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 + 1028 * q^11 - 122 * q^13 - 3768 * q^15 + 1206 * q^17 + 2360 * q^19 - 3592 * q^23 + 7817 * q^25 - 576 * q^27 - 2246 * q^29 + 7160 * q^31 - 1920 * q^33 + 7350 * q^35 - 14734 * q^37 + 9144 * q^39 - 7042 * q^41 + 2420 * q^43 + 15390 * q^45 + 34200 * q^47 + 36015 * q^49 - 16368 * q^51 - 32486 * q^53 + 70408 * q^55 - 68648 * q^57 - 29760 * q^59 + 93870 * q^61 - 19845 * q^63 + 55444 * q^65 - 91052 * q^67 - 22320 * q^69 - 138744 * q^71 - 66218 * q^73 + 195488 * q^75 - 3724 * q^77 - 188752 * q^79 + 160415 * q^81 - 122632 * q^83 - 197940 * q^85 - 157072 * q^87 - 248298 * q^89 + 49686 * q^91 + 181968 * q^93 + 14632 * q^95 + 154662 * q^97 + 120628 * q^99

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	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
112.6.a.a	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓				56.6.a.b	$1$	$1$	\(0\)	\(-30\)	\(32\)	\(-49\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-30q^{3}+2^{5}q^{5}-7^{2}q^{7}+657q^{9}+\cdots\)
112.6.a.b	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓				28.6.a.b	$1$	$1$	\(0\)	\(-26\)	\(16\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-26q^{3}+2^{4}q^{5}+7^{2}q^{7}+433q^{9}+\cdots\)
112.6.a.c	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓				14.6.a.a	$1$	$0$	\(0\)	\(-10\)	\(84\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{3}+84q^{5}-7^{2}q^{7}-143q^{9}+\cdots\)
112.6.a.d	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓				14.6.a.b	$1$	$1$	\(0\)	\(-8\)	\(10\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+10q^{5}+7^{2}q^{7}-179q^{9}+\cdots\)
112.6.a.e	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓				28.6.a.a	$1$	$0$	\(0\)	\(2\)	\(-96\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-96q^{5}-7^{2}q^{7}-239q^{9}+\cdots\)
112.6.a.f	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓				56.6.a.a	$1$	$1$	\(0\)	\(6\)	\(4\)	\(-49\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}+4q^{5}-7^{2}q^{7}-207q^{9}+240q^{11}+\cdots\)
112.6.a.g	$112$	$6$	112.a	1.a	$1$	$1$	$1$	$17.963$	\(\Q\)	None	✓				7.6.a.a	$1$	$1$	\(0\)	\(14\)	\(-56\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+14q^{3}-56q^{5}+7^{2}q^{7}-47q^{9}+\cdots\)
112.6.a.h	$112$	$6$	112.a	1.a	$1$	$2$	$2$	$17.963$	\(\Q(\sqrt{57}) \)	None	✓		✓		7.6.a.b	$1$	$0$	\(0\)	\(6\)	\(-18\)	\(-98\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+3\beta )q^{3}+(-9+5\beta )q^{5}-7^{2}q^{7}+\cdots\)
112.6.a.i	$112$	$6$	112.a	1.a	$1$	$2$	$2$	$17.963$	\(\Q(\sqrt{345}) \)	None	✓				56.6.a.e	$1$	$0$	\(0\)	\(6\)	\(82\)	\(98\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{3}+(41-3\beta )q^{5}+7^{2}q^{7}+\cdots\)
112.6.a.j	$112$	$6$	112.a	1.a	$1$	$2$	$2$	$17.963$	\(\Q(\sqrt{193}) \)	None	✓				56.6.a.d	$1$	$0$	\(0\)	\(14\)	\(42\)	\(98\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{3}+(21-5\beta )q^{5}+7^{2}q^{7}+\cdots\)
112.6.a.k	$112$	$6$	112.a	1.a	$1$	$2$	$2$	$17.963$	\(\Q(\sqrt{177}) \)	None	✓		✓		56.6.a.c	$1$	$1$	\(0\)	\(26\)	\(-62\)	\(-98\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(13-\beta )q^{3}+(-31+5\beta )q^{5}-7^{2}q^{7}+\cdots\)

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

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