Space of modular forms of level 209, weight 4, and trivial character

• Label: 209.4.a
• Level: $209$
• Weight: $4$
• Character orbit: 209.a
• Rep. character: $\chi_{209}(1,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $4$
• Sturm bound: $80$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 209 = 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	209.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(80\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(209))\).

	Total	New	Old
Modular forms	62	46	16
Cusp forms	58	46	12
Eisenstein series	4	0	4

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(19\)	\(13\)	\(6\)		\(18\)	\(13\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(12\)	\(8\)	\(4\)		\(11\)	\(8\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(14\)	\(10\)	\(4\)		\(13\)	\(10\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(17\)	\(15\)	\(2\)		\(16\)	\(15\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(36\)	\(28\)	\(8\)		\(34\)	\(28\)	\(6\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(26\)	\(18\)	\(8\)		\(24\)	\(18\)	\(6\)		\(2\)	\(0\)	\(2\)

Trace form

46 * q - 4 * q^2 + 4 * q^3 + 192 * q^4 - 4 * q^5 + 76 * q^6 - 60 * q^8 + 362 * q^9 + 8 * q^10 + 44 * q^11 + 128 * q^12 - 168 * q^13 + 28 * q^14 - 76 * q^15 + 888 * q^16 + 88 * q^17 - 692 * q^18 - 136 * q^20 + 64 * q^21 + 160 * q^23 + 172 * q^24 + 850 * q^25 - 460 * q^26 + 628 * q^27 + 864 * q^28 - 240 * q^29 + 1608 * q^30 + 212 * q^31 - 604 * q^32 - 176 * q^33 - 768 * q^34 + 56 * q^35 + 396 * q^36 + 704 * q^37 + 228 * q^38 - 180 * q^39 + 1368 * q^40 - 1784 * q^41 - 420 * q^42 + 760 * q^43 - 176 * q^44 - 1144 * q^45 + 2328 * q^46 + 568 * q^47 + 1088 * q^48 + 3342 * q^49 + 2692 * q^50 + 464 * q^51 + 1856 * q^52 + 276 * q^53 + 4 * q^54 + 44 * q^55 - 268 * q^56 - 1676 * q^58 - 1236 * q^59 - 3876 * q^60 - 1736 * q^61 - 2356 * q^62 + 1252 * q^63 + 5488 * q^64 + 1520 * q^65 + 528 * q^66 + 588 * q^67 - 2868 * q^68 - 2772 * q^69 - 6700 * q^70 - 76 * q^71 - 10460 * q^72 + 816 * q^73 - 2148 * q^74 - 1872 * q^75 + 440 * q^77 + 1212 * q^78 + 1088 * q^79 + 2576 * q^80 + 8958 * q^81 - 1760 * q^82 - 1008 * q^83 - 4448 * q^84 - 2008 * q^85 - 7168 * q^86 - 2948 * q^87 + 472 * q^89 - 5480 * q^90 - 208 * q^91 + 508 * q^92 + 2644 * q^93 - 1048 * q^94 - 6708 * q^96 - 1464 * q^97 + 2700 * q^98 + 968 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(209))\) 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
209.4.a.a	$209$	$4$	209.a	1.a	$1$	$8$	$8$	$12.331$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	209.4.a.a	$1$	$1$	\(0\)	\(-1\)	\(-12\)	\(-59\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{3}q^{3}+(3+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
209.4.a.b	$209$	$4$	209.a	1.a	$1$	$10$	$10$	$12.331$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓	✓	209.4.a.b	$1$	$1$	\(-6\)	\(-9\)	\(-10\)	\(-53\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1+\beta _{4})q^{3}+(2+\cdots)q^{4}+\cdots\)
209.4.a.c	$209$	$4$	209.a	1.a	$1$	$13$	$13$	$12.331$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	✓	✓	✓	209.4.a.c	$1$	$0$	\(-2\)	\(11\)	\(8\)	\(39\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{5})q^{3}+(6+\beta _{2})q^{4}+\cdots\)
209.4.a.d	$209$	$4$	209.a	1.a	$1$	$15$	$15$	$12.331$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	✓	✓	✓	209.4.a.d	$1$	$0$	\(4\)	\(3\)	\(10\)	\(73\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(5+\beta _{2})q^{4}+(1+\beta _{6}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(209))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(209)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 2}\)