Space of modular forms of level 1776, weight 2, and Character 1776.bz

• Label: 1776.2.bz
• Level: $1776$
• Weight: $2$
• Character orbit: 1776.bz
• Rep. character: $\chi_{1776}(529,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $76$
• Newform subspaces: $12$
• Sturm bound: $608$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1776.bz (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 37 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(608\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1776, [\chi])\).

	Total	New	Old
Modular forms	632	76	556
Cusp forms	584	76	508
Eisenstein series	48	0	48

Trace form

76 * q - 2 * q^3 - 6 * q^5 - 6 * q^7 - 38 * q^9 - 8 * q^11 - 6 * q^17 + 40 * q^25 + 4 * q^27 - 18 * q^37 + 6 * q^39 - 18 * q^41 - 50 * q^49 - 12 * q^53 + 6 * q^61 + 12 * q^63 - 6 * q^67 + 20 * q^71 - 32 * q^73 - 28 * q^75 + 8 * q^77 - 54 * q^79 - 38 * q^81 + 24 * q^83 + 28 * q^85 - 6 * q^89 + 42 * q^91 - 24 * q^93 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(1776, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1776.2.bz.a	$1776$	$2$	1776.bz	37.e	$6$	$2$	$1$	$14.181$	\(\Q(\sqrt{-3}) \)	None					444.2.r.a	$2$	$1$	\(0\)	\(1\)	\(-6\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{6}q^{3}+(-4+2\zeta_{6})q^{5}+\zeta_{6}q^{7}+\cdots\)
1776.2.bz.b	$1776$	$2$	1776.bz	37.e	$6$	$2$	$1$	$14.181$	\(\Q(\sqrt{-3}) \)	None					111.2.j.a	$2$	$1$	\(0\)	\(1\)	\(-6\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{6}q^{3}+(-4+2\zeta_{6})q^{5}+\zeta_{6}q^{7}+\cdots\)
1776.2.bz.c	$1776$	$2$	1776.bz	37.e	$6$	$2$	$1$	$14.181$	\(\Q(\sqrt{-3}) \)	None					444.2.r.b	$2$	$0$	\(0\)	\(1\)	\(-3\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\zeta_{6}q^{3}+(-2+\zeta_{6})q^{5}-2\zeta_{6}q^{7}+\cdots\)
1776.2.bz.d	$1776$	$2$	1776.bz	37.e	$6$	$4$	$2$	$14.181$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None					111.2.j.b	$2$	$1$	\(0\)	\(-2\)	\(3\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-\beta _{2}q^{3}+(1-\beta _{3})q^{5}-3\beta _{2}q^{7}+(-1+\cdots)q^{9}+\cdots\)
1776.2.bz.e	$1776$	$2$	1776.bz	37.e	$6$	$4$	$2$	$14.181$	\(\Q(\zeta_{12})\)	None					222.2.j.a	$2$	$0$	\(0\)	\(2\)	\(0\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\zeta_{12}^{2})q^{3}+\zeta_{12}q^{5}+(-3+\zeta_{12}+\cdots)q^{7}+\cdots\)
1776.2.bz.f	$1776$	$2$	1776.bz	37.e	$6$	$4$	$2$	$14.181$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None					444.2.r.c	$2$	$0$	\(0\)	\(2\)	\(9\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{3}+(3-2\beta _{2}+\beta _{3})q^{5}+(2-4\beta _{1}+\cdots)q^{7}+\cdots\)
1776.2.bz.g	$1776$	$2$	1776.bz	37.e	$6$	$6$	$3$	$14.181$	6.0.16553403.1	None					111.2.j.c	$2$	$0$	\(0\)	\(3\)	\(9\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{3}+(2-\beta _{2}+\beta _{3})q^{5}+\beta _{2}q^{7}+\cdots\)
1776.2.bz.h	$1776$	$2$	1776.bz	37.e	$6$	$8$	$4$	$14.181$	8.0.\(\cdots\).1	None					444.2.r.d	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{3}q^{3}-\beta _{7}q^{5}+(\beta _{2}-\beta _{4}+\beta _{6}-\beta _{7})q^{7}+\cdots\)
1776.2.bz.i	$1776$	$2$	1776.bz	37.e	$6$	$8$	$4$	$14.181$	\(\Q(i, \sqrt{3}, \sqrt{11})\)	None					222.2.j.b	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(4\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{3})q^{3}+(-\beta _{2}+\beta _{4}+\beta _{5}+\cdots)q^{5}+\cdots\)
1776.2.bz.j	$1776$	$2$	1776.bz	37.e	$6$	$8$	$4$	$14.181$	8.0.22581504.2	None					888.2.bj.a	$2$	$0$	\(0\)	\(4\)	\(-6\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\beta _{4})q^{3}+(-\beta _{3}+\beta _{4})q^{5}+(-1+\cdots)q^{7}+\cdots\)
1776.2.bz.k	$1776$	$2$	1776.bz	37.e	$6$	$8$	$4$	$14.181$	8.0.195105024.2	None					888.2.bj.b	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{4}q^{3}+(-\beta _{6}-\beta _{7})q^{5}+(-\beta _{4}+\beta _{6}+\cdots)q^{7}+\cdots\)
1776.2.bz.l	$1776$	$2$	1776.bz	37.e	$6$	$20$	$10$	$14.181$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None			✓		888.2.bj.c	$2$	$0$	\(0\)	\(-10\)	\(-6\)	\(-2\)	$2^{10}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1+\beta _{12})q^{3}+\beta _{7}q^{5}+\beta _{18}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1776, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1776, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(37, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(74, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(111, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(148, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(222, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(296, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(444, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(592, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(888, [\chi])\)\(^{\oplus 2}\)