Space of modular forms of level 1728, weight 4, and Character 1728.c

• Label: 1728.4.c
• Level: $1728$
• Weight: $4$
• Character orbit: 1728.c
• Rep. character: $\chi_{1728}(1727,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $12$
• Sturm bound: $1152$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 12 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(1152\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1728, [\chi])\).

	Total	New	Old
Modular forms	900	96	804
Cusp forms	828	96	732
Eisenstein series	72	0	72

Trace form

\( 96 q - 72 q^{13} - 2400 q^{25} - 504 q^{37} - 4704 q^{49} - 168 q^{61} + 240 q^{85} - 1488 q^{97}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1728, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1728.4.c.a	$1728$	$4$	1728.c	12.b	$2$	$2$	$2$	$101.955$	\(\Q(\sqrt{-3}) \)	None					432.4.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-12\beta q^{5}+17\beta q^{7}-36 q^{11}-19 q^{13}+\cdots\)
1728.4.c.b	$1728$	$4$	1728.c	12.b	$2$	$2$	$2$	$101.955$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					432.4.c.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{7}-19 q^{13}-17\beta q^{19}+125 q^{25}+\cdots\)
1728.4.c.c	$1728$	$4$	1728.c	12.b	$2$	$2$	$2$	$101.955$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)					432.4.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+19\beta q^{7}+89 q^{13}+73\beta q^{19}+125 q^{25}+\cdots\)
1728.4.c.d	$1728$	$4$	1728.c	12.b	$2$	$2$	$2$	$101.955$	\(\Q(\sqrt{-3}) \)	None					432.4.c.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-12\beta q^{5}-17\beta q^{7}+36 q^{11}-19 q^{13}+\cdots\)
1728.4.c.e	$1728$	$4$	1728.c	12.b	$2$	$4$	$4$	$101.955$	\(\Q(\zeta_{12})\)	None					432.4.c.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-6\beta_{2}+\beta_1)q^{5}+(\beta_{2}-6\beta_1)q^{7}+\cdots\)
1728.4.c.f	$1728$	$4$	1728.c	12.b	$2$	$4$	$4$	$101.955$	\(\Q(\zeta_{12})\)	None					432.4.c.g	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{5}+\beta_{2} q^{7}+\beta_{3} q^{11}-7 q^{13}+\cdots\)
1728.4.c.g	$1728$	$4$	1728.c	12.b	$2$	$4$	$4$	$101.955$	\(\Q(\sqrt{3}, \sqrt{-17})\)	None					432.4.c.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}-\beta _{3}q^{7}-\beta _{2}q^{11}+2q^{13}+\cdots\)
1728.4.c.h	$1728$	$4$	1728.c	12.b	$2$	$4$	$4$	$101.955$	\(\Q(\zeta_{12})\)	None					432.4.c.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-6\beta_{2}+\beta_1)q^{5}+(-\beta_{2}+6\beta_1)q^{7}+\cdots\)
1728.4.c.i	$1728$	$4$	1728.c	12.b	$2$	$12$	$12$	$101.955$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					108.4.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{36}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{5}-\beta _{1}q^{7}+\beta _{5}q^{11}+(-3-\beta _{2}+\cdots)q^{13}+\cdots\)
1728.4.c.j	$1728$	$4$	1728.c	12.b	$2$	$12$	$12$	$101.955$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					108.4.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{32}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{5}-\beta _{1}q^{7}-\beta _{6}q^{11}+(6-\beta _{5}+\cdots)q^{13}+\cdots\)
1728.4.c.k	$1728$	$4$	1728.c	12.b	$2$	$24$	$24$	$101.955$		None					864.4.c.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
1728.4.c.l	$1728$	$4$	1728.c	12.b	$2$	$24$	$24$	$101.955$		None					864.4.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{4}^{\mathrm{old}}(1728, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1728, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)