Space of modular forms of level 16, weight 42, and trivial character

• Label: 16.42.a
• Level: $16$
• Weight: $42$
• Character orbit: 16.a
• Rep. character: $\chi_{16}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $6$
• Sturm bound: $84$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 42 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(84\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{42}(\Gamma_0(16))\).

	Total	New	Old
Modular forms	85	21	64
Cusp forms	79	20	59
Eisenstein series	6	1	5

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

\(2\)	Dim
\(+\)	\(10\)
\(-\)	\(10\)

Trace form

\( 20 q - 3486784400 q^{3} + 105262218614520 q^{5} + 283370945575456480 q^{7} + 236216753458921118596 q^{9} + O(q^{10}) \)

Decomposition of \(S_{42}^{\mathrm{new}}(\Gamma_0(16))\) 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
16.42.a.a	$16$	$42$	16.a	1.a	$1$	$1$	$1$	$170.355$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-5043516516\)	\(-48\!\cdots\!50\)	\(11\!\cdots\!68\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5043516516q^{3}-48504195130650q^{5}+\cdots\)
16.42.a.b	$16$	$42$	16.a	1.a	$1$	$2$	$2$	$170.355$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-8863347528\)	\(97\!\cdots\!80\)	\(-21\!\cdots\!56\)		$-$	$-$	$2^{7}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-4431673764-\beta )q^{3}+(48799592162790+\cdots)q^{5}+\cdots\)
16.42.a.c	$16$	$42$	16.a	1.a	$1$	$3$	$3$	$170.355$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(10820953044\)	\(-21\!\cdots\!50\)	\(-57\!\cdots\!92\)		$-$	$-$	$2^{22}\cdot 3^{3}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(3606984348+\beta _{2})q^{3}+(-70767450093850+\cdots)q^{5}+\cdots\)
16.42.a.d	$16$	$42$	16.a	1.a	$1$	$4$	$4$	$170.355$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-1162346640\)	\(22\!\cdots\!24\)	\(40\!\cdots\!80\)		$-$	$-$	$2^{45}\cdot 3^{8}\cdot 5\cdot 7$	$\mathrm{SU}(2)$	\(q+(-290586660+\beta _{1})q^{3}+(55732604152806+\cdots)q^{5}+\cdots\)
16.42.a.e	$16$	$42$	16.a	1.a	$1$	$5$	$5$	$170.355$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-1810632404\)	\(-19\!\cdots\!10\)	\(10\!\cdots\!68\)		$+$	$+$	$2^{73}\cdot 3^{6}\cdot 5^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-362126481+\beta _{1})q^{3}+(-39240493433260+\cdots)q^{5}+\cdots\)
16.42.a.f	$16$	$42$	16.a	1.a	$1$	$5$	$5$	$170.355$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(2572105644\)	\(24\!\cdots\!26\)	\(-72\!\cdots\!88\)		$+$	$+$	$2^{72}\cdot 3^{10}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q+(514421129+\beta _{1})q^{3}+(48348326049989+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{42}^{\mathrm{old}}(\Gamma_0(16))\) into lower level spaces

\( S_{42}^{\mathrm{old}}(\Gamma_0(16)) \cong \) \(S_{42}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{42}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{42}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{42}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)