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

• Label: 637.2.f
• Level: $637$
• Weight: $2$
• Character orbit: 637.f
• Rep. character: $\chi_{637}(295,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $84$
• Newform subspaces: $12$
• Sturm bound: $130$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.f (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 12 \)
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	148	104	44
Cusp forms	116	84	32
Eisenstein series	32	20	12

Trace form

\( 84 q + 2 q^{2} - 34 q^{4} + 8 q^{5} - 6 q^{6} - 24 q^{8} - 26 q^{9} + O(q^{10}) \)

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
637.2.f.a	$637$	$2$	637.f	13.c	$3$	$2$	$1$	$5.086$	\(\Q(\sqrt{-3}) \)	None					91.2.g.a	$2$	$1$	\(-1\)	\(-3\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
637.2.f.b	$637$	$2$	637.f	13.c	$3$	$2$	$1$	$5.086$	\(\Q(\sqrt{-3}) \)	None					91.2.g.a	$2$	$0$	\(-1\)	\(3\)	\(6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
637.2.f.c	$637$	$2$	637.f	13.c	$3$	$4$	$2$	$5.086$	\(\Q(\sqrt{-3}, \sqrt{5})\)	None					91.2.f.a	$2$	$0$	\(-3\)	\(-3\)	\(-6\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1}-\beta _{3})q^{2}+(-1-\beta _{1}-\beta _{3})q^{3}+\cdots\)
637.2.f.d	$637$	$2$	637.f	13.c	$3$	$4$	$2$	$5.086$	\(\Q(\zeta_{12})\)	None					91.2.f.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta_{2} q^{2}+(\beta_{2}+\beta_1)q^{3}+(\beta_1-1)q^{4}+\cdots\)
637.2.f.e	$637$	$2$	637.f	13.c	$3$	$4$	$2$	$5.086$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			637.2.f.e	$2$	$0$	\(2\)	\(0\)	\(-4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{2})q^{2}-\beta _{1}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
637.2.f.f	$637$	$2$	637.f	13.c	$3$	$4$	$2$	$5.086$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None		✓			637.2.f.e	$2$	$0$	\(2\)	\(0\)	\(4\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1}+\beta _{2})q^{2}+\beta _{1}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
637.2.f.g	$637$	$2$	637.f	13.c	$3$	$8$	$4$	$5.086$	8.0.\(\cdots\).7	None		✓			637.2.f.g	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{2})q^{2}+\beta _{6}q^{3}+\beta _{2}q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
637.2.f.h	$637$	$2$	637.f	13.c	$3$	$8$	$4$	$5.086$	8.0.\(\cdots\).6	None		✓			637.2.f.h	$4$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{2}-\beta _{6})q^{2}+(-\beta _{3}+\beta _{7})q^{3}+\cdots\)
637.2.f.i	$637$	$2$	637.f	13.c	$3$	$8$	$4$	$5.086$	8.0.\(\cdots\).1	None					91.2.f.c	$2$	$0$	\(1\)	\(1\)	\(14\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
637.2.f.j	$637$	$2$	637.f	13.c	$3$	$12$	$6$	$5.086$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					91.2.g.b	$2$	$0$	\(2\)	\(-1\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{5}+\beta _{11})q^{2}-\beta _{11}q^{3}+\cdots\)
637.2.f.k	$637$	$2$	637.f	13.c	$3$	$12$	$6$	$5.086$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					91.2.g.b	$2$	$0$	\(2\)	\(1\)	\(-2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{5}+\beta _{11})q^{2}+\beta _{11}q^{3}+\cdots\)
637.2.f.l	$637$	$2$	637.f	13.c	$3$	$16$	$8$	$5.086$	16.0.\(\cdots\).2	None		✓	✓		637.2.f.l	$4$	$0$	\(4\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{4}+\beta _{10})q^{2}+(\beta _{8}-\beta _{15})q^{3}+\cdots\)

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

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