Space of modular forms of level 57, weight 4, and Character 57.f

• Label: 57.4.f
• Level: $57$
• Weight: $4$
• Character orbit: 57.f
• Rep. character: $\chi_{57}(8,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $36$
• Newform subspaces: $3$
• Sturm bound: $26$
• Trace bound: $3$

Defining parameters

Level:	$N$	$=$	$57 = 3 \cdot 19$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	57.f (of order $6$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$57$
Character field:			$\Q(\zeta_{6})$
Newform subspaces:			$3$
Sturm bound:			$26$
Trace bound:			$3$
Distinguishing $T_p$:			$2$, $7$

Dimensions

The following table gives the dimensions of various subspaces of $M_{4}(57, [\chi])$.

	Total	New	Old
Modular forms	44	44	0
Cusp forms	36	36	0
Eisenstein series	8	8	0

Trace form

36 * q - 72 * q^4 + 25 * q^6 - 16 * q^7 + 14 * q^9 - 96 * q^10 + 24 * q^13 - 9 * q^15 - 324 * q^16 + 286 * q^19 - 84 * q^21 + 150 * q^22 + 403 * q^24 + 332 * q^25 - 106 * q^28 + 904 * q^30 + 447 * q^33 - 1032 * q^34 - 195 * q^36 - 1514 * q^39 + 12 * q^40 + 898 * q^42 - 1062 * q^43 - 1774 * q^45 - 21 * q^48 + 312 * q^49 + 1833 * q^51 - 1734 * q^52 + 1520 * q^54 - 352 * q^55 - 91 * q^57 + 3672 * q^58 - 2994 * q^60 - 748 * q^61 - 2074 * q^63 + 8256 * q^64 + 1409 * q^66 + 2160 * q^67 - 7536 * q^70 - 2877 * q^72 - 2646 * q^73 - 2192 * q^76 + 4572 * q^78 + 1230 * q^79 - 422 * q^81 + 2526 * q^82 - 1378 * q^85 + 1706 * q^87 + 3720 * q^90 + 6948 * q^91 + 20 * q^93 - 17958 * q^96 - 2616 * q^97 + 4069 * q^99

Decomposition of $S_{4}^{\mathrm{new}}(57, [\chi])$ 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
57.4.f.a	$57$	$4$	57.f	57.f	$6$	$2$	$1$	$3.363$	$\Q(\sqrt{-3})$	$\Q(\sqrt{-3})$		✓			57.4.f.a	$4$	$0$	$0$	$-9$	$0$	$74$	$1$	$\mathrm{U}(1)[D_{6}]$	$q+(-3-3\zeta_{6})q^{3}+8\zeta_{6}q^{4}+37q^{7}+\cdots$
57.4.f.b	$57$	$4$	57.f	57.f	$6$	$2$	$1$	$3.363$	$\Q(\sqrt{-3})$	$\Q(\sqrt{-3})$		✓			57.4.f.b	$4$	$0$	$0$	$9$	$0$	$-34$	$1$	$\mathrm{U}(1)[D_{6}]$	$q+(3+3\zeta_{6})q^{3}+8\zeta_{6}q^{4}-17q^{7}+3^{3}\zeta_{6}q^{9}+\cdots$
57.4.f.c	$57$	$4$	57.f	57.f	$6$	$32$	$16$	$3.363$		None		✓	✓		57.4.f.c	$4$	$0$	$0$	$0$	$0$	$-56$		$\mathrm{SU}(2)[C_{6}]$