Space of modular forms of level 207, weight 6, and trivial character

• Label: 207.6.a
• Level: $207$
• Weight: $6$
• Character orbit: 207.a
• Rep. character: $\chi_{207}(1,\cdot)$
• Character field: $\Q$
• Dimension: $47$
• Newform subspaces: $9$
• Sturm bound: $144$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(144\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(207))\).

	Total	New	Old
Modular forms	124	47	77
Cusp forms	116	47	69
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
\(+\)	\(+\)	$+$	\(10\)
\(+\)	\(-\)	$-$	\(10\)
\(-\)	\(+\)	$-$	\(15\)
\(-\)	\(-\)	$+$	\(12\)
Plus space		\(+\)	\(22\)
Minus space		\(-\)	\(25\)

Trace form

\( 47 q - 8 q^{2} + 800 q^{4} - 16 q^{5} + 98 q^{7} + 75 q^{8} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(207))\) 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
207.6.a.a	$207$	$6$	207.a	1.a	$1$	$2$	$2$	$33.199$	\(\Q(\sqrt{29}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-94\)	\(-118\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-2\beta q^{2}+84q^{4}+(-47-\beta )q^{5}+(-59+\cdots)q^{7}+\cdots\)
207.6.a.b	$207$	$6$	207.a	1.a	$1$	$3$	$3$	$33.199$	3.3.7925.1	None	✓	$1$	$1$	\(4\)	\(0\)	\(58\)	\(-282\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-6\beta _{1}+4\beta _{2})q^{4}+\cdots\)
207.6.a.c	$207$	$6$	207.a	1.a	$1$	$3$	$3$	$33.199$	3.3.5333.1	None	✓	$1$	$0$	\(8\)	\(0\)	\(56\)	\(-114\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3-\beta _{2})q^{2}+(9-4\beta _{1}-9\beta _{2})q^{4}+\cdots\)
207.6.a.d	$207$	$6$	207.a	1.a	$1$	$4$	$4$	$33.199$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-4\)	\(0\)	\(122\)	\(62\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(-12+4\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
207.6.a.e	$207$	$6$	207.a	1.a	$1$	$4$	$4$	$33.199$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(-22\)	\(-62\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(7-\beta _{1}+\beta _{3})q^{4}+\cdots\)
207.6.a.f	$207$	$6$	207.a	1.a	$1$	$5$	$5$	$33.199$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-8\)	\(0\)	\(-94\)	\(272\)		$-$	$-$	$+$	$2^{5}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{2})q^{2}+(24-\beta _{2}+\beta _{3})q^{4}+\cdots\)
207.6.a.g	$207$	$6$	207.a	1.a	$1$	$6$	$6$	$33.199$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-4\)	\(0\)	\(-42\)	\(300\)		$-$	$+$	$-$	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(20-2\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
207.6.a.h	$207$	$6$	207.a	1.a	$1$	$10$	$10$	$33.199$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$1$	\(-8\)	\(0\)	\(-100\)	\(20\)		$+$	$+$	$+$	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(19-\beta _{1}+\beta _{2})q^{4}+\cdots\)
207.6.a.i	$207$	$6$	207.a	1.a	$1$	$10$	$10$	$33.199$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(0\)	\(100\)	\(20\)		$+$	$-$	$-$	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(19-\beta _{1}+\beta _{2})q^{4}+(9+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(207))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(207)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 2}\)