Space of modular forms of level 70, weight 8, and trivial character

• Label: 70.8.a
• Level: $70$
• Weight: $8$
• Character orbit: 70.a
• Rep. character: $\chi_{70}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $8$
• Sturm bound: $96$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 70 = 2 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	70.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(96\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(70))\).

	Total	New	Old
Modular forms	88	14	74
Cusp forms	80	14	66
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
\(+\)	\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(1\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(6\)

Trace form

\( 14 q - 16 q^{2} + 52 q^{3} + 896 q^{4} - 1376 q^{6} - 1024 q^{8} + 11294 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(70))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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
70.8.a.a	$70$	$8$	70.a	1.a	$1$	$1$	$1$	$21.867$	\(\Q\)	None	✓	✓	✓	✓	70.8.a.a	$1$	$1$	\(8\)	\(-93\)	\(125\)	\(343\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-93q^{3}+2^{6}q^{4}+5^{3}q^{5}-744q^{6}+\cdots\)
70.8.a.b	$70$	$8$	70.a	1.a	$1$	$1$	$1$	$21.867$	\(\Q\)	None	✓	✓	✓	✓	70.8.a.b	$1$	$1$	\(8\)	\(9\)	\(-125\)	\(-343\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+9q^{3}+2^{6}q^{4}-5^{3}q^{5}+72q^{6}+\cdots\)
70.8.a.c	$70$	$8$	70.a	1.a	$1$	$2$	$2$	$21.867$	\(\Q(\sqrt{9241}) \)	None	✓	✓	✓	✓	70.8.a.c	$1$	$1$	\(-16\)	\(11\)	\(-250\)	\(686\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+(6-\beta )q^{3}+2^{6}q^{4}-5^{3}q^{5}+\cdots\)
70.8.a.d	$70$	$8$	70.a	1.a	$1$	$2$	$2$	$21.867$	\(\Q(\sqrt{11761}) \)	None	✓	✓	✓	✓	70.8.a.d	$1$	$0$	\(-16\)	\(25\)	\(-250\)	\(-686\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+(13-\beta )q^{3}+2^{6}q^{4}-5^{3}q^{5}+\cdots\)
70.8.a.e	$70$	$8$	70.a	1.a	$1$	$2$	$2$	$21.867$	\(\Q(\sqrt{18481}) \)	None	✓	✓	✓	✓	70.8.a.e	$1$	$0$	\(-16\)	\(31\)	\(250\)	\(686\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+(2^{4}-\beta )q^{3}+2^{6}q^{4}+5^{3}q^{5}+\cdots\)
70.8.a.f	$70$	$8$	70.a	1.a	$1$	$2$	$2$	$21.867$	\(\Q(\sqrt{1401}) \)	None	✓	✓	✓	✓	70.8.a.f	$1$	$1$	\(-16\)	\(45\)	\(250\)	\(-686\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+(23-\beta )q^{3}+2^{6}q^{4}+5^{3}q^{5}+\cdots\)
70.8.a.g	$70$	$8$	70.a	1.a	$1$	$2$	$2$	$21.867$	\(\Q(\sqrt{8761}) \)	None	✓	✓	✓	✓	70.8.a.g	$1$	$0$	\(16\)	\(-5\)	\(-250\)	\(686\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+(-2-\beta )q^{3}+2^{6}q^{4}-5^{3}q^{5}+\cdots\)
70.8.a.h	$70$	$8$	70.a	1.a	$1$	$2$	$2$	$21.867$	\(\Q(\sqrt{12121}) \)	None	✓	✓	✓	✓	70.8.a.h	$1$	$0$	\(16\)	\(29\)	\(250\)	\(-686\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+(15-\beta )q^{3}+2^{6}q^{4}+5^{3}q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(70))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(70)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)