Space of modular forms of level 63, weight 6, and trivial character

• Label: 63.6.a
• Level: $63$
• Weight: $6$
• Character orbit: 63.a
• Rep. character: $\chi_{63}(1,\cdot)$
• Character field: $\Q$
• Dimension: $13$
• Newform subspaces: $8$
• Sturm bound: $48$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(48\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(63))\).

	Total	New	Old
Modular forms	44	13	31
Cusp forms	36	13	23
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
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(4\)
\(-\)	\(-\)	\(+\)	\(3\)
Plus space		\(+\)	\(5\)
Minus space		\(-\)	\(8\)

Trace form

13 * q - 9 * q^2 + 193 * q^4 + 42 * q^5 + 49 * q^7 + 99 * q^8 - 156 * q^10 - 876 * q^11 - 742 * q^13 + 147 * q^14 + 4645 * q^16 + 3090 * q^17 + 4868 * q^19 + 7788 * q^20 + 1788 * q^22 - 3168 * q^23 + 11083 * q^25 - 9372 * q^26 + 637 * q^28 - 3066 * q^29 - 9976 * q^31 - 741 * q^32 - 13230 * q^34 - 7938 * q^35 + 3902 * q^37 + 29862 * q^38 - 27144 * q^40 + 6138 * q^41 + 5276 * q^43 - 3168 * q^44 - 43020 * q^46 - 9840 * q^47 + 31213 * q^49 - 70131 * q^50 - 67084 * q^52 + 31110 * q^53 + 17856 * q^55 - 20433 * q^56 - 39414 * q^58 + 40092 * q^59 - 63190 * q^61 - 157404 * q^62 + 7201 * q^64 + 146412 * q^65 + 158996 * q^67 + 108798 * q^68 + 37044 * q^70 - 42864 * q^71 + 39146 * q^73 - 194946 * q^74 + 210194 * q^76 - 39396 * q^77 - 103504 * q^79 + 491748 * q^80 - 178806 * q^82 - 90108 * q^83 + 206004 * q^85 + 216840 * q^86 + 460572 * q^88 - 17070 * q^89 + 44198 * q^91 - 314760 * q^92 - 758436 * q^94 - 467832 * q^95 + 98282 * q^97 - 21609 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(63))\) 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
63.6.a.a	$63$	$6$	63.a	1.a	$1$	$1$	$1$	$10.104$	\(\Q\)	None	✓				21.6.a.d	$1$	$0$	\(-10\)	\(0\)	\(106\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-10q^{2}+68q^{4}+106q^{5}-7^{2}q^{7}+\cdots\)
63.6.a.b	$63$	$6$	63.a	1.a	$1$	$1$	$1$	$10.104$	\(\Q\)	None	✓				21.6.a.c	$1$	$0$	\(-5\)	\(0\)	\(-94\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}-7q^{4}-94q^{5}-7^{2}q^{7}+195q^{8}+\cdots\)
63.6.a.c	$63$	$6$	63.a	1.a	$1$	$1$	$1$	$10.104$	\(\Q\)	None	✓				21.6.a.b	$1$	$0$	\(-1\)	\(0\)	\(34\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-31q^{4}+34q^{5}-7^{2}q^{7}+63q^{8}+\cdots\)
63.6.a.d	$63$	$6$	63.a	1.a	$1$	$1$	$1$	$10.104$	\(\Q\)	None	✓				21.6.a.a	$1$	$1$	\(6\)	\(0\)	\(-78\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{2}+4q^{4}-78q^{5}+7^{2}q^{7}-168q^{8}+\cdots\)
63.6.a.e	$63$	$6$	63.a	1.a	$1$	$1$	$1$	$10.104$	\(\Q\)	None	✓				7.6.a.a	$1$	$0$	\(10\)	\(0\)	\(56\)	\(-49\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+10q^{2}+68q^{4}+56q^{5}-7^{2}q^{7}+\cdots\)
63.6.a.f	$63$	$6$	63.a	1.a	$1$	$2$	$2$	$10.104$	\(\Q(\sqrt{57}) \)	None	✓		✓		7.6.a.b	$1$	$1$	\(-9\)	\(0\)	\(18\)	\(98\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{2}+(-2+9\beta )q^{4}+(14+\cdots)q^{5}+\cdots\)
63.6.a.g	$63$	$6$	63.a	1.a	$1$	$2$	$2$	$10.104$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓	✓	63.6.a.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-98\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-4q^{4}-7\beta q^{5}-7^{2}q^{7}-6^{2}\beta q^{8}+\cdots\)
63.6.a.h	$63$	$6$	63.a	1.a	$1$	$4$	$4$	$10.104$	4.4.358541904.1	None	✓	✓	✓	✓	63.6.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(196\)		$+$	$-$	$-$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(24+\beta _{3})q^{4}+(3\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(63))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(63)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)