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	Dim
\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(-\)	$-$	\(5\)
\(-\)	\(+\)	$-$	\(7\)
\(-\)	\(-\)	$+$	\(10\)
Plus space		\(+\)	\(15\)
Minus space		\(-\)	\(12\)

Trace form

\( 27 q + 4 q^{2} + 104 q^{4} + 8 q^{5} - 32 q^{7} + 3 q^{8} + O(q^{10}) \)

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

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

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