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 - 88 * q^10 - 24 * q^12 - 116 * q^13 - 28 * q^14 + 60 * q^15 + 212 * q^16 + 152 * q^17 - 36 * q^18 + 8 * q^19 - 596 * q^20 - 84 * q^21 + 44 * q^22 + 244 * q^23 - 324 * q^24 + 146 * q^25 + 428 * q^26 - 192 * q^28 - 120 * q^29 + 336 * q^30 + 160 * q^31 + 364 * q^32 + 66 * q^33 - 208 * q^34 + 920 * q^35 + 396 * q^36 - 268 * q^37 - 696 * q^38 - 336 * q^39 - 168 * q^40 - 760 * q^41 + 852 * q^42 + 200 * q^43 - 616 * q^44 - 144 * q^45 - 464 * q^46 - 644 * q^47 - 816 * q^48 + 774 * q^49 + 148 * q^50 - 624 * q^51 + 264 * q^52 - 696 * q^53 - 108 * q^54 + 440 * q^55 + 228 * q^56 + 84 * q^57 + 840 * q^58 + 40 * q^59 + 12 * q^60 - 1180 * q^61 + 3032 * q^62 - 288 * q^63 + 1068 * q^64 - 880 * q^65 + 264 * q^66 - 1344 * q^67 + 1000 * q^68 + 1020 * q^69 - 2296 * q^70 + 756 * q^71 + 324 * q^72 + 724 * q^73 + 2256 * q^74 + 840 * q^75 + 2048 * q^76 - 176 * q^77 + 420 * q^78 - 144 * q^79 - 5732 * q^80 + 486 * q^81 - 2176 * q^82 + 264 * q^83 - 2880 * q^84 - 1840 * q^85 - 288 * q^86 + 1368 * q^87 + 132 * q^88 - 180 * q^89 - 792 * q^90 + 1680 * q^91 + 4268 * q^92 - 1632 * q^93 + 5104 * q^94 - 168 * q^95 - 3084 * q^96 - 2148 * q^97 - 8580 * q^98

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	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	11
33.4.a.a	$33$	$4$	33.a	1.a	$1$	$1$	$1$	$1.947$	\(\Q\)	None	✓	✓	✓	✓	33.4.a.a	$1$	$1$	\(-5\)	\(3\)	\(-14\)	\(-32\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+3q^{3}+17q^{4}-14q^{5}-15q^{6}+\cdots\)
33.4.a.b	$33$	$4$	33.a	1.a	$1$	$1$	$1$	$1.947$	\(\Q\)	None	✓	✓	✓	✓	33.4.a.b	$1$	$1$	\(-1\)	\(-3\)	\(-4\)	\(-26\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}-7q^{4}-4q^{5}+3q^{6}+\cdots\)
33.4.a.c	$33$	$4$	33.a	1.a	$1$	$2$	$2$	$1.947$	\(\Q(\sqrt{97}) \)	None	✓	✓	✓	✓	33.4.a.c	$1$	$0$	\(1\)	\(-6\)	\(-14\)	\(24\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-3q^{3}+(2^{4}+\beta )q^{4}+(-6-2\beta )q^{5}+\cdots\)
33.4.a.d	$33$	$4$	33.a	1.a	$1$	$2$	$2$	$1.947$	\(\Q(\sqrt{33}) \)	None	✓	✓	✓	✓	33.4.a.d	$1$	$0$	\(1\)	\(6\)	\(16\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+3q^{3}+\beta q^{4}+(10-4\beta )q^{5}+\cdots\)

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

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