Space of modular forms of level 1296, weight 3, and Character 1296.q

• Label: 1296.3.q
• Level: $1296$
• Weight: $3$
• Character orbit: 1296.q
• Rep. character: $\chi_{1296}(593,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $94$
• Newform subspaces: $17$
• Sturm bound: $648$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.q (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 9 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(648\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1296, [\chi])\).

	Total	New	Old
Modular forms	936	98	838
Cusp forms	792	94	698
Eisenstein series	144	4	140

Trace form

\( 94 q - 2 q^{7} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1296, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1296.3.q.a	$1296$	$3$	1296.q	9.d	$6$	$2$	$1$	$35.313$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-13\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-13+13\zeta_{6})q^{7}+\zeta_{6}q^{13}-11q^{19}+\cdots\)
1296.3.q.b	$1296$	$3$	1296.q	9.d	$6$	$2$	$1$	$35.313$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(2\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(2-2\zeta_{6})q^{7}+22\zeta_{6}q^{13}-26q^{19}+\cdots\)
1296.3.q.c	$1296$	$3$	1296.q	9.d	$6$	$2$	$1$	$35.313$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(11\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(11-11\zeta_{6})q^{7}-23\zeta_{6}q^{13}+37q^{19}+\cdots\)
1296.3.q.d	$1296$	$3$	1296.q	9.d	$6$	$4$	$2$	$35.313$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-14\)	$3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{5}-7\zeta_{12}^{2}q^{7}+(\zeta_{12}-\zeta_{12}^{3})q^{11}+\cdots\)
1296.3.q.e	$1296$	$3$	1296.q	9.d	$6$	$4$	$2$	$35.313$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}-6\beta _{2}q^{7}+(\beta _{1}-\beta _{3})q^{11}+\cdots\)
1296.3.q.f	$1296$	$3$	1296.q	9.d	$6$	$4$	$2$	$35.313$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}-4\beta _{2}q^{7}+(-4\beta _{1}+4\beta _{3})q^{11}+\cdots\)
1296.3.q.g	$1296$	$3$	1296.q	9.d	$6$	$4$	$2$	$35.313$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-6\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}-3\beta _{2}q^{7}+(7\beta _{1}-7\beta _{3})q^{11}+\cdots\)
1296.3.q.h	$1296$	$3$	1296.q	9.d	$6$	$4$	$2$	$35.313$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(6\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}+3\beta _{2}q^{7}+(\beta _{1}-\beta _{3})q^{11}+\cdots\)
1296.3.q.i	$1296$	$3$	1296.q	9.d	$6$	$4$	$2$	$35.313$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(10\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{5}+5\beta _{2}q^{7}+(-\beta _{1}+\beta _{3})q^{11}+\cdots\)
1296.3.q.j	$1296$	$3$	1296.q	9.d	$6$	$4$	$2$	$35.313$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(10\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{12}q^{5}+5\zeta_{12}^{2}q^{7}+(5\zeta_{12}-5\zeta_{12}^{3})q^{11}+\cdots\)
1296.3.q.k	$1296$	$3$	1296.q	9.d	$6$	$4$	$2$	$35.313$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(24\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+5\beta _{1}q^{5}+12\beta _{2}q^{7}+(-4\beta _{1}+4\beta _{3})q^{11}+\cdots\)
1296.3.q.l	$1296$	$3$	1296.q	9.d	$6$	$8$	$4$	$35.313$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\zeta_{24}-\zeta_{24}^{3}-\zeta_{24}^{4})q^{5}+(-3+3\zeta_{24}^{2}+\cdots)q^{7}+\cdots\)
1296.3.q.m	$1296$	$3$	1296.q	9.d	$6$	$8$	$4$	$35.313$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$3^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q-\zeta_{24}^{6}q^{5}+(-1+\zeta_{24}+\zeta_{24}^{2}-\zeta_{24}^{4}+\cdots)q^{7}+\cdots\)
1296.3.q.n	$1296$	$3$	1296.q	9.d	$6$	$8$	$4$	$35.313$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$3^{10}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{24}^{5}q^{5}+(-\zeta_{24}+\zeta_{24}^{2})q^{7}-\zeta_{24}^{7}q^{11}+\cdots\)
1296.3.q.o	$1296$	$3$	1296.q	9.d	$6$	$8$	$4$	$35.313$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$3^{8}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{24}^{6}q^{5}+(2\zeta_{24}^{2}+2\zeta_{24}^{3})q^{7}+\cdots\)
1296.3.q.p	$1296$	$3$	1296.q	9.d	$6$	$8$	$4$	$35.313$	\(\Q(\zeta_{24})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)	$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{24}^{6}q^{5}+(3\zeta_{24}^{2}-\zeta_{24}^{3})q^{7}+(\zeta_{24}+\cdots)q^{11}+\cdots\)
1296.3.q.q	$1296$	$3$	1296.q	9.d	$6$	$16$	$8$	$35.313$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)	$2^{4}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{11}+\beta _{12})q^{5}+(-2-2\beta _{2}+\beta _{7}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1296, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1296, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 15}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(648, [\chi])\)\(^{\oplus 2}\)