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	Dim
\(+\)	\(+\)	$+$	\(9\)
\(+\)	\(-\)	$-$	\(9\)
\(-\)	\(+\)	$-$	\(10\)
\(-\)	\(-\)	$+$	\(7\)
Plus space		\(+\)	\(16\)
Minus space		\(-\)	\(19\)

Trace form

\( 35 q - 2 q^{2} - 9 q^{3} + 624 q^{4} - 38 q^{5} - 126 q^{6} - 552 q^{8} + 2835 q^{9} + O(q^{10}) \)

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

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

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