Space of modular forms of level 207, weight 8, and trivial character

• Label: 207.8.a
• Level: $207$
• Weight: $8$
• Character orbit: 207.a
• Rep. character: $\chi_{207}(1,\cdot)$
• Character field: $\Q$
• Dimension: $63$
• Newform subspaces: $8$
• Sturm bound: $192$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(192\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(207))\).

	Total	New	Old
Modular forms	172	63	109
Cusp forms	164	63	101
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
\(+\)	\(+\)	$+$	\(12\)
\(+\)	\(-\)	$-$	\(12\)
\(-\)	\(+\)	$-$	\(18\)
\(-\)	\(-\)	$+$	\(21\)
Plus space		\(+\)	\(33\)
Minus space		\(-\)	\(30\)

Trace form

\( 63 q + 3840 q^{4} + 388 q^{5} - 622 q^{7} - 2301 q^{8} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(207))\) 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
207.8.a.a	$207$	$8$	207.a	1.a	$1$	$5$	$5$	$64.664$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(266\)	\(-496\)		$-$	$+$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(54+2\beta _{1}+\beta _{3}+\beta _{4})q^{4}+\cdots\)
207.8.a.b	$207$	$8$	207.a	1.a	$1$	$5$	$5$	$64.664$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(16\)	\(0\)	\(56\)	\(-1156\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{2}+(51-7\beta _{1}-3\beta _{2}+\beta _{4})q^{4}+\cdots\)
207.8.a.c	$207$	$8$	207.a	1.a	$1$	$6$	$6$	$64.664$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(0\)	\(372\)	\(-1104\)		$-$	$-$	$+$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(29+3\beta _{1}+\beta _{2})q^{4}+\cdots\)
207.8.a.d	$207$	$8$	207.a	1.a	$1$	$7$	$7$	$64.664$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(516\)	\(1018\)		$-$	$-$	$+$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(93+\beta _{1}+\beta _{2})q^{4}+(74+10\beta _{1}+\cdots)q^{5}+\cdots\)
207.8.a.e	$207$	$8$	207.a	1.a	$1$	$8$	$8$	$64.664$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-24\)	\(0\)	\(-378\)	\(126\)		$-$	$+$	$-$	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+(70-4\beta _{1}+\beta _{2})q^{4}+\cdots\)
207.8.a.f	$207$	$8$	207.a	1.a	$1$	$8$	$8$	$64.664$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(0\)	\(-444\)	\(1446\)		$-$	$-$	$+$	$2^{4}\cdot 3^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(80+2\beta _{1}+\beta _{2})q^{4}+(-55+\cdots)q^{5}+\cdots\)
207.8.a.g	$207$	$8$	207.a	1.a	$1$	$12$	$12$	$64.664$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$1$	\(-16\)	\(0\)	\(-500\)	\(-228\)		$+$	$-$	$-$	$2^{9}\cdot 3^{7}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(53+2\beta _{1}+\beta _{2})q^{4}+\cdots\)
207.8.a.h	$207$	$8$	207.a	1.a	$1$	$12$	$12$	$64.664$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	$1$	$0$	\(16\)	\(0\)	\(500\)	\(-228\)		$+$	$+$	$+$	$2^{9}\cdot 3^{7}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(53+2\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(207))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(207)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 2}\)