Space of modular forms of level 114, weight 4, and trivial character

• Label: 114.4.a
• Level: $114$
• Weight: $4$
• Character orbit: 114.a
• Rep. character: $\chi_{114}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $6$
• Sturm bound: $80$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	114.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(80\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(114))\).

	Total	New	Old
Modular forms	64	8	56
Cusp forms	56	8	48
Eisenstein series	8	0	8

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

\(2\)	\(3\)	\(19\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(10\)	\(1\)	\(9\)		\(9\)	\(1\)	\(8\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(7\)	\(0\)	\(7\)		\(6\)	\(0\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(8\)	\(1\)	\(7\)		\(7\)	\(1\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(8\)	\(1\)	\(7\)		\(7\)	\(1\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(9\)	\(2\)	\(7\)		\(8\)	\(2\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(7\)	\(2\)	\(5\)		\(6\)	\(2\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(8\)	\(0\)	\(8\)		\(7\)	\(0\)	\(7\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(34\)	\(6\)	\(28\)		\(30\)	\(6\)	\(24\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(30\)	\(2\)	\(28\)		\(26\)	\(2\)	\(24\)		\(4\)	\(0\)	\(4\)

Trace form

8 * q + 4 * q^2 + 32 * q^4 + 4 * q^5 - 12 * q^6 - 4 * q^7 + 16 * q^8 + 72 * q^9 + 64 * q^10 - 24 * q^11 + 20 * q^13 + 84 * q^15 + 128 * q^16 + 260 * q^17 + 36 * q^18 - 38 * q^19 + 16 * q^20 + 168 * q^22 + 180 * q^23 - 48 * q^24 + 340 * q^25 - 72 * q^26 - 16 * q^28 - 436 * q^29 - 120 * q^30 - 132 * q^31 + 64 * q^32 - 180 * q^33 + 8 * q^34 - 516 * q^35 + 288 * q^36 - 620 * q^37 - 24 * q^39 + 256 * q^40 + 412 * q^41 + 240 * q^42 - 876 * q^43 - 96 * q^44 + 36 * q^45 - 96 * q^46 - 624 * q^47 - 788 * q^49 - 36 * q^50 - 552 * q^51 + 80 * q^52 + 68 * q^53 - 108 * q^54 + 1196 * q^55 - 114 * q^57 - 688 * q^58 - 496 * q^59 + 336 * q^60 + 268 * q^61 - 1760 * q^62 - 36 * q^63 + 512 * q^64 + 32 * q^65 - 24 * q^66 - 32 * q^67 + 1040 * q^68 - 288 * q^69 - 960 * q^70 - 64 * q^71 + 144 * q^72 - 1948 * q^73 + 488 * q^74 - 1680 * q^75 - 152 * q^76 + 932 * q^77 + 528 * q^78 + 1140 * q^79 + 64 * q^80 + 648 * q^81 - 1280 * q^82 + 1084 * q^83 + 732 * q^85 - 2096 * q^86 + 132 * q^87 + 672 * q^88 + 1916 * q^89 + 576 * q^90 + 2744 * q^91 + 720 * q^92 - 360 * q^93 + 48 * q^94 + 1064 * q^95 - 192 * q^96 - 208 * q^97 + 1252 * q^98 - 216 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(114))\) 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	3	19
114.4.a.a	$114$	$4$	114.a	1.a	$1$	$1$	$1$	$6.726$	\(\Q\)	None	✓	✓	✓	✓	114.4.a.a	$1$	$0$	\(-2\)	\(-3\)	\(-19\)	\(9\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}-19q^{5}+6q^{6}+\cdots\)
114.4.a.b	$114$	$4$	114.a	1.a	$1$	$1$	$1$	$6.726$	\(\Q\)	None	✓	✓	✓	✓	114.4.a.b	$1$	$1$	\(-2\)	\(3\)	\(-7\)	\(-15\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-7q^{5}-6q^{6}+\cdots\)
114.4.a.c	$114$	$4$	114.a	1.a	$1$	$1$	$1$	$6.726$	\(\Q\)	None	✓	✓	✓	✓	114.4.a.c	$1$	$0$	\(-2\)	\(3\)	\(12\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+12q^{5}-6q^{6}+\cdots\)
114.4.a.d	$114$	$4$	114.a	1.a	$1$	$1$	$1$	$6.726$	\(\Q\)	None	✓	✓	✓	✓	114.4.a.d	$1$	$1$	\(2\)	\(-3\)	\(-11\)	\(-15\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-11q^{5}-6q^{6}+\cdots\)
114.4.a.e	$114$	$4$	114.a	1.a	$1$	$2$	$2$	$6.726$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	114.4.a.e	$1$	$0$	\(4\)	\(-6\)	\(18\)	\(4\)		$-$	$+$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+(9+\beta )q^{5}-6q^{6}+\cdots\)
114.4.a.f	$114$	$4$	114.a	1.a	$1$	$2$	$2$	$6.726$	\(\Q(\sqrt{273}) \)	None	✓	✓	✓	✓	114.4.a.f	$1$	$0$	\(4\)	\(6\)	\(11\)	\(9\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+(6-\beta )q^{5}+6q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(114))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(114)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 2}\)