Space of modular forms of level 637, weight 2, and trivial character

• Label: 637.2.a
• Level: $637$
• Weight: $2$
• Character orbit: 637.a
• Rep. character: $\chi_{637}(1,\cdot)$
• Character field: $\Q$
• Dimension: $41$
• Newform subspaces: $14$
• Sturm bound: $130$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(130\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(2\), \(3\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(637))\).

	Total	New	Old
Modular forms	72	41	31
Cusp forms	57	41	16
Eisenstein series	15	0	15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(7\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(9\)
\(+\)	\(-\)	$-$	\(11\)
\(-\)	\(+\)	$-$	\(12\)
\(-\)	\(-\)	$+$	\(9\)
Plus space		\(+\)	\(18\)
Minus space		\(-\)	\(23\)

Trace form

\( 41 q - q^{2} + 4 q^{3} + 43 q^{4} - 2 q^{5} - 9 q^{8} + 45 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(637))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7	13
637.2.a.a	$637$	$2$	637.a	1.a	$1$	$1$	$1$	$5.086$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+3q^{5}-3q^{9}-6q^{10}+\cdots\)
637.2.a.b	$637$	$2$	637.a	1.a	$1$	$1$	$1$	$5.086$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{4}+3q^{5}+q^{9}-4q^{12}+\cdots\)
637.2.a.c	$637$	$2$	637.a	1.a	$1$	$1$	$1$	$5.086$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{8}-3q^{9}-3q^{11}+\cdots\)
637.2.a.d	$637$	$2$	637.a	1.a	$1$	$1$	$1$	$5.086$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{8}-3q^{9}-3q^{11}+\cdots\)
637.2.a.e	$637$	$2$	637.a	1.a	$1$	$2$	$2$	$5.086$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(-1+2\beta )q^{3}+3\beta q^{4}+\cdots\)
637.2.a.f	$637$	$2$	637.a	1.a	$1$	$2$	$2$	$5.086$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(1-2\beta )q^{3}+3\beta q^{4}+\cdots\)
637.2.a.g	$637$	$2$	637.a	1.a	$1$	$2$	$2$	$5.086$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+\beta q^{3}+(-3-\beta )q^{5}+2q^{6}+\cdots\)
637.2.a.h	$637$	$2$	637.a	1.a	$1$	$3$	$3$	$5.086$	3.3.404.1	None	✓	$1$	$1$	\(-2\)	\(-4\)	\(-5\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{2})q^{3}+(2+\cdots)q^{4}+\cdots\)
637.2.a.i	$637$	$2$	637.a	1.a	$1$	$3$	$3$	$5.086$	3.3.404.1	None	✓	$1$	$0$	\(-2\)	\(4\)	\(5\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1+\beta _{2})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
637.2.a.j	$637$	$2$	637.a	1.a	$1$	$3$	$3$	$5.086$	3.3.316.1	None	✓	$1$	$0$	\(1\)	\(2\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{1}+\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
637.2.a.k	$637$	$2$	637.a	1.a	$1$	$5$	$5$	$5.086$	5.5.746052.1	None	✓	$1$	$0$	\(4\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+\beta _{4}q^{3}+(2-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
637.2.a.l	$637$	$2$	637.a	1.a	$1$	$5$	$5$	$5.086$	5.5.746052.1	None	✓	$1$	$0$	\(4\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-\beta _{4}q^{3}+(2-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
637.2.a.m	$637$	$2$	637.a	1.a	$1$	$6$	$6$	$5.086$	6.6.4507648.1	None	✓	$1$	$1$	\(0\)	\(-8\)	\(-6\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(-1-\beta _{1})q^{3}+(1-\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
637.2.a.n	$637$	$2$	637.a	1.a	$1$	$6$	$6$	$5.086$	6.6.4507648.1	None	✓	$1$	$0$	\(0\)	\(8\)	\(6\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1+\beta _{1})q^{3}+(1-\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(637))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(637)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 2}\)