Space of modular forms of level 112, weight 8, and trivial character

• Label: 112.8.a
• Level: $112$
• Weight: $8$
• Character orbit: 112.a
• Rep. character: $\chi_{112}(1,\cdot)$
• Character field: $\Q$
• Dimension: $21$
• Newform subspaces: $12$
• Sturm bound: $128$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(128\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(112))\).

	Total	New	Old
Modular forms	118	21	97
Cusp forms	106	21	85
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
\(+\)	\(+\)	\(+\)		\(31\)	\(5\)	\(26\)		\(28\)	\(5\)	\(23\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(28\)	\(5\)	\(23\)		\(25\)	\(5\)	\(20\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(30\)	\(5\)	\(25\)		\(27\)	\(5\)	\(22\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(29\)	\(6\)	\(23\)		\(26\)	\(6\)	\(20\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(60\)	\(11\)	\(49\)		\(54\)	\(11\)	\(43\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(58\)	\(10\)	\(48\)		\(52\)	\(10\)	\(42\)		\(6\)	\(0\)	\(6\)

Trace form

21 * q - 278 * q^5 + 343 * q^7 + 13073 * q^9 - 8588 * q^11 + 6554 * q^13 + 50952 * q^15 + 25426 * q^17 - 60584 * q^19 + 108856 * q^23 + 336187 * q^25 - 604800 * q^27 - 86266 * q^29 + 268024 * q^31 + 80640 * q^33 - 257250 * q^35 + 414718 * q^37 + 1107000 * q^39 + 79962 * q^41 - 1321436 * q^43 - 1013310 * q^45 - 360552 * q^47 + 2470629 * q^49 + 910416 * q^51 - 1461514 * q^53 - 710200 * q^55 - 315896 * q^57 + 6168960 * q^59 - 1836078 * q^61 + 1250235 * q^63 - 4414436 * q^65 + 5222180 * q^67 + 4790448 * q^69 - 845752 * q^71 + 6559394 * q^73 + 9928160 * q^75 - 2978612 * q^77 - 4082576 * q^79 + 50261 * q^81 + 7882456 * q^83 + 14604180 * q^85 - 14906512 * q^87 + 2186242 * q^89 - 4521426 * q^91 - 18562512 * q^93 - 6079128 * q^95 - 13940286 * q^97 - 33743612 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(112))\) 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
112.8.a.a	$112$	$8$	112.a	1.a	$1$	$1$	$1$	$34.987$	\(\Q\)	None	✓				56.8.a.b	$1$	$1$	\(0\)	\(-46\)	\(-160\)	\(343\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-46q^{3}-160q^{5}+7^{3}q^{7}-71q^{9}+\cdots\)
112.8.a.b	$112$	$8$	112.a	1.a	$1$	$1$	$1$	$34.987$	\(\Q\)	None	✓				56.8.a.a	$1$	$1$	\(0\)	\(18\)	\(160\)	\(343\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+18q^{3}+160q^{5}+7^{3}q^{7}-1863q^{9}+\cdots\)
112.8.a.c	$112$	$8$	112.a	1.a	$1$	$1$	$1$	$34.987$	\(\Q\)	None	✓				7.8.a.a	$1$	$1$	\(0\)	\(42\)	\(-84\)	\(-343\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+42q^{3}-84q^{5}-7^{3}q^{7}-423q^{9}+\cdots\)
112.8.a.d	$112$	$8$	112.a	1.a	$1$	$1$	$1$	$34.987$	\(\Q\)	None	✓				14.8.a.b	$1$	$0$	\(0\)	\(66\)	\(-400\)	\(343\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+66q^{3}-20^{2}q^{5}+7^{3}q^{7}+2169q^{9}+\cdots\)
112.8.a.e	$112$	$8$	112.a	1.a	$1$	$1$	$1$	$34.987$	\(\Q\)	None	✓				14.8.a.a	$1$	$0$	\(0\)	\(82\)	\(448\)	\(343\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+82q^{3}+448q^{5}+7^{3}q^{7}+4537q^{9}+\cdots\)
112.8.a.f	$112$	$8$	112.a	1.a	$1$	$2$	$2$	$34.987$	\(\Q(\sqrt{865}) \)	None	✓				7.8.a.b	$1$	$0$	\(0\)	\(-94\)	\(330\)	\(686\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-47-\beta )q^{3}+(165+5\beta )q^{5}+7^{3}q^{7}+\cdots\)
112.8.a.g	$112$	$8$	112.a	1.a	$1$	$2$	$2$	$34.987$	\(\Q(\sqrt{1969}) \)	None	✓				14.8.a.c	$1$	$1$	\(0\)	\(-70\)	\(126\)	\(-686\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-35-\beta )q^{3}+(63-9\beta )q^{5}-7^{3}q^{7}+\cdots\)
112.8.a.h	$112$	$8$	112.a	1.a	$1$	$2$	$2$	$34.987$	\(\Q(\sqrt{1009}) \)	None	✓				28.8.a.b	$1$	$0$	\(0\)	\(-14\)	\(-294\)	\(686\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{3}+(-147-11\beta )q^{5}+\cdots\)
112.8.a.i	$112$	$8$	112.a	1.a	$1$	$2$	$2$	$34.987$	\(\Q(\sqrt{3529}) \)	None	✓				28.8.a.a	$1$	$1$	\(0\)	\(14\)	\(42\)	\(-686\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{3}+(21+3\beta )q^{5}-7^{3}q^{7}+\cdots\)
112.8.a.j	$112$	$8$	112.a	1.a	$1$	$2$	$2$	$34.987$	\(\Q(\sqrt{249}) \)	None	✓				56.8.a.c	$1$	$0$	\(0\)	\(42\)	\(14\)	\(-686\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(21-3\beta )q^{3}+(7-11\beta )q^{5}-7^{3}q^{7}+\cdots\)
112.8.a.k	$112$	$8$	112.a	1.a	$1$	$3$	$3$	$34.987$	3.3.3109313.1	None	✓		✓		56.8.a.e	$1$	$0$	\(0\)	\(-28\)	\(138\)	\(-1029\)		$+$	$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-9+\beta _{1})q^{3}+(46-\beta _{1}-\beta _{2})q^{5}+\cdots\)
112.8.a.l	$112$	$8$	112.a	1.a	$1$	$3$	$3$	$34.987$	3.3.294792.1	None	✓		✓		56.8.a.d	$1$	$1$	\(0\)	\(-12\)	\(-598\)	\(1029\)		$+$	$-$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{3}+(-199-\beta _{1}-\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(112))\) into lower level spaces

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