Space of modular forms of level 756, weight 4, and Character 756.f

• Label: 756.4.f
• Level: $756$
• Weight: $4$
• Character orbit: 756.f
• Rep. character: $\chi_{756}(377,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $5$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(576\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(756, [\chi])\).

	Total	New	Old
Modular forms	450	32	418
Cusp forms	414	32	382
Eisenstein series	36	0	36

Trace form

\( 32 q + 10 q^{7} + 920 q^{25} - 760 q^{37} - 1276 q^{43} + 476 q^{49} + 916 q^{67} + 1120 q^{79} - 2352 q^{85} + 2964 q^{91}+O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(756, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
756.4.f.a	$756$	$4$	756.f	21.c	$2$	$2$	$2$	$44.605$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			756.4.f.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-17\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(1-19\zeta_{6})q^{7}+(-17+34\zeta_{6})q^{13}+\cdots\)
756.4.f.b	$756$	$4$	756.f	21.c	$2$	$2$	$2$	$44.605$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			756.4.f.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(37\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(19-\zeta_{6})q^{7}+(-53+106\zeta_{6})q^{13}+\cdots\)
756.4.f.c	$756$	$4$	756.f	21.c	$2$	$4$	$4$	$44.605$	\(\Q(\sqrt{-3}, \sqrt{35})\)	None		✓			756.4.f.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(56\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}+(14-7\beta _{2})q^{7}-\beta _{3}q^{11}+\cdots\)
756.4.f.d	$756$	$4$	756.f	21.c	$2$	$8$	$8$	$44.605$	8.0.\(\cdots\).1	None		✓			756.4.f.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-80\)	$2^{12}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{5}+(-10+\beta _{1}+\beta _{4}-2\beta _{5}+\cdots)q^{7}+\cdots\)
756.4.f.e	$756$	$4$	756.f	21.c	$2$	$16$	$16$	$44.605$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓	✓		756.4.f.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(14\)	$2^{8}\cdot 3^{26}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}+(1-\beta _{6})q^{7}-\beta _{2}q^{11}+\beta _{10}q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(756, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(756, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 2}\)