Space of modular forms of level 21, weight 12, and trivial character

• Label: 21.12.a
• Level: $21$
• Weight: $12$
• Character orbit: 21.a
• Rep. character: $\chi_{21}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $5$
• Sturm bound: $32$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(32\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(21))\).

	Total	New	Old
Modular forms	32	12	20
Cusp forms	28	12	16
Eisenstein series	4	0	4

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

\(3\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(-\)	$-$	\(3\)
\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(-\)	$+$	\(4\)
Plus space		\(+\)	\(7\)
Minus space		\(-\)	\(5\)

Trace form

\( 12 q - 110 q^{2} + 7298 q^{4} + 20864 q^{5} + 22356 q^{6} + 33614 q^{7} - 178530 q^{8} + 708588 q^{9} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(21))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7
21.12.a.a	$21$	$12$	21.a	1.a	$1$	$1$	$1$	$16.135$	\(\Q\)	None	✓	$1$	$1$	\(-62\)	\(243\)	\(-3310\)	\(-16807\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-62q^{2}+3^{5}q^{3}+1796q^{4}-3310q^{5}+\cdots\)
21.12.a.b	$21$	$12$	21.a	1.a	$1$	$1$	$1$	$16.135$	\(\Q\)	None	✓	$1$	$1$	\(8\)	\(243\)	\(4390\)	\(-16807\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{5}q^{3}-1984q^{4}+4390q^{5}+\cdots\)
21.12.a.c	$21$	$12$	21.a	1.a	$1$	$3$	$3$	$16.135$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(-68\)	\(-729\)	\(3326\)	\(-50421\)		$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-23-\beta _{1})q^{2}-3^{5}q^{3}+(664+72\beta _{1}+\cdots)q^{4}+\cdots\)
21.12.a.d	$21$	$12$	21.a	1.a	$1$	$3$	$3$	$16.135$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(-33\)	\(-729\)	\(3102\)	\(50421\)		$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-11-\beta _{2})q^{2}-3^{5}q^{3}+(255+13\beta _{1}+\cdots)q^{4}+\cdots\)
21.12.a.e	$21$	$12$	21.a	1.a	$1$	$4$	$4$	$16.135$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(45\)	\(972\)	\(13356\)	\(67228\)		$-$	$-$	$+$	$2^{2}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(11+\beta _{1})q^{2}+3^{5}q^{3}+(1196+18\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)