Space of modular forms of level 70, weight 6, and trivial character

• Label: 70.6.a
• Level: $70$
• Weight: $6$
• Character orbit: 70.a
• Rep. character: $\chi_{70}(1,\cdot)$
• Character field: $\Q$
• Dimension: $10$
• Newform subspaces: $8$
• Sturm bound: $72$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 70 = 2 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	70.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(72\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(70))\).

	Total	New	Old
Modular forms	64	10	54
Cusp forms	56	10	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.

\(2\)	\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(2\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(2\)
Plus space			\(+\)	\(4\)
Minus space			\(-\)	\(6\)

Trace form

\( 10 q + 8 q^{2} - 44 q^{3} + 160 q^{4} + 16 q^{6} + 128 q^{8} + 986 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(70))\) 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	5	7
70.6.a.a	$70$	$6$	70.a	1.a	$1$	$1$	$1$	$11.227$	\(\Q\)	None	✓	$1$	$1$	\(-4\)	\(-23\)	\(25\)	\(49\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-23q^{3}+2^{4}q^{4}+5^{2}q^{5}+92q^{6}+\cdots\)
70.6.a.b	$70$	$6$	70.a	1.a	$1$	$1$	$1$	$11.227$	\(\Q\)	None	✓	$1$	$0$	\(-4\)	\(-9\)	\(25\)	\(-49\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-9q^{3}+2^{4}q^{4}+5^{2}q^{5}+6^{2}q^{6}+\cdots\)
70.6.a.c	$70$	$6$	70.a	1.a	$1$	$1$	$1$	$11.227$	\(\Q\)	None	✓	$1$	$0$	\(-4\)	\(-3\)	\(-25\)	\(49\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}-3q^{3}+2^{4}q^{4}-5^{2}q^{5}+12q^{6}+\cdots\)
70.6.a.d	$70$	$6$	70.a	1.a	$1$	$1$	$1$	$11.227$	\(\Q\)	None	✓	$1$	$1$	\(-4\)	\(11\)	\(-25\)	\(-49\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+11q^{3}+2^{4}q^{4}-5^{2}q^{5}-44q^{6}+\cdots\)
70.6.a.e	$70$	$6$	70.a	1.a	$1$	$1$	$1$	$11.227$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(-17\)	\(25\)	\(-49\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-17q^{3}+2^{4}q^{4}+5^{2}q^{5}-68q^{6}+\cdots\)
70.6.a.f	$70$	$6$	70.a	1.a	$1$	$1$	$1$	$11.227$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(-11\)	\(-25\)	\(49\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-11q^{3}+2^{4}q^{4}-5^{2}q^{5}-44q^{6}+\cdots\)
70.6.a.g	$70$	$6$	70.a	1.a	$1$	$2$	$2$	$11.227$	\(\Q(\sqrt{3369}) \)	None	✓	$1$	$0$	\(8\)	\(3\)	\(-50\)	\(-98\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+(2-\beta )q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
70.6.a.h	$70$	$6$	70.a	1.a	$1$	$2$	$2$	$11.227$	\(\Q(\sqrt{1129}) \)	None	✓	$1$	$0$	\(8\)	\(5\)	\(50\)	\(98\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+(3-\beta )q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(70))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(70)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)