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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)		\(0\)	\(0\)	\(0\)
\(+\)	\(+\)	\(-\)	\(-\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(4\)	\(1\)	\(3\)		\(3\)	\(1\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(12\)	\(1\)	\(11\)		\(9\)	\(1\)	\(8\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(18\)	\(6\)	\(12\)		\(14\)	\(6\)	\(8\)		\(4\)	\(0\)	\(4\)

Trace form

7 * q - q^2 + 7 * q^4 + q^5 + 8 * q^7 - q^8 + 7 * q^9 + q^10 - 8 * q^11 + 2 * q^13 - 4 * q^15 + 7 * q^16 + 3 * q^17 - 5 * q^18 + q^20 - 8 * q^21 - 8 * q^22 - 16 * q^23 + 7 * q^25 + 6 * q^26 + 8 * q^28 - 2 * q^29 - 8 * q^31 - q^32 - 16 * q^33 - q^34 - 4 * q^35 + 7 * q^36 - 2 * q^37 - 4 * q^38 - 8 * q^39 + q^40 - 18 * q^41 - 24 * q^42 - 4 * q^43 - 8 * q^44 - 3 * q^45 + 7 * q^49 - q^50 - 4 * q^51 + 2 * q^52 - 22 * q^53 - 24 * q^54 - 12 * q^55 + 16 * q^57 - 2 * q^58 + 8 * q^59 - 4 * q^60 + 22 * q^61 + 24 * q^63 + 7 * q^64 + 6 * q^65 + 24 * q^66 + 20 * q^67 + 3 * q^68 + 16 * q^69 + 4 * q^70 + 8 * q^71 - 5 * q^72 + 38 * q^73 + 6 * q^74 - 16 * q^78 - 8 * q^79 + q^80 - 25 * q^81 + 14 * q^82 + 12 * q^83 - 8 * q^84 + q^85 - 4 * q^86 + 40 * q^87 - 8 * q^88 + 2 * q^89 + 13 * q^90 + 40 * q^91 - 16 * q^92 + 8 * q^93 + 12 * q^94 - 20 * q^95 - 10 * q^97 + 31 * q^98 - 32 * q^99

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

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

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