Space of modular forms of level 1386, weight 2, and Character 1386.k

• Label: 1386.2.k
• Level: $1386$
• Weight: $2$
• Character orbit: 1386.k
• Rep. character: $\chi_{1386}(793,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $64$
• Newform subspaces: $23$
• Sturm bound: $576$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.k (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(576\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(13\), \(17\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1386, [\chi])\).

	Total	New	Old
Modular forms	608	64	544
Cusp forms	544	64	480
Eisenstein series	64	0	64

Trace form

\( 64 q - 32 q^{4} - 4 q^{5} - 8 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1386, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1386.2.k.a	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None					154.2.e.d	$2$	$1$	\(-1\)	\(0\)	\(-4\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-4\zeta_{6}q^{5}+\cdots\)
1386.2.k.b	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None		✓			1386.2.k.b	$2$	$0$	\(-1\)	\(0\)	\(-3\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots\)
1386.2.k.c	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None					462.2.i.c	$2$	$0$	\(-1\)	\(0\)	\(-3\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots\)
1386.2.k.d	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None					462.2.i.b	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-3+2\zeta_{6})q^{7}+\cdots\)
1386.2.k.e	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None					154.2.e.c	$2$	$1$	\(-1\)	\(0\)	\(0\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(-2+3\zeta_{6})q^{7}+\cdots\)
1386.2.k.f	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None					462.2.i.d	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(3-2\zeta_{6})q^{7}+\cdots\)
1386.2.k.g	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None		✓			1386.2.k.g	$2$	$0$	\(-1\)	\(0\)	\(1\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
1386.2.k.h	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None		✓			1386.2.k.h	$2$	$0$	\(-1\)	\(0\)	\(2\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots\)
1386.2.k.i	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None		✓			1386.2.k.i	$2$	$0$	\(-1\)	\(0\)	\(2\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots\)
1386.2.k.j	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None					462.2.i.a	$2$	$0$	\(-1\)	\(0\)	\(3\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
1386.2.k.k	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None		✓			1386.2.k.i	$2$	$0$	\(1\)	\(0\)	\(-2\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots\)
1386.2.k.l	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None		✓			1386.2.k.h	$2$	$0$	\(1\)	\(0\)	\(-2\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots\)
1386.2.k.m	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None		✓			1386.2.k.g	$2$	$0$	\(1\)	\(0\)	\(-1\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}-\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
1386.2.k.n	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None					154.2.e.b	$2$	$0$	\(1\)	\(0\)	\(0\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+(2+\zeta_{6})q^{7}+\cdots\)
1386.2.k.o	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None					154.2.e.a	$2$	$1$	\(1\)	\(0\)	\(2\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots\)
1386.2.k.p	$1386$	$2$	1386.k	7.c	$3$	$2$	$1$	$11.067$	\(\Q(\sqrt{-3}) \)	None		✓			1386.2.k.b	$2$	$0$	\(1\)	\(0\)	\(3\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\zeta_{6}q^{2}+(-1+\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
1386.2.k.q	$1386$	$2$	1386.k	7.c	$3$	$4$	$2$	$11.067$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			1386.2.k.q	$2$	$0$	\(-2\)	\(0\)	\(-2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)
1386.2.k.r	$1386$	$2$	1386.k	7.c	$3$	$4$	$2$	$11.067$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None					462.2.i.e	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+\beta _{1}q^{5}-\beta _{1}q^{7}+\cdots\)
1386.2.k.s	$1386$	$2$	1386.k	7.c	$3$	$4$	$2$	$11.067$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None					154.2.e.f	$2$	$0$	\(-2\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{2})q^{2}+\beta _{2}q^{4}+(1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
1386.2.k.t	$1386$	$2$	1386.k	7.c	$3$	$4$	$2$	$11.067$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					154.2.e.e	$2$	$0$	\(2\)	\(0\)	\(-4\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}+(-1-\beta _{2})q^{4}+(-\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots\)
1386.2.k.u	$1386$	$2$	1386.k	7.c	$3$	$4$	$2$	$11.067$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			1386.2.k.q	$2$	$0$	\(2\)	\(0\)	\(2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{2})q^{2}+\beta _{2}q^{4}+(1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
1386.2.k.v	$1386$	$2$	1386.k	7.c	$3$	$6$	$3$	$11.067$	6.0.21870000.1	None					462.2.i.g	$2$	$0$	\(3\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}+(-1-\beta _{2})q^{4}+(\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)
1386.2.k.w	$1386$	$2$	1386.k	7.c	$3$	$6$	$3$	$11.067$	6.0.1156923.1	None					462.2.i.f	$2$	$0$	\(3\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}+(-1-\beta _{2})q^{4}+\beta _{4}q^{5}+(\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1386, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1386, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(154, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(231, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(462, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(693, [\chi])\)\(^{\oplus 2}\)