Space of modular forms of level 147, weight 2, and trivial character

• Label: 147.2.a
• Level: $147$
• Weight: $2$
• Character orbit: 147.a
• Rep. character: $\chi_{147}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $5$
• Sturm bound: $37$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(37\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(147))\).

	Total	New	Old
Modular forms	26	7	19
Cusp forms	11	7	4
Eisenstein series	15	0	15

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
\(+\)	\(+\)	\(+\)		\(5\)	\(2\)	\(3\)		\(2\)	\(2\)	\(0\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(7\)	\(2\)	\(5\)		\(3\)	\(2\)	\(1\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(8\)	\(3\)	\(5\)		\(4\)	\(3\)	\(1\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(6\)	\(0\)	\(6\)		\(2\)	\(0\)	\(2\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(11\)	\(2\)	\(9\)		\(4\)	\(2\)	\(2\)		\(7\)	\(0\)	\(7\)
Minus space		\(-\)		\(15\)	\(5\)	\(10\)		\(7\)	\(5\)	\(2\)		\(8\)	\(0\)	\(8\)

Trace form

7 * q - q^2 - q^3 + 7 * q^4 + 2 * q^5 + q^6 - 9 * q^8 + 7 * q^9 - 2 * q^10 - 8 * q^11 + q^12 + 2 * q^13 + 2 * q^15 + 3 * q^16 + 6 * q^17 - q^18 - 4 * q^19 - 2 * q^20 - 4 * q^22 - 8 * q^23 - 3 * q^24 + q^25 - 2 * q^26 - q^27 - 10 * q^29 - 6 * q^30 - 9 * q^32 - 4 * q^33 - 6 * q^34 + 7 * q^36 - 4 * q^37 + 4 * q^38 + 16 * q^39 + 6 * q^40 - 2 * q^41 + 6 * q^43 - 20 * q^44 + 2 * q^45 - 24 * q^46 + q^48 + 25 * q^50 + 2 * q^51 - 2 * q^52 + 22 * q^53 + q^54 + 8 * q^55 + 6 * q^57 + 18 * q^58 - 12 * q^59 - 14 * q^60 + 2 * q^61 - 37 * q^64 + 24 * q^65 + 4 * q^66 - 6 * q^67 - 6 * q^68 - 20 * q^71 - 9 * q^72 + 6 * q^73 + 22 * q^74 + q^75 + 4 * q^76 - 18 * q^78 + 14 * q^79 - 2 * q^80 + 7 * q^81 + 2 * q^82 + 12 * q^83 + 4 * q^85 + 56 * q^86 + 2 * q^87 + 36 * q^88 + 14 * q^89 - 2 * q^90 + 56 * q^92 + 2 * q^93 - 28 * q^95 + 5 * q^96 - 18 * q^97 - 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(147))\) 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
147.2.a.a	$147$	$2$	147.a	1.a	$1$	$1$	$1$	$1.174$	\(\Q\)	None	✓				21.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+2q^{5}+q^{6}+3q^{8}+\cdots\)
147.2.a.b	$147$	$2$	147.a	1.a	$1$	$1$	$1$	$1.174$	\(\Q\)	None	✓				21.2.e.a	$1$	$0$	\(2\)	\(-1\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}+2q^{5}-2q^{6}+\cdots\)
147.2.a.c	$147$	$2$	147.a	1.a	$1$	$1$	$1$	$1.174$	\(\Q\)	None	✓				21.2.e.a	$1$	$0$	\(2\)	\(1\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}-2q^{5}+2q^{6}+\cdots\)
147.2.a.d	$147$	$2$	147.a	1.a	$1$	$2$	$2$	$1.174$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	147.2.a.d	$1$	$1$	\(-2\)	\(-2\)	\(-4\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+(-2+\cdots)q^{5}+\cdots\)
147.2.a.e	$147$	$2$	147.a	1.a	$1$	$2$	$2$	$1.174$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		147.2.a.d	$1$	$0$	\(-2\)	\(2\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+(2+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(147))\) into lower level spaces

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