Space of modular forms of level 63, weight 9, and Character 63.d

• Label: 63.9.d
• Level: $63$
• Weight: $9$
• Character orbit: 63.d
• Rep. character: $\chi_{63}(55,\cdot)$
• Character field: $\Q$
• Dimension: $25$
• Newform subspaces: $5$
• Sturm bound: $72$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	63.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(72\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(63, [\chi])\).

	Total	New	Old
Modular forms	68	27	41
Cusp forms	60	25	35
Eisenstein series	8	2	6

Trace form

25 * q + 5 * q^2 + 2721 * q^4 + 205 * q^7 - 9563 * q^8 - 20398 * q^11 + 70397 * q^14 + 261125 * q^16 + 365926 * q^22 + 13058 * q^23 - 588215 * q^25 - 778539 * q^28 + 1475762 * q^29 + 74517 * q^32 - 701496 * q^35 + 104474 * q^37 - 910342 * q^43 - 21621270 * q^44 - 5349146 * q^46 - 5465351 * q^49 + 25277381 * q^50 - 21969214 * q^53 + 54336709 * q^56 + 12792310 * q^58 - 1478831 * q^64 - 95169744 * q^65 + 28309058 * q^67 + 8352384 * q^70 + 63593426 * q^71 + 31956778 * q^74 + 104910002 * q^77 + 38390594 * q^79 - 102090672 * q^85 - 356444102 * q^86 + 39671894 * q^88 - 87045936 * q^91 + 58194762 * q^92 - 71912016 * q^95 + 227412965 * q^98

Decomposition of \(S_{9}^{\mathrm{new}}(63, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
63.9.d.a	$63$	$9$	63.d	7.b	$2$	$1$	$1$	$25.665$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				7.9.b.a	$2$	$0$	\(31\)	\(0\)	\(0\)	\(2401\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+31q^{2}+705q^{4}+7^{4}q^{7}+13919q^{8}+\cdots\)
63.9.d.b	$63$	$9$	63.d	7.b	$2$	$2$	$2$	$25.665$	\(\Q(\sqrt{7}) \)	\(\Q(\sqrt{-7}) \)	✓	✓			63.9.d.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-4802\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{2}-193q^{4}-7^{4}q^{7}-449\beta q^{8}+\cdots\)
63.9.d.c	$63$	$9$	63.d	7.b	$2$	$4$	$4$	$25.665$	\(\mathbb{Q}[x]/(x^{4} + \cdots)\)	None					7.9.b.b	$2$	$0$	\(-32\)	\(0\)	\(0\)	\(1428\)	$2^{4}\cdot 3^{3}\cdot 7$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-8-\beta _{2})q^{2}+(-8+2^{4}\beta _{2})q^{4}+\cdots\)
63.9.d.d	$63$	$9$	63.d	7.b	$2$	$8$	$8$	$25.665$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		✓			63.9.d.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(5180\)	$2^{10}\cdot 3^{6}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+(153+\beta _{2})q^{4}-\beta _{3}q^{5}+(648+\cdots)q^{7}+\cdots\)
63.9.d.e	$63$	$9$	63.d	7.b	$2$	$10$	$10$	$25.665$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None			✓		21.9.d.a	$2$	$0$	\(6\)	\(0\)	\(0\)	\(-4002\)	$2^{13}\cdot 3^{12}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{1})q^{2}+(119-5\beta _{1}-\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(63, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(63, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)