Space of modular forms of level 63, weight 10, and trivial character

• Label: 63.10.a
• Level: $63$
• Weight: $10$
• Character orbit: 63.a
• Rep. character: $\chi_{63}(1,\cdot)$
• Character field: $\Q$
• Dimension: $23$
• Newform subspaces: $8$
• Sturm bound: $80$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	63.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(80\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(63))\).

	Total	New	Old
Modular forms	76	23	53
Cusp forms	68	23	45
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.

\(3\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(-\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(7\)
\(-\)	\(-\)	\(+\)	\(6\)
Plus space		\(+\)	\(10\)
Minus space		\(-\)	\(13\)

Trace form

\( 23 q - 17 q^{2} + 6913 q^{4} - 3868 q^{5} + 2401 q^{7} - 26061 q^{8} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(63))\) 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	7
63.10.a.a	$63$	$10$	63.a	1.a	$1$	$1$	$1$	$32.447$	\(\Q\)	None	✓				21.10.a.a	$1$	$1$	\(24\)	\(0\)	\(144\)	\(2401\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+24q^{2}+2^{6}q^{4}+12^{2}q^{5}+7^{4}q^{7}+\cdots\)
63.10.a.b	$63$	$10$	63.a	1.a	$1$	$2$	$2$	$32.447$	\(\Q(\sqrt{345}) \)	None	✓				21.10.a.c	$1$	$1$	\(-30\)	\(0\)	\(-1128\)	\(4802\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-15-\beta )q^{2}+(58+30\beta )q^{4}+(-564+\cdots)q^{5}+\cdots\)
63.10.a.c	$63$	$10$	63.a	1.a	$1$	$2$	$2$	$32.447$	\(\Q(\sqrt{2353}) \)	None	✓				21.10.a.b	$1$	$0$	\(-9\)	\(0\)	\(-1170\)	\(-4802\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{2}+(92+9\beta )q^{4}+(-620+\cdots)q^{5}+\cdots\)
63.10.a.d	$63$	$10$	63.a	1.a	$1$	$2$	$2$	$32.447$	\(\Q(\sqrt{193}) \)	None	✓				7.10.a.a	$1$	$0$	\(6\)	\(0\)	\(2238\)	\(-4802\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{2}+(-310+6\beta )q^{4}+(1119+\cdots)q^{5}+\cdots\)
63.10.a.e	$63$	$10$	63.a	1.a	$1$	$3$	$3$	$32.447$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		7.10.a.b	$1$	$1$	\(-21\)	\(0\)	\(-1554\)	\(7203\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-7+\beta _{2})q^{2}+(519+7\beta _{1}-8\beta _{2})q^{4}+\cdots\)
63.10.a.f	$63$	$10$	63.a	1.a	$1$	$3$	$3$	$32.447$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		21.10.a.d	$1$	$0$	\(13\)	\(0\)	\(-2398\)	\(-7203\)		$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1})q^{2}+(555+7\beta _{1}-5\beta _{2})q^{4}+\cdots\)
63.10.a.g	$63$	$10$	63.a	1.a	$1$	$4$	$4$	$32.447$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	63.10.a.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-9604\)		$+$	$+$	$+$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(446+\beta _{3})q^{4}+(-20\beta _{1}+\cdots)q^{5}+\cdots\)
63.10.a.h	$63$	$10$	63.a	1.a	$1$	$6$	$6$	$32.447$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	63.10.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(14406\)		$+$	$-$	$-$	$2^{5}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(357+\beta _{3})q^{4}+(24\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(63))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(63)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)