Space of modular forms of level 207, weight 4, and trivial character

• Label: 207.4.a
• Level: $207$
• Weight: $4$
• Character orbit: 207.a
• Rep. character: $\chi_{207}(1,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $8$
• Sturm bound: $96$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(96\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(207))\).

	Total	New	Old
Modular forms	76	27	49
Cusp forms	68	27	41
Eisenstein series	8	0	8

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

\(3\)	\(23\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(22\)	\(5\)	\(17\)		\(20\)	\(5\)	\(15\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(16\)	\(5\)	\(11\)		\(14\)	\(5\)	\(9\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(19\)	\(7\)	\(12\)		\(17\)	\(7\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(19\)	\(10\)	\(9\)		\(17\)	\(10\)	\(7\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(41\)	\(15\)	\(26\)		\(37\)	\(15\)	\(22\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(35\)	\(12\)	\(23\)		\(31\)	\(12\)	\(19\)		\(4\)	\(0\)	\(4\)

Trace form

27 * q + 4 * q^2 + 104 * q^4 + 8 * q^5 - 32 * q^7 + 3 * q^8 + 62 * q^10 - 42 * q^11 + 98 * q^13 - 28 * q^14 + 256 * q^16 - 114 * q^17 - 86 * q^19 + 472 * q^20 - 112 * q^22 + 69 * q^23 + 993 * q^25 + 411 * q^26 - 62 * q^28 + 198 * q^29 - 610 * q^31 - 252 * q^32 + 202 * q^34 - 200 * q^35 - 252 * q^37 - 692 * q^38 - 602 * q^40 + 766 * q^41 + 794 * q^43 + 106 * q^44 - 92 * q^46 + 666 * q^47 + 1283 * q^49 + 500 * q^50 - 1165 * q^52 - 1204 * q^53 - 1320 * q^55 + 1066 * q^56 + 195 * q^58 - 508 * q^59 - 1524 * q^61 + 627 * q^62 + 2275 * q^64 - 1836 * q^65 + 246 * q^67 - 3128 * q^68 + 524 * q^70 - 1142 * q^71 - 3978 * q^73 - 2658 * q^74 + 1616 * q^76 + 640 * q^77 + 104 * q^79 + 6102 * q^80 - 11 * q^82 - 2114 * q^83 - 704 * q^85 + 702 * q^86 - 1676 * q^88 - 2058 * q^89 + 3788 * q^91 + 552 * q^92 - 767 * q^94 + 3128 * q^95 + 3574 * q^97 - 968 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(207))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	23
207.4.a.a	$207$	$4$	207.a	1.a	$1$	$1$	$1$	$12.213$	\(\Q\)	None	✓				23.4.a.a	$1$	$1$	\(2\)	\(0\)	\(6\)	\(-8\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-4q^{4}+6q^{5}-8q^{7}-24q^{8}+\cdots\)
207.4.a.b	$207$	$4$	207.a	1.a	$1$	$2$	$2$	$12.213$	\(\Q(\sqrt{2}) \)	None	✓				69.4.a.b	$1$	$1$	\(2\)	\(0\)	\(-8\)	\(-28\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(-4-4\beta )q^{5}+\cdots\)
207.4.a.c	$207$	$4$	207.a	1.a	$1$	$2$	$2$	$12.213$	\(\Q(\sqrt{5}) \)	None	✓				69.4.a.a	$1$	$0$	\(4\)	\(0\)	\(26\)	\(-10\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{2}+(1+4\beta )q^{4}+(13-\beta )q^{5}+\cdots\)
207.4.a.d	$207$	$4$	207.a	1.a	$1$	$4$	$4$	$12.213$	4.4.2009704.1	None	✓		✓		69.4.a.d	$1$	$1$	\(-4\)	\(0\)	\(-4\)	\(-14\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(7-2\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
207.4.a.e	$207$	$4$	207.a	1.a	$1$	$4$	$4$	$12.213$	4.4.334189.1	None	✓				23.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(-14\)	\(16\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{3})q^{2}+(6-3\beta _{1}-2\beta _{2}-2\beta _{3})q^{4}+\cdots\)
207.4.a.f	$207$	$4$	207.a	1.a	$1$	$4$	$4$	$12.213$	4.4.1140200.1	None	✓				69.4.a.c	$1$	$0$	\(2\)	\(0\)	\(2\)	\(32\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(6-\beta _{1}-2\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
207.4.a.g	$207$	$4$	207.a	1.a	$1$	$5$	$5$	$12.213$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	207.4.a.g	$1$	$1$	\(-4\)	\(0\)	\(-20\)	\(-10\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(3-\beta _{1}+\beta _{2})q^{4}+\cdots\)
207.4.a.h	$207$	$4$	207.a	1.a	$1$	$5$	$5$	$12.213$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	207.4.a.g	$1$	$0$	\(4\)	\(0\)	\(20\)	\(-10\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(3-\beta _{1}+\beta _{2})q^{4}+(4+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(207))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(207)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 2}\)