Space of modular forms of level 6, weight 18, and trivial character

• Label: 6.18.a
• Level: $6$
• Weight: $18$
• Character orbit: 6.a
• Rep. character: $\chi_{6}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $18$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 6 = 2 \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 18 \)
Character orbit:	\([\chi]\)	\(=\)	6.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(18\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{18}(\Gamma_0(6))\).

	Total	New	Old
Modular forms	19	3	16
Cusp forms	15	3	12
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
\(+\)	\(+\)	\(+\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(10\)	\(2\)	\(8\)		\(8\)	\(2\)	\(6\)		\(2\)	\(0\)	\(2\)

Trace form

3 * q - 256 * q^2 - 6561 * q^3 + 196608 * q^4 + 373314 * q^5 - 1679616 * q^6 + 20293512 * q^7 - 16777216 * q^8 + 129140163 * q^9 - 197789184 * q^10 + 784792212 * q^11 - 429981696 * q^12 + 1602597066 * q^13 + 7584004096 * q^14 - 3396537846 * q^15 + 12884901888 * q^16 + 63064451862 * q^17 - 11019960576 * q^18 - 112465967988 * q^19 + 24465506304 * q^20 - 246522226680 * q^21 - 136621919232 * q^22 + 502204534536 * q^23 - 110075314176 * q^24 - 1827528895779 * q^25 + 1754059256320 * q^26 - 282429536481 * q^27 + 1329955602432 * q^28 - 3268241365782 * q^29 + 1540184357376 * q^30 + 4428127804848 * q^31 - 1099511627776 * q^32 + 10063371421188 * q^33 + 2058013011456 * q^34 - 1795311930576 * q^35 + 8463329722368 * q^36 - 34265998294158 * q^37 - 4158262154240 * q^38 + 26240907525138 * q^39 - 12962311962624 * q^40 - 49334969807970 * q^41 - 20734268295168 * q^42 - 78330110674764 * q^43 + 51432142405632 * q^44 + 16069943603394 * q^45 - 254423840495616 * q^46 + 609853163061552 * q^47 - 28179280429056 * q^48 + 15522206736555 * q^49 + 97630780039424 * q^50 + 18993259062990 * q^51 + 105027801317376 * q^52 - 201393768300174 * q^53 - 72301961339136 * q^54 - 431372192162760 * q^55 + 497025292435456 * q^56 + 813714677695260 * q^57 - 330891509767680 * q^58 - 3978739340006460 * q^59 - 222595504275456 * q^60 + 2545999496260602 * q^61 + 1093581265543168 * q^62 + 873569149174152 * q^63 + 844424930131968 * q^64 - 4546522505652132 * q^65 - 2997996227693568 * q^66 + 3306215142940092 * q^67 + 4132991917228032 * q^68 - 1075810753198296 * q^69 - 2091756079263744 * q^70 - 718407280036008 * q^71 - 722204136308736 * q^72 + 5920503858987486 * q^73 + 19406546444491264 * q^74 + 2047501942916481 * q^75 - 7370569678061568 * q^76 - 8870296929042336 * q^77 - 20917802780757504 * q^78 + 24057256930293696 * q^79 + 1603371421138944 * q^80 + 5559060566555523 * q^81 - 37354157530163712 * q^82 - 33050904413718324 * q^83 - 16156080647700480 * q^84 - 13005613799191836 * q^85 + 59034957775885312 * q^86 + 69098564553893010 * q^87 - 8953654098788352 * q^88 - 114944772828400434 * q^89 - 8514175820465664 * q^90 + 59742068442608496 * q^91 + 32912476375351296 * q^92 + 61239254144690448 * q^93 - 4534883594514432 * q^94 - 22335787055851896 * q^95 - 7213895789838336 * q^96 - 3753776454789018 * q^97 + 195877499937900288 * q^98 + 33782731392936852 * q^99

Decomposition of \(S_{18}^{\mathrm{new}}(\Gamma_0(6))\) 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
6.18.a.a	$6$	$18$	6.a	1.a	$1$	$1$	$1$	$10.993$	\(\Q\)	None	✓	✓	✓	✓	6.18.a.a	$1$	$1$	\(-256\)	\(-6561\)	\(645150\)	\(3974432\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}-3^{8}q^{3}+2^{16}q^{4}+645150q^{5}+\cdots\)
6.18.a.b	$6$	$18$	6.a	1.a	$1$	$1$	$1$	$10.993$	\(\Q\)	None	✓	✓	✓	✓	6.18.a.b	$1$	$0$	\(-256\)	\(6561\)	\(-72186\)	\(-8640184\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{8}q^{2}+3^{8}q^{3}+2^{16}q^{4}-72186q^{5}+\cdots\)
6.18.a.c	$6$	$18$	6.a	1.a	$1$	$1$	$1$	$10.993$	\(\Q\)	None	✓	✓	✓	✓	6.18.a.c	$1$	$0$	\(256\)	\(-6561\)	\(-199650\)	\(24959264\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{8}q^{2}-3^{8}q^{3}+2^{16}q^{4}-199650q^{5}+\cdots\)

Decomposition of \(S_{18}^{\mathrm{old}}(\Gamma_0(6))\) into lower level spaces

\( S_{18}^{\mathrm{old}}(\Gamma_0(6)) \simeq \) \(S_{18}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{18}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)