Space of modular forms of level 147, weight 6, and trivial character

• Label: 147.6.a
• Level: $147$
• Weight: $6$
• Character orbit: 147.a
• Rep. character: $\chi_{147}(1,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $15$
• Sturm bound: $112$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(112\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(147))\).

	Total	New	Old
Modular forms	102	35	67
Cusp forms	86	35	51
Eisenstein series	16	0	16

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

\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(24\)	\(9\)	\(15\)		\(20\)	\(9\)	\(11\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(26\)	\(9\)	\(17\)		\(22\)	\(9\)	\(13\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(27\)	\(10\)	\(17\)		\(23\)	\(10\)	\(13\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(25\)	\(7\)	\(18\)		\(21\)	\(7\)	\(14\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(49\)	\(16\)	\(33\)		\(41\)	\(16\)	\(25\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(53\)	\(19\)	\(34\)		\(45\)	\(19\)	\(26\)		\(8\)	\(0\)	\(8\)

Trace form

35 * q - 2 * q^2 - 9 * q^3 + 624 * q^4 - 38 * q^5 - 126 * q^6 - 552 * q^8 + 2835 * q^9 + 1128 * q^10 + 920 * q^11 - 828 * q^12 - 550 * q^13 + 450 * q^15 + 11196 * q^16 + 2286 * q^17 - 162 * q^18 - 3844 * q^19 + 6476 * q^20 - 4820 * q^22 - 1096 * q^23 - 2052 * q^24 + 14429 * q^25 - 2128 * q^26 - 729 * q^27 + 1174 * q^29 + 5292 * q^30 - 12016 * q^31 - 22776 * q^32 + 4716 * q^33 + 17664 * q^34 + 50544 * q^36 + 36032 * q^37 + 12296 * q^38 + 2952 * q^39 + 40236 * q^40 - 11002 * q^41 - 6394 * q^43 - 38272 * q^44 - 3078 * q^45 + 30584 * q^46 + 29856 * q^47 + 2304 * q^48 - 23086 * q^50 + 10386 * q^51 - 37660 * q^52 + 61814 * q^53 - 10206 * q^54 - 37944 * q^55 - 24246 * q^57 - 157816 * q^58 + 11508 * q^59 + 58140 * q^60 - 47590 * q^61 - 39264 * q^62 + 195740 * q^64 - 8448 * q^65 - 68112 * q^66 - 60726 * q^67 - 26820 * q^68 + 32400 * q^69 - 100324 * q^71 - 44712 * q^72 - 60706 * q^73 - 303640 * q^74 - 87687 * q^75 + 93008 * q^76 + 139968 * q^78 - 23594 * q^79 + 348548 * q^80 + 229635 * q^81 + 69648 * q^82 - 75540 * q^83 - 256044 * q^85 + 748444 * q^86 - 224622 * q^87 + 387924 * q^88 + 110710 * q^89 + 91368 * q^90 - 80024 * q^92 - 199458 * q^93 - 138240 * q^94 - 388652 * q^95 - 20340 * q^96 + 230150 * q^97 + 74520 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(147))\) 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	7
147.6.a.a	$147$	$6$	147.a	1.a	$1$	$1$	$1$	$23.576$	\(\Q\)	None	✓				3.6.a.a	$1$	$0$	\(-6\)	\(-9\)	\(-6\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{2}-9q^{3}+4q^{4}-6q^{5}+54q^{6}+\cdots\)
147.6.a.b	$147$	$6$	147.a	1.a	$1$	$1$	$1$	$23.576$	\(\Q\)	None	✓				21.6.a.a	$1$	$1$	\(-6\)	\(9\)	\(-78\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{2}+9q^{3}+4q^{4}-78q^{5}-54q^{6}+\cdots\)
147.6.a.c	$147$	$6$	147.a	1.a	$1$	$1$	$1$	$23.576$	\(\Q\)	None	✓				21.6.e.a	$1$	$1$	\(-2\)	\(-9\)	\(11\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-9q^{3}-28q^{4}+11q^{5}+18q^{6}+\cdots\)
147.6.a.d	$147$	$6$	147.a	1.a	$1$	$1$	$1$	$23.576$	\(\Q\)	None	✓				21.6.e.a	$1$	$1$	\(-2\)	\(9\)	\(-11\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+9q^{3}-28q^{4}-11q^{5}-18q^{6}+\cdots\)
147.6.a.e	$147$	$6$	147.a	1.a	$1$	$1$	$1$	$23.576$	\(\Q\)	None	✓				21.6.a.b	$1$	$1$	\(1\)	\(9\)	\(34\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+9q^{3}-31q^{4}+34q^{5}+9q^{6}+\cdots\)
147.6.a.f	$147$	$6$	147.a	1.a	$1$	$1$	$1$	$23.576$	\(\Q\)	None	✓				21.6.a.c	$1$	$0$	\(5\)	\(-9\)	\(-94\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}-9q^{3}-7q^{4}-94q^{5}-45q^{6}+\cdots\)
147.6.a.g	$147$	$6$	147.a	1.a	$1$	$1$	$1$	$23.576$	\(\Q\)	None	✓				21.6.a.d	$1$	$0$	\(10\)	\(-9\)	\(106\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+10q^{2}-9q^{3}+68q^{4}+106q^{5}+\cdots\)
147.6.a.h	$147$	$6$	147.a	1.a	$1$	$2$	$2$	$23.576$	\(\Q(\sqrt{193}) \)	None	✓	✓			147.6.a.h	$1$	$0$	\(-3\)	\(-18\)	\(72\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-9q^{3}+(17+3\beta )q^{4}+\cdots\)
147.6.a.i	$147$	$6$	147.a	1.a	$1$	$2$	$2$	$23.576$	\(\Q(\sqrt{249}) \)	None	✓				21.6.e.b	$1$	$1$	\(-3\)	\(-18\)	\(-33\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-9q^{3}+(31+3\beta )q^{4}+\cdots\)
147.6.a.j	$147$	$6$	147.a	1.a	$1$	$2$	$2$	$23.576$	\(\Q(\sqrt{193}) \)	None	✓	✓			147.6.a.h	$1$	$1$	\(-3\)	\(18\)	\(-72\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+9q^{3}+(17+3\beta )q^{4}+\cdots\)
147.6.a.k	$147$	$6$	147.a	1.a	$1$	$2$	$2$	$23.576$	\(\Q(\sqrt{249}) \)	None	✓				21.6.e.b	$1$	$1$	\(-3\)	\(18\)	\(33\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+9q^{3}+(31+3\beta )q^{4}+\cdots\)
147.6.a.l	$147$	$6$	147.a	1.a	$1$	$4$	$4$	$23.576$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		21.6.e.c	$1$	$0$	\(3\)	\(-36\)	\(0\)	\(0\)		$+$	$-$	$-$	$7$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-9q^{3}+(18-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
147.6.a.m	$147$	$6$	147.a	1.a	$1$	$4$	$4$	$23.576$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				21.6.e.c	$1$	$0$	\(3\)	\(36\)	\(0\)	\(0\)		$-$	$+$	$-$	$7$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+9q^{3}+(18-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
147.6.a.n	$147$	$6$	147.a	1.a	$1$	$6$	$6$	$23.576$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		147.6.a.n	$1$	$1$	\(2\)	\(-54\)	\(-100\)	\(0\)		$+$	$+$	$+$	$2^{4}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}-9q^{3}+(5^{2}-\beta _{1}+\beta _{4})q^{4}+\cdots\)
147.6.a.o	$147$	$6$	147.a	1.a	$1$	$6$	$6$	$23.576$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		147.6.a.n	$1$	$0$	\(2\)	\(54\)	\(100\)	\(0\)		$-$	$+$	$-$	$2^{4}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+9q^{3}+(5^{2}-\beta _{1}+\beta _{4})q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(147))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(147)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)