Space of modular forms of level 648, weight 4, and Character 648.i

• Label: 648.4.i
• Level: $648$
• Weight: $4$
• Character orbit: 648.i
• Rep. character: $\chi_{648}(217,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $72$
• Newform subspaces: $22$
• Sturm bound: $432$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 22 \)
Sturm bound:			\(432\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(648, [\chi])\).

	Total	New	Old
Modular forms	696	72	624
Cusp forms	600	72	528
Eisenstein series	96	0	96

Trace form

\( 72 q + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(648, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
648.4.i.a	$648$	$4$	648.i	9.c	$3$	$2$	$1$	$38.233$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-16\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2^{4}\zeta_{6}q^{5}+(12-12\zeta_{6})q^{7}+(-2^{6}+\cdots)q^{11}+\cdots\)
648.4.i.b	$648$	$4$	648.i	9.c	$3$	$2$	$1$	$38.233$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-14\)	\(24\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-14\zeta_{6}q^{5}+(24-24\zeta_{6})q^{7}+(28-28\zeta_{6})q^{11}+\cdots\)
648.4.i.c	$648$	$4$	648.i	9.c	$3$	$2$	$1$	$38.233$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(0\)	\(-5\)	\(-36\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-5\zeta_{6}q^{5}+(-6^{2}+6^{2}\zeta_{6})q^{7}+(2^{6}+\cdots)q^{11}+\cdots\)
648.4.i.d	$648$	$4$	648.i	9.c	$3$	$2$	$1$	$38.233$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(-28+\cdots)q^{11}+\cdots\)
648.4.i.e	$648$	$4$	648.i	9.c	$3$	$2$	$1$	$38.233$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-2\)	\(-24\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{5}+(-24+24\zeta_{6})q^{7}+(-44+\cdots)q^{11}+\cdots\)
648.4.i.f	$648$	$4$	648.i	9.c	$3$	$2$	$1$	$38.233$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(9\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{5}+(9-9\zeta_{6})q^{7}+(17-17\zeta_{6})q^{11}+\cdots\)
648.4.i.g	$648$	$4$	648.i	9.c	$3$	$2$	$1$	$38.233$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(9\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{5}+(9-9\zeta_{6})q^{7}+(-17+17\zeta_{6})q^{11}+\cdots\)
648.4.i.h	$648$	$4$	648.i	9.c	$3$	$2$	$1$	$38.233$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(-24\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{5}+(-24+24\zeta_{6})q^{7}+(44+\cdots)q^{11}+\cdots\)
648.4.i.i	$648$	$4$	648.i	9.c	$3$	$2$	$1$	$38.233$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(28-28\zeta_{6})q^{11}+\cdots\)
648.4.i.j	$648$	$4$	648.i	9.c	$3$	$2$	$1$	$38.233$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(5\)	\(-36\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+5\zeta_{6}q^{5}+(-6^{2}+6^{2}\zeta_{6})q^{7}+(-2^{6}+\cdots)q^{11}+\cdots\)
648.4.i.k	$648$	$4$	648.i	9.c	$3$	$2$	$1$	$38.233$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(14\)	\(24\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+14\zeta_{6}q^{5}+(24-24\zeta_{6})q^{7}+(-28+\cdots)q^{11}+\cdots\)
648.4.i.l	$648$	$4$	648.i	9.c	$3$	$2$	$1$	$38.233$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(16\)	\(12\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2^{4}\zeta_{6}q^{5}+(12-12\zeta_{6})q^{7}+(2^{6}-2^{6}\zeta_{6})q^{11}+\cdots\)
648.4.i.m	$648$	$4$	648.i	9.c	$3$	$4$	$2$	$38.233$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(-24\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4-4\beta _{1}-\beta _{3})q^{5}+(12\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
648.4.i.n	$648$	$4$	648.i	9.c	$3$	$4$	$2$	$38.233$	\(\Q(\sqrt{-3}, \sqrt{-67})\)	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(30\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-4-4\beta _{1}+\beta _{3})q^{5}+(-15\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
648.4.i.o	$648$	$4$	648.i	9.c	$3$	$4$	$2$	$38.233$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(6\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2\beta _{1}-\beta _{2})q^{5}+(3-3\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
648.4.i.p	$648$	$4$	648.i	9.c	$3$	$4$	$2$	$38.233$	\(\Q(\sqrt{-3}, \sqrt{-43})\)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(6\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2-2\beta _{1}+\beta _{3})q^{5}+(-3\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
648.4.i.q	$648$	$4$	648.i	9.c	$3$	$4$	$2$	$38.233$	\(\Q(\sqrt{-3}, \sqrt{-43})\)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(6\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2+2\beta _{1}+\beta _{3})q^{5}+(-3\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
648.4.i.r	$648$	$4$	648.i	9.c	$3$	$4$	$2$	$38.233$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(6\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2\beta _{1}+\beta _{2})q^{5}+(3-3\beta _{1}+2\beta _{2}+2\beta _{3})q^{7}+\cdots\)
648.4.i.s	$648$	$4$	648.i	9.c	$3$	$4$	$2$	$38.233$	\(\Q(\sqrt{-3}, \sqrt{-11})\)	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(-24\)	$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4+4\beta _{1}+\beta _{3})q^{5}+(12\beta _{1}-\beta _{2}+\beta _{3})q^{7}+\cdots\)
648.4.i.t	$648$	$4$	648.i	9.c	$3$	$4$	$2$	$38.233$	\(\Q(\sqrt{-3}, \sqrt{-67})\)	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(30\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(4+4\beta _{1}-\beta _{3})q^{5}+(-15\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
648.4.i.u	$648$	$4$	648.i	9.c	$3$	$8$	$4$	$38.233$	8.0.897122304.10	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$2^{6}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2\beta _{1}+\beta _{7})q^{5}+(\beta _{2}+\beta _{3}-\beta _{5}-\beta _{7})q^{7}+\cdots\)
648.4.i.v	$648$	$4$	648.i	9.c	$3$	$8$	$4$	$38.233$	8.0.897122304.10	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$2^{6}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2\beta _{1}+\beta _{7})q^{5}+(-\beta _{2}-\beta _{3}+\beta _{5}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(648, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(648, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 2}\)