Space of modular forms of level 170, weight 4, and trivial character

• Label: 170.4.a
• Level: $170$
• Weight: $4$
• Character orbit: 170.a
• Rep. character: $\chi_{170}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $8$
• Sturm bound: $108$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 170 = 2 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	170.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(108\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(170))\).

	Total	New	Old
Modular forms	86	16	70
Cusp forms	78	16	62
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.

\(2\)	\(5\)	\(17\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(12\)	\(3\)	\(9\)		\(11\)	\(3\)	\(8\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(9\)	\(1\)	\(8\)		\(8\)	\(1\)	\(7\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(10\)	\(2\)	\(8\)		\(9\)	\(2\)	\(7\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(11\)	\(2\)	\(9\)		\(10\)	\(2\)	\(8\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(10\)	\(2\)	\(8\)		\(9\)	\(2\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(12\)	\(3\)	\(9\)		\(11\)	\(3\)	\(8\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(11\)	\(3\)	\(8\)		\(10\)	\(3\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(11\)	\(0\)	\(11\)		\(10\)	\(0\)	\(10\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(46\)	\(11\)	\(35\)		\(42\)	\(11\)	\(31\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(40\)	\(5\)	\(35\)		\(36\)	\(5\)	\(31\)		\(4\)	\(0\)	\(4\)

Trace form

16 * q + 12 * q^3 + 64 * q^4 - 10 * q^5 - 8 * q^6 + 8 * q^7 + 172 * q^9 - 20 * q^10 - 28 * q^11 + 48 * q^12 + 200 * q^13 + 96 * q^14 + 256 * q^16 - 68 * q^17 - 112 * q^18 + 460 * q^19 - 40 * q^20 + 232 * q^21 - 88 * q^22 - 152 * q^23 - 32 * q^24 + 400 * q^25 + 152 * q^26 - 192 * q^27 + 32 * q^28 - 452 * q^29 + 280 * q^30 + 168 * q^31 + 56 * q^33 + 260 * q^35 + 688 * q^36 + 244 * q^37 - 64 * q^38 + 72 * q^39 - 80 * q^40 - 256 * q^41 + 464 * q^42 - 48 * q^43 - 112 * q^44 - 50 * q^45 - 192 * q^46 + 696 * q^47 + 192 * q^48 + 1368 * q^49 + 800 * q^52 + 560 * q^53 + 352 * q^54 + 384 * q^56 - 320 * q^57 - 408 * q^58 - 2436 * q^59 - 2372 * q^61 - 1056 * q^62 - 2688 * q^63 + 1024 * q^64 - 700 * q^65 - 2032 * q^66 - 1136 * q^67 - 272 * q^68 + 528 * q^69 + 360 * q^70 - 1576 * q^71 - 448 * q^72 - 2648 * q^73 - 1080 * q^74 + 300 * q^75 + 1840 * q^76 - 2040 * q^77 - 1072 * q^78 + 2088 * q^79 - 160 * q^80 + 1592 * q^81 - 2160 * q^82 + 968 * q^83 + 928 * q^84 - 170 * q^85 + 1744 * q^86 - 3568 * q^87 - 352 * q^88 - 3460 * q^89 - 740 * q^90 - 1072 * q^91 - 608 * q^92 - 2384 * q^93 - 648 * q^94 - 128 * q^96 + 728 * q^97 + 1264 * q^98 + 4100 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(170))\) 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
170.4.a.a	$170$	$4$	170.a	1.a	$1$	$1$	$1$	$10.030$	\(\Q\)	None	✓	✓	✓	✓	170.4.a.a	$1$	$1$	\(-2\)	\(4\)	\(-5\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{3}+4q^{4}-5q^{5}-8q^{6}+\cdots\)
170.4.a.b	$170$	$4$	170.a	1.a	$1$	$1$	$1$	$10.030$	\(\Q\)	None	✓	✓			170.4.a.b	$1$	$0$	\(2\)	\(7\)	\(5\)	\(-10\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+7q^{3}+4q^{4}+5q^{5}+14q^{6}+\cdots\)
170.4.a.c	$170$	$4$	170.a	1.a	$1$	$2$	$2$	$10.030$	\(\Q(\sqrt{43}) \)	None	✓	✓	✓	✓	170.4.a.c	$1$	$1$	\(-4\)	\(-6\)	\(10\)	\(-34\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-3+\beta )q^{3}+4q^{4}+5q^{5}+\cdots\)
170.4.a.d	$170$	$4$	170.a	1.a	$1$	$2$	$2$	$10.030$	\(\Q(\sqrt{145}) \)	None	✓	✓	✓	✓	170.4.a.d	$1$	$0$	\(-4\)	\(3\)	\(10\)	\(28\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(1+\beta )q^{3}+4q^{4}+5q^{5}+(-2+\cdots)q^{6}+\cdots\)
170.4.a.e	$170$	$4$	170.a	1.a	$1$	$2$	$2$	$10.030$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓	✓	170.4.a.e	$1$	$1$	\(4\)	\(-4\)	\(-10\)	\(-28\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-2+\beta )q^{3}+4q^{4}-5q^{5}+\cdots\)
170.4.a.f	$170$	$4$	170.a	1.a	$1$	$2$	$2$	$10.030$	\(\Q(\sqrt{19}) \)	None	✓	✓	✓		170.4.a.f	$1$	$0$	\(4\)	\(2\)	\(10\)	\(46\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(1+\beta )q^{3}+4q^{4}+5q^{5}+(2+\cdots)q^{6}+\cdots\)
170.4.a.g	$170$	$4$	170.a	1.a	$1$	$3$	$3$	$10.030$	3.3.701288.1	None	✓	✓	✓	✓	170.4.a.g	$1$	$0$	\(-6\)	\(7\)	\(-15\)	\(-10\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(2+\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)
170.4.a.h	$170$	$4$	170.a	1.a	$1$	$3$	$3$	$10.030$	3.3.62013.1	None	✓	✓	✓	✓	170.4.a.h	$1$	$0$	\(6\)	\(-1\)	\(-15\)	\(20\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+\beta _{1}q^{3}+4q^{4}-5q^{5}+2\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(170))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(170)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 2}\)