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

• Label: 1728.4.d
• Level: $1728$
• Weight: $4$
• Character orbit: 1728.d
• Rep. character: $\chi_{1728}(865,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $11$
• Sturm bound: $1152$
• Trace bound: $49$

Defining parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(1152\)
Trace bound:			\(49\)
Distinguishing \(T_p\):			\(5\), \(7\), \(17\), \(23\)

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

\( 96 q - 2400 q^{25} + 5424 q^{49} + 432 q^{73} - 6048 q^{97}+O(q^{100}) \)

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.d.a	$1728$	$4$	1728.d	8.b	$2$	$4$	$4$	$101.955$	\(\Q(i, \sqrt{19})\)	None		✓			1728.4.d.a	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-12\beta _{1}q^{5}-\beta _{2}q^{7}+4\beta _{3}q^{11}-5\beta _{3}q^{13}+\cdots\)
1728.4.d.b	$1728$	$4$	1728.d	8.b	$2$	$4$	$4$	$101.955$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		✓			1728.4.d.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta_{3} q^{7}-53\beta_{2} q^{13}+163\beta_1 q^{19}+\cdots\)
1728.4.d.c	$1728$	$4$	1728.d	8.b	$2$	$4$	$4$	$101.955$	\(\Q(\zeta_{12})\)	\(\Q(\sqrt{-3}) \)		✓			1728.4.d.c	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q-19\beta_{3} q^{7}+17\beta_{2} q^{13}+107\beta_1 q^{19}+\cdots\)
1728.4.d.d	$1728$	$4$	1728.d	8.b	$2$	$4$	$4$	$101.955$	\(\Q(i, \sqrt{19})\)	None		✓			1728.4.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-12\beta _{1}q^{5}+\beta _{2}q^{7}-4\beta _{3}q^{11}-5\beta _{3}q^{13}+\cdots\)
1728.4.d.e	$1728$	$4$	1728.d	8.b	$2$	$8$	$8$	$101.955$	8.0.\(\cdots\).6	None		✓			1728.4.d.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{5}+(\beta _{4}+\beta _{6})q^{7}+(-12\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
1728.4.d.f	$1728$	$4$	1728.d	8.b	$2$	$8$	$8$	$101.955$	8.0.592240896.1	None		✓			1728.4.d.f	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{6}q^{5}+7\beta _{1}q^{7}-\beta _{7}q^{11}-9\beta _{2}q^{13}+\cdots\)
1728.4.d.g	$1728$	$4$	1728.d	8.b	$2$	$8$	$8$	$101.955$	8.0.1731891456.1	None		✓			1728.4.d.g	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{5}+\beta _{6}q^{7}-\beta _{7}q^{11}+\beta _{5}q^{13}+\cdots\)
1728.4.d.h	$1728$	$4$	1728.d	8.b	$2$	$8$	$8$	$101.955$	8.0.\(\cdots\).6	None		✓			1728.4.d.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{5}q^{5}+(\beta _{4}+\beta _{6})q^{7}+(-12\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
1728.4.d.i	$1728$	$4$	1728.d	8.b	$2$	$16$	$16$	$101.955$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓			1728.4.d.i	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{40}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{8}q^{5}+\beta _{7}q^{7}+(-6\beta _{4}+\beta _{12})q^{11}+\cdots\)
1728.4.d.j	$1728$	$4$	1728.d	8.b	$2$	$16$	$16$	$101.955$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			1728.4.d.j	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{44}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{11}q^{5}+(-\beta _{5}+\beta _{8})q^{7}-\beta _{7}q^{11}+\cdots\)
1728.4.d.k	$1728$	$4$	1728.d	8.b	$2$	$16$	$16$	$101.955$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓			1728.4.d.i	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{40}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{8}q^{5}-\beta _{7}q^{7}+(6\beta _{4}-\beta _{12})q^{11}+\cdots\)

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

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