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

• Label: 1296.3.g
• Level: $1296$
• Weight: $3$
• Character orbit: 1296.g
• Rep. character: $\chi_{1296}(1135,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $11$
• Sturm bound: $648$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.g (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(648\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(17\)

Dimensions

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

	Total	New	Old
Modular forms	468	48	420
Cusp forms	396	48	348
Eisenstein series	72	0	72

Trace form

\( 48 q + O(q^{10}) \)

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.g.a	$1296$	$3$	1296.g	4.b	$2$	$2$	$2$	$35.313$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(-3+\beta )q^{5}+(-5+3\beta )q^{13}+(-15+\cdots)q^{17}+\cdots\)
1296.3.g.b	$1296$	$3$	1296.g	4.b	$2$	$2$	$2$	$35.313$	\(\Q(\sqrt{3}) \)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+(3+\beta )q^{5}+(-5-3\beta )q^{13}+(15+2\beta )q^{17}+\cdots\)
1296.3.g.c	$1296$	$3$	1296.g	4.b	$2$	$4$	$4$	$35.313$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(0\)	\(-24\)	\(0\)	$2^{5}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-6+\zeta_{12})q^{5}-\zeta_{12}^{3}q^{7}+(\zeta_{12}^{2}+\cdots)q^{11}+\cdots\)
1296.3.g.d	$1296$	$3$	1296.g	4.b	$2$	$4$	$4$	$35.313$	\(\Q(\sqrt{-7}, \sqrt{-15})\)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2+\beta _{3})q^{5}+\beta _{1}q^{7}+(\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
1296.3.g.e	$1296$	$3$	1296.g	4.b	$2$	$4$	$4$	$35.313$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{4}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{12}q^{5}-\zeta_{12}^{2}q^{7}-\zeta_{12}^{3}q^{11}+\cdots\)
1296.3.g.f	$1296$	$3$	1296.g	4.b	$2$	$4$	$4$	$35.313$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{5}-\beta _{3}q^{7}+(-\beta _{2}-\beta _{3})q^{11}+\cdots\)
1296.3.g.g	$1296$	$3$	1296.g	4.b	$2$	$4$	$4$	$35.313$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}+\beta _{3}q^{7}+(-\beta _{2}-\beta _{3})q^{11}+\cdots\)
1296.3.g.h	$1296$	$3$	1296.g	4.b	$2$	$4$	$4$	$35.313$	\(\Q(\sqrt{-7}, \sqrt{-15})\)	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2-\beta _{3})q^{5}+\beta _{1}q^{7}+(-\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
1296.3.g.i	$1296$	$3$	1296.g	4.b	$2$	$4$	$4$	$35.313$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(0\)	\(24\)	\(0\)	$2^{5}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(6+\zeta_{12})q^{5}+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{7}+\cdots\)
1296.3.g.j	$1296$	$3$	1296.g	4.b	$2$	$8$	$8$	$35.313$	8.0.856615824.2	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$2^{8}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{3})q^{5}+\beta _{6}q^{7}+(-\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)
1296.3.g.k	$1296$	$3$	1296.g	4.b	$2$	$8$	$8$	$35.313$	8.0.856615824.2	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2^{8}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{3})q^{5}-\beta _{6}q^{7}+(-\beta _{1}+\beta _{2}+\cdots)q^{11}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1296, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(648, [\chi])\)\(^{\oplus 2}\)