Space of modular forms of level 170, weight 2, and trivial character

• Label: 170.2.a
• Level: $170$
• Weight: $2$
• Character orbit: 170.a
• Rep. character: $\chi_{170}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $6$
• Sturm bound: $54$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 170 = 2 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	170.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(54\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(170))\).

	Total	New	Old
Modular forms	30	7	23
Cusp forms	23	7	16
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.

\(2\)	\(5\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	$+$	\(1\)
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(2\)
Plus space			\(+\)	\(1\)
Minus space			\(-\)	\(6\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(170))\) 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	17
170.2.a.a	$170$	$2$	170.a	1.a	$1$	$1$	$1$	$1.357$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(-2\)	\(-1\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}-q^{5}+2q^{6}+2q^{7}+\cdots\)
170.2.a.b	$170$	$2$	170.a	1.a	$1$	$1$	$1$	$1.357$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-2\)	\(1\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-2q^{7}+\cdots\)
170.2.a.c	$170$	$2$	170.a	1.a	$1$	$1$	$1$	$1.357$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(1\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+2q^{7}+\cdots\)
170.2.a.d	$170$	$2$	170.a	1.a	$1$	$1$	$1$	$1.357$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(3\)	\(-1\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}+q^{4}-q^{5}-3q^{6}+2q^{7}+\cdots\)
170.2.a.e	$170$	$2$	170.a	1.a	$1$	$1$	$1$	$1.357$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
170.2.a.f	$170$	$2$	170.a	1.a	$1$	$2$	$2$	$1.357$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(2\)	\(-1\)	\(2\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta q^{3}+q^{4}+q^{5}-\beta q^{6}+2\beta q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(170))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(170)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 2}\)