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

• Label: 770.2.n
• Level: $770$
• Weight: $2$
• Character orbit: 770.n
• Rep. character: $\chi_{770}(71,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $96$
• Newform subspaces: $11$
• Sturm bound: $288$
• Trace bound: $3$

Defining parameters

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

Dimensions

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

	Total	New	Old
Modular forms	608	96	512
Cusp forms	544	96	448
Eisenstein series	64	0	64

Trace form

\( 96 q - 4 q^{2} - 8 q^{3} - 24 q^{4} + 12 q^{6} - 4 q^{8} - 20 q^{9} + O(q^{10}) \)

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.n.a	$770$	$2$	770.n	11.c	$5$	$4$	$1$	$6.148$	\(\Q(\zeta_{10})\)	None		$2$	$0$	\(-1\)	\(-3\)	\(-1\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
770.2.n.b	$770$	$2$	770.n	11.c	$5$	$4$	$1$	$6.148$	\(\Q(\zeta_{10})\)	None		$2$	$0$	\(1\)	\(-4\)	\(1\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-\zeta_{10}+\cdots)q^{3}+\cdots\)
770.2.n.c	$770$	$2$	770.n	11.c	$5$	$4$	$1$	$6.148$	\(\Q(\zeta_{10})\)	None		$2$	$0$	\(1\)	\(-2\)	\(1\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-\zeta_{10}+\cdots)q^{3}+\cdots\)
770.2.n.d	$770$	$2$	770.n	11.c	$5$	$4$	$1$	$6.148$	\(\Q(\zeta_{10})\)	None		$2$	$0$	\(1\)	\(2\)	\(1\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(\zeta_{10}+\cdots)q^{3}+\cdots\)
770.2.n.e	$770$	$2$	770.n	11.c	$5$	$8$	$2$	$6.148$	8.0.682515625.5	None		$2$	$0$	\(2\)	\(-1\)	\(2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{2}q^{2}+(\beta _{1}+\beta _{4}-\beta _{6})q^{3}+\beta _{3}q^{4}+\cdots\)
770.2.n.f	$770$	$2$	770.n	11.c	$5$	$8$	$2$	$6.148$	8.0.484000000.9	None		$2$	$0$	\(2\)	\(2\)	\(-2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{5}q^{2}+(-\beta _{3}-\beta _{4}-\beta _{7})q^{3}+(-1+\cdots)q^{4}+\cdots\)
770.2.n.g	$770$	$2$	770.n	11.c	$5$	$12$	$3$	$6.148$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(-3\)	\(-2\)	\(-3\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{8}q^{2}+(-\beta _{4}-\beta _{5})q^{3}+\beta _{6}q^{4}+\cdots\)
770.2.n.h	$770$	$2$	770.n	11.c	$5$	$12$	$3$	$6.148$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(-3\)	\(0\)	\(3\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1-\beta _{2}-\beta _{3}-\beta _{10})q^{2}+(-\beta _{5}+\cdots)q^{3}+\cdots\)
770.2.n.i	$770$	$2$	770.n	11.c	$5$	$12$	$3$	$6.148$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(-3\)	\(2\)	\(-3\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q-\beta _{8}q^{2}+(1-\beta _{2}+\beta _{6}+\beta _{7}-\beta _{8}+\cdots)q^{3}+\cdots\)
770.2.n.j	$770$	$2$	770.n	11.c	$5$	$12$	$3$	$6.148$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(3\)	\(3\)	\(-3\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+\beta _{6}q^{2}+\beta _{4}q^{3}-\beta _{8}q^{4}+\beta _{7}q^{5}+\cdots\)
770.2.n.k	$770$	$2$	770.n	11.c	$5$	$16$	$4$	$6.148$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		$2$	$0$	\(-4\)	\(-5\)	\(4\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1-\beta _{3}+\beta _{8}+\beta _{9})q^{2}-\beta _{6}q^{3}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(770, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(22, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(55, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(110, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(154, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(385, [\chi])\)\(^{\oplus 2}\)