Space of modular forms of level 112, weight 10, and trivial character

• Label: 112.10.a
• Level: $112$
• Weight: $10$
• Character orbit: 112.a
• Rep. character: $\chi_{112}(1,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $11$
• Sturm bound: $160$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(160\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(112))\).

	Total	New	Old
Modular forms	150	27	123
Cusp forms	138	27	111
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
\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(-\)	\(-\)	\(7\)
\(-\)	\(+\)	\(-\)	\(7\)
\(-\)	\(-\)	\(+\)	\(6\)
Plus space		\(+\)	\(13\)
Minus space		\(-\)	\(14\)

Trace form

27 * q + 718 * q^5 - 2401 * q^7 + 204887 * q^9 + 54916 * q^11 + 86158 * q^13 - 81528 * q^15 - 344210 * q^17 + 480888 * q^19 - 812552 * q^23 + 11407757 * q^25 + 5798016 * q^27 + 746738 * q^29 - 6344712 * q^31 + 9585408 * q^33 + 9003750 * q^35 - 2210198 * q^37 - 83320200 * q^39 - 5961738 * q^41 + 83940724 * q^43 + 23553990 * q^45 - 173719272 * q^47 + 155649627 * q^49 + 229586448 * q^51 + 3292082 * q^53 - 246809592 * q^55 + 58487224 * q^57 + 86244480 * q^59 - 90027306 * q^61 - 78764805 * q^63 + 154041124 * q^65 + 181960980 * q^67 - 382360176 * q^69 - 285033976 * q^71 - 471146066 * q^73 - 501618592 * q^75 - 103022108 * q^77 + 978029168 * q^79 + 1432922699 * q^81 + 28717816 * q^83 - 324885860 * q^85 + 4293732080 * q^87 - 388851410 * q^89 + 411449766 * q^91 + 1674528144 * q^93 - 181551768 * q^95 + 344333406 * q^97 + 409926388 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(112))\) 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
112.10.a.a	$112$	$10$	112.a	1.a	$1$	$1$	$1$	$57.684$	\(\Q\)	None	✓				14.10.a.b	$1$	$1$	\(0\)	\(-170\)	\(544\)	\(2401\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-170q^{3}+544q^{5}+7^{4}q^{7}+9217q^{9}+\cdots\)
112.10.a.b	$112$	$10$	112.a	1.a	$1$	$1$	$1$	$57.684$	\(\Q\)	None	✓				14.10.a.a	$1$	$1$	\(0\)	\(6\)	\(560\)	\(2401\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}+560q^{5}+7^{4}q^{7}-19647q^{9}+\cdots\)
112.10.a.c	$112$	$10$	112.a	1.a	$1$	$2$	$2$	$57.684$	\(\Q(\sqrt{2305}) \)	None	✓				14.10.a.c	$1$	$0$	\(0\)	\(14\)	\(-2730\)	\(-4802\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(7+5\beta )q^{3}+(-1365-21\beta )q^{5}+\cdots\)
112.10.a.d	$112$	$10$	112.a	1.a	$1$	$2$	$2$	$57.684$	\(\Q(\sqrt{11209}) \)	None	✓				28.10.a.b	$1$	$1$	\(0\)	\(70\)	\(1554\)	\(4802\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(35-\beta )q^{3}+(777+19\beta )q^{5}+7^{4}q^{7}+\cdots\)
112.10.a.e	$112$	$10$	112.a	1.a	$1$	$2$	$2$	$57.684$	\(\Q(\sqrt{193}) \)	None	✓				7.10.a.a	$1$	$1$	\(0\)	\(86\)	\(-2238\)	\(4802\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(43+11\beta )q^{3}+(-1119+95\beta )q^{5}+\cdots\)
112.10.a.f	$112$	$10$	112.a	1.a	$1$	$2$	$2$	$57.684$	\(\Q(\sqrt{4561}) \)	None	✓				28.10.a.a	$1$	$0$	\(0\)	\(224\)	\(1596\)	\(-4802\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(112-\beta )q^{3}+(798-9\beta )q^{5}-7^{4}q^{7}+\cdots\)
112.10.a.g	$112$	$10$	112.a	1.a	$1$	$3$	$3$	$57.684$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				56.10.a.b	$1$	$1$	\(0\)	\(-84\)	\(-2958\)	\(-7203\)		$+$	$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-28+\beta _{1})q^{3}+(-986+5\beta _{1}-\beta _{2})q^{5}+\cdots\)
112.10.a.h	$112$	$10$	112.a	1.a	$1$	$3$	$3$	$57.684$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		7.10.a.b	$1$	$0$	\(0\)	\(-84\)	\(1554\)	\(-7203\)		$-$	$+$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-28+\beta _{1})q^{3}+(518-6\beta _{1}+7\beta _{2})q^{5}+\cdots\)
112.10.a.i	$112$	$10$	112.a	1.a	$1$	$3$	$3$	$57.684$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				56.10.a.a	$1$	$0$	\(0\)	\(92\)	\(274\)	\(7203\)		$+$	$-$	$-$	$2^{6}\cdot 3$	$\mathrm{SU}(2)$	\(q+(31+\beta _{1})q^{3}+(92+\beta _{1}-\beta _{2})q^{5}+\cdots\)
112.10.a.j	$112$	$10$	112.a	1.a	$1$	$4$	$4$	$57.684$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		56.10.a.d	$1$	$0$	\(0\)	\(-84\)	\(1540\)	\(9604\)		$+$	$-$	$-$	$2^{13}\cdot 3^{3}\cdot 17$	$\mathrm{SU}(2)$	\(q+(-21+\beta _{1})q^{3}+(385-2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
112.10.a.k	$112$	$10$	112.a	1.a	$1$	$4$	$4$	$57.684$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		56.10.a.c	$1$	$1$	\(0\)	\(-70\)	\(1022\)	\(-9604\)		$+$	$+$	$+$	$2^{11}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-17+\beta _{1})q^{3}+(255-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(112))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(112)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)