Space of modular forms of level 670, weight 2, and Character 670.e

• Label: 670.2.e
• Level: $670$
• Weight: $2$
• Character orbit: 670.e
• Rep. character: $\chi_{670}(171,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $48$
• Newform subspaces: $10$
• Sturm bound: $204$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 670 = 2 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	670.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 67 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(204\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(670, [\chi])\).

	Total	New	Old
Modular forms	212	48	164
Cusp forms	196	48	148
Eisenstein series	16	0	16

Trace form

\( 48 q + 2 q^{2} + 4 q^{3} - 24 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{7} - 4 q^{8} + 64 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(670, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
670.2.e.a	$670$	$2$	670.e	67.c	$3$	$2$	$1$	$5.350$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(-1\)	\(-6\)	\(-2\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-3q^{3}-\zeta_{6}q^{4}-q^{5}+\cdots\)
670.2.e.b	$670$	$2$	670.e	67.c	$3$	$2$	$1$	$5.350$	\(\Q(\sqrt{-3}) \)	None		$2$	$1$	\(-1\)	\(-6\)	\(2\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}-3q^{3}-\zeta_{6}q^{4}+q^{5}+\cdots\)
670.2.e.c	$670$	$2$	670.e	67.c	$3$	$2$	$1$	$5.350$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-1\)	\(2\)	\(-2\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+q^{3}-\zeta_{6}q^{4}-q^{5}+\cdots\)
670.2.e.d	$670$	$2$	670.e	67.c	$3$	$2$	$1$	$5.350$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(1\)	\(2\)	\(-2\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}+q^{3}-\zeta_{6}q^{4}-q^{5}+(1+\cdots)q^{6}+\cdots\)
670.2.e.e	$670$	$2$	670.e	67.c	$3$	$4$	$2$	$5.350$	\(\Q(\sqrt{-3}, \sqrt{193})\)	None		$2$	$0$	\(-2\)	\(4\)	\(-4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}+q^{3}+(-1+\beta _{2})q^{4}-q^{5}+\cdots\)
670.2.e.f	$670$	$2$	670.e	67.c	$3$	$4$	$2$	$5.350$	\(\Q(\sqrt{-2}, \sqrt{-3})\)	None		$2$	$0$	\(-2\)	\(4\)	\(-4\)	\(-4\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(-1+\beta _{1})q^{4}+\cdots\)
670.2.e.g	$670$	$2$	670.e	67.c	$3$	$6$	$3$	$5.350$	6.0.954288.1	None		$2$	$0$	\(3\)	\(-2\)	\(6\)	\(-2\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{3})q^{2}+\beta _{4}q^{3}-\beta _{3}q^{4}+q^{5}+\cdots\)
670.2.e.h	$670$	$2$	670.e	67.c	$3$	$6$	$3$	$5.350$	6.0.2696112.1	None		$2$	$0$	\(3\)	\(2\)	\(6\)	\(2\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{2}-\beta _{2}q^{3}+(-1+\beta _{3})q^{4}+q^{5}+\cdots\)
670.2.e.i	$670$	$2$	670.e	67.c	$3$	$8$	$4$	$5.350$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(-4\)	\(8\)	\(8\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{2})q^{2}+(1+\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots\)
670.2.e.j	$670$	$2$	670.e	67.c	$3$	$12$	$6$	$5.350$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(6\)	\(-4\)	\(-12\)	\(3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{5}q^{2}-\beta _{2}q^{3}+(-1-\beta _{5})q^{4}-q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(670, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(670, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(67, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(134, [\chi])\)\(^{\oplus 2}\)