Space of modular forms of level 1296, weight 2, and Character 1296.i

• Label: 1296.2.i
• Level: $1296$
• Weight: $2$
• Character orbit: 1296.i
• Rep. character: $\chi_{1296}(433,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $46$
• Newform subspaces: $20$
• Sturm bound: $432$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1296.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(432\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1296, [\chi])\).

	Total	New	Old
Modular forms	504	50	454
Cusp forms	360	46	314
Eisenstein series	144	4	140

Trace form

\( 46 q - 2 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1296, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1296.2.i.a	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(0\)	\(-4\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(4-4\zeta_{6})q^{11}+\cdots\)
1296.2.i.b	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-3\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{7}+\zeta_{6}q^{13}+\cdots\)
1296.2.i.c	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-3\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-3\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+(-3+3\zeta_{6})q^{11}+\cdots\)
1296.2.i.d	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(0\)	\(0\)	\(-3\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-3\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+(-6+6\zeta_{6})q^{11}+\cdots\)
1296.2.i.e	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}+2\zeta_{6}q^{13}+\cdots\)
1296.2.i.f	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+5\zeta_{6}q^{13}+\cdots\)
1296.2.i.g	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-1\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{5}+(3-3\zeta_{6})q^{7}+(-5+5\zeta_{6})q^{11}+\cdots\)
1296.2.i.h	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(-4+4\zeta_{6})q^{7}-2\zeta_{6}q^{13}-8q^{19}+\cdots\)
1296.2.i.i	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-1\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(-1+\zeta_{6})q^{7}-5\zeta_{6}q^{13}+7q^{19}+\cdots\)
1296.2.i.j	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(5\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+(5-5\zeta_{6})q^{7}+7\zeta_{6}q^{13}+q^{19}+(5+\cdots)q^{25}+\cdots\)
1296.2.i.k	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}+5\zeta_{6}q^{13}+\cdots\)
1296.2.i.l	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(1\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{5}+(3-3\zeta_{6})q^{7}+(5-5\zeta_{6})q^{11}+\cdots\)
1296.2.i.m	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+2\zeta_{6}q^{13}+\cdots\)
1296.2.i.n	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+3\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{7}+\zeta_{6}q^{13}+\cdots\)
1296.2.i.o	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+3\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+(3-3\zeta_{6})q^{11}+\cdots\)
1296.2.i.p	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+3\zeta_{6}q^{5}+(2-2\zeta_{6})q^{7}+(6-6\zeta_{6})q^{11}+\cdots\)
1296.2.i.q	$1296$	$2$	1296.i	9.c	$3$	$2$	$1$	$10.349$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(-4+4\zeta_{6})q^{11}+\cdots\)
1296.2.i.r	$1296$	$2$	1296.i	9.c	$3$	$4$	$2$	$10.349$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2\zeta_{12}+\zeta_{12}^{2})q^{5}+(2\zeta_{12}^{2}-2\zeta_{12}^{3})q^{7}+\cdots\)
1296.2.i.s	$1296$	$2$	1296.i	9.c	$3$	$4$	$2$	$10.349$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{12}^{2}q^{5}+(2-2\zeta_{12})q^{7}+(-2\zeta_{12}^{2}+\cdots)q^{11}+\cdots\)
1296.2.i.t	$1296$	$2$	1296.i	9.c	$3$	$4$	$2$	$10.349$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{5}+2\zeta_{12}^{2}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1296, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1296, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(648, [\chi])\)\(^{\oplus 2}\)