Space of modular forms of level 22, weight 10, and trivial character

• Label: 22.10.a
• Level: $22$
• Weight: $10$
• Character orbit: 22.a
• Rep. character: $\chi_{22}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $5$
• Sturm bound: $30$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 22 = 2 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	22.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(30\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(22))\).

	Total	New	Old
Modular forms	29	7	22
Cusp forms	25	7	18
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.

\(2\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(-\)	$+$	\(1\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(4\)

Trace form

\( 7 q - 16 q^{2} + 310 q^{3} + 1792 q^{4} - 1284 q^{5} + 4544 q^{6} - 228 q^{7} - 4096 q^{8} + 19137 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(22))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	11
22.10.a.a	$22$	$10$	22.a	1.a	$1$	$1$	$1$	$11.331$	\(\Q\)	None	✓	$1$	$1$	\(16\)	\(-41\)	\(-1039\)	\(-3482\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}-41q^{3}+2^{8}q^{4}-1039q^{5}+\cdots\)
22.10.a.b	$22$	$10$	22.a	1.a	$1$	$1$	$1$	$11.331$	\(\Q\)	None	✓	$1$	$0$	\(16\)	\(137\)	\(-595\)	\(11354\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+137q^{3}+2^{8}q^{4}-595q^{5}+\cdots\)
22.10.a.c	$22$	$10$	22.a	1.a	$1$	$1$	$1$	$11.331$	\(\Q\)	None	✓	$1$	$0$	\(16\)	\(201\)	\(2349\)	\(-8806\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+201q^{3}+2^{8}q^{4}+2349q^{5}+\cdots\)
22.10.a.d	$22$	$10$	22.a	1.a	$1$	$2$	$2$	$11.331$	\(\Q(\sqrt{889}) \)	None	✓	$1$	$0$	\(-32\)	\(-21\)	\(-521\)	\(-7490\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(-6-9\beta )q^{3}+2^{8}q^{4}+(-212+\cdots)q^{5}+\cdots\)
22.10.a.e	$22$	$10$	22.a	1.a	$1$	$2$	$2$	$11.331$	\(\Q(\sqrt{463}) \)	None	✓	$1$	$1$	\(-32\)	\(34\)	\(-1478\)	\(8196\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+(17+\beta )q^{3}+2^{8}q^{4}+(-739+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(22))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(22)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)