Space of modular forms of level 770, weight 2, and Character 770.m

• Label: 770.2.m
• Level: $770$
• Weight: $2$
• Character orbit: 770.m
• Rep. character: $\chi_{770}(43,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $72$
• Newform subspaces: $6$
• Sturm bound: $288$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	770.m (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 55 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(288\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(770, [\chi])\).

	Total	New	Old
Modular forms	304	72	232
Cusp forms	272	72	200
Eisenstein series	32	0	32

Trace form

\( 72 q - 8 q^{3} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(770, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
770.2.m.a	$770$	$2$	770.m	55.e	$4$	$4$	$2$	$6.148$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(-8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}+(-2-\zeta_{8}^{2})q^{5}+\cdots\)
770.2.m.b	$770$	$2$	770.m	55.e	$4$	$4$	$2$	$6.148$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}+(1+2\zeta_{8}^{2})q^{5}+\cdots\)
770.2.m.c	$770$	$2$	770.m	55.e	$4$	$4$	$2$	$6.148$	\(\Q(\zeta_{8})\)	None		$4$	$0$	\(0\)	\(8\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}q^{2}+(2+2\zeta_{8}^{2})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
770.2.m.d	$770$	$2$	770.m	55.e	$4$	$8$	$4$	$6.148$	8.0.40960000.1	None		$4$	$0$	\(0\)	\(-4\)	\(16\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{7}q^{2}+\beta _{5}q^{3}-\beta _{2}q^{4}+(2-\beta _{2}+\cdots)q^{5}+\cdots\)
770.2.m.e	$770$	$2$	770.m	55.e	$4$	$16$	$8$	$6.148$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(0\)	\(-8\)	\(-8\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{10}q^{2}+(-1-\beta _{3}+\beta _{7}-\beta _{15})q^{3}+\cdots\)
770.2.m.f	$770$	$2$	770.m	55.e	$4$	$36$	$18$	$6.148$		None		$4$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{2}^{\mathrm{old}}(770, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(770, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(55, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(110, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(385, [\chi])\)\(^{\oplus 2}\)