Space of modular forms of level 138, weight 4, and trivial character

• Label: 138.4.a
• Level: $138$
• Weight: $4$
• Character orbit: 138.a
• Rep. character: $\chi_{138}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $6$
• Sturm bound: $96$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	138.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(96\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(138))\).

	Total	New	Old
Modular forms	76	10	66
Cusp forms	68	10	58
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.

\(2\)	\(3\)	\(23\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(12\)	\(0\)	\(12\)		\(11\)	\(0\)	\(11\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(10\)	\(1\)	\(9\)		\(9\)	\(1\)	\(8\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(9\)	\(2\)	\(7\)		\(8\)	\(2\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(11\)	\(1\)	\(10\)		\(10\)	\(1\)	\(9\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(8\)	\(2\)	\(6\)		\(7\)	\(2\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(11\)	\(3\)	\(8\)		\(10\)	\(3\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(8\)	\(0\)	\(8\)		\(7\)	\(0\)	\(7\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(40\)	\(7\)	\(33\)		\(36\)	\(7\)	\(29\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(36\)	\(3\)	\(33\)		\(32\)	\(3\)	\(29\)		\(4\)	\(0\)	\(4\)

Trace form

10 * q + 4 * q^2 + 6 * q^3 + 40 * q^4 + 20 * q^5 - 12 * q^6 + 16 * q^7 + 16 * q^8 + 90 * q^9 + 64 * q^10 + 24 * q^12 + 52 * q^13 - 80 * q^14 + 84 * q^15 + 160 * q^16 + 60 * q^17 + 36 * q^18 - 68 * q^19 + 80 * q^20 - 12 * q^21 + 280 * q^22 - 48 * q^24 + 166 * q^25 - 72 * q^26 + 54 * q^27 + 64 * q^28 + 68 * q^29 + 128 * q^31 + 64 * q^32 - 180 * q^33 + 152 * q^34 - 896 * q^35 + 360 * q^36 - 760 * q^37 + 496 * q^38 + 252 * q^39 + 256 * q^40 - 292 * q^41 + 408 * q^42 - 196 * q^43 + 180 * q^45 - 184 * q^46 - 1208 * q^47 + 96 * q^48 + 1450 * q^49 - 308 * q^50 + 240 * q^51 + 208 * q^52 - 2500 * q^53 - 108 * q^54 - 1136 * q^55 - 320 * q^56 - 348 * q^57 - 520 * q^58 - 920 * q^59 + 336 * q^60 + 1880 * q^61 + 720 * q^62 + 144 * q^63 + 640 * q^64 - 584 * q^65 - 1468 * q^67 + 240 * q^68 - 276 * q^69 - 368 * q^70 - 1128 * q^71 + 144 * q^72 + 996 * q^73 - 440 * q^74 + 1146 * q^75 - 272 * q^76 + 832 * q^77 - 648 * q^78 - 2352 * q^79 + 320 * q^80 + 810 * q^81 - 1096 * q^82 - 2888 * q^83 - 48 * q^84 + 520 * q^85 - 2784 * q^86 + 1644 * q^87 + 1120 * q^88 - 1092 * q^89 + 576 * q^90 + 2992 * q^91 - 288 * q^93 - 1104 * q^94 - 520 * q^95 - 192 * q^96 + 396 * q^97 - 1372 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(138))\) 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	23
138.4.a.a	$138$	$4$	138.a	1.a	$1$	$1$	$1$	$8.142$	\(\Q\)	None	✓	✓	✓	✓	138.4.a.a	$1$	$1$	\(-2\)	\(-3\)	\(-10\)	\(32\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+4q^{4}-10q^{5}+6q^{6}+\cdots\)
138.4.a.b	$138$	$4$	138.a	1.a	$1$	$1$	$1$	$8.142$	\(\Q\)	None	✓	✓	✓	✓	138.4.a.b	$1$	$1$	\(-2\)	\(3\)	\(2\)	\(-32\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+2q^{5}-6q^{6}+\cdots\)
138.4.a.c	$138$	$4$	138.a	1.a	$1$	$1$	$1$	$8.142$	\(\Q\)	None	✓	✓	✓	✓	138.4.a.c	$1$	$1$	\(2\)	\(-3\)	\(-2\)	\(-34\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}-2q^{5}-6q^{6}+\cdots\)
138.4.a.d	$138$	$4$	138.a	1.a	$1$	$2$	$2$	$8.142$	\(\Q(\sqrt{277}) \)	None	✓	✓	✓	✓	138.4.a.d	$1$	$0$	\(-4\)	\(6\)	\(2\)	\(28\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}+(1-\beta )q^{5}-6q^{6}+\cdots\)
138.4.a.e	$138$	$4$	138.a	1.a	$1$	$2$	$2$	$8.142$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	138.4.a.e	$1$	$0$	\(4\)	\(-6\)	\(8\)	\(12\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+(4+\beta )q^{5}-6q^{6}+\cdots\)
138.4.a.f	$138$	$4$	138.a	1.a	$1$	$3$	$3$	$8.142$	3.3.16372.1	None	✓	✓	✓	✓	138.4.a.f	$1$	$0$	\(6\)	\(9\)	\(20\)	\(10\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+(7+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(138))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(138)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 2}\)