Space of modular forms of level 33, weight 6, and trivial character

• Label: 33.6.a
• Level: $33$
• Weight: $6$
• Character orbit: 33.a
• Rep. character: $\chi_{33}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $5$
• Sturm bound: $24$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	33.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(24\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(33))\).

	Total	New	Old
Modular forms	22	8	14
Cusp forms	18	8	10
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(7\)	\(3\)	\(4\)		\(6\)	\(3\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(10\)	\(3\)	\(7\)		\(8\)	\(3\)	\(5\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(12\)	\(5\)	\(7\)		\(10\)	\(5\)	\(5\)		\(2\)	\(0\)	\(2\)

Trace form

8 * q + 8 * q^2 - 18 * q^3 + 108 * q^4 + 32 * q^5 + 180 * q^6 - 36 * q^7 - 108 * q^8 + 648 * q^9 + 544 * q^10 - 864 * q^12 - 480 * q^13 + 1508 * q^14 - 900 * q^15 + 948 * q^16 - 4612 * q^17 + 648 * q^18 - 916 * q^19 + 5788 * q^20 + 2484 * q^21 - 968 * q^22 - 2480 * q^23 + 540 * q^24 + 15816 * q^25 - 14308 * q^26 - 1458 * q^27 - 24176 * q^28 - 6180 * q^29 - 2736 * q^30 + 1752 * q^31 + 13612 * q^32 - 2178 * q^33 - 3392 * q^34 + 9992 * q^35 + 8748 * q^36 - 27904 * q^37 + 8304 * q^38 - 10476 * q^39 + 10968 * q^40 + 39188 * q^41 - 15372 * q^42 + 30356 * q^43 - 4840 * q^44 + 2592 * q^45 + 54896 * q^46 + 68152 * q^47 + 12528 * q^48 + 12744 * q^49 - 137624 * q^50 + 2448 * q^51 - 20696 * q^52 - 16776 * q^53 + 14580 * q^54 - 24200 * q^55 + 5508 * q^56 + 32220 * q^57 + 7320 * q^58 + 33280 * q^59 - 20052 * q^60 - 51208 * q^61 + 49592 * q^62 - 2916 * q^63 - 180532 * q^64 - 128992 * q^65 - 17424 * q^66 + 58648 * q^67 - 111848 * q^68 - 10440 * q^69 + 217864 * q^70 + 35064 * q^71 - 8748 * q^72 + 30456 * q^73 + 23304 * q^74 - 98766 * q^75 - 117712 * q^76 + 29524 * q^77 + 123084 * q^78 - 128372 * q^79 + 59164 * q^80 + 52488 * q^81 + 86560 * q^82 + 51576 * q^83 + 41328 * q^84 + 87160 * q^85 + 220296 * q^86 - 31320 * q^87 + 68244 * q^88 + 40368 * q^89 + 44064 * q^90 + 42536 * q^91 + 210140 * q^92 - 118152 * q^93 - 128896 * q^94 - 234168 * q^95 - 44460 * q^96 - 267584 * q^97 - 228120 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(33))\) 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
33.6.a.a	$33$	$6$	33.a	1.a	$1$	$1$	$1$	$5.293$	\(\Q\)	None	✓	✓			33.6.a.a	$1$	$0$	\(-2\)	\(-9\)	\(46\)	\(148\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-9q^{3}-28q^{4}+46q^{5}+18q^{6}+\cdots\)
33.6.a.b	$33$	$6$	33.a	1.a	$1$	$1$	$1$	$5.293$	\(\Q\)	None	✓	✓	✓	✓	33.6.a.b	$1$	$1$	\(1\)	\(9\)	\(-92\)	\(-26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+9q^{3}-31q^{4}-92q^{5}+9q^{6}+\cdots\)
33.6.a.c	$33$	$6$	33.a	1.a	$1$	$2$	$2$	$5.293$	\(\Q(\sqrt{177}) \)	None	✓	✓	✓	✓	33.6.a.c	$1$	$1$	\(-5\)	\(-18\)	\(58\)	\(-286\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{2}-9q^{3}+(2^{4}+5\beta )q^{4}+\cdots\)
33.6.a.d	$33$	$6$	33.a	1.a	$1$	$2$	$2$	$5.293$	\(\Q(\sqrt{313}) \)	None	✓	✓	✓		33.6.a.d	$1$	$0$	\(1\)	\(-18\)	\(-38\)	\(-18\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-9q^{3}+(46+\beta )q^{4}+(-24+\cdots)q^{5}+\cdots\)
33.6.a.e	$33$	$6$	33.a	1.a	$1$	$2$	$2$	$5.293$	\(\Q(\sqrt{33}) \)	None	✓	✓	✓	✓	33.6.a.e	$1$	$0$	\(13\)	\(18\)	\(58\)	\(146\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{2}+9q^{3}+(5^{2}-13\beta )q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(33))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(33)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)