Space of modular forms of level 1764, weight 4, and Character 1764.k

• Label: 1764.4.k
• Level: $1764$
• Weight: $4$
• Character orbit: 1764.k
• Rep. character: $\chi_{1764}(361,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $100$
• Newform subspaces: $30$
• Sturm bound: $1344$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(1344\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(1764, [\chi])\).

	Total	New	Old
Modular forms	2112	100	2012
Cusp forms	1920	100	1820
Eisenstein series	192	0	192

Trace form

\( 100 q + 6 q^{5} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(1764, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1764.4.k.a	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-20\zeta_{6}q^{5}+(44-44\zeta_{6})q^{11}-44q^{13}+\cdots\)
1764.4.k.b	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-18\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-18\zeta_{6}q^{5}+(6^{2}-6^{2}\zeta_{6})q^{11}-10q^{13}+\cdots\)
1764.4.k.c	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-14\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-14\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}-54q^{13}+\cdots\)
1764.4.k.d	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-8\zeta_{6}q^{5}+(-40+40\zeta_{6})q^{11}-12q^{13}+\cdots\)
1764.4.k.e	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-6\zeta_{6}q^{5}+(-12+12\zeta_{6})q^{11}+82q^{13}+\cdots\)
1764.4.k.f	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-6\zeta_{6}q^{5}+(6^{2}-6^{2}\zeta_{6})q^{11}-62q^{13}+\cdots\)
1764.4.k.g	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-4\zeta_{6}q^{5}+(-20+20\zeta_{6})q^{11}+4q^{13}+\cdots\)
1764.4.k.h	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q-89q^{13}-163\zeta_{6}q^{19}+(5^{3}-5^{3}\zeta_{6})q^{25}+\cdots\)
1764.4.k.i	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+19q^{13}+107\zeta_{6}q^{19}+(5^{3}-5^{3}\zeta_{6})q^{25}+\cdots\)
1764.4.k.j	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+4\zeta_{6}q^{5}+(-20+20\zeta_{6})q^{11}-4q^{13}+\cdots\)
1764.4.k.k	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+6\zeta_{6}q^{5}+(-12+12\zeta_{6})q^{11}-82q^{13}+\cdots\)
1764.4.k.l	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+6\zeta_{6}q^{5}+(6^{2}-6^{2}\zeta_{6})q^{11}+62q^{13}+\cdots\)
1764.4.k.m	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+8\zeta_{6}q^{5}+(-40+40\zeta_{6})q^{11}+12q^{13}+\cdots\)
1764.4.k.n	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(14\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+14\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+54q^{13}+\cdots\)
1764.4.k.o	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(18\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+18\zeta_{6}q^{5}+(6^{2}-6^{2}\zeta_{6})q^{11}+10q^{13}+\cdots\)
1764.4.k.p	$1764$	$4$	1764.k	7.c	$3$	$2$	$1$	$104.079$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(20\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+20\zeta_{6}q^{5}+(44-44\zeta_{6})q^{11}+44q^{13}+\cdots\)
1764.4.k.q	$1764$	$4$	1764.k	7.c	$3$	$4$	$2$	$104.079$	\(\Q(\sqrt{-3}, \sqrt{193})\)	None		$2$	$0$	\(0\)	\(0\)	\(-11\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-6\beta _{2})q^{5}+(1-7\beta _{1}+6\beta _{2}-7\beta _{3})q^{11}+\cdots\)
1764.4.k.r	$1764$	$4$	1764.k	7.c	$3$	$4$	$2$	$104.079$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+7\beta _{1}q^{5}+28\beta _{2}q^{11}+3\beta _{3}q^{13}+\cdots\)
1764.4.k.s	$1764$	$4$	1764.k	7.c	$3$	$4$	$2$	$104.079$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2\beta _{1}q^{5}+26\beta _{2}q^{11}-24\beta _{3}q^{13}+\cdots\)
1764.4.k.t	$1764$	$4$	1764.k	7.c	$3$	$4$	$2$	$104.079$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{5}+(13\beta _{1}+13\beta _{3})q^{11}-30q^{13}+\cdots\)
1764.4.k.u	$1764$	$4$	1764.k	7.c	$3$	$4$	$2$	$104.079$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{5}+(\beta _{1}+\beta _{3})q^{11}-26q^{13}+\cdots\)
1764.4.k.v	$1764$	$4$	1764.k	7.c	$3$	$4$	$2$	$104.079$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{5}+(-\beta _{1}-\beta _{3})q^{11}+26q^{13}+\cdots\)
1764.4.k.w	$1764$	$4$	1764.k	7.c	$3$	$4$	$2$	$104.079$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{5}+(-13\beta _{1}-13\beta _{3})q^{11}+30q^{13}+\cdots\)
1764.4.k.x	$1764$	$4$	1764.k	7.c	$3$	$4$	$2$	$104.079$	\(\Q(\sqrt{-3}, \sqrt{385})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{5}+(\beta _{2}+\beta _{3})q^{11}+54q^{13}+\cdots\)
1764.4.k.y	$1764$	$4$	1764.k	7.c	$3$	$4$	$2$	$104.079$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+7\beta _{1}q^{5}-28\beta _{2}q^{11}-3\beta _{3}q^{13}+\cdots\)
1764.4.k.z	$1764$	$4$	1764.k	7.c	$3$	$4$	$2$	$104.079$	\(\Q(\sqrt{-3}, \sqrt{-19})\)	None		$2$	$0$	\(0\)	\(0\)	\(3\)	\(0\)	$3^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{3})q^{5}+(-1+5^{2}\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots\)
1764.4.k.ba	$1764$	$4$	1764.k	7.c	$3$	$4$	$2$	$104.079$	\(\Q(\sqrt{-3}, \sqrt{37})\)	None		$2$	$0$	\(0\)	\(0\)	\(14\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(7\beta _{1}+2\beta _{2})q^{5}+(2^{4}-2^{4}\beta _{1}+7\beta _{2}+\cdots)q^{11}+\cdots\)
1764.4.k.bb	$1764$	$4$	1764.k	7.c	$3$	$8$	$4$	$104.079$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{6}q^{5}+(-3\beta _{2}-\beta _{4})q^{11}+(3\beta _{4}+\cdots)q^{13}+\cdots\)
1764.4.k.bc	$1764$	$4$	1764.k	7.c	$3$	$8$	$4$	$104.079$	8.0.\(\cdots\).42	None		$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{16}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{5}q^{5}+(\beta _{2}+\beta _{3})q^{11}+\beta _{6}q^{13}+\cdots\)
1764.4.k.bd	$1764$	$4$	1764.k	7.c	$3$	$8$	$4$	$104.079$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{6}q^{5}+(-3\beta _{2}-\beta _{4})q^{11}+(-3\beta _{4}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(1764, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(1764, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(882, [\chi])\)\(^{\oplus 2}\)