Space of modular forms of level 2, weight 22, and trivial character

• Label: 2.22.a
• Level: $2$
• Weight: $22$
• Character orbit: 2.a
• Rep. character: $\chi_{2}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $5$
• Trace bound: $2$

Defining parameters

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

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(2))\).

	Total	New	Old
Modular forms	6	2	4
Cusp forms	4	2	2
Eisenstein series	2	0	2

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
\(+\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)

Trace form

2 * q + 130920 * q^3 + 2097152 * q^4 - 23718420 * q^5 - 12582912 * q^6 + 574223440 * q^7 - 12275185734 * q^9 + 34477178880 * q^10 - 20036715336 * q^11 + 137279569920 * q^12 - 1045474008260 * q^13 + 2335363301376 * q^14 - 1759470846480 * q^15 + 2199023255552 * q^16 - 7946413934940 * q^17 - 1647354839040 * q^18 + 34056523599880 * q^19 - 24870565969920 * q^20 + 23576486574144 * q^21 - 198143052349440 * q^22 + 275998560315120 * q^23 - 13194139533312 * q^24 - 105587771970850 * q^25 + 763057026367488 * q^26 - 2163118970450160 * q^27 + 602116917821440 * q^28 + 1648319193086940 * q^29 + 2406099525304320 * q^30 - 4025217119435456 * q^31 - 122745071797920 * q^33 - 14808624685645824 * q^34 + 31583536312739040 * q^35 - 12871465156214784 * q^36 + 18166192255567660 * q^37 - 12296613569495040 * q^38 - 73015070738904528 * q^39 + 36151942321274880 * q^40 + 21101722895757204 * q^41 + 149260180201144320 * q^42 - 157459727760364040 * q^43 - 21010018820161536 * q^44 + 118491491854692540 * q^45 - 17400697687375872 * q^46 - 642515922620105760 * q^47 + 143948062308433920 * q^48 + 1648407099408845874 * q^49 - 817744209090969600 * q^50 - 431320508067297456 * q^51 - 1096258953685237760 * q^52 + 1470670856960699340 * q^53 + 101014647015997440 * q^54 - 3019852165744949040 * q^55 + 2448805909103640576 * q^56 + 2303119716265115040 * q^57 + 3191216599172382720 * q^58 - 6736451234462070120 * q^59 - 1844938902318612480 * q^60 + 364336743223117084 * q^61 + 2322545873591992320 * q^62 - 5358824269905078000 * q^63 + 2305843009213693952 * q^64 + 24943153326978577320 * q^65 - 12844384093873373184 * q^66 - 13954116530216070680 * q^67 - 8332418938243645440 * q^68 + 18171269944352010432 * q^69 - 17796761688326799360 * q^70 + 4322530771479236304 * q^71 - 1727376747701207040 * q^72 + 11112960893524425940 * q^73 + 45152071124533641216 * q^74 - 2005310298666023400 * q^75 + 35710853290267770880 * q^76 - 226402501182977528640 * q^77 + 56527266668127191040 * q^78 + 133177052109561844000 * q^79 - 26078678582474833920 * q^80 - 13801077277696652238 * q^81 - 72015436582087557120 * q^82 + 207172716519439614600 * q^83 + 24721737985969618944 * q^84 - 149215598230637549160 * q^85 - 111936012458259382272 * q^86 + 88751674784436796080 * q^87 - 207768049260366397440 * q^88 + 77778919424044267380 * q^89 - 192070560187479490560 * q^90 + 549563307151695419744 * q^91 + 289405466380987269120 * q^92 - 277425987879796903680 * q^93 + 263647117032718073856 * q^94 - 606039792323431352400 * q^95 - 13835058055282163712 * q^96 - 1296645454449453731900 * q^97 + 1341020348565883453440 * q^98 + 278622531605875217112 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(2))\) 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
2.22.a.a	$2$	$22$	2.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓	✓	✓	✓	2.22.a.a	$1$	$1$	\(-1024\)	\(71604\)	\(-28693770\)	\(-853202392\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{10}q^{2}+71604q^{3}+2^{20}q^{4}-28693770q^{5}+\cdots\)
2.22.a.b	$2$	$22$	2.a	1.a	$1$	$1$	$1$	$5.590$	\(\Q\)	None	✓	✓	✓	✓	2.22.a.b	$1$	$0$	\(1024\)	\(59316\)	\(4975350\)	\(1427425832\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{10}q^{2}+59316q^{3}+2^{20}q^{4}+4975350q^{5}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(2)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)