Space of modular forms of level 663, weight 4, and trivial character

• Label: 663.4.a
• Level: $663$
• Weight: $4$
• Character orbit: 663.a
• Rep. character: $\chi_{663}(1,\cdot)$
• Character field: $\Q$
• Dimension: $96$
• Newform subspaces: $10$
• Sturm bound: $336$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 663 = 3 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	663.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(336\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(663))\).

	Total	New	Old
Modular forms	256	96	160
Cusp forms	248	96	152
Eisenstein series	8	0	8

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

\(3\)	\(13\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(13\)
\(+\)	\(+\)	\(-\)	$-$	\(9\)
\(+\)	\(-\)	\(+\)	$-$	\(11\)
\(+\)	\(-\)	\(-\)	$+$	\(15\)
\(-\)	\(+\)	\(+\)	$-$	\(11\)
\(-\)	\(+\)	\(-\)	$+$	\(15\)
\(-\)	\(-\)	\(+\)	$+$	\(13\)
\(-\)	\(-\)	\(-\)	$-$	\(9\)
Plus space			\(+\)	\(56\)
Minus space			\(-\)	\(40\)

Trace form

\( 96 q - 8 q^{2} + 424 q^{4} - 24 q^{6} - 80 q^{7} - 96 q^{8} + 864 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(663))\) 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	13	17
663.4.a.a	$663$	$4$	663.a	1.a	$1$	$1$	$1$	$39.118$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(3\)	\(-10\)	\(-10\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+3q^{3}+8q^{4}-10q^{5}+12q^{6}+\cdots\)
663.4.a.b	$663$	$4$	663.a	1.a	$1$	$2$	$2$	$39.118$	\(\Q(\sqrt{205}) \)	None	✓	$1$	$1$	\(2\)	\(-6\)	\(-1\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}-7q^{4}+(-1+\beta )q^{5}+\cdots\)
663.4.a.c	$663$	$4$	663.a	1.a	$1$	$7$	$7$	$39.118$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-7\)	\(-21\)	\(-4\)	\(-12\)		$+$	$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-3q^{3}+(5+\beta _{2})q^{4}+\cdots\)
663.4.a.d	$663$	$4$	663.a	1.a	$1$	$9$	$9$	$39.118$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(-13\)	\(27\)	\(-25\)	\(-49\)		$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+3q^{3}+(3+2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
663.4.a.e	$663$	$4$	663.a	1.a	$1$	$10$	$10$	$39.118$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-9\)	\(30\)	\(9\)	\(-17\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+3q^{3}+(3-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
663.4.a.f	$663$	$4$	663.a	1.a	$1$	$11$	$11$	$39.118$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(-33\)	\(-9\)	\(-69\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(5+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
663.4.a.g	$663$	$4$	663.a	1.a	$1$	$13$	$13$	$39.118$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(-39\)	\(-25\)	\(15\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
663.4.a.h	$663$	$4$	663.a	1.a	$1$	$13$	$13$	$39.118$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$0$	\(7\)	\(39\)	\(55\)	\(1\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+3q^{3}+(5+\beta _{2})q^{4}+(4+\cdots)q^{5}+\cdots\)
663.4.a.i	$663$	$4$	663.a	1.a	$1$	$15$	$15$	$39.118$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(-45\)	\(11\)	\(21\)		$+$	$-$	$-$	$+$	$2^{6}\cdot 7$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(6+\beta _{2})q^{4}+(1+\beta _{5}+\cdots)q^{5}+\cdots\)
663.4.a.j	$663$	$4$	663.a	1.a	$1$	$15$	$15$	$39.118$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(45\)	\(-1\)	\(35\)		$-$	$+$	$-$	$+$	$2^{10}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(663))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(663)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(221))\)\(^{\oplus 2}\)