Space of modular forms of level 111, weight 6, and trivial character

• Label: 111.6.a
• Level: $111$
• Weight: $6$
• Character orbit: 111.a
• Rep. character: $\chi_{111}(1,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $4$
• Sturm bound: $76$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 111 = 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	111.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(76\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(111))\).

	Total	New	Old
Modular forms	66	30	36
Cusp forms	62	30	32
Eisenstein series	4	0	4

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(37\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(14\)	\(6\)	\(8\)		\(13\)	\(6\)	\(7\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(18\)	\(9\)	\(9\)		\(17\)	\(9\)	\(8\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(19\)	\(10\)	\(9\)		\(18\)	\(10\)	\(8\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(15\)	\(5\)	\(10\)		\(14\)	\(5\)	\(9\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(29\)	\(11\)	\(18\)		\(27\)	\(11\)	\(16\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(37\)	\(19\)	\(18\)		\(35\)	\(19\)	\(16\)		\(2\)	\(0\)	\(2\)

Trace form

30 * q + 460 * q^4 - 44 * q^5 + 108 * q^6 + 312 * q^7 - 492 * q^8 + 2430 * q^9 + 1096 * q^10 - 40 * q^11 + 792 * q^12 - 796 * q^13 - 84 * q^14 - 108 * q^15 + 9892 * q^16 + 2848 * q^17 + 1228 * q^19 - 368 * q^20 - 248 * q^22 + 1068 * q^23 - 3024 * q^24 + 12562 * q^25 + 15448 * q^26 + 12764 * q^28 + 10720 * q^29 - 2736 * q^30 - 2244 * q^31 + 24856 * q^32 + 15624 * q^33 + 14124 * q^34 + 26668 * q^35 + 37260 * q^36 - 2738 * q^37 + 17008 * q^38 - 12168 * q^39 + 78552 * q^40 - 11868 * q^41 - 1908 * q^42 - 23412 * q^43 - 12240 * q^44 - 3564 * q^45 + 2828 * q^46 + 35488 * q^47 + 30312 * q^48 + 77190 * q^49 - 132988 * q^50 - 18360 * q^51 - 101544 * q^52 - 28484 * q^53 + 8748 * q^54 - 53744 * q^55 + 5624 * q^56 - 12996 * q^57 + 80808 * q^58 + 26944 * q^59 - 39348 * q^60 - 115180 * q^61 - 18328 * q^62 + 25272 * q^63 + 70884 * q^64 - 63360 * q^65 - 24552 * q^66 + 106440 * q^67 + 173404 * q^68 - 62928 * q^69 - 82840 * q^70 + 117816 * q^71 - 39852 * q^72 - 42492 * q^73 + 86040 * q^75 - 152 * q^76 - 30304 * q^77 + 23004 * q^78 - 42220 * q^79 - 239512 * q^80 + 196830 * q^81 - 409176 * q^82 - 451912 * q^83 - 124704 * q^84 + 133896 * q^85 - 215256 * q^86 - 89856 * q^87 + 45264 * q^88 - 162700 * q^89 + 88776 * q^90 + 267496 * q^91 + 188512 * q^92 + 110844 * q^93 - 675288 * q^94 - 388240 * q^95 - 276120 * q^96 - 10612 * q^97 - 786912 * q^98 - 3240 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(111))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	37
111.6.a.a	$111$	$6$	111.a	1.a	$1$	$5$	$5$	$17.803$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	111.6.a.a	$1$	$1$	\(-3\)	\(45\)	\(-64\)	\(-216\)		$-$	$-$	$+$	$2\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+9q^{3}+(11-4\beta _{1}+\cdots)q^{4}+\cdots\)
111.6.a.b	$111$	$6$	111.a	1.a	$1$	$6$	$6$	$17.803$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	111.6.a.b	$1$	$1$	\(-9\)	\(-54\)	\(-58\)	\(176\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}-9q^{3}+(10-2\beta _{1}+\cdots)q^{4}+\cdots\)
111.6.a.c	$111$	$6$	111.a	1.a	$1$	$9$	$9$	$17.803$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓	✓	111.6.a.c	$1$	$0$	\(3\)	\(-81\)	\(42\)	\(-20\)		$+$	$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-9q^{3}+(14+\beta _{1}+\beta _{2})q^{4}+\cdots\)
111.6.a.d	$111$	$6$	111.a	1.a	$1$	$10$	$10$	$17.803$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓	✓	111.6.a.d	$1$	$0$	\(9\)	\(90\)	\(36\)	\(372\)		$-$	$+$	$-$	$2^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+9q^{3}+(23+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(111))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(111)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 2}\)