Space of modular forms of level 784, weight 4, and Character 784.f

• Label: 784.4.f
• Level: $784$
• Weight: $4$
• Character orbit: 784.f
• Rep. character: $\chi_{784}(783,\cdot)$
• Character field: $\Q$
• Dimension: $60$
• Newform subspaces: $10$
• Sturm bound: $448$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 28 \)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(448\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(784, [\chi])\).

	Total	New	Old
Modular forms	360	60	300
Cusp forms	312	60	252
Eisenstein series	48	0	48

Trace form

\( 60q + 540q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(784, [\chi])\) into newform subspaces

Label	Dim.	\(A\)	Field	CM	Traces				$q$-expansion
					\(a_2\)	\(a_3\)	\(a_5\)	\(a_7\)
784.4.f.a	\(2\)	\(46.257\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(-14\)	\(0\)	\(0\)	\(q-7q^{3}+9\zeta_{6}q^{5}+22q^{9}-7\zeta_{6}q^{11}+\cdots\)
784.4.f.b	\(2\)	\(46.257\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(-2\)	\(0\)	\(0\)	\(q-q^{3}+3\zeta_{6}q^{5}-26q^{9}-15\zeta_{6}q^{11}+\cdots\)
784.4.f.c	\(2\)	\(46.257\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(2\)	\(0\)	\(0\)	\(q+q^{3}+3\zeta_{6}q^{5}-26q^{9}+15\zeta_{6}q^{11}+\cdots\)
784.4.f.d	\(2\)	\(46.257\)	\(\Q(\sqrt{-3}) \)	None	\(0\)	\(14\)	\(0\)	\(0\)	\(q+7q^{3}-9\zeta_{6}q^{5}+22q^{9}-7\zeta_{6}q^{11}+\cdots\)
784.4.f.e	\(4\)	\(46.257\)	4.0.2048.2	\(\Q(\sqrt{-1}) \)	\(0\)	\(0\)	\(0\)	\(0\)	\(q+(-\beta _{1}-5\beta _{2})q^{5}-3^{3}q^{9}+(4\beta _{1}+21\beta _{2}+\cdots)q^{13}+\cdots\)
784.4.f.f	\(4\)	\(46.257\)	\(\Q(\sqrt{-3}, \sqrt{7})\)	None	\(0\)	\(0\)	\(0\)	\(0\)	\(q+\beta _{1}q^{3}-5\beta _{2}q^{5}-20q^{9}-15\beta _{3}q^{11}+\cdots\)
784.4.f.g	\(6\)	\(46.257\)	6.0.\(\cdots\).1	None	\(0\)	\(-14\)	\(0\)	\(0\)	\(q+(-2+\beta _{2})q^{3}-\beta _{3}q^{5}+(26-\beta _{1}+\cdots)q^{9}+\cdots\)
784.4.f.h	\(6\)	\(46.257\)	6.0.\(\cdots\).1	None	\(0\)	\(14\)	\(0\)	\(0\)	\(q+(2-\beta _{2})q^{3}+\beta _{3}q^{5}+(26-\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
784.4.f.i	\(8\)	\(46.257\)	8.0.\(\cdots\).32	None	\(0\)	\(0\)	\(0\)	\(0\)	\(q+\beta _{3}q^{3}+(2\beta _{2}+\beta _{4})q^{5}+(43+3\beta _{5}+\cdots)q^{9}+\cdots\)
784.4.f.j	\(24\)	\(46.257\)		None	\(0\)	\(0\)	\(0\)	\(0\)

Decomposition of \(S_{4}^{\mathrm{old}}(784, [\chi])\) into lower level spaces

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