Space of modular forms of level 972, weight 2, and Character 972.e

• Label: 972.2.e
• Level: $972$
• Weight: $2$
• Character orbit: 972.e
• Rep. character: $\chi_{972}(325,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $24$
• 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.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
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 \(M_{2}(972, [\chi])\).

	Total	New	Old
Modular forms	378	24	354
Cusp forms	270	24	246
Eisenstein series	108	0	108

Trace form

\( 24 q + 3 q^{7} + 3 q^{13} - 6 q^{19} - 12 q^{25} - 6 q^{31} - 6 q^{37} - 6 q^{43} - 9 q^{49} + 36 q^{55} - 24 q^{61} - 24 q^{67} + 48 q^{73} - 15 q^{79} + 72 q^{85} + 60 q^{91} + 75 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(972, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
972.2.e.a	$972$	$2$	972.e	9.c	$3$	$2$	$1$	$7.761$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			972.2.a.d	$4$	$1$	\(0\)	\(0\)	\(0\)	\(-5\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(-5+5\zeta_{6})q^{7}-5\zeta_{6}q^{13}-7q^{19}+\cdots\)
972.2.e.b	$972$	$2$	972.e	9.c	$3$	$2$	$1$	$7.761$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			972.2.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(1\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(1-\zeta_{6})q^{7}-2\zeta_{6}q^{13}+8q^{19}+(5+\cdots)q^{25}+\cdots\)
972.2.e.c	$972$	$2$	972.e	9.c	$3$	$2$	$1$	$7.761$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			972.2.a.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(1\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(1-\zeta_{6})q^{7}+7\zeta_{6}q^{13}-q^{19}+(5+\cdots)q^{25}+\cdots\)
972.2.e.d	$972$	$2$	972.e	9.c	$3$	$2$	$1$	$7.761$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			972.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(4-4\zeta_{6})q^{7}-5\zeta_{6}q^{13}-7q^{19}+\cdots\)
972.2.e.e	$972$	$2$	972.e	9.c	$3$	$4$	$2$	$7.761$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			972.2.a.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{5}+2\beta _{2}q^{7}+(-\beta _{1}-\beta _{3})q^{11}+\cdots\)
972.2.e.f	$972$	$2$	972.e	9.c	$3$	$6$	$3$	$7.761$	\(\Q(\zeta_{18})\)	None		✓			972.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(3\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta_{4} q^{5}+(\beta_{5}-\beta_{2}+\beta_1)q^{7}+(-\beta_{5}-3\beta_1)q^{11}+\cdots\)
972.2.e.g	$972$	$2$	972.e	9.c	$3$	$6$	$3$	$7.761$	\(\Q(\zeta_{18})\)	None		✓			972.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(3\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta_{2} q^{5}+(\beta_{5}-\beta_{4}+\beta_{3}+\cdots+1)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(972, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(972, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(243, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(486, [\chi])\)\(^{\oplus 2}\)