Space of modular forms of level 209, weight 2, and trivial character

• Label: 209.2.a
• Level: $209$
• Weight: $2$
• Character orbit: 209.a
• Rep. character: $\chi_{209}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $4$
• Sturm bound: $40$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 209 = 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	209.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(40\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(209))\).

	Total	New	Old
Modular forms	22	15	7
Cusp forms	19	15	4
Eisenstein series	3	0	3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(11\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(7\)
\(-\)	\(+\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(1\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(12\)

Trace form

15 * q + q^2 + 2 * q^3 + 19 * q^4 - 8 * q^5 - 8 * q^6 + 8 * q^7 - 3 * q^8 + 13 * q^9 + 6 * q^10 - 3 * q^11 - 12 * q^12 - 2 * q^13 - 12 * q^14 + 14 * q^15 + 31 * q^16 + 2 * q^17 - 3 * q^18 + q^19 - 6 * q^20 + 3 * q^22 + 10 * q^23 - 8 * q^24 + 11 * q^25 - 2 * q^26 - 22 * q^27 + 24 * q^28 - 18 * q^29 - 32 * q^30 + 18 * q^31 - 35 * q^32 + 2 * q^33 - 6 * q^34 + 16 * q^35 + 7 * q^36 - 3 * q^38 + 20 * q^39 - 18 * q^40 - 2 * q^41 - 48 * q^42 + 24 * q^43 - 11 * q^44 - 26 * q^45 - 8 * q^46 - 56 * q^48 + 27 * q^49 - 65 * q^50 - 8 * q^51 - 58 * q^52 + 2 * q^53 - 32 * q^54 - 8 * q^55 - 4 * q^56 + 4 * q^57 - 26 * q^58 - 10 * q^59 + 8 * q^60 - 14 * q^61 + 4 * q^62 + 24 * q^63 + 63 * q^64 - 16 * q^65 + 4 * q^66 + 10 * q^67 + 6 * q^68 - 2 * q^69 - 12 * q^70 + 26 * q^71 + 25 * q^72 + 10 * q^73 + 54 * q^74 + 20 * q^75 + 7 * q^76 - 4 * q^77 - 4 * q^78 + 48 * q^79 - 34 * q^80 - q^81 - 22 * q^82 - 20 * q^83 + 16 * q^84 - 28 * q^85 + 44 * q^86 + 44 * q^87 + 15 * q^88 - 28 * q^89 + 38 * q^90 + 24 * q^91 + 52 * q^92 - 18 * q^93 + 24 * q^94 + 6 * q^95 + 56 * q^96 - 8 * q^97 + 25 * q^98 - 9 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(209))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		11	19
209.2.a.a	$209$	$2$	209.a	1.a	$1$	$1$	$1$	$1.669$	\(\Q\)	None	✓	✓	✓	✓	209.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-3q^{5}-4q^{7}-2q^{9}+\cdots\)
209.2.a.b	$209$	$2$	209.a	1.a	$1$	$2$	$2$	$1.669$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	209.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1-\beta )q^{3}-q^{5}+(-2-\beta )q^{6}+\cdots\)
209.2.a.c	$209$	$2$	209.a	1.a	$1$	$5$	$5$	$1.669$	5.5.246832.1	None	✓	✓	✓	✓	209.2.a.c	$1$	$0$	\(2\)	\(1\)	\(-5\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4})q^{4}+\cdots\)
209.2.a.d	$209$	$2$	209.a	1.a	$1$	$7$	$7$	$1.669$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	209.2.a.d	$1$	$0$	\(-1\)	\(2\)	\(2\)	\(10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{2}q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(209))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(209)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)