Space of modular forms of level 663, weight 2, and trivial character

• Label: 663.2.a
• Level: $663$
• Weight: $2$
• Character orbit: 663.a
• Rep. character: $\chi_{663}(1,\cdot)$
• Character field: $\Q$
• Dimension: $31$
• Newform subspaces: $9$
• Sturm bound: $168$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 663 = 3 \cdot 13 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	663.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(168\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(663))\).

	Total	New	Old
Modular forms	88	31	57
Cusp forms	81	31	50
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.

\(3\)	\(13\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(-\)	$-$	\(7\)
\(+\)	\(-\)	\(+\)	$-$	\(5\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(-\)	$-$	\(6\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(23\)

Trace form

\( 31 q + 5 q^{2} - q^{3} + 33 q^{4} - 6 q^{5} + q^{6} + 8 q^{7} + 33 q^{8} + 31 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(663))\) 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
663.2.a.a	$663$	$2$	663.a	1.a	$1$	$1$	$1$	$5.294$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(-2\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}-2q^{5}+q^{6}+3q^{8}+\cdots\)
663.2.a.b	$663$	$2$	663.a	1.a	$1$	$1$	$1$	$5.294$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{6}-2q^{7}+3q^{8}+\cdots\)
663.2.a.c	$663$	$2$	663.a	1.a	$1$	$1$	$1$	$5.294$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-1\)	\(-4\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-4q^{5}-q^{6}+2q^{7}+\cdots\)
663.2.a.d	$663$	$2$	663.a	1.a	$1$	$3$	$3$	$5.294$	3.3.148.1	None	✓	$1$	$1$	\(-3\)	\(3\)	\(-6\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
663.2.a.e	$663$	$2$	663.a	1.a	$1$	$3$	$3$	$5.294$	3.3.148.1	None	✓	$1$	$1$	\(-1\)	\(-3\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
663.2.a.f	$663$	$2$	663.a	1.a	$1$	$5$	$5$	$5.294$	5.5.1004368.1	None	✓	$1$	$0$	\(1\)	\(-5\)	\(2\)	\(8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots\)
663.2.a.g	$663$	$2$	663.a	1.a	$1$	$5$	$5$	$5.294$	5.5.153424.1	None	✓	$1$	$0$	\(3\)	\(5\)	\(0\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{3})q^{2}+q^{3}+(2+\beta _{3}-\beta _{4})q^{4}+\cdots\)
663.2.a.h	$663$	$2$	663.a	1.a	$1$	$6$	$6$	$5.294$	6.6.83831632.1	None	✓	$1$	$0$	\(2\)	\(-6\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)
663.2.a.i	$663$	$2$	663.a	1.a	$1$	$6$	$6$	$5.294$	6.6.15187408.1	None	✓	$1$	$0$	\(4\)	\(6\)	\(4\)	\(2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(663))\) into lower level spaces

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