Space of modular forms of level 147, weight 4, and trivial character

• Label: 147.4.a
• Level: $147$
• Weight: $4$
• Character orbit: 147.a
• Rep. character: $\chi_{147}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $13$
• Sturm bound: $74$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(74\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(147))\).

	Total	New	Old
Modular forms	64	20	44
Cusp forms	48	20	28
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
\(+\)	\(+\)	\(+\)		\(17\)	\(5\)	\(12\)		\(13\)	\(5\)	\(8\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(15\)	\(5\)	\(10\)		\(11\)	\(5\)	\(6\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(15\)	\(3\)	\(12\)		\(11\)	\(3\)	\(8\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(17\)	\(7\)	\(10\)		\(13\)	\(7\)	\(6\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(34\)	\(12\)	\(22\)		\(26\)	\(12\)	\(14\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(30\)	\(8\)	\(22\)		\(22\)	\(8\)	\(14\)		\(8\)	\(0\)	\(8\)

Trace form

20 * q + 4 * q^2 + 60 * q^4 + 16 * q^5 + 12 * q^6 + 48 * q^8 + 180 * q^9 + 28 * q^10 - 40 * q^11 - 24 * q^12 + 80 * q^13 - 84 * q^15 + 332 * q^16 - 120 * q^17 + 36 * q^18 - 40 * q^19 - 172 * q^20 - 132 * q^22 - 40 * q^23 + 324 * q^24 + 728 * q^25 - 172 * q^26 + 376 * q^29 + 288 * q^30 + 168 * q^31 + 168 * q^32 + 96 * q^33 - 276 * q^34 + 540 * q^36 - 508 * q^37 - 1360 * q^38 - 756 * q^39 + 924 * q^40 + 1016 * q^41 - 828 * q^43 - 1072 * q^44 + 144 * q^45 - 520 * q^46 - 552 * q^47 - 816 * q^48 - 292 * q^50 + 456 * q^51 + 1420 * q^52 - 784 * q^53 + 108 * q^54 - 952 * q^55 - 276 * q^57 - 124 * q^58 - 792 * q^59 - 204 * q^60 - 592 * q^61 + 1824 * q^62 + 156 * q^64 - 168 * q^65 + 1896 * q^66 + 148 * q^67 - 492 * q^68 - 144 * q^69 + 608 * q^71 + 432 * q^72 + 72 * q^73 + 1196 * q^74 + 624 * q^75 + 832 * q^76 - 1692 * q^78 + 1936 * q^79 - 3484 * q^80 + 1620 * q^81 - 1492 * q^82 - 2208 * q^83 + 3904 * q^85 - 9836 * q^86 + 1032 * q^87 - 5484 * q^88 - 920 * q^89 + 252 * q^90 - 2744 * q^92 + 1788 * q^93 + 1344 * q^94 - 416 * q^95 + 204 * q^96 + 744 * q^97 - 360 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(147))\) 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
147.4.a.a	$147$	$4$	147.a	1.a	$1$	$1$	$1$	$8.673$	\(\Q\)	None	✓				21.4.e.a	$1$	$1$	\(-3\)	\(-3\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}-3q^{3}+q^{4}+3q^{5}+9q^{6}+\cdots\)
147.4.a.b	$147$	$4$	147.a	1.a	$1$	$1$	$1$	$8.673$	\(\Q\)	None	✓				21.4.e.a	$1$	$1$	\(-3\)	\(3\)	\(-3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+3q^{3}+q^{4}-3q^{5}-9q^{6}+\cdots\)
147.4.a.c	$147$	$4$	147.a	1.a	$1$	$1$	$1$	$8.673$	\(\Q\)	None	✓				21.4.a.a	$1$	$0$	\(-3\)	\(3\)	\(18\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}+3q^{3}+q^{4}+18q^{5}-9q^{6}+\cdots\)
147.4.a.d	$147$	$4$	147.a	1.a	$1$	$1$	$1$	$8.673$	\(\Q\)	None	✓	✓			147.4.a.d	$1$	$1$	\(-1\)	\(-3\)	\(12\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}-7q^{4}+12q^{5}+3q^{6}+\cdots\)
147.4.a.e	$147$	$4$	147.a	1.a	$1$	$1$	$1$	$8.673$	\(\Q\)	None	✓	✓			147.4.a.d	$1$	$0$	\(-1\)	\(3\)	\(-12\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}-7q^{4}-12q^{5}-3q^{6}+\cdots\)
147.4.a.f	$147$	$4$	147.a	1.a	$1$	$1$	$1$	$8.673$	\(\Q\)	None	✓	✓			147.4.a.f	$1$	$1$	\(4\)	\(-3\)	\(-18\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-3q^{3}+8q^{4}-18q^{5}-12q^{6}+\cdots\)
147.4.a.g	$147$	$4$	147.a	1.a	$1$	$1$	$1$	$8.673$	\(\Q\)	None	✓				21.4.a.b	$1$	$0$	\(4\)	\(3\)	\(4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+3q^{3}+8q^{4}+4q^{5}+12q^{6}+\cdots\)
147.4.a.h	$147$	$4$	147.a	1.a	$1$	$1$	$1$	$8.673$	\(\Q\)	None	✓	✓			147.4.a.f	$1$	$0$	\(4\)	\(3\)	\(18\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+3q^{3}+8q^{4}+18q^{5}+12q^{6}+\cdots\)
147.4.a.i	$147$	$4$	147.a	1.a	$1$	$2$	$2$	$8.673$	\(\Q(\sqrt{57}) \)	None	✓		✓		21.4.a.c	$1$	$1$	\(-3\)	\(-6\)	\(-6\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}-3q^{3}+(7+3\beta )q^{4}+\cdots\)
147.4.a.j	$147$	$4$	147.a	1.a	$1$	$2$	$2$	$8.673$	\(\Q(\sqrt{2}) \)	None	✓	✓			147.4.a.j	$1$	$0$	\(2\)	\(-6\)	\(20\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-3q^{3}+(-5+2\beta )q^{4}+\cdots\)
147.4.a.k	$147$	$4$	147.a	1.a	$1$	$2$	$2$	$8.673$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		147.4.a.j	$1$	$1$	\(2\)	\(6\)	\(-20\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3q^{3}+(-5+2\beta )q^{4}+\cdots\)
147.4.a.l	$147$	$4$	147.a	1.a	$1$	$3$	$3$	$8.673$	3.3.57516.1	None	✓		✓		21.4.e.b	$1$	$0$	\(1\)	\(-9\)	\(11\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(8+\beta _{1}+\beta _{2})q^{4}+\cdots\)
147.4.a.m	$147$	$4$	147.a	1.a	$1$	$3$	$3$	$8.673$	3.3.57516.1	None	✓		✓		21.4.e.b	$1$	$0$	\(1\)	\(9\)	\(-11\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(8+\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(147))\) into lower level spaces

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