Space of modular forms of level 6, weight 16, and trivial character

• Label: 6.16.a
• Level: $6$
• Weight: $16$
• Character orbit: 6.a
• Rep. character: $\chi_{6}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $16$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 6 = 2 \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	6.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(16\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_0(6))\).

	Total	New	Old
Modular forms	17	3	14
Cusp forms	13	3	10
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
\(+\)	\(+\)	\(+\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(9\)	\(2\)	\(7\)		\(7\)	\(2\)	\(5\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(8\)	\(1\)	\(7\)		\(6\)	\(1\)	\(5\)		\(2\)	\(0\)	\(2\)

Trace form

3 * q + 128 * q^2 - 2187 * q^3 + 49152 * q^4 - 351654 * q^5 + 279936 * q^6 - 247368 * q^7 + 2097152 * q^8 + 14348907 * q^9 + 35497728 * q^10 + 54814524 * q^11 - 35831808 * q^12 + 236812146 * q^13 - 550077440 * q^14 + 1108690902 * q^15 + 805306368 * q^16 - 4106086218 * q^17 + 612220032 * q^18 + 1202087364 * q^19 - 5761499136 * q^20 + 3874436712 * q^21 - 21209034240 * q^22 + 14220884280 * q^23 + 4586471424 * q^24 + 26561463141 * q^25 + 15963702016 * q^26 - 10460353203 * q^27 - 4052877312 * q^28 + 286157354226 * q^29 - 34161709824 * q^30 - 542755745328 * q^31 + 34359738368 * q^32 + 90121502340 * q^33 - 31472305920 * q^34 - 229291374576 * q^35 + 235092492288 * q^36 - 167027835126 * q^37 - 399908991488 * q^38 + 729250972830 * q^39 + 581594775552 * q^40 - 1366519968402 * q^41 + 1629700051968 * q^42 - 489890179812 * q^43 + 898081161216 * q^44 - 1681950180726 * q^45 + 225655723008 * q^46 + 9009642578640 * q^47 - 587068342272 * q^48 - 352670037093 * q^49 - 14107046503552 * q^50 - 4901628870918 * q^51 + 3879930200064 * q^52 + 12546949293162 * q^53 + 1338925209984 * q^54 - 19068946165368 * q^55 - 9012468776960 * q^56 - 28414203586308 * q^57 + 20053749259008 * q^58 + 23389468339980 * q^59 + 18164791738368 * q^60 + 62057091114114 * q^61 - 64291906973696 * q^62 - 1183153475592 * q^63 + 13194139533312 * q^64 + 16494962423388 * q^65 + 73264268772864 * q^66 - 64428771786348 * q^67 - 67274116595712 * q^68 - 32557661374968 * q^69 + 133686828582912 * q^70 + 19829570466024 * q^71 + 10030613004288 * q^72 - 221540395581714 * q^73 - 146555267074304 * q^74 - 165203392251933 * q^75 + 19694999371776 * q^76 + 573789241106400 * q^77 + 124723753115904 * q^78 - 414153951555648 * q^79 - 94396401844224 * q^80 + 68630377364883 * q^81 + 22809377279232 * q^82 + 377513484920196 * q^83 + 63478771089408 * q^84 + 246030873170004 * q^85 - 397238123314688 * q^86 - 112819827728322 * q^87 - 347488816988160 * q^88 + 92197563215742 * q^89 + 169784532594432 * q^90 + 647499908194896 * q^91 + 232994968043520 * q^92 + 199263780625104 * q^93 + 295167948171264 * q^94 - 1704507560977800 * q^95 + 75144747810816 * q^96 - 461982087578586 * q^97 + 120415920194688 * q^98 + 262176169041756 * q^99

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_0(6))\) 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
6.16.a.a	$6$	$16$	6.a	1.a	$1$	$1$	$1$	$8.562$	\(\Q\)	None	✓	✓	✓	✓	6.16.a.a	$1$	$0$	\(-128\)	\(-2187\)	\(-314490\)	\(2025056\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}-314490q^{5}+\cdots\)
6.16.a.b	$6$	$16$	6.a	1.a	$1$	$1$	$1$	$8.562$	\(\Q\)	None	✓	✓	✓	✓	6.16.a.b	$1$	$1$	\(128\)	\(-2187\)	\(-114810\)	\(-3034528\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{7}q^{2}-3^{7}q^{3}+2^{14}q^{4}-114810q^{5}+\cdots\)
6.16.a.c	$6$	$16$	6.a	1.a	$1$	$1$	$1$	$8.562$	\(\Q\)	None	✓	✓	✓	✓	6.16.a.c	$1$	$0$	\(128\)	\(2187\)	\(77646\)	\(762104\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{7}q^{2}+3^{7}q^{3}+2^{14}q^{4}+77646q^{5}+\cdots\)

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_0(6))\) into lower level spaces

\( S_{16}^{\mathrm{old}}(\Gamma_0(6)) \simeq \) \(S_{16}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 2}\)\(\oplus\)\(S_{16}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)