Space of modular forms of level 784, weight 2, and Character 784.m

• Label: 784.2.m
• Level: $784$
• Weight: $2$
• Character orbit: 784.m
• Rep. character: $\chi_{784}(197,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $154$
• Newform subspaces: $12$
• Sturm bound: $224$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.m (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 16 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(224\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(784, [\chi])\).

	Total	New	Old
Modular forms	240	174	66
Cusp forms	208	154	54
Eisenstein series	32	20	12

Trace form

\( 154 q + 2 q^{2} + 2 q^{3} + 2 q^{5} + 8 q^{6} - 16 q^{8} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(784, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
784.2.m.a	$784$	$2$	784.m	16.e	$4$	$2$	$1$	$6.260$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(-2\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+i)q^{2}-2iq^{4}+(-2-2i)q^{5}+\cdots\)
784.2.m.b	$784$	$2$	784.m	16.e	$4$	$2$	$1$	$6.260$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(-2\)	\(2\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1-i)q^{2}+(1-i)q^{3}+2iq^{4}+\cdots\)
784.2.m.c	$784$	$2$	784.m	16.e	$4$	$2$	$1$	$6.260$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(-2\)	\(4\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+i)q^{2}+(2-2i)q^{3}-2iq^{4}+\cdots\)
784.2.m.d	$784$	$2$	784.m	16.e	$4$	$4$	$2$	$6.260$	\(\Q(\zeta_{12})\)	None		$2$	$1$	\(-2\)	\(-2\)	\(-6\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+\zeta_{12}^{3}q^{3}+\cdots\)
784.2.m.e	$784$	$2$	784.m	16.e	$4$	$4$	$2$	$6.260$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(-2\)	\(2\)	\(6\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-1+\zeta_{12})q^{2}+(1-\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
784.2.m.f	$784$	$2$	784.m	16.e	$4$	$4$	$2$	$6.260$	\(\Q(i, \sqrt{7})\)	\(\Q(\sqrt{-7}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{4}]$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}-2\beta _{3})q^{8}+\cdots\)
784.2.m.g	$784$	$2$	784.m	16.e	$4$	$8$	$4$	$6.260$	8.0.214798336.3	None		$2$	$0$	\(2\)	\(0\)	\(4\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{3}q^{2}+(\beta _{1}+\beta _{4})q^{3}+(1+\beta _{4}-\beta _{6}+\cdots)q^{4}+\cdots\)
784.2.m.h	$784$	$2$	784.m	16.e	$4$	$12$	$6$	$6.260$	12.0.\(\cdots\).1	None		$2$	$0$	\(2\)	\(-4\)	\(-4\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{2}+(\beta _{2}-\beta _{10})q^{3}+\beta _{1}q^{4}+(-1+\cdots)q^{5}+\cdots\)
784.2.m.i	$784$	$2$	784.m	16.e	$4$	$20$	$10$	$6.260$	\(\mathbb{Q}[x]/(x^{20} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{11}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{4}q^{2}+\beta _{9}q^{3}-\beta _{17}q^{4}+\beta _{2}q^{5}+\cdots\)
784.2.m.j	$784$	$2$	784.m	16.e	$4$	$24$	$12$	$6.260$		None		$2$	$0$	\(4\)	\(0\)	\(-4\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$
784.2.m.k	$784$	$2$	784.m	16.e	$4$	$24$	$12$	$6.260$		None		$2$	$0$	\(4\)	\(0\)	\(4\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$
784.2.m.l	$784$	$2$	784.m	16.e	$4$	$48$	$24$	$6.260$		None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{2}^{\mathrm{old}}(784, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(784, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)