Space of modular forms of level 756, weight 2, and Character 756.k

• Label: 756.2.k
• Level: $756$
• Weight: $2$
• Character orbit: 756.k
• Rep. character: $\chi_{756}(109,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $22$
• Newform subspaces: $6$
• Sturm bound: $288$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.k (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(288\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(756, [\chi])\).

	Total	New	Old
Modular forms	324	22	302
Cusp forms	252	22	230
Eisenstein series	72	0	72

Trace form

\( 22 q + 3 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(756, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
756.2.k.a	$756$	$2$	756.k	7.c	$3$	$2$	$1$	$6.037$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(-3+2\zeta_{6})q^{7}-7q^{13}-8\zeta_{6}q^{19}+\cdots\)
756.2.k.b	$756$	$2$	756.k	7.c	$3$	$2$	$1$	$6.037$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(-1-2\zeta_{6})q^{7}+5q^{13}-8\zeta_{6}q^{19}+\cdots\)
756.2.k.c	$756$	$2$	756.k	7.c	$3$	$2$	$1$	$6.037$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(5\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(3-\zeta_{6})q^{7}+2q^{13}+\zeta_{6}q^{19}+(5+\cdots)q^{25}+\cdots\)
756.2.k.d	$756$	$2$	756.k	7.c	$3$	$4$	$2$	$6.037$	\(\Q(\sqrt{-3}, \sqrt{10})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{5}+(2+3\beta _{2})q^{7}+(-2\beta _{1}-2\beta _{3})q^{11}+\cdots\)
756.2.k.e	$756$	$2$	756.k	7.c	$3$	$6$	$3$	$6.037$	6.0.309123.1	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(2\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{2}+\beta _{4}-\beta _{5})q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
756.2.k.f	$756$	$2$	756.k	7.c	$3$	$6$	$3$	$6.037$	6.0.309123.1	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(2\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{2}-\beta _{4}+\beta _{5})q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(756, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(756, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 2}\)