\) 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
112.4.a.a	$112$	$4$	112.a	1.a	$1$	$1$	$1$	$6.608$	\(\Q\)	None	✓				14.4.a.a	$1$	$0$	\(0\)	\(-8\)	\(-14\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-14q^{5}+7q^{7}+37q^{9}+28q^{11}+\cdots\)
112.4.a.b	$112$	$4$	112.a	1.a	$1$	$1$	$1$	$6.608$	\(\Q\)	None	✓				56.4.a.b	$1$	$1$	\(0\)	\(-6\)	\(8\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}+8q^{5}+7q^{7}+9q^{9}-56q^{11}+\cdots\)
112.4.a.c	$112$	$4$	112.a	1.a	$1$	$1$	$1$	$6.608$	\(\Q\)	None	✓				28.4.a.b	$1$	$1$	\(0\)	\(-4\)	\(6\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{3}+6q^{5}-7q^{7}-11q^{9}+12q^{11}+\cdots\)
112.4.a.d	$112$	$4$	112.a	1.a	$1$	$1$	$1$	$6.608$	\(\Q\)	None	✓				56.4.a.a	$1$	$1$	\(0\)	\(2\)	\(-16\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2^{4}q^{5}+7q^{7}-23q^{9}-24q^{11}+\cdots\)
112.4.a.e	$112$	$4$	112.a	1.a	$1$	$1$	$1$	$6.608$	\(\Q\)	None	✓				14.4.a.b	$1$	$1$	\(0\)	\(2\)	\(-12\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-12q^{5}-7q^{7}-23q^{9}-48q^{11}+\cdots\)
112.4.a.f	$112$	$4$	112.a	1.a	$1$	$1$	$1$	$6.608$	\(\Q\)	None	✓				7.4.a.a	$1$	$0$	\(0\)	\(2\)	\(16\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2^{4}q^{5}+7q^{7}-23q^{9}+8q^{11}+\cdots\)
112.4.a.g	$112$	$4$	112.a	1.a	$1$	$1$	$1$	$6.608$	\(\Q\)	None	✓				28.4.a.a	$1$	$0$	\(0\)	\(10\)	\(-8\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{3}-8q^{5}+7q^{7}+73q^{9}+40q^{11}+\cdots\)
112.4.a.h	$112$	$4$	112.a	1.a	$1$	$2$	$2$	$6.608$	\(\Q(\sqrt{57}) \)	None	✓		✓	✓	56.4.a.c	$1$	$0$	\(0\)	\(2\)	\(22\)	\(-14\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(11+\beta )q^{5}-7q^{7}+(31+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(112))\) into lower level spaces

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