Space of modular forms of level 147, weight 12, and trivial character

• Label: 147.12.a
• Level: $147$
• Weight: $12$
• Character orbit: 147.a
• Rep. character: $\chi_{147}(1,\cdot)$
• Character field: $\Q$
• Dimension: $75$
• Newform subspaces: $14$
• Sturm bound: $224$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 147 = 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	147.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(224\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(147))\).

	Total	New	Old
Modular forms	214	75	139
Cusp forms	198	75	123
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
\(+\)	\(+\)	$+$	\(18\)
\(+\)	\(-\)	$-$	\(19\)
\(-\)	\(+\)	$-$	\(17\)
\(-\)	\(-\)	$+$	\(21\)
Plus space		\(+\)	\(39\)
Minus space		\(-\)	\(36\)

Trace form

\( 75 q - 14 q^{2} + 243 q^{3} + 75696 q^{4} - 15494 q^{5} - 3402 q^{6} + 41544 q^{8} + 4428675 q^{9} + O(q^{10}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(147))\) 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
147.12.a.a	$147$	$12$	147.a	1.a	$1$	$1$	$1$	$112.946$	\(\Q\)	None	✓	$1$	$1$	\(-62\)	\(-243\)	\(3310\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-62q^{2}-3^{5}q^{3}+1796q^{4}+3310q^{5}+\cdots\)
147.12.a.b	$147$	$12$	147.a	1.a	$1$	$1$	$1$	$112.946$	\(\Q\)	None	✓	$1$	$1$	\(8\)	\(-243\)	\(-4390\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{5}q^{3}-1984q^{4}-4390q^{5}+\cdots\)
147.12.a.c	$147$	$12$	147.a	1.a	$1$	$1$	$1$	$112.946$	\(\Q\)	None	✓	$1$	$0$	\(78\)	\(243\)	\(5370\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+78q^{2}+3^{5}q^{3}+4036q^{4}+5370q^{5}+\cdots\)
147.12.a.d	$147$	$12$	147.a	1.a	$1$	$3$	$3$	$112.946$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(-68\)	\(729\)	\(-3326\)	\(0\)		$-$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+(-23-\beta _{1})q^{2}+3^{5}q^{3}+(664+72\beta _{1}+\cdots)q^{4}+\cdots\)
147.12.a.e	$147$	$12$	147.a	1.a	$1$	$3$	$3$	$112.946$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(-33\)	\(729\)	\(-3102\)	\(0\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-11-\beta _{2})q^{2}+3^{5}q^{3}+(255+13\beta _{1}+\cdots)q^{4}+\cdots\)
147.12.a.f	$147$	$12$	147.a	1.a	$1$	$4$	$4$	$112.946$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(45\)	\(-972\)	\(-13356\)	\(0\)		$+$	$-$	$-$	$2^{2}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(11+\beta _{1})q^{2}-3^{5}q^{3}+(1196+18\beta _{1}+\cdots)q^{4}+\cdots\)
147.12.a.g	$147$	$12$	147.a	1.a	$1$	$6$	$6$	$112.946$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(9\)	\(-1458\)	\(-8496\)	\(0\)		$+$	$-$	$-$	$2^{7}\cdot 3^{5}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}-3^{5}q^{3}+(1265+6\beta _{1}+\cdots)q^{4}+\cdots\)
147.12.a.h	$147$	$12$	147.a	1.a	$1$	$6$	$6$	$112.946$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(9\)	\(1458\)	\(8496\)	\(0\)		$-$	$-$	$+$	$2^{7}\cdot 3^{5}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+3^{5}q^{3}+(1265+6\beta _{1}+\cdots)q^{4}+\cdots\)
147.12.a.i	$147$	$12$	147.a	1.a	$1$	$7$	$7$	$112.946$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-9\)	\(-1701\)	\(7218\)	\(0\)		$+$	$-$	$-$	$2^{6}\cdot 3^{3}\cdot 7^{4}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}-3^{5}q^{3}+(1241+11\beta _{1}+\cdots)q^{4}+\cdots\)
147.12.a.j	$147$	$12$	147.a	1.a	$1$	$7$	$7$	$112.946$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-9\)	\(1701\)	\(-7218\)	\(0\)		$-$	$+$	$-$	$2^{6}\cdot 3^{3}\cdot 7^{4}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+3^{5}q^{3}+(1241+11\beta _{1}+\cdots)q^{4}+\cdots\)
147.12.a.k	$147$	$12$	147.a	1.a	$1$	$8$	$8$	$112.946$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(55\)	\(-1944\)	\(2156\)	\(0\)		$+$	$+$	$+$	$2^{5}\cdot 3^{4}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q+(7-\beta _{1})q^{2}-3^{5}q^{3}+(1346-7\beta _{1}+\cdots)q^{4}+\cdots\)
147.12.a.l	$147$	$12$	147.a	1.a	$1$	$8$	$8$	$112.946$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(55\)	\(1944\)	\(-2156\)	\(0\)		$-$	$-$	$+$	$2^{5}\cdot 3^{4}\cdot 7^{6}$	$\mathrm{SU}(2)$	\(q+(7-\beta _{1})q^{2}+3^{5}q^{3}+(1346-7\beta _{1}+\cdots)q^{4}+\cdots\)
147.12.a.m	$147$	$12$	147.a	1.a	$1$	$10$	$10$	$112.946$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(-46\)	\(-2430\)	\(12500\)	\(0\)		$+$	$+$	$+$	$2^{12}\cdot 3^{2}\cdot 7^{10}$	$\mathrm{SU}(2)$	\(q+(-5-\beta _{1})q^{2}-3^{5}q^{3}+(2^{9}+2\beta _{1}+\cdots)q^{4}+\cdots\)
147.12.a.n	$147$	$12$	147.a	1.a	$1$	$10$	$10$	$112.946$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-46\)	\(2430\)	\(-12500\)	\(0\)		$-$	$+$	$-$	$2^{12}\cdot 3^{2}\cdot 7^{10}$	$\mathrm{SU}(2)$	\(q+(-5-\beta _{1})q^{2}+3^{5}q^{3}+(2^{9}+2\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(147))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(147)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)