Space of modular forms of level 529, weight 8, and trivial character

• Label: 529.8.a
• Level: $529$
• Weight: $8$
• Character orbit: 529.a
• Rep. character: $\chi_{529}(1,\cdot)$
• Character field: $\Q$
• Dimension: $284$
• Newform subspaces: $11$
• Sturm bound: $368$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 529 = 23^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	529.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(368\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(529))\).

	Total	New	Old
Modular forms	334	305	29
Cusp forms	310	284	26
Eisenstein series	24	21	3

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

\(23\)	Dim
\(+\)	\(145\)
\(-\)	\(139\)

Trace form

\( 284 q + 16 q^{2} + 28 q^{3} + 17408 q^{4} - 388 q^{5} + 1120 q^{6} - 290 q^{7} + 4074 q^{8} + 188792 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(529))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		23
529.8.a.a	$529$	$8$	529.a	1.a	$1$	$3$	$3$	$165.252$	3.3.621.1	\(\Q(\sqrt{-23}) \)	✓	✓			529.8.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+(-5\beta _{1}-6\beta _{2})q^{2}+(24\beta _{1}+23\beta _{2})q^{3}+\cdots\)
529.8.a.b	$529$	$8$	529.a	1.a	$1$	$5$	$5$	$165.252$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				23.8.a.a	$1$	$1$	\(-16\)	\(-68\)	\(56\)	\(1156\)		$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+(-14+\beta _{2})q^{3}+(51+\cdots)q^{4}+\cdots\)
529.8.a.c	$529$	$8$	529.a	1.a	$1$	$8$	$8$	$165.252$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				23.8.a.b	$1$	$1$	\(0\)	\(40\)	\(-444\)	\(-1446\)		$-$	$-$	$2^{4}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(5-\beta _{1}+\beta _{2})q^{3}+(80+2\beta _{1}+\cdots)q^{4}+\cdots\)
529.8.a.d	$529$	$8$	529.a	1.a	$1$	$10$	$10$	$165.252$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			529.8.a.d	$2$	$1$	\(0\)	\(-80\)	\(0\)	\(0\)		$-$	$-$	$2^{13}\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(-8+2\beta _{2}-\beta _{5})q^{3}+(51+\cdots)q^{4}+\cdots\)
529.8.a.e	$529$	$8$	529.a	1.a	$1$	$12$	$12$	$165.252$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			529.8.a.e	$1$	$1$	\(-16\)	\(-40\)	\(-194\)	\(612\)		$-$	$-$	$2^{9}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(-3+\beta _{2})q^{3}+(53+\cdots)q^{4}+\cdots\)
529.8.a.f	$529$	$8$	529.a	1.a	$1$	$12$	$12$	$165.252$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			529.8.a.e	$1$	$1$	\(-16\)	\(-40\)	\(194\)	\(-612\)		$-$	$-$	$2^{9}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(-3+\beta _{2})q^{3}+(53+\cdots)q^{4}+\cdots\)
529.8.a.g	$529$	$8$	529.a	1.a	$1$	$24$	$24$	$165.252$		None	✓	✓			529.8.a.g	$2$	$1$	\(0\)	\(-28\)	\(0\)	\(0\)		$-$	$-$		$\mathrm{SU}(2)$
529.8.a.h	$529$	$8$	529.a	1.a	$1$	$28$	$28$	$165.252$		None	✓	✓			529.8.a.h	$2$	$0$	\(16\)	\(108\)	\(0\)	\(0\)		$+$	$+$		$\mathrm{SU}(2)$
529.8.a.i	$529$	$8$	529.a	1.a	$1$	$52$	$52$	$165.252$		None	✓	✓			529.8.a.i	$2$	$0$	\(32\)	\(108\)	\(0\)	\(0\)		$+$	$+$		$\mathrm{SU}(2)$
529.8.a.j	$529$	$8$	529.a	1.a	$1$	$65$	$65$	$165.252$		None	✓		✓		23.8.c.a	$1$	$1$	\(8\)	\(14\)	\(-1181\)	\(-3628\)		$-$	$-$		$\mathrm{SU}(2)$
529.8.a.k	$529$	$8$	529.a	1.a	$1$	$65$	$65$	$165.252$		None	✓		✓		23.8.c.a	$1$	$0$	\(8\)	\(14\)	\(1181\)	\(3628\)		$+$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(529))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(529)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)