Space of modular forms of level 16, weight 22, and trivial character

• Label: 16.22.a
• Level: $16$
• Weight: $22$
• Character orbit: 16.a
• Rep. character: $\chi_{16}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $6$
• Sturm bound: $44$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(44\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(16))\).

	Total	New	Old
Modular forms	45	11	34
Cusp forms	39	10	29
Eisenstein series	6	1	5

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

\(2\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(22\)	\(5\)	\(17\)		\(19\)	\(5\)	\(14\)		\(3\)	\(0\)	\(3\)
\(-\)		\(23\)	\(6\)	\(17\)		\(20\)	\(5\)	\(15\)		\(3\)	\(1\)	\(2\)

Trace form

10 * q - 59048 * q^3 - 10391780 * q^5 - 286396624 * q^7 + 36243817378 * q^9 - 44145613432 * q^11 - 53827631700 * q^13 - 409034126192 * q^15 - 1849642826188 * q^17 - 64445589026888 * q^19 - 6335038461120 * q^21 + 333505906879376 * q^23 + 512198372079702 * q^25 - 3179712712147856 * q^27 + 4339955132705484 * q^29 - 544217111281472 * q^31 + 277341305837024 * q^33 + 16343014300785696 * q^35 - 19463787357296804 * q^37 - 61106929702818224 * q^39 - 54372406343966748 * q^41 + 104523248903382600 * q^43 - 105167698877440724 * q^45 + 874784109581930784 * q^47 + 281167627130179834 * q^49 - 1699907057117920720 * q^51 - 10180307078564228 * q^53 - 1215488655761389904 * q^55 + 1041229908777309472 * q^57 + 1660839230536236200 * q^59 + 1888770384295213836 * q^61 - 12251256396388940688 * q^63 + 7397352098231670856 * q^65 + 26576033182092256984 * q^67 - 28834712558382134848 * q^69 + 41200641038922856624 * q^71 - 37765020242610577116 * q^73 - 73979495734498489048 * q^75 + 24391643967118988736 * q^77 + 13899556848849002976 * q^79 + 234173317447560005626 * q^81 - 47026879772027908808 * q^83 - 244084099503461631752 * q^85 - 227604522934813134384 * q^87 - 240584055283923843516 * q^89 + 402710460549494674336 * q^91 + 913461579600302447872 * q^93 + 23588040578229740752 * q^95 + 710462713306615232468 * q^97 - 775806929716913573272 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(16))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2
16.22.a.a	$16$	$22$	16.a	1.a	$1$	$1$	$1$	$44.716$	\(\Q\)	None	✓				2.22.a.a	$1$	$0$	\(0\)	\(-71604\)	\(-28693770\)	\(853202392\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-71604q^{3}-28693770q^{5}+853202392q^{7}+\cdots\)
16.22.a.b	$16$	$22$	16.a	1.a	$1$	$1$	$1$	$44.716$	\(\Q\)	None	✓				2.22.a.b	$1$	$0$	\(0\)	\(-59316\)	\(4975350\)	\(-1427425832\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-59316q^{3}+4975350q^{5}-1427425832q^{7}+\cdots\)
16.22.a.c	$16$	$22$	16.a	1.a	$1$	$1$	$1$	$44.716$	\(\Q\)	None	✓				1.22.a.a	$1$	$0$	\(0\)	\(128844\)	\(21640950\)	\(768078808\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+128844q^{3}+21640950q^{5}+768078808q^{7}+\cdots\)
16.22.a.d	$16$	$22$	16.a	1.a	$1$	$2$	$2$	$44.716$	\(\Q(\sqrt{2161}) \)	None	✓		✓		4.22.a.a	$1$	$0$	\(0\)	\(-65640\)	\(13689324\)	\(260508080\)		$-$	$-$	$2^{8}\cdot 3\cdot 7$	$\mathrm{SU}(2)$	\(q+(-32820-\beta )q^{3}+(6844662+204\beta )q^{5}+\cdots\)
16.22.a.e	$16$	$22$	16.a	1.a	$1$	$2$	$2$	$44.716$	\(\Q(\sqrt{358549}) \)	None	✓				8.22.a.a	$1$	$1$	\(0\)	\(105432\)	\(2108140\)	\(-444771792\)		$+$	$+$	$2^{7}\cdot 3$	$\mathrm{SU}(2)$	\(q+(52716-\beta )q^{3}+(1054070-20\beta )q^{5}+\cdots\)
16.22.a.f	$16$	$22$	16.a	1.a	$1$	$3$	$3$	$44.716$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		8.22.a.b	$1$	$1$	\(0\)	\(-96764\)	\(-24111774\)	\(-295988280\)		$+$	$+$	$2^{21}\cdot 3\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(-32255+\beta _{1})q^{3}+(-8037261+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(16))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(16)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)