Space of modular forms of level 6, weight 22, and trivial character

• Label: 6.22.a
• Level: $6$
• Weight: $22$
• Character orbit: 6.a
• Rep. character: $\chi_{6}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $22$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 6 = 2 \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	6.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(22\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{22}(\Gamma_0(6))\).

	Total	New	Old
Modular forms	23	3	20
Cusp forms	19	3	16
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.

\(2\)	\(3\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(11\)	\(1\)	\(10\)		\(9\)	\(1\)	\(8\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(12\)	\(2\)	\(10\)		\(10\)	\(2\)	\(8\)		\(2\)	\(0\)	\(2\)

Trace form

3 * q + 1024 * q^2 + 59049 * q^3 + 3145728 * q^4 + 16153674 * q^5 - 60466176 * q^6 - 1636375008 * q^7 + 1073741824 * q^8 + 10460353203 * q^9 - 37617076224 * q^10 - 98395423908 * q^11 + 61917364224 * q^12 + 1099574688426 * q^13 - 2015853297664 * q^14 - 576003745026 * q^15 + 3298534883328 * q^16 + 23877926268102 * q^17 + 3570467226624 * q^18 - 11567072167068 * q^19 + 16938354868224 * q^20 - 39996769291200 * q^21 + 114034790043648 * q^22 + 340893203414856 * q^23 - 63403380965376 * q^24 - 23054276382099 * q^25 + 438673445795840 * q^26 + 205891132094649 * q^27 - 1715863560388608 * q^28 + 4030132752182658 * q^29 - 3787829463988224 * q^30 - 7075669005317352 * q^31 + 1125899906842624 * q^32 - 19469104089550092 * q^33 - 5441907977385984 * q^34 + 28933847452893504 * q^35 + 10968475320188928 * q^36 - 5837601526121118 * q^37 - 19155079287869440 * q^38 + 30012138157854078 * q^39 - 39444363318657024 * q^40 + 64202623305887310 * q^41 - 61045473892220928 * q^42 + 425104435306891356 * q^43 - 103175080019755008 * q^44 + 56324378522039274 * q^45 - 106337564428640256 * q^46 - 557743059082102848 * q^47 + 64925062108545024 * q^48 + 332159879552591115 * q^49 - 479240611205989376 * q^50 + 627334178863920210 * q^51 + 1152987628490981376 * q^52 - 2728003506188961654 * q^53 - 210832519264920576 * q^54 + 1262544774270568200 * q^55 - 2113775387451326464 * q^56 - 4057814241887276340 * q^57 - 366680730067200000 * q^58 + 5653138491641034780 * q^59 - 603983702944382976 * q^60 - 408089737914454038 * q^61 + 10620503418739417088 * q^62 - 5705686852080650208 * q^63 + 3458764513820540928 * q^64 + 1818329229000884748 * q^65 - 7253127250796187648 * q^66 + 30566343163574783652 * q^67 + 25037820414501322752 * q^68 - 19478918202179686056 * q^69 + 20631684004056170496 * q^70 - 69525201378432548808 * q^71 + 3743906242624487424 * q^72 - 33536250911384785794 * q^73 - 82718090089237129216 * q^74 + 35134083667603341351 * q^75 - 12128954264655495168 * q^76 + 71553913985152541184 * q^77 - 9851418861070546944 * q^78 + 66909092022784482216 * q^79 + 17761152394302849024 * q^80 + 36472996377170786403 * q^81 - 111647600565496584192 * q^82 - 384466770624937646364 * q^83 - 41939652356289331200 * q^84 + 410124662544577652244 * q^85 + 166165071545072095232 * q^86 + 320552290165541526630 * q^87 + 119574144004808245248 * q^88 + 130769230072423245966 * q^89 - 131162634589071181824 * q^90 - 705608677189118001984 * q^91 + 357452431663936045056 * q^92 - 6244696289600779032 * q^93 - 1265485229912200937472 * q^94 + 1480681379169368573304 * q^95 - 66483263599150104576 * q^96 - 11286510703838866938 * q^97 + 1427520150706407564288 * q^98 - 343083629212196859108 * q^99

Decomposition of \(S_{22}^{\mathrm{new}}(\Gamma_0(6))\) 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	3
6.22.a.a	$6$	$22$	6.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓	✓	✓	✓	6.22.a.a	$1$	$0$	\(-1024\)	\(59049\)	\(26444550\)	\(166115864\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{10}q^{2}+3^{10}q^{3}+2^{20}q^{4}+26444550q^{5}+\cdots\)
6.22.a.b	$6$	$22$	6.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓	✓	✓	✓	6.22.a.b	$1$	$0$	\(1024\)	\(-59049\)	\(12954174\)	\(-479513104\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{10}q^{2}-3^{10}q^{3}+2^{20}q^{4}+12954174q^{5}+\cdots\)
6.22.a.c	$6$	$22$	6.a	1.a	$1$	$1$	$1$	$16.769$	\(\Q\)	None	✓	✓	✓	✓	6.22.a.c	$1$	$1$	\(1024\)	\(59049\)	\(-23245050\)	\(-1322977768\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{10}q^{2}+3^{10}q^{3}+2^{20}q^{4}-23245050q^{5}+\cdots\)

Decomposition of \(S_{22}^{\mathrm{old}}(\Gamma_0(6))\) into lower level spaces

\( S_{22}^{\mathrm{old}}(\Gamma_0(6)) \simeq \) \(S_{22}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{22}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)