Space of modular forms of level 63, weight 4, and trivial character

• Label: 63.4.a
• Level: $63$
• Weight: $4$
• Character orbit: 63.a
• Rep. character: $\chi_{63}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $5$
• Sturm bound: $32$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	63.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(32\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(63))\).

	Total	New	Old
Modular forms	28	7	21
Cusp forms	20	7	13
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(8\)	\(2\)	\(6\)		\(6\)	\(2\)	\(4\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(6\)	\(0\)	\(6\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(6\)	\(2\)	\(4\)		\(4\)	\(2\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(8\)	\(3\)	\(5\)		\(6\)	\(3\)	\(3\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(16\)	\(5\)	\(11\)		\(12\)	\(5\)	\(7\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(12\)	\(2\)	\(10\)		\(8\)	\(2\)	\(6\)		\(4\)	\(0\)	\(4\)

Trace form

7 * q + 3 * q^2 + 41 * q^4 - 7 * q^7 + 51 * q^8 + 32 * q^10 - 12 * q^11 + 112 * q^13 + 63 * q^14 - 19 * q^16 - 174 * q^17 - 110 * q^19 - 60 * q^20 - 292 * q^22 - 12 * q^23 + 5 * q^25 - 144 * q^26 - 35 * q^28 + 270 * q^29 + 156 * q^31 + 651 * q^32 - 462 * q^34 + 168 * q^35 + 30 * q^37 - 1470 * q^38 - 456 * q^40 + 834 * q^41 + 632 * q^43 - 1320 * q^44 - 396 * q^46 - 876 * q^47 + 343 * q^49 + 1329 * q^50 + 188 * q^52 + 1506 * q^53 + 64 * q^55 + 567 * q^56 + 1586 * q^58 - 1602 * q^59 + 1684 * q^61 + 1836 * q^62 - 551 * q^64 - 672 * q^65 + 1300 * q^67 - 114 * q^68 - 392 * q^70 + 1212 * q^71 - 522 * q^73 + 534 * q^74 - 502 * q^76 + 672 * q^77 - 1840 * q^79 - 4140 * q^80 - 134 * q^82 - 906 * q^83 - 1728 * q^85 - 1752 * q^86 - 3180 * q^88 - 1650 * q^89 - 1036 * q^91 + 3528 * q^92 + 2436 * q^94 + 1104 * q^95 + 1146 * q^97 + 147 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(63))\) 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
63.4.a.a	$63$	$4$	63.a	1.a	$1$	$1$	$1$	$3.717$	\(\Q\)	None	✓				21.4.a.b	$1$	$1$	\(-4\)	\(0\)	\(4\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+8q^{4}+4q^{5}-7q^{7}-2^{4}q^{10}+\cdots\)
63.4.a.b	$63$	$4$	63.a	1.a	$1$	$1$	$1$	$3.717$	\(\Q\)	None	✓				7.4.a.a	$1$	$1$	\(1\)	\(0\)	\(-16\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}-2^{4}q^{5}-7q^{7}-15q^{8}+\cdots\)
63.4.a.c	$63$	$4$	63.a	1.a	$1$	$1$	$1$	$3.717$	\(\Q\)	None	✓				21.4.a.a	$1$	$0$	\(3\)	\(0\)	\(18\)	\(7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}+18q^{5}+7q^{7}-21q^{8}+\cdots\)
63.4.a.d	$63$	$4$	63.a	1.a	$1$	$2$	$2$	$3.717$	\(\Q(\sqrt{19}) \)	None	✓	✓	✓	✓	63.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-14\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+11q^{4}+2\beta q^{5}-7q^{7}+3\beta q^{8}+\cdots\)
63.4.a.e	$63$	$4$	63.a	1.a	$1$	$2$	$2$	$3.717$	\(\Q(\sqrt{57}) \)	None	✓		✓		21.4.a.c	$1$	$0$	\(3\)	\(0\)	\(-6\)	\(14\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(7+3\beta )q^{4}+(-2-2\beta )q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(63))\) into lower level spaces

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