Space of modular forms of level 627, weight 2, and trivial character

• Label: 627.2.a
• Level: $627$
• Weight: $2$
• Character orbit: 627.a
• Rep. character: $\chi_{627}(1,\cdot)$
• Character field: $\Q$
• Dimension: $31$
• Newform subspaces: $10$
• Sturm bound: $160$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 627 = 3 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	627.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(160\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(627))\).

	Total	New	Old
Modular forms	84	31	53
Cusp forms	77	31	46
Eisenstein series	7	0	7

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

\(3\)	\(11\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(-\)	$-$	\(5\)
\(+\)	\(-\)	\(+\)	$-$	\(5\)
\(+\)	\(-\)	\(-\)	$+$	\(3\)
\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(-\)	\(+\)	\(-\)	$+$	\(3\)
\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(-\)	$-$	\(4\)
Plus space			\(+\)	\(12\)
Minus space			\(-\)	\(19\)

Trace form

\( 31 q + q^{2} - q^{3} + 29 q^{4} + 10 q^{5} - 3 q^{6} - 8 q^{7} - 3 q^{8} + 31 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(627))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	11	19
627.2.a.a	$627$	$2$	627.a	1.a	$1$	$1$	$1$	$5.007$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+2q^{7}+q^{9}+q^{11}-2q^{12}+\cdots\)
627.2.a.b	$627$	$2$	627.a	1.a	$1$	$1$	$1$	$5.007$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(1\)	\(4\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+4q^{5}+2q^{7}+q^{9}-q^{11}+\cdots\)
627.2.a.c	$627$	$2$	627.a	1.a	$1$	$3$	$3$	$5.007$	3.3.169.1	None	✓	$1$	$1$	\(-2\)	\(-3\)	\(1\)	\(-5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
627.2.a.d	$627$	$2$	627.a	1.a	$1$	$3$	$3$	$5.007$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(-2\)	\(3\)	\(-7\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
627.2.a.e	$627$	$2$	627.a	1.a	$1$	$3$	$3$	$5.007$	3.3.321.1	None	✓	$1$	$1$	\(-2\)	\(3\)	\(-3\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
627.2.a.f	$627$	$2$	627.a	1.a	$1$	$3$	$3$	$5.007$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(2\)	\(-3\)	\(-3\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
627.2.a.g	$627$	$2$	627.a	1.a	$1$	$3$	$3$	$5.007$	3.3.169.1	None	✓	$1$	$0$	\(2\)	\(3\)	\(5\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
627.2.a.h	$627$	$2$	627.a	1.a	$1$	$4$	$4$	$5.007$	4.4.23377.1	None	✓	$1$	$0$	\(1\)	\(4\)	\(3\)	\(-5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{2}+\cdots)q^{5}+\cdots\)
627.2.a.i	$627$	$2$	627.a	1.a	$1$	$5$	$5$	$5.007$	5.5.2179633.1	None	✓	$1$	$0$	\(1\)	\(-5\)	\(3\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{4})q^{5}+\cdots\)
627.2.a.j	$627$	$2$	627.a	1.a	$1$	$5$	$5$	$5.007$	5.5.1920025.1	None	✓	$1$	$0$	\(1\)	\(-5\)	\(7\)	\(7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+(1-\beta _{4})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(627))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(627)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 2}\)