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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(15\)	\(9\)	\(6\)		\(12\)	\(9\)	\(3\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(19\)	\(11\)	\(8\)		\(15\)	\(11\)	\(4\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(20\)	\(12\)	\(8\)		\(16\)	\(12\)	\(4\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(18\)	\(9\)	\(9\)		\(14\)	\(9\)	\(5\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(33\)	\(18\)	\(15\)		\(26\)	\(18\)	\(8\)		\(7\)	\(0\)	\(7\)
Minus space		\(-\)		\(39\)	\(23\)	\(16\)		\(31\)	\(23\)	\(8\)		\(8\)	\(0\)	\(8\)

Trace form

41 * q - q^2 + 4 * q^3 + 43 * q^4 - 2 * q^5 - 9 * q^8 + 45 * q^9 - 2 * q^10 + 8 * q^12 - q^13 + 4 * q^15 + 43 * q^16 - 2 * q^17 - q^18 + 12 * q^19 + 2 * q^20 - 16 * q^22 - 16 * q^23 + 4 * q^24 + 23 * q^25 - 3 * q^26 + 4 * q^27 - 26 * q^29 - 36 * q^30 + 4 * q^31 - 49 * q^32 + 4 * q^33 - 2 * q^34 + 27 * q^36 - 10 * q^37 + 8 * q^38 + 4 * q^39 + 10 * q^40 - 2 * q^41 - 4 * q^43 - 24 * q^44 - 22 * q^45 - 16 * q^46 - 8 * q^47 - 12 * q^48 - 35 * q^50 - 16 * q^51 + q^52 - 2 * q^53 - 48 * q^54 - 12 * q^55 - 16 * q^57 - 42 * q^58 - 16 * q^59 + 14 * q^61 + 8 * q^62 + 23 * q^64 + 2 * q^65 - 12 * q^66 - 4 * q^67 - 42 * q^68 + 20 * q^69 + 12 * q^71 - 49 * q^72 + 22 * q^73 - 50 * q^74 + 48 * q^75 - 16 * q^76 - 8 * q^78 - 32 * q^79 + 38 * q^80 + 49 * q^81 - 14 * q^82 - 24 * q^83 - 24 * q^85 + 64 * q^86 + 16 * q^87 - 12 * q^88 - 26 * q^89 + 50 * q^90 + 76 * q^92 - 28 * q^93 + 20 * q^94 - 40 * q^95 + 4 * q^96 + 6 * q^97 + 12 * q^99

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	Twist minimal	Largest	Maximal	Minimal twist	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	✓				91.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+3q^{5}-3q^{9}-6q^{10}+\cdots\)
637.2.a.b	$637$	$2$	637.a	1.a	$1$	$1$	$1$	$5.086$	\(\Q\)	None	✓				91.2.a.b	$1$	$0$	\(0\)	\(2\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{4}+3q^{5}+q^{9}-4q^{12}+\cdots\)
637.2.a.c	$637$	$2$	637.a	1.a	$1$	$1$	$1$	$5.086$	\(\Q\)	None	✓				91.2.e.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{8}-3q^{9}-3q^{11}+\cdots\)
637.2.a.d	$637$	$2$	637.a	1.a	$1$	$1$	$1$	$5.086$	\(\Q\)	None	✓				91.2.e.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-3q^{8}-3q^{9}-3q^{11}+\cdots\)
637.2.a.e	$637$	$2$	637.a	1.a	$1$	$2$	$2$	$5.086$	\(\Q(\sqrt{5}) \)	None	✓				91.2.e.b	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(-1+2\beta )q^{3}+3\beta q^{4}+\cdots\)
637.2.a.f	$637$	$2$	637.a	1.a	$1$	$2$	$2$	$5.086$	\(\Q(\sqrt{5}) \)	None	✓				91.2.e.b	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(1-2\beta )q^{3}+3\beta q^{4}+\cdots\)
637.2.a.g	$637$	$2$	637.a	1.a	$1$	$2$	$2$	$5.086$	\(\Q(\sqrt{2}) \)	None	✓				91.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+\beta q^{3}+(-3-\beta )q^{5}+2q^{6}+\cdots\)
637.2.a.h	$637$	$2$	637.a	1.a	$1$	$3$	$3$	$5.086$	3.3.404.1	None	✓	✓	✓		637.2.a.h	$1$	$1$	\(-2\)	\(-4\)	\(-5\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{2})q^{3}+(2+\cdots)q^{4}+\cdots\)
637.2.a.i	$637$	$2$	637.a	1.a	$1$	$3$	$3$	$5.086$	3.3.404.1	None	✓	✓			637.2.a.h	$1$	$0$	\(-2\)	\(4\)	\(5\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1+\beta _{2})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
637.2.a.j	$637$	$2$	637.a	1.a	$1$	$3$	$3$	$5.086$	3.3.316.1	None	✓				91.2.a.d	$1$	$0$	\(1\)	\(2\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{1}+\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
637.2.a.k	$637$	$2$	637.a	1.a	$1$	$5$	$5$	$5.086$	5.5.746052.1	None	✓		✓		91.2.e.c	$1$	$0$	\(4\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+\beta _{4}q^{3}+(2-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
637.2.a.l	$637$	$2$	637.a	1.a	$1$	$5$	$5$	$5.086$	5.5.746052.1	None	✓				91.2.e.c	$1$	$0$	\(4\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-\beta _{4}q^{3}+(2-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
637.2.a.m	$637$	$2$	637.a	1.a	$1$	$6$	$6$	$5.086$	6.6.4507648.1	None	✓	✓	✓		637.2.a.m	$1$	$1$	\(0\)	\(-8\)	\(-6\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(-1-\beta _{1})q^{3}+(1-\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
637.2.a.n	$637$	$2$	637.a	1.a	$1$	$6$	$6$	$5.086$	6.6.4507648.1	None	✓	✓	✓		637.2.a.m	$1$	$0$	\(0\)	\(8\)	\(6\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(1+\beta _{1})q^{3}+(1-\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)

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

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