Space of modular forms of level 784, weight 5, and Character 784.c

• Label: 784.5.c
• Level: $784$
• Weight: $5$
• Character orbit: 784.c
• Rep. character: $\chi_{784}(97,\cdot)$
• Character field: $\Q$
• Dimension: $78$
• Newform subspaces: $10$
• Sturm bound: $560$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	784.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(560\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(784, [\chi])\).

	Total	New	Old
Modular forms	472	82	390
Cusp forms	424	78	346
Eisenstein series	48	4	44

Trace form

\( 78 q - 1996 q^{9} + O(q^{10}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(784, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
784.5.c.a	$784$	$5$	784.c	7.b	$2$	$4$	$4$	$81.042$	4.0.2048.2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-4\beta _{1}-7\beta _{3})q^{3}+(-5\beta _{1}-13\beta _{3})q^{5}+\cdots\)
784.5.c.b	$784$	$5$	784.c	7.b	$2$	$4$	$4$	$81.042$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-3\beta _{2})q^{3}+(\beta _{1}+9\beta _{2})q^{5}+\cdots\)
784.5.c.c	$784$	$5$	784.c	7.b	$2$	$4$	$4$	$81.042$	\(\Q(\sqrt{-3}, \sqrt{22})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 7$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-2\beta _{2}-\beta _{3})q^{5}+(12-6\beta _{1}+\cdots)q^{9}+\cdots\)
784.5.c.d	$784$	$5$	784.c	7.b	$2$	$4$	$4$	$81.042$	4.0.2048.2	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}+(-19\beta _{1}-13\beta _{3})q^{5}+(63+\cdots)q^{9}+\cdots\)
784.5.c.e	$784$	$5$	784.c	7.b	$2$	$6$	$6$	$81.042$	6.0.11337408.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{5}\cdot 3^{3}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(\beta _{2}+\beta _{3})q^{5}+(-30+\beta _{5})q^{9}+\cdots\)
784.5.c.f	$784$	$5$	784.c	7.b	$2$	$8$	$8$	$81.042$	8.0.\(\cdots\).9	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 7^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+(-44-\beta _{5}+\cdots)q^{9}+\cdots\)
784.5.c.g	$784$	$5$	784.c	7.b	$2$	$8$	$8$	$81.042$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 7^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{3}q^{3}+(\beta _{3}-\beta _{6})q^{5}+(-44+9\beta _{1}+\cdots)q^{9}+\cdots\)
784.5.c.h	$784$	$5$	784.c	7.b	$2$	$12$	$12$	$81.042$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{5}q^{3}+(\beta _{2}+2\beta _{3}-\beta _{4}-\beta _{5})q^{5}+\cdots\)
784.5.c.i	$784$	$5$	784.c	7.b	$2$	$12$	$12$	$81.042$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{7}q^{3}+\beta _{11}q^{5}+(-8+\beta _{1}+\beta _{2}+\cdots)q^{9}+\cdots\)
784.5.c.j	$784$	$5$	784.c	7.b	$2$	$16$	$16$	$81.042$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{46}\cdot 7^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-\beta _{3}q^{5}+(-5^{2}+\beta _{11})q^{9}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(784, [\chi])\) into lower level spaces

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