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

• Label: 784.3.c
• Level: $784$
• Weight: $3$
• Character orbit: 784.c
• Rep. character: $\chi_{784}(97,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $8$
• Sturm bound: $336$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	784.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(336\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\)

Dimensions

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

	Total	New	Old
Modular forms	248	42	206
Cusp forms	200	38	162
Eisenstein series	48	4	44

Trace form

\( 38 q - 100 q^{9} + O(q^{10}) \)

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	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
784.3.c.a	$784$	$3$	784.c	7.b	$2$	$2$	$2$	$21.362$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\zeta_{6}q^{3}-\zeta_{6}q^{5}+6q^{9}-15q^{11}+\cdots\)
784.3.c.b	$784$	$3$	784.c	7.b	$2$	$4$	$4$	$21.362$	4.0.2048.2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2\beta _{1}+2\beta _{3})q^{3}+(4\beta _{1}-\beta _{3})q^{5}+(-7+\cdots)q^{9}+\cdots\)
784.3.c.c	$784$	$3$	784.c	7.b	$2$	$4$	$4$	$21.362$	4.0.2048.2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$7$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+\beta _{3}q^{5}+(-1-\beta _{2})q^{9}+(-2+\cdots)q^{11}+\cdots\)
784.3.c.d	$784$	$3$	784.c	7.b	$2$	$4$	$4$	$21.362$	4.0.2048.2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(-\beta _{1}-2\beta _{2})q^{5}+(-1-\beta _{3})q^{9}+\cdots\)
784.3.c.e	$784$	$3$	784.c	7.b	$2$	$4$	$4$	$21.362$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}+\beta _{2})q^{3}+(2\beta _{1}+\beta _{2})q^{5}+2\beta _{3}q^{9}+\cdots\)
784.3.c.f	$784$	$3$	784.c	7.b	$2$	$4$	$4$	$21.362$	4.0.2048.2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(7+\beta _{2}+\cdots)q^{9}+\cdots\)
784.3.c.g	$784$	$3$	784.c	7.b	$2$	$8$	$8$	$21.362$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{4})q^{3}+(\beta _{4}+\beta _{5}+\beta _{6})q^{5}+\cdots\)
784.3.c.h	$784$	$3$	784.c	7.b	$2$	$8$	$8$	$21.362$	8.0.\(\cdots\).2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{6}q^{3}-\beta _{7}q^{5}+(-5+\beta _{1})q^{9}+(1+\cdots)q^{11}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(784, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(98, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(392, [\chi])\)\(^{\oplus 2}\)