Space of modular forms of level 637, weight 2, and Character 637.g

• Label: 637.2.g
• Level: $637$
• Weight: $2$
• Character orbit: 637.g
• Rep. character: $\chi_{637}(263,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $86$
• Newform subspaces: $13$
• Sturm bound: $130$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.g (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 91 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(130\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(637, [\chi])\).

	Total	New	Old
Modular forms	146	102	44
Cusp forms	114	86	28
Eisenstein series	32	16	16

Trace form

\( 86 q - q^{2} + 8 q^{3} - 41 q^{4} + 2 q^{5} + 6 q^{6} - 12 q^{8} + 66 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(637, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
637.2.g.a	$637$	$2$	637.g	91.g	$3$	$2$	$1$	$5.086$	\(\Q(\sqrt{-3}) \)	None		$2$	$0$	\(-1\)	\(6\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{6}q^{2}+3q^{3}+(1-\zeta_{6})q^{4}+(3-3\zeta_{6})q^{5}+\cdots\)
637.2.g.b	$637$	$2$	637.g	91.g	$3$	$4$	$2$	$5.086$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$1$	\(-3\)	\(-6\)	\(-3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1}-\beta _{3})q^{2}+(-1+\beta _{2})q^{3}+\cdots\)
637.2.g.c	$637$	$2$	637.g	91.g	$3$	$4$	$2$	$5.086$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None		$2$	$0$	\(-3\)	\(6\)	\(3\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1}-\beta _{3})q^{2}+(1-\beta _{2})q^{3}+\cdots\)
637.2.g.d	$637$	$2$	637.g	91.g	$3$	$4$	$2$	$5.086$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{12}^{2}q^{2}+(-1-\zeta_{12}^{3})q^{3}+(-1+\cdots)q^{4}+\cdots\)
637.2.g.e	$637$	$2$	637.g	91.g	$3$	$4$	$2$	$5.086$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\zeta_{12}^{2}q^{2}+(1+\zeta_{12}^{3})q^{3}+(-1+\zeta_{12}+\cdots)q^{4}+\cdots\)
637.2.g.f	$637$	$2$	637.g	91.g	$3$	$4$	$2$	$5.086$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(2\)	\(0\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{2})q^{2}+\beta _{3}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
637.2.g.g	$637$	$2$	637.g	91.g	$3$	$4$	$2$	$5.086$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		$2$	$0$	\(2\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{2})q^{2}-\beta _{3}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
637.2.g.h	$637$	$2$	637.g	91.g	$3$	$8$	$4$	$5.086$	8.0.\(\cdots\).7	None		$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{2})q^{2}+(-\beta _{5}+\beta _{6})q^{3}+\beta _{2}q^{4}+\cdots\)
637.2.g.i	$637$	$2$	637.g	91.g	$3$	$8$	$4$	$5.086$	8.0.\(\cdots\).6	None		$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{2}+\beta _{5})q^{2}+\beta _{3}q^{3}+(-2-2\beta _{2}+\cdots)q^{4}+\cdots\)
637.2.g.j	$637$	$2$	637.g	91.g	$3$	$8$	$4$	$5.086$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(1\)	\(-2\)	\(-7\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
637.2.g.k	$637$	$2$	637.g	91.g	$3$	$8$	$4$	$5.086$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$2$	$0$	\(1\)	\(2\)	\(7\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
637.2.g.l	$637$	$2$	637.g	91.g	$3$	$12$	$6$	$5.086$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		$2$	$0$	\(2\)	\(2\)	\(-1\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{5}+\beta _{11})q^{2}+(-\beta _{3}+\beta _{11})q^{3}+\cdots\)
637.2.g.m	$637$	$2$	637.g	91.g	$3$	$16$	$8$	$5.086$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{4}+\beta _{10})q^{2}+(\beta _{1}-\beta _{5}+\beta _{8}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(637, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(637, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(91, [\chi])\)\(^{\oplus 2}\)