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

• Label: 756.2.t
• Level: $756$
• Weight: $2$
• Character orbit: 756.t
• Rep. character: $\chi_{756}(269,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $22$
• Newform subspaces: $5$
• Sturm bound: $288$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.t (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 21 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(288\)
Trace bound:			\(19\)
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

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

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.t.a	$756$	$2$	756.t	21.g	$6$	$2$	$1$	$6.037$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-5\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-3+\zeta_{6})q^{7}+(4-8\zeta_{6})q^{13}+(10+\cdots)q^{19}+\cdots\)
756.2.t.b	$756$	$2$	756.t	21.g	$6$	$2$	$1$	$6.037$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(1+2\zeta_{6})q^{7}+(-3+6\zeta_{6})q^{13}+(-4+\cdots)q^{19}+\cdots\)
756.2.t.c	$756$	$2$	756.t	21.g	$6$	$2$	$1$	$6.037$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(3-2\zeta_{6})q^{7}+(1-2\zeta_{6})q^{13}+(4-2\zeta_{6})q^{19}+\cdots\)
756.2.t.d	$756$	$2$	756.t	21.g	$6$	$4$	$2$	$6.037$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-10\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}+(-2+\beta _{2})q^{7}+(2+4\beta _{2}+\cdots)q^{13}+\cdots\)
756.2.t.e	$756$	$2$	756.t	21.g	$6$	$12$	$6$	$6.037$	12.0.\(\cdots\).1	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{9}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{2}q^{5}+\beta _{10}q^{7}+(-\beta _{3}-\beta _{5})q^{11}+\cdots\)

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

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