Space of modular forms of level 99, weight 6, and trivial character

• Label: 99.6.a
• Level: $99$
• Weight: $6$
• Character orbit: 99.a
• Rep. character: $\chi_{99}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $9$
• Sturm bound: $72$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	99.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(72\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(99))\).

	Total	New	Old
Modular forms	64	22	42
Cusp forms	56	22	34
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.

\(3\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(14\)	\(5\)	\(9\)		\(12\)	\(5\)	\(7\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(18\)	\(5\)	\(13\)		\(16\)	\(5\)	\(11\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(16\)	\(7\)	\(9\)		\(14\)	\(7\)	\(7\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(16\)	\(5\)	\(11\)		\(14\)	\(5\)	\(9\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(30\)	\(10\)	\(20\)		\(26\)	\(10\)	\(16\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(34\)	\(12\)	\(22\)		\(30\)	\(12\)	\(18\)		\(4\)	\(0\)	\(4\)

Trace form

22 * q - 4 * q^2 + 380 * q^4 - 37 * q^5 + 22 * q^7 + 480 * q^8 - 122 * q^10 - 242 * q^11 - 890 * q^13 - 448 * q^14 + 5696 * q^16 + 2840 * q^17 - 1404 * q^19 - 2612 * q^20 + 484 * q^22 + 1837 * q^23 + 13745 * q^25 - 2416 * q^26 - 7456 * q^28 + 15030 * q^29 + 10579 * q^31 + 3916 * q^32 + 1168 * q^34 + 14426 * q^35 + 927 * q^37 - 600 * q^38 - 4668 * q^40 - 43654 * q^41 - 43962 * q^43 - 7260 * q^44 - 46786 * q^46 - 58088 * q^47 + 62682 * q^49 + 85498 * q^50 + 68432 * q^52 - 3948 * q^53 + 5203 * q^55 + 34884 * q^56 + 73908 * q^58 - 23081 * q^59 - 140658 * q^61 - 55390 * q^62 + 115552 * q^64 + 114848 * q^65 - 72211 * q^67 + 135424 * q^68 - 22940 * q^70 - 105369 * q^71 - 31382 * q^73 - 128346 * q^74 - 261528 * q^76 - 38478 * q^77 - 143962 * q^79 - 182708 * q^80 + 104260 * q^82 + 195066 * q^83 + 224758 * q^85 - 479460 * q^86 + 137940 * q^88 + 49041 * q^89 + 80440 * q^91 + 185732 * q^92 - 555160 * q^94 + 249072 * q^95 + 381665 * q^97 + 157812 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(99))\) 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
99.6.a.a	$99$	$6$	99.a	1.a	$1$	$1$	$1$	$15.878$	\(\Q\)	None	✓				33.6.a.b	$1$	$0$	\(-1\)	\(0\)	\(92\)	\(-26\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-31q^{4}+92q^{5}-26q^{7}+63q^{8}+\cdots\)
99.6.a.b	$99$	$6$	99.a	1.a	$1$	$1$	$1$	$15.878$	\(\Q\)	None	✓				33.6.a.a	$1$	$0$	\(2\)	\(0\)	\(-46\)	\(148\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-28q^{4}-46q^{5}+148q^{7}+\cdots\)
99.6.a.c	$99$	$6$	99.a	1.a	$1$	$1$	$1$	$15.878$	\(\Q\)	None	✓				11.6.a.a	$1$	$1$	\(4\)	\(0\)	\(19\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-2^{4}q^{4}+19q^{5}+10q^{7}-192q^{8}+\cdots\)
99.6.a.d	$99$	$6$	99.a	1.a	$1$	$2$	$2$	$15.878$	\(\Q(\sqrt{33}) \)	None	✓				33.6.a.e	$1$	$1$	\(-13\)	\(0\)	\(-58\)	\(146\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-6-\beta )q^{2}+(12+13\beta )q^{4}+(-34+\cdots)q^{5}+\cdots\)
99.6.a.e	$99$	$6$	99.a	1.a	$1$	$2$	$2$	$15.878$	\(\Q(\sqrt{313}) \)	None	✓				33.6.a.d	$1$	$0$	\(-1\)	\(0\)	\(38\)	\(-18\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(46+\beta )q^{4}+(24-10\beta )q^{5}+\cdots\)
99.6.a.f	$99$	$6$	99.a	1.a	$1$	$2$	$2$	$15.878$	\(\Q(\sqrt{177}) \)	None	✓				33.6.a.c	$1$	$1$	\(5\)	\(0\)	\(-58\)	\(-286\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{2}+(21-5\beta )q^{4}+(-34+10\beta )q^{5}+\cdots\)
99.6.a.g	$99$	$6$	99.a	1.a	$1$	$3$	$3$	$15.878$	3.3.54492.1	None	✓		✓		11.6.a.b	$1$	$0$	\(0\)	\(0\)	\(-24\)	\(84\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(28-2\beta _{1}-4\beta _{2})q^{4}+(-8+\cdots)q^{5}+\cdots\)
99.6.a.h	$99$	$6$	99.a	1.a	$1$	$5$	$5$	$15.878$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	99.6.a.h	$1$	$1$	\(-4\)	\(0\)	\(-100\)	\(-18\)		$+$	$+$	$+$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(21-3\beta _{1}+\beta _{4})q^{4}+\cdots\)
99.6.a.i	$99$	$6$	99.a	1.a	$1$	$5$	$5$	$15.878$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	99.6.a.h	$1$	$0$	\(4\)	\(0\)	\(100\)	\(-18\)		$+$	$-$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(21-3\beta _{1}+\beta _{4})q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(99))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(99)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)