Space of modular forms of level 670, weight 2, and trivial character

• Label: 670.2.a
• Level: $670$
• Weight: $2$
• Character orbit: 670.a
• Rep. character: $\chi_{670}(1,\cdot)$
• Character field: $\Q$
• Dimension: $21$
• Newform subspaces: $10$
• Sturm bound: $204$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 670 = 2 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	670.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(204\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(670))\).

	Total	New	Old
Modular forms	106	21	85
Cusp forms	99	21	78
Eisenstein series	7	0	7

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

\(2\)	\(5\)	\(67\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(-\)	$-$	\(4\)
\(+\)	\(-\)	\(+\)	$-$	\(3\)
\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(3\)
\(-\)	\(+\)	\(-\)	$+$	\(2\)
\(-\)	\(-\)	\(+\)	$+$	\(2\)
\(-\)	\(-\)	\(-\)	$-$	\(3\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(13\)

Trace form

\( 21 q - q^{2} - 4 q^{3} + 21 q^{4} - q^{5} - 4 q^{6} - 8 q^{7} - q^{8} + 17 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(670))\) 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	67
670.2.a.a	$670$	$2$	670.a	1.a	$1$	$1$	$1$	$5.350$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-2\)	\(1\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-q^{7}+\cdots\)
670.2.a.b	$670$	$2$	670.a	1.a	$1$	$1$	$1$	$5.350$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-3q^{9}+\cdots\)
670.2.a.c	$670$	$2$	670.a	1.a	$1$	$1$	$1$	$5.350$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-2\)	\(-1\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}+q^{7}+\cdots\)
670.2.a.d	$670$	$2$	670.a	1.a	$1$	$1$	$1$	$5.350$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-5\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}-5q^{7}+q^{8}-3q^{9}+\cdots\)
670.2.a.e	$670$	$2$	670.a	1.a	$1$	$2$	$2$	$5.350$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
670.2.a.f	$670$	$2$	670.a	1.a	$1$	$2$	$2$	$5.350$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(2\)	\(-4\)	\(2\)	\(-6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-2+\beta )q^{3}+q^{4}+q^{5}+(-2+\cdots)q^{6}+\cdots\)
670.2.a.g	$670$	$2$	670.a	1.a	$1$	$3$	$3$	$5.350$	3.3.148.1	None	✓	$1$	$0$	\(-3\)	\(4\)	\(3\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
670.2.a.h	$670$	$2$	670.a	1.a	$1$	$3$	$3$	$5.350$	3.3.756.1	None	✓	$1$	$0$	\(3\)	\(0\)	\(-3\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
670.2.a.i	$670$	$2$	670.a	1.a	$1$	$3$	$3$	$5.350$	3.3.404.1	None	✓	$1$	$0$	\(3\)	\(2\)	\(3\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
670.2.a.j	$670$	$2$	670.a	1.a	$1$	$4$	$4$	$5.350$	4.4.15188.1	None	✓	$1$	$0$	\(-4\)	\(-2\)	\(-4\)	\(-5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta _{2})q^{3}+q^{4}-q^{5}+(1+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(670))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(670)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(335))\)\(^{\oplus 2}\)