Space of modular forms of level 972, weight 2, and trivial character

• Label: 972.2.a
• Level: $972$
• Weight: $2$
• Character orbit: 972.a
• Rep. character: $\chi_{972}(1,\cdot)$
• Character field: $\Q$
• Dimension: $12$
• Newform subspaces: $7$
• Sturm bound: $324$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 972 = 2^{2} \cdot 3^{5} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	972.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(324\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(972))\).

	Total	New	Old
Modular forms	189	12	177
Cusp forms	136	12	124
Eisenstein series	53	0	53

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

\(2\)	\(3\)	Fricke	Dim
\(-\)	\(+\)	$-$	\(7\)
\(-\)	\(-\)	$+$	\(5\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(7\)

Trace form

\( 12 q - 3 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(972))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3
972.2.a.a	$972$	$2$	972.a	1.a	$1$	$1$	$1$	$7.761$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{7}+5q^{13}-7q^{19}-5q^{25}-7q^{31}+\cdots\)
972.2.a.b	$972$	$2$	972.a	1.a	$1$	$1$	$1$	$7.761$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-q^{7}-7q^{13}-q^{19}-5q^{25}-7q^{31}+\cdots\)
972.2.a.c	$972$	$2$	972.a	1.a	$1$	$1$	$1$	$7.761$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-q^{7}+2q^{13}+8q^{19}-5q^{25}+11q^{31}+\cdots\)
972.2.a.d	$972$	$2$	972.a	1.a	$1$	$1$	$1$	$7.761$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(5\)		$-$	$+$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+5q^{7}+5q^{13}-7q^{19}-5q^{25}+11q^{31}+\cdots\)
972.2.a.e	$972$	$2$	972.a	1.a	$1$	$2$	$2$	$7.761$	\(\Q(\sqrt{2}) \)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+2q^{7}+\beta q^{11}-q^{13}-\beta q^{17}+\cdots\)
972.2.a.f	$972$	$2$	972.a	1.a	$1$	$3$	$3$	$7.761$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{5}+(-1+\beta _{2})q^{7}+(-3+\beta _{1}+\cdots)q^{11}+\cdots\)
972.2.a.g	$972$	$2$	972.a	1.a	$1$	$3$	$3$	$7.761$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(-1-\beta _{1})q^{7}+(3+\beta _{1}-\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(972))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(972)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(243))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(486))\)\(^{\oplus 2}\)