Space of modular forms of level 4056, weight 2, and Character 4056.c

• Label: 4056.2.c
• Level: $4056$
• Weight: $2$
• Character orbit: 4056.c
• Rep. character: $\chi_{4056}(337,\cdot)$
• Character field: $\Q$
• Dimension: $78$
• Newform subspaces: $18$
• Sturm bound: $1456$
• Trace bound: $23$

Defining parameters

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(1456\)
Trace bound:			\(23\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(4056, [\chi])\).

	Total	New	Old
Modular forms	784	78	706
Cusp forms	672	78	594
Eisenstein series	112	0	112

Trace form

\( 78 q - 2 q^{3} + 78 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(4056, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
4056.2.c.a	$4056$	$2$	4056.c	13.b	$2$	$2$	$2$	$32.387$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+2iq^{5}+iq^{7}+q^{9}-6iq^{11}+\cdots\)
4056.2.c.b	$4056$	$2$	4056.c	13.b	$2$	$2$	$2$	$32.387$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+3iq^{5}+4iq^{7}+q^{9}-4iq^{11}+\cdots\)
4056.2.c.c	$4056$	$2$	4056.c	13.b	$2$	$2$	$2$	$32.387$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+iq^{5}+2iq^{7}+q^{9}-iq^{15}+\cdots\)
4056.2.c.d	$4056$	$2$	4056.c	13.b	$2$	$2$	$2$	$32.387$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+2iq^{5}+q^{9}+iq^{11}-2iq^{15}+\cdots\)
4056.2.c.e	$4056$	$2$	4056.c	13.b	$2$	$2$	$2$	$32.387$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+iq^{5}+q^{9}+2iq^{11}-iq^{15}+\cdots\)
4056.2.c.f	$4056$	$2$	4056.c	13.b	$2$	$2$	$2$	$32.387$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+2iq^{7}+q^{9}+iq^{11}+6q^{17}+\cdots\)
4056.2.c.g	$4056$	$2$	4056.c	13.b	$2$	$2$	$2$	$32.387$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+q^{9}+3iq^{11}-2q^{17}-4q^{23}+\cdots\)
4056.2.c.h	$4056$	$2$	4056.c	13.b	$2$	$2$	$2$	$32.387$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+iq^{5}+q^{9}+iq^{15}-2q^{17}+\cdots\)
4056.2.c.i	$4056$	$2$	4056.c	13.b	$2$	$2$	$2$	$32.387$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+3iq^{5}+q^{9}+3iq^{15}+q^{17}+\cdots\)
4056.2.c.j	$4056$	$2$	4056.c	13.b	$2$	$2$	$2$	$32.387$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+2iq^{5}-2iq^{7}+q^{9}-iq^{11}+\cdots\)
4056.2.c.k	$4056$	$2$	4056.c	13.b	$2$	$4$	$4$	$32.387$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}-\zeta_{12}^{2}q^{5}+(-\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{7}+\cdots\)
4056.2.c.l	$4056$	$2$	4056.c	13.b	$2$	$4$	$4$	$32.387$	\(\Q(i, \sqrt{13})\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+\beta _{1}q^{5}+(\beta _{1}-\beta _{2})q^{7}+q^{9}+\cdots\)
4056.2.c.m	$4056$	$2$	4056.c	13.b	$2$	$6$	$6$	$32.387$	6.0.44836416.1	None		$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}-\beta _{4}q^{5}+(\beta _{1}+\beta _{5})q^{7}+q^{9}+\cdots\)
4056.2.c.n	$4056$	$2$	4056.c	13.b	$2$	$6$	$6$	$32.387$	6.0.153664.1	None		$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+(\beta _{3}+\beta _{5})q^{5}+(-2\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots\)
4056.2.c.o	$4056$	$2$	4056.c	13.b	$2$	$6$	$6$	$32.387$	6.0.153664.1	None		$2$	$0$	\(0\)	\(6\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+(\beta _{1}+\beta _{3})q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots\)
4056.2.c.p	$4056$	$2$	4056.c	13.b	$2$	$8$	$8$	$32.387$	8.0.649638144.4	None		$2$	$0$	\(0\)	\(8\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}-\beta _{1}q^{5}+(-\beta _{1}-\beta _{4})q^{7}+q^{9}+\cdots\)
4056.2.c.q	$4056$	$2$	4056.c	13.b	$2$	$12$	$12$	$32.387$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(-12\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}-\beta _{10}q^{5}+(-\beta _{1}+\beta _{9})q^{7}+q^{9}+\cdots\)
4056.2.c.r	$4056$	$2$	4056.c	13.b	$2$	$12$	$12$	$32.387$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(12\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+\beta _{1}q^{5}+(-\beta _{9}+\beta _{11})q^{7}+q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(4056, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(4056, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(312, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(338, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(507, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(676, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1014, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1352, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(2028, [\chi])\)\(^{\oplus 2}\)