Space of modular forms of level 63, weight 8, and Character 63.e

• Label: 63.8.e
• Level: $63$
• Weight: $8$
• Character orbit: 63.e
• Rep. character: $\chi_{63}(37,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $44$
• Newform subspaces: $5$
• Sturm bound: $64$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	63.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(64\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(63, [\chi])\).

	Total	New	Old
Modular forms	120	48	72
Cusp forms	104	44	60
Eisenstein series	16	4	12

Trace form

44 * q + 8 * q^2 - 1372 * q^4 + 250 * q^5 + 50 * q^7 - 2472 * q^8 - 5190 * q^10 - 1934 * q^11 + 12532 * q^13 - 17516 * q^14 - 95860 * q^16 + 10164 * q^17 - 36422 * q^19 + 66704 * q^20 + 47196 * q^22 - 82572 * q^23 - 313234 * q^25 + 150446 * q^26 - 963184 * q^28 - 600488 * q^29 - 212438 * q^31 - 254504 * q^32 - 517524 * q^34 + 1169366 * q^35 + 343642 * q^37 + 795020 * q^38 - 2131332 * q^40 - 1546924 * q^41 + 796228 * q^43 - 61972 * q^44 + 1701294 * q^46 + 1991382 * q^47 - 318922 * q^49 - 5287556 * q^50 - 5775044 * q^52 - 971136 * q^53 + 1144368 * q^55 + 7855344 * q^56 + 1751496 * q^58 + 3771168 * q^59 - 6565604 * q^61 - 10641744 * q^62 + 27256232 * q^64 - 3772778 * q^65 + 1919014 * q^67 + 13057296 * q^68 - 13170390 * q^70 - 9179652 * q^71 + 11634262 * q^73 + 6023892 * q^74 + 45223840 * q^76 + 7926500 * q^77 + 23170 * q^79 + 27476 * q^80 + 3205152 * q^82 - 15172764 * q^83 - 20246040 * q^85 - 9150230 * q^86 - 19290204 * q^88 + 2588140 * q^89 + 1231162 * q^91 + 33631056 * q^92 - 26602494 * q^94 + 19731970 * q^95 + 40377136 * q^97 - 4086898 * q^98

Decomposition of \(S_{8}^{\mathrm{new}}(63, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
63.8.e.a	$63$	$8$	63.e	7.c	$3$	$2$	$1$	$19.680$	\(\Q(\sqrt{-3}) \)	\(\Q(\sqrt{-3}) \)		✓			63.8.e.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(1763\)	$1$	$\mathrm{U}(1)[D_{3}]$	\(q+2^{7}\zeta_{6}q^{4}+(757+249\zeta_{6})q^{7}+2009q^{13}+\cdots\)
63.8.e.b	$63$	$8$	63.e	7.c	$3$	$8$	$4$	$19.680$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					7.8.c.a	$2$	$0$	\(-6\)	\(0\)	\(252\)	\(672\)	$2^{8}\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-2\beta _{2}-\beta _{3}+\beta _{4})q^{2}+(-88+88\beta _{2}+\cdots)q^{4}+\cdots\)
63.8.e.c	$63$	$8$	63.e	7.c	$3$	$8$	$4$	$19.680$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					21.8.e.a	$2$	$0$	\(-1\)	\(0\)	\(196\)	\(154\)	$2^{2}\cdot 3^{3}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{6}q^{2}+(-2\beta _{1}+5^{2}\beta _{4}+2\beta _{6}-\beta _{7})q^{4}+\cdots\)
63.8.e.d	$63$	$8$	63.e	7.c	$3$	$10$	$5$	$19.680$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None					21.8.e.b	$2$	$0$	\(15\)	\(0\)	\(-198\)	\(-859\)	$2^{2}\cdot 3^{3}\cdot 7^{3}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}-\beta _{2}-3\beta _{4})q^{2}+(-46-6\beta _{1}+\cdots)q^{4}+\cdots\)
63.8.e.e	$63$	$8$	63.e	7.c	$3$	$16$	$8$	$19.680$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓	✓		63.8.e.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-1680\)	$2^{10}\cdot 3^{9}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{2}+(-102-102\beta _{2}+\beta _{6}+\beta _{11}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(63, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(63, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)