Space of modular forms of level 1728, weight 3, and Character 1728.b

• Label: 1728.3.b
• Level: $1728$
• Weight: $3$
• Character orbit: 1728.b
• Rep. character: $\chi_{1728}(1567,\cdot)$
• Character field: $\Q$
• Dimension: $64$
• Newform subspaces: $10$
• Sturm bound: $864$
• Trace bound: $49$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(864\)
Trace bound:			\(49\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1728, [\chi])\).

	Total	New	Old
Modular forms	612	64	548
Cusp forms	540	64	476
Eisenstein series	72	0	72

Trace form

\( 64 q - 320 q^{25} - 480 q^{49} + 160 q^{73} + 320 q^{97}+O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1728, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1728.3.b.a	$1728$	$3$	1728.b	8.d	$2$	$4$	$4$	$47.085$	\(\Q(\zeta_{12})\)	None		✓			1728.3.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+6\beta_1 q^{5}-5\beta_1 q^{7}-6 q^{11}-\beta_{2} q^{13}+\cdots\)
1728.3.b.b	$1728$	$3$	1728.b	8.d	$2$	$4$	$4$	$47.085$	\(\Q(\zeta_{12})\)	None		✓			1728.3.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta_{2} q^{5}+\beta_1 q^{7}-10\beta_{3} q^{11}+\cdots\)
1728.3.b.c	$1728$	$3$	1728.b	8.d	$2$	$4$	$4$	$47.085$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		✓			1728.3.b.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+13\beta_1 q^{7}-5\beta_{2} q^{13}-7\beta_{3} q^{19}+\cdots\)
1728.3.b.d	$1728$	$3$	1728.b	8.d	$2$	$4$	$4$	$47.085$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		✓			1728.3.b.d	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+11\beta_1 q^{7}-7\beta_{2} q^{13}+5\beta_{3} q^{19}+\cdots\)
1728.3.b.e	$1728$	$3$	1728.b	8.d	$2$	$4$	$4$	$47.085$	\(\Q(\zeta_{12})\)	None		✓			1728.3.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta_{2} q^{5}-\beta_1 q^{7}+10\beta_{3} q^{11}+\cdots\)
1728.3.b.f	$1728$	$3$	1728.b	8.d	$2$	$4$	$4$	$47.085$	\(\Q(\zeta_{12})\)	None		✓			1728.3.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+6\beta_1 q^{5}+5\beta_1 q^{7}+6 q^{11}+\beta_{2} q^{13}+\cdots\)
1728.3.b.g	$1728$	$3$	1728.b	8.d	$2$	$8$	$8$	$47.085$	\(\Q(\zeta_{24})\)	None		✓			1728.3.b.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{3} q^{5}-7\beta_1 q^{7}-\beta_{6} q^{11}+11\beta_{2} q^{13}+\cdots\)
1728.3.b.h	$1728$	$3$	1728.b	8.d	$2$	$8$	$8$	$47.085$	\(\Q(\zeta_{24})\)	\(\Q(\sqrt{-6}) \)		✓			1728.3.b.h	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}\cdot 3^{4}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta_{3} q^{5}+(-\beta_{4}+2\beta_1)q^{7}+(-\beta_{6}+2\beta_{2})q^{11}+\cdots\)
1728.3.b.i	$1728$	$3$	1728.b	8.d	$2$	$12$	$12$	$47.085$	12.0.\(\cdots\).2	None		✓			1728.3.b.i	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{5}+\beta _{2}q^{7}+(-2\beta _{1}+\beta _{9})q^{11}+\cdots\)
1728.3.b.j	$1728$	$3$	1728.b	8.d	$2$	$12$	$12$	$47.085$	12.0.\(\cdots\).2	None		✓			1728.3.b.i	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{22}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{5}-\beta _{2}q^{7}+(2\beta _{1}-\beta _{9})q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1728, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1728, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 16}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)