Space of modular forms of level 42, weight 10, and trivial character

• Label: 42.10.a
• Level: $42$
• Weight: $10$
• Character orbit: 42.a
• Rep. character: $\chi_{42}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $8$
• Sturm bound: $80$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(80\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(42))\).

	Total	New	Old
Modular forms	76	10	66
Cusp forms	68	10	58
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
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(2\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(6\)

Trace form

\( 10 q + 32 q^{2} + 2560 q^{4} - 3116 q^{5} + 8192 q^{8} + 65610 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(42))\) 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
42.10.a.a	$42$	$10$	42.a	1.a	$1$	$1$	$1$	$21.632$	\(\Q\)	None	✓	$1$	$1$	\(-16\)	\(-81\)	\(-76\)	\(-2401\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}-3^{4}q^{3}+2^{8}q^{4}-76q^{5}+\cdots\)
42.10.a.b	$42$	$10$	42.a	1.a	$1$	$1$	$1$	$21.632$	\(\Q\)	None	✓	$1$	$0$	\(-16\)	\(-81\)	\(1590\)	\(2401\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}-3^{4}q^{3}+2^{8}q^{4}+1590q^{5}+\cdots\)
42.10.a.c	$42$	$10$	42.a	1.a	$1$	$1$	$1$	$21.632$	\(\Q\)	None	✓	$1$	$0$	\(-16\)	\(81\)	\(-2290\)	\(-2401\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+3^{4}q^{3}+2^{8}q^{4}-2290q^{5}+\cdots\)
42.10.a.d	$42$	$10$	42.a	1.a	$1$	$1$	$1$	$21.632$	\(\Q\)	None	✓	$1$	$1$	\(-16\)	\(81\)	\(-624\)	\(2401\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+3^{4}q^{3}+2^{8}q^{4}-624q^{5}+\cdots\)
42.10.a.e	$42$	$10$	42.a	1.a	$1$	$1$	$1$	$21.632$	\(\Q\)	None	✓	$1$	$1$	\(16\)	\(-81\)	\(-474\)	\(2401\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}-3^{4}q^{3}+2^{8}q^{4}-474q^{5}+\cdots\)
42.10.a.f	$42$	$10$	42.a	1.a	$1$	$1$	$1$	$21.632$	\(\Q\)	None	✓	$1$	$1$	\(16\)	\(81\)	\(-1634\)	\(-2401\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+3^{4}q^{3}+2^{8}q^{4}-1634q^{5}+\cdots\)
42.10.a.g	$42$	$10$	42.a	1.a	$1$	$2$	$2$	$21.632$	\(\Q(\sqrt{474769}) \)	None	✓	$1$	$0$	\(32\)	\(-162\)	\(-142\)	\(-4802\)		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}-3^{4}q^{3}+2^{8}q^{4}+(-71-\beta )q^{5}+\cdots\)
42.10.a.h	$42$	$10$	42.a	1.a	$1$	$2$	$2$	$21.632$	\(\Q(\sqrt{243601}) \)	None	✓	$1$	$0$	\(32\)	\(162\)	\(534\)	\(4802\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+3^{4}q^{3}+2^{8}q^{4}+(267-\beta )q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(42))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(42)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)