Space of modular forms of level 333, weight 2, and trivial character

• Label: 333.2.a
• Level: $333$
• Weight: $2$
• Character orbit: 333.a
• Rep. character: $\chi_{333}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $7$
• Sturm bound: $76$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	333.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(76\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(333))\).

	Total	New	Old
Modular forms	42	15	27
Cusp forms	35	15	20
Eisenstein series	7	0	7

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

\(3\)	\(37\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(9\)	\(2\)	\(7\)		\(8\)	\(2\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(11\)	\(4\)	\(7\)		\(9\)	\(4\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(12\)	\(5\)	\(7\)		\(10\)	\(5\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(10\)	\(4\)	\(6\)		\(8\)	\(4\)	\(4\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(19\)	\(6\)	\(13\)		\(16\)	\(6\)	\(10\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(23\)	\(9\)	\(14\)		\(19\)	\(9\)	\(10\)		\(4\)	\(0\)	\(4\)

Trace form

15 * q - q^2 + 11 * q^4 - 2 * q^7 - 3 * q^8 - 10 * q^10 - 2 * q^11 + 10 * q^14 + 3 * q^16 - 8 * q^17 - 2 * q^19 + 26 * q^20 - 2 * q^22 + 4 * q^23 + 5 * q^25 + 18 * q^26 - 8 * q^28 - 14 * q^29 - 7 * q^32 - 2 * q^34 + 2 * q^35 + q^37 - 8 * q^38 - 22 * q^40 - 12 * q^41 - 10 * q^43 - 20 * q^44 + 4 * q^46 + 22 * q^47 + 25 * q^49 - 27 * q^50 - 2 * q^52 - 14 * q^55 + 32 * q^56 - 10 * q^58 - 16 * q^59 + 14 * q^61 - 12 * q^62 + 11 * q^64 - 40 * q^65 - 16 * q^67 - 38 * q^68 + 20 * q^70 + 6 * q^71 - 20 * q^73 - 5 * q^74 + 30 * q^77 + 26 * q^79 + 30 * q^80 + 8 * q^82 + 26 * q^83 - 12 * q^85 - 20 * q^86 - 20 * q^88 - 32 * q^89 - 2 * q^91 + 48 * q^92 - 30 * q^94 + 56 * q^95 - 2 * q^97 - 87 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(333))\) 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	37
333.2.a.a	$333$	$2$	333.a	1.a	$1$	$1$	$1$	$2.659$	\(\Q\)	None	✓	✓			333.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(2\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+2q^{5}-4q^{7}+3q^{8}-2q^{10}+\cdots\)
333.2.a.b	$333$	$2$	333.a	1.a	$1$	$1$	$1$	$2.659$	\(\Q\)	None	✓				37.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{4}-q^{7}-3q^{11}-4q^{13}+4q^{16}+\cdots\)
333.2.a.c	$333$	$2$	333.a	1.a	$1$	$1$	$1$	$2.659$	\(\Q\)	None	✓	✓			333.2.a.a	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-2q^{5}-4q^{7}-3q^{8}-2q^{10}+\cdots\)
333.2.a.d	$333$	$2$	333.a	1.a	$1$	$1$	$1$	$2.659$	\(\Q\)	None	✓				37.2.a.a	$1$	$0$	\(2\)	\(0\)	\(2\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+2q^{5}-q^{7}+4q^{10}+\cdots\)
333.2.a.e	$333$	$2$	333.a	1.a	$1$	$3$	$3$	$2.659$	3.3.148.1	None	✓		✓		111.2.a.a	$1$	$1$	\(-3\)	\(0\)	\(-4\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
333.2.a.f	$333$	$2$	333.a	1.a	$1$	$4$	$4$	$2.659$	4.4.27648.1	None	✓	✓	✓	✓	333.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+2q^{7}+\cdots\)
333.2.a.g	$333$	$2$	333.a	1.a	$1$	$4$	$4$	$2.659$	4.4.6224.1	None	✓		✓		111.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(1+\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(333))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(333)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(111))\)\(^{\oplus 2}\)