Space of modular forms of level 42, weight 8, and trivial character

• Label: 42.8.a
• Level: $42$
• Weight: $8$
• Character orbit: 42.a
• Rep. character: $\chi_{42}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $6$
• Sturm bound: $64$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(64\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(42))\).

	Total	New	Old
Modular forms	60	6	54
Cusp forms	52	6	46
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.

\(2\)	\(3\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(2\)

Trace form

\( 6 q - 16 q^{2} + 384 q^{4} + 220 q^{5} - 1024 q^{8} + 4374 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(42))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	7
42.8.a.a	$42$	$8$	42.a	1.a	$1$	$1$	$1$	$13.120$	\(\Q\)	None	✓	$1$	$0$	\(-8\)	\(-27\)	\(-410\)	\(-343\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}-410q^{5}+\cdots\)
42.8.a.b	$42$	$8$	42.a	1.a	$1$	$1$	$1$	$13.120$	\(\Q\)	None	✓	$1$	$1$	\(-8\)	\(-27\)	\(-18\)	\(343\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}-18q^{5}+6^{3}q^{6}+\cdots\)
42.8.a.c	$42$	$8$	42.a	1.a	$1$	$1$	$1$	$13.120$	\(\Q\)	None	✓	$1$	$1$	\(-8\)	\(27\)	\(-122\)	\(-343\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-122q^{5}+\cdots\)
42.8.a.d	$42$	$8$	42.a	1.a	$1$	$1$	$1$	$13.120$	\(\Q\)	None	✓	$1$	$0$	\(-8\)	\(27\)	\(270\)	\(343\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+270q^{5}+\cdots\)
42.8.a.e	$42$	$8$	42.a	1.a	$1$	$1$	$1$	$13.120$	\(\Q\)	None	✓	$1$	$0$	\(8\)	\(-27\)	\(30\)	\(343\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+30q^{5}-6^{3}q^{6}+\cdots\)
42.8.a.f	$42$	$8$	42.a	1.a	$1$	$1$	$1$	$13.120$	\(\Q\)	None	✓	$1$	$0$	\(8\)	\(27\)	\(470\)	\(-343\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+470q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(42))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(42)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)