Space of modular forms of level 121, weight 12, and trivial character

• Label: 121.12.a
• Level: $121$
• Weight: $12$
• Character orbit: 121.a
• Rep. character: $\chi_{121}(1,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $10$
• Sturm bound: $132$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	121.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(132\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(121))\).

	Total	New	Old
Modular forms	127	105	22
Cusp forms	115	96	19
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.

\(11\)	Dim
\(+\)	\(49\)
\(-\)	\(47\)

Trace form

\( 96 q - 8 q^{2} + 234 q^{3} + 94796 q^{4} + 11852 q^{5} - 4894 q^{6} - 57214 q^{7} - 27804 q^{8} + 5432546 q^{9} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(121))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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
121.12.a.a	$121$	$12$	121.a	1.a	$1$	$1$	$1$	$92.970$	\(\Q\)	\(\Q(\sqrt{-11}) \)	✓	$2$	$0$	\(0\)	\(-67\)	\(13593\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-67q^{3}-2^{11}q^{4}+13593q^{5}-172658q^{9}+\cdots\)
121.12.a.b	$121$	$12$	121.a	1.a	$1$	$1$	$1$	$92.970$	\(\Q\)	None	✓	$1$	$1$	\(24\)	\(252\)	\(4830\)	\(16744\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+24q^{2}+252q^{3}-1472q^{4}+4830q^{5}+\cdots\)
121.12.a.c	$121$	$12$	121.a	1.a	$1$	$3$	$3$	$92.970$	3.3.202533.1	None	✓	$1$	$1$	\(0\)	\(-393\)	\(-7305\)	\(5082\)		$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-131-2\beta _{1}-4\beta _{2})q^{3}+\cdots\)
121.12.a.d	$121$	$12$	121.a	1.a	$1$	$5$	$5$	$92.970$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-32\)	\(160\)	\(-8398\)	\(-79040\)		$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-6-\beta _{1})q^{2}+(2^{5}-\beta _{3})q^{3}+(1239+\cdots)q^{4}+\cdots\)
121.12.a.e	$121$	$12$	121.a	1.a	$1$	$8$	$8$	$92.970$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(806\)	\(-8870\)	\(0\)		$+$	$+$	$2^{8}\cdot 3^{2}\cdot 5\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(101+\beta _{4})q^{3}+(786+\beta _{2}+\cdots)q^{4}+\cdots\)
121.12.a.f	$121$	$12$	121.a	1.a	$1$	$9$	$9$	$92.970$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(-32\)	\(10\)	\(-55\)	\(62786\)		$-$	$-$	$2^{14}\cdot 3^{3}\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q+(-4+\beta _{1})q^{2}+(2-2\beta _{1}-\beta _{3})q^{3}+\cdots\)
121.12.a.g	$121$	$12$	121.a	1.a	$1$	$9$	$9$	$92.970$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(32\)	\(10\)	\(-55\)	\(-62786\)		$-$	$-$	$2^{14}\cdot 3^{3}\cdot 11^{2}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{1})q^{2}+(2-2\beta _{1}-\beta _{3})q^{3}+(808+\cdots)q^{4}+\cdots\)
121.12.a.h	$121$	$12$	121.a	1.a	$1$	$20$	$20$	$92.970$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$1$	$1$	\(-88\)	\(-768\)	\(574\)	\(-63800\)		$-$	$-$	$2^{24}\cdot 3^{4}\cdot 5\cdot 11^{16}$	$\mathrm{SU}(2)$	\(q+(-4-\beta _{1})q^{2}+(-39+\beta _{1}+\beta _{5}+\cdots)q^{3}+\cdots\)
121.12.a.i	$121$	$12$	121.a	1.a	$1$	$20$	$20$	$92.970$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(992\)	\(16964\)	\(0\)		$+$	$+$	$2^{25}\cdot 3^{2}\cdot 11^{16}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(50-\beta _{3})q^{3}+(1427-\beta _{3}+\cdots)q^{4}+\cdots\)
121.12.a.j	$121$	$12$	121.a	1.a	$1$	$20$	$20$	$92.970$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$1$	$0$	\(88\)	\(-768\)	\(574\)	\(63800\)		$+$	$+$	$2^{24}\cdot 3^{4}\cdot 5\cdot 11^{16}$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1})q^{2}+(-39+\beta _{1}+\beta _{5})q^{3}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(121))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(121)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)