Space of modular forms of level 209, weight 2, and Character 209.f

• Label: 209.2.f
• Level: $209$
• Weight: $2$
• Character orbit: 209.f
• Rep. character: $\chi_{209}(20,\cdot)$
• Character field: $\Q(\zeta_{5})$
• Dimension: $72$
• Newform subspaces: $3$
• Sturm bound: $40$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 209 = 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	209.f (of order \(5\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 11 \)
Character field:			\(\Q(\zeta_{5})\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(40\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(209, [\chi])\).

	Total	New	Old
Modular forms	88	72	16
Cusp forms	72	72	0
Eisenstein series	16	0	16

Trace form

72 * q - 4 * q^2 - 2 * q^3 - 22 * q^4 - 4 * q^5 - 4 * q^6 - 9 * q^7 + 8 * q^8 - 8 * q^9 - 24 * q^10 - q^11 + 16 * q^12 + 6 * q^13 - 2 * q^14 + 14 * q^15 - 10 * q^16 - 9 * q^17 + 14 * q^18 - 4 * q^19 - 24 * q^20 - 36 * q^21 + 24 * q^22 - 24 * q^23 + 8 * q^24 - 50 * q^25 + 4 * q^27 + 26 * q^28 - 24 * q^29 - 2 * q^31 - 8 * q^32 + 10 * q^33 - 8 * q^34 + 12 * q^35 - 54 * q^36 + 18 * q^37 - 12 * q^39 - 12 * q^40 + 2 * q^41 + 12 * q^42 - 26 * q^43 + 30 * q^44 + 82 * q^45 + 46 * q^46 + 28 * q^47 + 28 * q^48 - 47 * q^49 - 28 * q^50 + 44 * q^51 - 16 * q^52 + 72 * q^54 - 7 * q^55 + 4 * q^56 - 4 * q^57 + 26 * q^58 - 38 * q^59 - 60 * q^60 + 19 * q^61 + 50 * q^62 + 9 * q^63 - 54 * q^64 + 16 * q^65 + 34 * q^66 + 60 * q^67 + 6 * q^68 - 94 * q^69 - 52 * q^70 - 10 * q^71 + 100 * q^72 + 40 * q^73 + 12 * q^74 - 48 * q^75 + 40 * q^76 - 81 * q^77 - 84 * q^78 - 4 * q^79 + 24 * q^80 - 38 * q^81 - 24 * q^82 - 58 * q^83 + 88 * q^84 - 37 * q^85 - 36 * q^86 - 20 * q^87 + 80 * q^88 - 16 * q^89 - 12 * q^90 + 66 * q^91 + 20 * q^92 - 46 * q^93 - 48 * q^94 - 13 * q^95 - 88 * q^96 + 4 * q^97 + 24 * q^98 + 63 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(209, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
209.2.f.a	$209$	$2$	209.f	11.c	$5$	$4$	$1$	$1.669$	\(\Q(\zeta_{10})\)	None		✓			209.2.f.a	$2$	$1$	\(-5\)	\(-6\)	\(-2\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{5}]$	\(q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
209.2.f.b	$209$	$2$	209.f	11.c	$5$	$28$	$7$	$1.669$		None		✓			209.2.f.b	$2$	$0$	\(-2\)	\(1\)	\(7\)	\(-12\)		$\mathrm{SU}(2)[C_{5}]$
209.2.f.c	$209$	$2$	209.f	11.c	$5$	$40$	$10$	$1.669$		None		✓	✓		209.2.f.c	$2$	$0$	\(3\)	\(3\)	\(-9\)	\(4\)		$\mathrm{SU}(2)[C_{5}]$