Space of modular forms of level 784, weight 3, and Character 784.r

• Label: 784.3.r
• Level: $784$
• Weight: $3$
• Character orbit: 784.r
• Rep. character: $\chi_{784}(79,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $80$
• Newform subspaces: $20$
• Sturm bound: $336$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	784.r (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(336\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(784, [\chi])\).

	Total	New	Old
Modular forms	496	80	416
Cusp forms	400	80	320
Eisenstein series	96	0	96

Trace form

\( 80 q + 120 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(784, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
784.3.r.a	$784$	$3$	784.r	28.g	$6$	$2$	$1$	$21.362$	\(\Q(\sqrt{-3}) \)	None		✓			784.3.d.e	$2$	$0$	\(0\)	\(-6\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4+2\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{5}+\cdots\)
784.3.r.b	$784$	$3$	784.r	28.g	$6$	$2$	$1$	$21.362$	\(\Q(\sqrt{-3}) \)	None		✓			784.3.d.e	$2$	$0$	\(0\)	\(-6\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4+2\zeta_{6})q^{3}+(2-2\zeta_{6})q^{5}+(3+\cdots)q^{9}+\cdots\)
784.3.r.c	$784$	$3$	784.r	28.g	$6$	$2$	$1$	$21.362$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-1}) \)		✓			784.3.d.a	$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-8+8\zeta_{6})q^{5}+(-9+9\zeta_{6})q^{9}+\cdots\)
784.3.r.d	$784$	$3$	784.r	28.g	$6$	$2$	$1$	$21.362$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-1}) \)					16.3.c.a	$4$	$0$	\(0\)	\(0\)	\(-6\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(-6+6\zeta_{6})q^{5}+(-9+9\zeta_{6})q^{9}+\cdots\)
784.3.r.e	$784$	$3$	784.r	28.g	$6$	$2$	$1$	$21.362$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-1}) \)					16.3.c.a	$4$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(6-6\zeta_{6})q^{5}+(-9+9\zeta_{6})q^{9}+10q^{13}+\cdots\)
784.3.r.f	$784$	$3$	784.r	28.g	$6$	$2$	$1$	$21.362$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-1}) \)		✓			784.3.d.a	$4$	$0$	\(0\)	\(0\)	\(8\)	\(0\)	$1$	$\mathrm{U}(1)[D_{6}]$	\(q+(8-8\zeta_{6})q^{5}+(-9+9\zeta_{6})q^{9}-24q^{13}+\cdots\)
784.3.r.g	$784$	$3$	784.r	28.g	$6$	$2$	$1$	$21.362$	\(\Q(\sqrt{-3}) \)	None		✓			784.3.d.e	$2$	$0$	\(0\)	\(6\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(4-2\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{5}+(3+\cdots)q^{9}+\cdots\)
784.3.r.h	$784$	$3$	784.r	28.g	$6$	$2$	$1$	$21.362$	\(\Q(\sqrt{-3}) \)	None		✓			784.3.d.e	$2$	$0$	\(0\)	\(6\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(4-2\zeta_{6})q^{3}+(2-2\zeta_{6})q^{5}+(3-3\zeta_{6})q^{9}+\cdots\)
784.3.r.i	$784$	$3$	784.r	28.g	$6$	$4$	$2$	$21.362$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None					112.3.d.b	$2$	$0$	\(0\)	\(-6\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\beta _{2}+\beta _{3})q^{3}+(-1-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
784.3.r.j	$784$	$3$	784.r	28.g	$6$	$4$	$2$	$21.362$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None					112.3.d.b	$2$	$0$	\(0\)	\(-6\)	\(2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-2+\beta _{2}+\beta _{3})q^{3}+(1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
784.3.r.k	$784$	$3$	784.r	28.g	$6$	$4$	$2$	$21.362$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None					112.3.d.a	$4$	$0$	\(0\)	\(0\)	\(-16\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{3}+(-8-8\beta _{1})q^{5}+(19+19\beta _{1}+\cdots)q^{9}+\cdots\)
784.3.r.l	$784$	$3$	784.r	28.g	$6$	$4$	$2$	$21.362$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None					112.3.r.a	$4$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{3}-\beta _{2}q^{5}-2\beta _{2}q^{9}-5\beta _{1}q^{11}+\cdots\)
784.3.r.m	$784$	$3$	784.r	28.g	$6$	$4$	$2$	$21.362$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None					112.3.d.a	$4$	$0$	\(0\)	\(0\)	\(16\)	\(0\)	$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{3}+(8+8\beta _{1})q^{5}+(19+19\beta _{1}+\cdots)q^{9}+\cdots\)
784.3.r.n	$784$	$3$	784.r	28.g	$6$	$4$	$2$	$21.362$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None					112.3.d.b	$2$	$0$	\(0\)	\(6\)	\(-2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\beta _{2}-\beta _{3})q^{3}+(-1-\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
784.3.r.o	$784$	$3$	784.r	28.g	$6$	$4$	$2$	$21.362$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None					112.3.d.b	$2$	$0$	\(0\)	\(6\)	\(2\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(2-\beta _{2}-\beta _{3})q^{3}+(1+\beta _{1}-\beta _{2}-2\beta _{3})q^{5}+\cdots\)
784.3.r.p	$784$	$3$	784.r	28.g	$6$	$6$	$3$	$21.362$	6.0.259470000.1	None					112.3.r.b	$2$	$0$	\(0\)	\(-3\)	\(1\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-1-\beta _{5})q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(7+\cdots)q^{9}+\cdots\)
784.3.r.q	$784$	$3$	784.r	28.g	$6$	$6$	$3$	$21.362$	6.0.259470000.1	None					112.3.r.b	$2$	$0$	\(0\)	\(3\)	\(1\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\beta _{5})q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(7+\beta _{1}+\cdots)q^{9}+\cdots\)
784.3.r.r	$784$	$3$	784.r	28.g	$6$	$8$	$4$	$21.362$	8.0.\(\cdots\).5	None		✓			784.3.d.m	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{4}+\beta _{5})q^{3}+(2\beta _{2}-\beta _{3}-\beta _{4}-3\beta _{5}+\cdots)q^{5}+\cdots\)
784.3.r.s	$784$	$3$	784.r	28.g	$6$	$8$	$4$	$21.362$	8.0.\(\cdots\).5	None		✓			784.3.d.m	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{4}+\beta _{5})q^{3}+(-2\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{5}+\cdots\)
784.3.r.t	$784$	$3$	784.r	28.g	$6$	$8$	$4$	$21.362$	8.0.796594176.2	None		✓			784.3.d.i	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{3}+(-2\beta _{4}-2\beta _{5})q^{5}+(5-5\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(784, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(784, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 2}\)