Space of modular forms of level 121, weight 8, and trivial character

• Label: 121.8.a
• Level: $121$
• Weight: $8$
• Character orbit: 121.a
• Rep. character: $\chi_{121}(1,\cdot)$
• Character field: $\Q$
• Dimension: $59$
• Newform subspaces: $9$
• Sturm bound: $88$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	121.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(88\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(121))\).

	Total	New	Old
Modular forms	83	68	15
Cusp forms	71	59	12
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
\(+\)		\(43\)	\(35\)	\(8\)		\(37\)	\(31\)	\(6\)		\(6\)	\(4\)	\(2\)
\(-\)		\(40\)	\(33\)	\(7\)		\(34\)	\(28\)	\(6\)		\(6\)	\(5\)	\(1\)

Trace form

59 * q + 8 * q^2 + 54 * q^3 + 3340 * q^4 - 138 * q^5 - 862 * q^6 + 1058 * q^7 - 60 * q^8 + 37229 * q^9 - 5750 * q^10 + 31606 * q^12 - 4594 * q^13 + 8688 * q^14 + 18982 * q^15 + 174332 * q^16 - 45832 * q^17 + 66886 * q^18 - 32564 * q^19 - 18804 * q^20 - 46906 * q^21 + 89850 * q^23 - 245964 * q^24 + 642397 * q^25 + 232656 * q^26 + 41874 * q^27 + 490704 * q^28 - 413118 * q^29 + 256826 * q^30 + 13758 * q^31 - 11192 * q^32 - 303030 * q^34 - 639478 * q^35 + 384334 * q^36 + 975722 * q^37 + 852966 * q^38 + 1384652 * q^39 - 1781268 * q^40 - 6226 * q^41 - 628236 * q^42 - 980414 * q^43 + 1598804 * q^45 - 4279846 * q^46 - 700776 * q^47 + 8669026 * q^48 + 4301623 * q^49 + 3252718 * q^50 - 617266 * q^51 - 1514952 * q^52 - 859368 * q^53 - 6445090 * q^54 + 4989084 * q^56 - 3819120 * q^57 - 1995376 * q^58 + 8983890 * q^59 + 627980 * q^60 + 5708566 * q^61 - 6940534 * q^62 - 859244 * q^63 + 1307212 * q^64 - 5688520 * q^65 + 915014 * q^67 - 18917816 * q^68 - 6502886 * q^69 + 22916292 * q^70 + 6010890 * q^71 + 26184720 * q^72 - 12352174 * q^73 + 13520406 * q^74 - 2328724 * q^75 - 6867104 * q^76 - 4820048 * q^78 - 4367742 * q^79 - 7286736 * q^80 + 23823851 * q^81 + 992970 * q^82 + 6503706 * q^83 - 14049200 * q^84 - 3504482 * q^85 + 11662206 * q^86 + 9064920 * q^87 - 5244414 * q^89 - 13795712 * q^90 + 26542548 * q^91 + 7172148 * q^92 + 16305366 * q^93 + 15632 * q^94 + 4528968 * q^95 - 19662920 * q^96 - 24508538 * q^97 - 24721536 * q^98

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(121))\) 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
121.8.a.a	$121$	$8$	121.a	1.a	$1$	$1$	$1$	$37.799$	\(\Q\)	\(\Q(\sqrt{-11}) \)	✓	✓			121.8.a.a	$2$	$0$	\(0\)	\(83\)	\(-507\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+83q^{3}-2^{7}q^{4}-507q^{5}+4702q^{9}+\cdots\)
121.8.a.b	$121$	$8$	121.a	1.a	$1$	$2$	$2$	$37.799$	\(\Q(\sqrt{15}) \)	None	✓				11.8.a.a	$1$	$1$	\(8\)	\(-6\)	\(-470\)	\(1228\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{2}+(-3+6\beta )q^{3}+(-52+\cdots)q^{4}+\cdots\)
121.8.a.c	$121$	$8$	121.a	1.a	$1$	$4$	$4$	$37.799$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				11.8.a.b	$1$	$1$	\(0\)	\(-35\)	\(537\)	\(-170\)		$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(-9-\beta _{1}-\beta _{2}-\beta _{3})q^{3}+\cdots\)
121.8.a.d	$121$	$8$	121.a	1.a	$1$	$5$	$5$	$37.799$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			121.8.a.d	$1$	$1$	\(-8\)	\(-14\)	\(-515\)	\(1286\)		$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(-2-\beta _{1}+\beta _{2})q^{3}+\cdots\)
121.8.a.e	$121$	$8$	121.a	1.a	$1$	$5$	$5$	$37.799$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓			121.8.a.d	$1$	$1$	\(8\)	\(-14\)	\(-515\)	\(-1286\)		$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(-2-\beta _{1}+\beta _{2})q^{3}+\cdots\)
121.8.a.f	$121$	$8$	121.a	1.a	$1$	$6$	$6$	$37.799$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			121.8.a.f	$2$	$0$	\(0\)	\(-16\)	\(240\)	\(0\)		$+$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-3+\beta _{3})q^{3}+(119+2\beta _{3}+\cdots)q^{4}+\cdots\)
121.8.a.g	$121$	$8$	121.a	1.a	$1$	$12$	$12$	$37.799$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		11.8.c.a	$1$	$1$	\(-24\)	\(-12\)	\(144\)	\(-2244\)		$-$	$-$	$2^{8}\cdot 11^{5}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(-1-\beta _{5})q^{3}+(46+\cdots)q^{4}+\cdots\)
121.8.a.h	$121$	$8$	121.a	1.a	$1$	$12$	$12$	$37.799$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			121.8.a.h	$2$	$0$	\(0\)	\(80\)	\(804\)	\(0\)		$+$	$+$	$2^{9}\cdot 11^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(7-\beta _{3})q^{3}+(68+\beta _{2})q^{4}+\cdots\)
121.8.a.i	$121$	$8$	121.a	1.a	$1$	$12$	$12$	$37.799$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				11.8.c.a	$1$	$0$	\(24\)	\(-12\)	\(144\)	\(2244\)		$+$	$+$	$2^{8}\cdot 11^{5}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(-1-\beta _{5})q^{3}+(46-3\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(121))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(121)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)