Space of modular forms of level 21, weight 22, and trivial character

• Label: 21.22.a
• Level: $21$
• Weight: $22$
• Character orbit: 21.a
• Rep. character: $\chi_{21}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $4$
• Sturm bound: $58$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(58\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(21))\).

	Total	New	Old
Modular forms	58	20	38
Cusp forms	54	20	34
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
\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(-\)	$-$	\(5\)
\(-\)	\(+\)	$-$	\(6\)
\(-\)	\(-\)	$+$	\(4\)
Plus space		\(+\)	\(9\)
Minus space		\(-\)	\(11\)

Trace form

\( 20 q + 1474 q^{2} + 14845954 q^{4} + 75749312 q^{5} - 89518284 q^{6} - 564950498 q^{7} + 8519022798 q^{8} + 69735688020 q^{9} + O(q^{10}) \)

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(21))\) 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
21.22.a.a	$21$	$22$	21.a	1.a	$1$	$4$	$4$	$58.690$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-1920\)	\(236196\)	\(-4581570\)	\(1129900996\)		$-$	$-$	$+$	$2^{8}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-480+\beta _{1})q^{2}+3^{10}q^{3}+(1004777+\cdots)q^{4}+\cdots\)
21.22.a.b	$21$	$22$	21.a	1.a	$1$	$5$	$5$	$58.690$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-138\)	\(-295245\)	\(24367158\)	\(1412376245\)		$+$	$-$	$-$	$2^{11}\cdot 3^{4}\cdot 5\cdot 7^{2}\cdot 19$	$\mathrm{SU}(2)$	\(q+(-28+\beta _{1})q^{2}-3^{10}q^{3}+(502232+\cdots)q^{4}+\cdots\)
21.22.a.c	$21$	$22$	21.a	1.a	$1$	$5$	$5$	$58.690$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(1633\)	\(-295245\)	\(22924976\)	\(-1412376245\)		$+$	$+$	$+$	$2^{8}\cdot 3^{6}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(327-\beta _{1})q^{2}-3^{10}q^{3}+(51773+\cdots)q^{4}+\cdots\)
21.22.a.d	$21$	$22$	21.a	1.a	$1$	$6$	$6$	$58.690$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(1899\)	\(354294\)	\(33038748\)	\(-1694851494\)		$-$	$+$	$-$	$2^{11}\cdot 3^{8}\cdot 5^{2}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+(317-\beta _{1})q^{2}+3^{10}q^{3}+(1342811+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(21))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(21)) \cong \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)