Space of modular forms of level 5, weight 22, and trivial character

• Label: 5.22.a
• Level: $5$
• Weight: $22$
• Character orbit: 5.a
• Rep. character: $\chi_{5}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $2$
• Sturm bound: $11$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	5.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(11\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(5))\).

	Total	New	Old
Modular forms	11	7	4
Cusp forms	9	7	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.

\(5\)	Dim
\(+\)	\(3\)
\(-\)	\(4\)

Trace form

7 * q + 1598 * q^2 + 135434 * q^3 + 10156404 * q^4 + 9765625 * q^5 - 134284136 * q^6 + 1197030358 * q^7 + 6286638600 * q^8 + 23738514151 * q^9 + 41230468750 * q^10 + 36915589184 * q^11 - 503308196432 * q^12 + 57357098574 * q^13 + 442007689008 * q^14 + 303183593750 * q^15 - 1311300484848 * q^16 + 733529966678 * q^17 + 81647171463494 * q^18 + 20282453842260 * q^19 + 79825507812500 * q^20 - 393096502255116 * q^21 - 203004688093624 * q^22 + 388545505306914 * q^23 - 60402224293920 * q^24 + 667572021484375 * q^25 - 393923079662716 * q^26 - 1611730473858100 * q^27 - 247898758582784 * q^28 + 2205150977956090 * q^29 - 1784817265625000 * q^30 + 1314904316573524 * q^31 + 6022308475794208 * q^32 - 47425047428324192 * q^33 + 22647614408032988 * q^34 - 1677761308593750 * q^35 + 108438862405790372 * q^36 + 26872991516371118 * q^37 + 38176781255684600 * q^38 - 101643150674642668 * q^39 + 39532105078125000 * q^40 - 200890308543164006 * q^41 - 334487502048658944 * q^42 + 395111109322505394 * q^43 - 374190571247623952 * q^44 + 192649051884765625 * q^45 - 66245447433411696 * q^46 + 47044210186534238 * q^47 + 192063027362083264 * q^48 + 46967908925791099 * q^49 + 152397155761718750 * q^50 + 1799568223710501124 * q^51 + 1711589438388353048 * q^52 - 2532372504975675466 * q^53 - 4745878732696436240 * q^54 + 298228161250000000 * q^55 + 3579690160855388160 * q^56 + 6067235229019091000 * q^57 - 14098088367269025500 * q^58 + 9343397897212744580 * q^59 + 2133178135468750000 * q^60 - 10173783479756547666 * q^61 + 28533087758085932736 * q^62 + 8967507041593209174 * q^63 - 38086581234543958976 * q^64 + 16305734683457031250 * q^65 - 55052939083404862432 * q^66 + 10473435607038416278 * q^67 + 9049459099955001256 * q^68 - 99535341024990147108 * q^69 + 48914097521718750000 * q^70 + 9719710503077767804 * q^71 + 273453111111150138600 * q^72 - 34103167714234877586 * q^73 - 151516771487276236652 * q^74 + 12915992736816406250 * q^75 + 20327109513274986320 * q^76 + 392698243802630342496 * q^77 - 452912683312301552432 * q^78 - 326331098506183861160 * q^79 + 191870807912656250000 * q^80 + 112233362070071869027 * q^81 + 787498996898908950316 * q^82 - 324246264182686907646 * q^83 - 1444856778433938010752 * q^84 + 338436056994941406250 * q^85 - 308758989954091563256 * q^86 + 1011767514843797750300 * q^87 - 624045884838102832800 * q^88 - 519584921854713878730 * q^89 + 1316318276168808593750 * q^90 - 283308006346291986196 * q^91 + 3458344050732703820928 * q^92 - 430585694785886094312 * q^93 - 3480710271939349970272 * q^94 + 1075710350170117187500 * q^95 - 1756686888340402147456 * q^96 + 1183125700288881568238 * q^97 - 1026968381183437208114 * q^98 - 2826043630301167165888 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(5))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5
5.22.a.a	$5$	$22$	5.a	1.a	$1$	$3$	$3$	$13.974$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	5.22.a.a	$1$	$1$	\(-1312\)	\(52194\)	\(-29296875\)	\(684416558\)		$+$	$+$	$2^{7}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-437+\beta _{1})q^{2}+(17400+8\beta _{1}+\cdots)q^{3}+\cdots\)
5.22.a.b	$5$	$22$	5.a	1.a	$1$	$4$	$4$	$13.974$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	5.22.a.b	$1$	$0$	\(2910\)	\(83240\)	\(39062500\)	\(512613800\)		$-$	$-$	$2^{6}\cdot 3\cdot 5^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(728-\beta _{1})q^{2}+(20803+14\beta _{1}+\beta _{3})q^{3}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(5))\) into lower level spaces

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