Space of modular forms of level 171, weight 4, and Character 171.f

• Label: 171.4.f
• Level: $171$
• Weight: $4$
• Character orbit: 171.f
• Rep. character: $\chi_{171}(64,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $48$
• Newform subspaces: $8$
• Sturm bound: $80$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	171.f (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(80\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(171, [\chi])\).

	Total	New	Old
Modular forms	128	52	76
Cusp forms	112	48	64
Eisenstein series	16	4	12

Trace form

\( 48 q + 3 q^{2} - 99 q^{4} - 3 q^{5} - 52 q^{7} - 102 q^{8} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(171, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
171.4.f.a	$171$	$4$	171.f	19.c	$3$	$2$	$1$	$10.089$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$1$	\(0\)	\(0\)	\(0\)	\(-74\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+8\zeta_{6}q^{4}-37q^{7}-89\zeta_{6}q^{13}+(-2^{6}+\cdots)q^{16}+\cdots\)
171.4.f.b	$171$	$4$	171.f	19.c	$3$	$2$	$1$	$10.089$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(34\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+8\zeta_{6}q^{4}+17q^{7}+19\zeta_{6}q^{13}+(-2^{6}+\cdots)q^{16}+\cdots\)
171.4.f.c	$171$	$4$	171.f	19.c	$3$	$4$	$2$	$10.089$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(60\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}+(-5+5\beta _{1})q^{4}+2\beta _{2}q^{5}+\cdots\)
171.4.f.d	$171$	$4$	171.f	19.c	$3$	$4$	$2$	$10.089$	\(\Q(\sqrt{-3}, \sqrt{73})\)	None		$2$	$0$	\(1\)	\(0\)	\(19\)	\(40\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(-11+\beta _{1}+10\beta _{2}+\beta _{3})q^{4}+\cdots\)
171.4.f.e	$171$	$4$	171.f	19.c	$3$	$4$	$2$	$10.089$	\(\Q(\sqrt{-3}, \sqrt{55})\)	None		$2$	$0$	\(2\)	\(0\)	\(-14\)	\(-28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{2}+(7+7\beta _{2})q^{4}+(\beta _{1}+7\beta _{2}+\cdots)q^{5}+\cdots\)
171.4.f.f	$171$	$4$	171.f	19.c	$3$	$10$	$5$	$10.089$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(-1\)	\(0\)	\(-12\)	\(-54\)	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}+(-\beta _{1}-\beta _{2}+6\beta _{4}+\beta _{7}+\cdots)q^{4}+\cdots\)
171.4.f.g	$171$	$4$	171.f	19.c	$3$	$10$	$5$	$10.089$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None		$2$	$0$	\(1\)	\(0\)	\(4\)	\(30\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(-4-4\beta _{5}+\beta _{7})q^{4}+(-\beta _{5}+\cdots)q^{5}+\cdots\)
171.4.f.h	$171$	$4$	171.f	19.c	$3$	$12$	$6$	$10.089$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(-60\)	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(-9+9\beta _{2}+\beta _{4}+\beta _{6})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(171, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(171, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(57, [\chi])\)\(^{\oplus 2}\)