Space of modular forms of level 33, weight 4, and trivial character

• Label: 33.4.a
• Level: $33$
• Weight: $4$
• Character orbit: 33.a
• Rep. character: $\chi_{33}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $4$
• Sturm bound: $16$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	33.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(16\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(33))\).

	Total	New	Old
Modular forms	14	6	8
Cusp forms	10	6	4
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.

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

Trace form

\( 6 q - 4 q^{2} + 44 q^{4} - 16 q^{5} - 12 q^{6} - 32 q^{7} + 36 q^{8} + 54 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(33))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	11
33.4.a.a	$33$	$4$	33.a	1.a	$1$	$1$	$1$	$1.947$	\(\Q\)	None	✓	$1$	$1$	\(-5\)	\(3\)	\(-14\)	\(-32\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+3q^{3}+17q^{4}-14q^{5}-15q^{6}+\cdots\)
33.4.a.b	$33$	$4$	33.a	1.a	$1$	$1$	$1$	$1.947$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-3\)	\(-4\)	\(-26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}-7q^{4}-4q^{5}+3q^{6}+\cdots\)
33.4.a.c	$33$	$4$	33.a	1.a	$1$	$2$	$2$	$1.947$	\(\Q(\sqrt{97}) \)	None	✓	$1$	$0$	\(1\)	\(-6\)	\(-14\)	\(24\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-3q^{3}+(2^{4}+\beta )q^{4}+(-6-2\beta )q^{5}+\cdots\)
33.4.a.d	$33$	$4$	33.a	1.a	$1$	$2$	$2$	$1.947$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$0$	\(1\)	\(6\)	\(16\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+3q^{3}+\beta q^{4}+(10-4\beta )q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(33))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(33)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)