Space of modular forms of level 42, weight 14, and trivial character

• Label: 42.14.a
• Level: $42$
• Weight: $14$
• Character orbit: 42.a
• Rep. character: $\chi_{42}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $9$
• Sturm bound: $112$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(112\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{14}(\Gamma_0(42))\).

	Total	New	Old
Modular forms	108	14	94
Cusp forms	100	14	86
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
\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	$-$	\(2\)
\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(2\)
Plus space			\(+\)	\(6\)
Minus space			\(-\)	\(8\)

Trace form

\( 14 q - 128 q^{2} + 57344 q^{4} + 16124 q^{5} - 524288 q^{8} + 7440174 q^{9} + O(q^{10}) \)

Decomposition of \(S_{14}^{\mathrm{new}}(\Gamma_0(42))\) 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
42.14.a.a	$42$	$14$	42.a	1.a	$1$	$1$	$1$	$45.037$	\(\Q\)	None	✓	$1$	$1$	\(-64\)	\(729\)	\(-51720\)	\(117649\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{6}q^{2}+3^{6}q^{3}+2^{12}q^{4}-51720q^{5}+\cdots\)
42.14.a.b	$42$	$14$	42.a	1.a	$1$	$1$	$1$	$45.037$	\(\Q\)	None	✓	$1$	$1$	\(-64\)	\(729\)	\(34758\)	\(117649\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{6}q^{2}+3^{6}q^{3}+2^{12}q^{4}+34758q^{5}+\cdots\)
42.14.a.c	$42$	$14$	42.a	1.a	$1$	$1$	$1$	$45.037$	\(\Q\)	None	✓	$1$	$1$	\(64\)	\(-729\)	\(-30330\)	\(117649\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{6}q^{2}-3^{6}q^{3}+2^{12}q^{4}-30330q^{5}+\cdots\)
42.14.a.d	$42$	$14$	42.a	1.a	$1$	$1$	$1$	$45.037$	\(\Q\)	None	✓	$1$	$1$	\(64\)	\(729\)	\(-22370\)	\(-117649\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{6}q^{2}+3^{6}q^{3}+2^{12}q^{4}-22370q^{5}+\cdots\)
42.14.a.e	$42$	$14$	42.a	1.a	$1$	$2$	$2$	$45.037$	\(\Q(\sqrt{601441}) \)	None	✓	$1$	$0$	\(-128\)	\(-1458\)	\(-10464\)	\(235298\)		$+$	$+$	$-$	$-$	$2^{2}\cdot 3\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{6}q^{2}-3^{6}q^{3}+2^{12}q^{4}+(-5232+\cdots)q^{5}+\cdots\)
42.14.a.f	$42$	$14$	42.a	1.a	$1$	$2$	$2$	$45.037$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	$1$	$1$	\(-128\)	\(-1458\)	\(46474\)	\(-235298\)		$+$	$+$	$+$	$+$	$2\cdot 3\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{6}q^{2}-3^{6}q^{3}+2^{12}q^{4}+(23237+\cdots)q^{5}+\cdots\)
42.14.a.g	$42$	$14$	42.a	1.a	$1$	$2$	$2$	$45.037$	\(\Q(\sqrt{407521}) \)	None	✓	$1$	$0$	\(-128\)	\(1458\)	\(39976\)	\(-235298\)		$+$	$-$	$+$	$-$	$2^{2}\cdot 3\cdot 7$	$\mathrm{SU}(2)$	\(q-2^{6}q^{2}+3^{6}q^{3}+2^{12}q^{4}+(19988+\cdots)q^{5}+\cdots\)
42.14.a.h	$42$	$14$	42.a	1.a	$1$	$2$	$2$	$45.037$	\(\Q(\sqrt{305281}) \)	None	✓	$1$	$0$	\(128\)	\(-1458\)	\(-27574\)	\(-235298\)		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{6}q^{2}-3^{6}q^{3}+2^{12}q^{4}+(-13787+\cdots)q^{5}+\cdots\)
42.14.a.i	$42$	$14$	42.a	1.a	$1$	$2$	$2$	$45.037$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	$1$	$0$	\(128\)	\(1458\)	\(37374\)	\(235298\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{6}q^{2}+3^{6}q^{3}+2^{12}q^{4}+(18687+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{14}^{\mathrm{old}}(\Gamma_0(42))\) into lower level spaces

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