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

• Label: 209.2.e
• Level: $209$
• Weight: $2$
• Character orbit: 209.e
• Rep. character: $\chi_{209}(45,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $36$
• Newform subspaces: $2$
• Sturm bound: $40$
• Trace bound: $2$

Defining parameters

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

Dimensions

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

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

Trace form

36 * q - 4 * q^3 - 18 * q^4 + 4 * q^6 - 12 * q^7 - 12 * q^8 - 22 * q^9 + 10 * q^10 + 24 * q^12 + 10 * q^13 - 2 * q^14 + 6 * q^15 - 18 * q^16 + 8 * q^17 + 4 * q^18 + 10 * q^19 - 40 * q^20 - 2 * q^22 + 10 * q^23 + 14 * q^24 - 22 * q^25 - 24 * q^26 + 20 * q^27 + 6 * q^28 + 40 * q^30 - 16 * q^31 + 22 * q^32 - 4 * q^33 + 4 * q^34 - 6 * q^35 - 12 * q^36 - 8 * q^37 + 16 * q^39 - 12 * q^40 - 12 * q^41 + 2 * q^42 + 16 * q^43 - 12 * q^45 - 8 * q^46 - 16 * q^47 - 34 * q^48 - 8 * q^50 + 10 * q^51 + 28 * q^52 - 20 * q^53 + 16 * q^54 - 44 * q^56 - 50 * q^57 + 8 * q^58 - 6 * q^59 + 26 * q^60 - 12 * q^61 + 6 * q^62 + 26 * q^63 + 96 * q^64 - 16 * q^65 + 10 * q^66 - 16 * q^67 - 4 * q^68 + 48 * q^69 - 32 * q^70 - 20 * q^71 + 38 * q^72 - 8 * q^73 + 42 * q^74 + 100 * q^75 - 92 * q^76 - 6 * q^78 + 30 * q^79 + 54 * q^80 - 74 * q^81 + 38 * q^82 + 20 * q^83 - 104 * q^84 + 20 * q^85 + 22 * q^86 - 104 * q^87 - 12 * q^88 - 14 * q^89 + 26 * q^90 - 48 * q^91 + 42 * q^92 + 12 * q^93 - 112 * q^94 + 12 * q^95 - 40 * q^96 - 28 * q^97 - 40 * q^98

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.e.a	$209$	$2$	209.e	19.c	$3$	$18$	$9$	$1.669$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		✓			209.2.e.a	$2$	$0$	\(-1\)	\(-4\)	\(0\)	\(-6\)	$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{2}-\beta _{7}q^{3}+(-1-\beta _{4}-\beta _{9}+\cdots)q^{4}+\cdots\)
209.2.e.b	$209$	$2$	209.e	19.c	$3$	$18$	$9$	$1.669$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		✓			209.2.e.b	$2$	$0$	\(1\)	\(0\)	\(0\)	\(-6\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(\beta _{4}-\beta _{12})q^{3}+(-1+\beta _{3}+\cdots)q^{4}+\cdots\)