Space of modular forms of level 3, weight 16, and trivial character

• Label: 3.16.a
• Level: $3$
• Weight: $16$
• Character orbit: 3.a
• Rep. character: $\chi_{3}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $5$
• Trace bound: $2$

Defining parameters

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

Dimensions

The following table gives the dimensions of various subspaces of \(M_{16}(\Gamma_0(3))\).

	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.

\(3\)		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 - 306 * q^2 - 5596 * q^4 + 59220 * q^5 + 354294 * q^6 - 3522344 * q^7 + 6867864 * q^8 + 9565938 * q^9 - 49738860 * q^10 + 71200368 * q^11 - 108413964 * q^12 + 104047996 * q^13 + 476090496 * q^14 - 1098311400 * q^15 - 719764720 * q^16 + 3752120340 * q^17 - 1463588514 * q^18 - 7168853264 * q^19 + 12281831640 * q^20 - 1696359672 * q^21 - 10639456920 * q^22 - 7183099728 * q^23 + 3986516088 * q^24 + 66820767950 * q^25 - 69703062156 * q^26 + 29080928128 * q^28 - 164366907948 * q^29 + 178532289540 * q^30 + 74360242552 * q^31 + 39122031456 * q^32 + 6863383368 * q^33 - 474144293412 * q^34 + 90470615760 * q^35 - 26765494524 * q^36 + 414541609276 * q^37 + 921006940248 * q^38 - 1452160406736 * q^39 - 254353653360 * q^40 + 1229001516612 * q^41 - 364429642752 * q^42 - 703402249136 * q^43 - 277003641168 * q^44 + 283247424180 * q^45 - 1310959874448 * q^46 + 3596817155472 * q^47 + 4159193314896 * q^48 - 2990848277550 * q^49 - 12632540500350 * q^50 + 2698113202416 * q^51 + 16166691650200 * q^52 - 9402324365628 * q^53 + 1694577218886 * q^54 + 1320224806080 * q^55 - 12802431962880 * q^56 - 4747345446888 * q^57 + 19677397654116 * q^58 + 30487277052864 * q^59 - 137062176840 * q^60 - 51165426961940 * q^61 - 26502594515568 * q^62 - 16847262159336 * q^63 - 15529712956864 * q^64 + 169810389342360 * q^65 + 11562833934792 * q^66 - 68739036837440 * q^67 - 41077049005368 * q^68 - 65069301586464 * q^69 + 55938961532160 * q^70 - 74160484607376 * q^71 + 32848780608216 * q^72 + 99993522076852 * q^73 + 2589181575876 * q^74 - 65042001108000 * q^75 + 73861699830736 * q^76 - 126613201161888 * q^77 + 240612332578020 * q^78 + 104253009464920 * q^79 - 498849243588000 * q^80 + 45753584909922 * q^81 - 198509068516788 * q^82 + 435753307340208 * q^83 + 195682052168064 * q^84 - 198683084417400 * q^85 + 5368846365288 * q^86 - 147709960072056 * q^87 + 247357473555744 * q^88 - 142972544516364 * q^89 - 237899425475340 * q^90 + 74270028249872 * q^91 + 757550397685536 * q^92 - 408387889938024 * q^93 - 505319834071296 * q^94 + 332795843199360 * q^95 - 894490897204512 * q^96 + 163536037017700 * q^97 + 678902094335934 * q^98 + 340549152932592 * q^99

Decomposition of \(S_{16}^{\mathrm{new}}(\Gamma_0(3))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3
3.16.a.a	$3$	$16$	3.a	1.a	$1$	$1$	$1$	$4.281$	\(\Q\)	None	✓	✓	✓	✓	3.16.a.a	$1$	$0$	\(-234\)	\(-2187\)	\(280710\)	\(-1373344\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-234q^{2}-3^{7}q^{3}+21988q^{4}+280710q^{5}+\cdots\)
3.16.a.b	$3$	$16$	3.a	1.a	$1$	$1$	$1$	$4.281$	\(\Q\)	None	✓	✓	✓	✓	3.16.a.b	$1$	$1$	\(-72\)	\(2187\)	\(-221490\)	\(-2149000\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-72q^{2}+3^{7}q^{3}-27584q^{4}-221490q^{5}+\cdots\)

Decomposition of \(S_{16}^{\mathrm{old}}(\Gamma_0(3))\) into lower level spaces

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