Space of modular forms of level 66, weight 4, and trivial character

• Label: 66.4.a
• Level: $66$
• Weight: $4$
• Character orbit: 66.a
• Rep. character: $\chi_{66}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $3$
• Sturm bound: $48$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 66 = 2 \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	66.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(48\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(66))\).

	Total	New	Old
Modular forms	40	4	36
Cusp forms	32	4	28
Eisenstein series	8	0	8

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(11\)	Fricke	Dim
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(0\)

Trace form

4 * q + 4 * q^2 + 6 * q^3 + 16 * q^4 + 20 * q^5 + 28 * q^7 + 16 * q^8 + 36 * q^9 + 40 * q^10 + 24 * q^12 + 104 * q^13 + 64 * q^16 - 60 * q^17 + 36 * q^18 - 148 * q^19 + 80 * q^20 - 12 * q^21 - 44 * q^22 - 412 * q^23 - 156 * q^25 - 112 * q^26 + 54 * q^27 + 112 * q^28 - 340 * q^29 - 64 * q^31 + 64 * q^32 - 66 * q^33 - 240 * q^34 - 432 * q^35 + 144 * q^36 + 128 * q^37 - 520 * q^38 + 252 * q^39 + 160 * q^40 - 44 * q^41 - 192 * q^42 - 60 * q^43 + 180 * q^45 - 320 * q^46 + 260 * q^47 + 96 * q^48 + 828 * q^49 + 188 * q^50 - 120 * q^51 + 416 * q^52 + 540 * q^53 + 420 * q^57 + 208 * q^58 + 1080 * q^59 + 1344 * q^61 - 64 * q^62 + 252 * q^63 + 256 * q^64 + 1528 * q^65 - 264 * q^66 - 472 * q^67 - 240 * q^68 - 732 * q^69 - 864 * q^70 + 604 * q^71 + 144 * q^72 - 80 * q^73 + 680 * q^74 - 318 * q^75 - 592 * q^76 + 352 * q^77 - 456 * q^78 + 508 * q^79 + 320 * q^80 + 324 * q^81 - 544 * q^82 - 752 * q^83 - 48 * q^84 - 888 * q^85 + 88 * q^86 - 2328 * q^87 - 176 * q^88 - 248 * q^89 + 360 * q^90 - 2808 * q^91 - 1648 * q^92 + 864 * q^93 - 464 * q^94 - 3680 * q^95 - 2304 * q^97 + 2244 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(66))\) 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	11
66.4.a.a	$66$	$4$	66.a	1.a	$1$	$1$	$1$	$3.894$	\(\Q\)	None	✓	✓	✓	✓	66.4.a.a	$1$	$0$	\(-2\)	\(3\)	\(0\)	\(14\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+14q^{7}+\cdots\)
66.4.a.b	$66$	$4$	66.a	1.a	$1$	$1$	$1$	$3.894$	\(\Q\)	None	✓	✓	✓	✓	66.4.a.b	$1$	$0$	\(2\)	\(-3\)	\(10\)	\(16\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+4q^{4}+10q^{5}-6q^{6}+\cdots\)
66.4.a.c	$66$	$4$	66.a	1.a	$1$	$2$	$2$	$3.894$	\(\Q(\sqrt{97}) \)	None	✓	✓	✓	✓	66.4.a.c	$1$	$0$	\(4\)	\(6\)	\(10\)	\(-2\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+4q^{4}+(5-\beta )q^{5}+6q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(66))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(66)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)