Space of modular forms of level 121, weight 4, and trivial character

• Label: 121.4.a
• Level: $121$
• Weight: $4$
• Character orbit: 121.a
• Rep. character: $\chi_{121}(1,\cdot)$
• Character field: $\Q$
• Dimension: $23$
• Newform subspaces: $8$
• Sturm bound: $44$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	121.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(44\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(121))\).

	Total	New	Old
Modular forms	39	32	7
Cusp forms	27	23	4
Eisenstein series	12	9	3

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

\(11\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(21\)	\(17\)	\(4\)		\(15\)	\(13\)	\(2\)		\(6\)	\(4\)	\(2\)
\(-\)		\(18\)	\(15\)	\(3\)		\(12\)	\(10\)	\(2\)		\(6\)	\(5\)	\(1\)

Trace form

23 * q - 2 * q^2 + 88 * q^4 + 2 * q^5 + 26 * q^6 - 20 * q^7 + 12 * q^8 + 107 * q^9 - 50 * q^10 - 2 * q^12 - 80 * q^13 - 48 * q^14 + 232 * q^15 + 140 * q^16 + 124 * q^17 - 92 * q^18 - 72 * q^19 - 4 * q^20 - 76 * q^21 + 108 * q^23 - 252 * q^24 - 83 * q^25 + 32 * q^26 + 276 * q^27 + 128 * q^28 - 144 * q^29 + 266 * q^30 + 26 * q^31 + 104 * q^32 + 110 * q^34 + 172 * q^35 - 14 * q^36 - 102 * q^37 - 670 * q^38 - 400 * q^39 + 492 * q^40 - 536 * q^41 - 324 * q^42 + 60 * q^43 - 616 * q^45 + 314 * q^46 + 674 * q^47 - 1166 * q^48 + 125 * q^49 - 232 * q^50 + 164 * q^51 + 560 * q^52 + 266 * q^53 + 110 * q^54 + 156 * q^56 + 1512 * q^57 - 152 * q^58 - 832 * q^59 - 1300 * q^60 - 840 * q^61 - 134 * q^62 - 248 * q^63 - 1556 * q^64 + 880 * q^65 - 934 * q^67 - 640 * q^68 - 692 * q^69 - 428 * q^70 + 562 * q^71 + 744 * q^72 + 400 * q^73 - 6 * q^74 + 626 * q^75 - 432 * q^76 + 784 * q^78 - 316 * q^79 + 304 * q^80 - 169 * q^81 - 930 * q^82 - 468 * q^83 + 736 * q^84 - 452 * q^85 + 298 * q^86 - 1200 * q^87 + 1442 * q^89 - 1532 * q^90 + 2456 * q^91 + 1332 * q^92 + 4728 * q^93 + 992 * q^94 - 2952 * q^95 + 952 * q^96 + 1244 * q^97 + 870 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(121))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		11
121.4.a.a	$121$	$4$	121.a	1.a	$1$	$1$	$1$	$7.139$	\(\Q\)	\(\Q(\sqrt{-11}) \)	✓	✓			121.4.a.a	$2$	$0$	\(0\)	\(8\)	\(18\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+8q^{3}-8q^{4}+18q^{5}+37q^{9}-2^{6}q^{12}+\cdots\)
121.4.a.b	$121$	$4$	121.a	1.a	$1$	$2$	$2$	$7.139$	\(\Q(\sqrt{3}) \)	None	✓	✓			121.4.a.b	$1$	$1$	\(-2\)	\(-8\)	\(-10\)	\(-8\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(-4+\beta )q^{3}+(5-2\beta )q^{4}+\cdots\)
121.4.a.c	$121$	$4$	121.a	1.a	$1$	$2$	$2$	$7.139$	\(\Q(\sqrt{3}) \)	None	✓				11.4.a.a	$1$	$1$	\(-2\)	\(-2\)	\(2\)	\(-20\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(-1+4\beta )q^{3}+(-4+\cdots)q^{4}+\cdots\)
121.4.a.d	$121$	$4$	121.a	1.a	$1$	$2$	$2$	$7.139$	\(\Q(\sqrt{26}) \)	None	✓	✓			121.4.a.d	$2$	$0$	\(0\)	\(-10\)	\(10\)	\(0\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-5q^{3}+18q^{4}+5q^{5}-5\beta q^{6}+\cdots\)
121.4.a.e	$121$	$4$	121.a	1.a	$1$	$2$	$2$	$7.139$	\(\Q(\sqrt{3}) \)	None	✓	✓			121.4.a.b	$1$	$1$	\(2\)	\(-8\)	\(-10\)	\(8\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-4-\beta )q^{3}+(5+2\beta )q^{4}+\cdots\)
121.4.a.f	$121$	$4$	121.a	1.a	$1$	$4$	$4$	$7.139$	\(\Q(\sqrt{5}, \sqrt{37})\)	None	✓		✓		11.4.c.a	$1$	$1$	\(-4\)	\(6\)	\(-11\)	\(-25\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}+\beta _{2})q^{2}+(1-\beta _{2})q^{3}+\cdots\)
121.4.a.g	$121$	$4$	121.a	1.a	$1$	$4$	$4$	$7.139$	\(\Q(\sqrt{5}, \sqrt{37})\)	None	✓				11.4.c.a	$1$	$0$	\(4\)	\(6\)	\(-11\)	\(25\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1}+\beta _{2})q^{2}+(2+\beta _{2})q^{3}+(3+\cdots)q^{4}+\cdots\)
121.4.a.h	$121$	$4$	121.a	1.a	$1$	$6$	$6$	$7.139$	6.6.\(\cdots\).1	None	✓	✓	✓		121.4.a.h	$2$	$0$	\(0\)	\(8\)	\(14\)	\(0\)		$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{4})q^{3}+(3+\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(121))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(121)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)