Space of modular forms of level 63, weight 7, and Character 63.m

• Label: 63.7.m
• Level: $63$
• Weight: $7$
• Character orbit: 63.m
• Rep. character: $\chi_{63}(10,\cdot)$
• Character field: $\Q(\zeta_{6})$
• Dimension: $38$
• Newform subspaces: $5$
• Sturm bound: $56$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	63.m (of order \(6\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{6})\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(56\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(63, [\chi])\).

	Total	New	Old
Modular forms	104	42	62
Cusp forms	88	38	50
Eisenstein series	16	4	12

Trace form

38 * q + 6 * q^2 - 532 * q^4 + 171 * q^5 + 66 * q^7 - 1560 * q^8 - 1452 * q^10 + 981 * q^11 + 5226 * q^14 - 16140 * q^16 - 9909 * q^17 - 4287 * q^19 + 4432 * q^22 + 2913 * q^23 + 17174 * q^25 - 27090 * q^26 - 31160 * q^28 + 2364 * q^29 - 4959 * q^31 + 155076 * q^32 + 221991 * q^35 - 54237 * q^37 - 487134 * q^38 - 28668 * q^40 + 164076 * q^43 + 218952 * q^44 - 172472 * q^46 - 158337 * q^47 - 39298 * q^49 - 327132 * q^50 + 274596 * q^52 + 388965 * q^53 + 329304 * q^56 - 2132 * q^58 - 1192293 * q^59 - 155823 * q^61 + 398760 * q^64 + 792708 * q^65 - 239801 * q^67 + 1638000 * q^68 + 800532 * q^70 - 390348 * q^71 + 2078745 * q^73 - 63690 * q^74 - 1888287 * q^77 - 937049 * q^79 - 1406808 * q^80 - 5327028 * q^82 + 2608338 * q^85 + 2433858 * q^86 + 175820 * q^88 + 5750055 * q^89 + 3721992 * q^91 - 1744032 * q^92 - 1538652 * q^94 - 1526805 * q^95 - 10292460 * q^98

Decomposition of \(S_{7}^{\mathrm{new}}(63, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
63.7.m.a	$63$	$7$	63.m	7.d	$6$	$2$	$1$	$14.493$	\(\Q(\sqrt{-3}) \)	None					7.7.d.a	$2$	$0$	\(-12\)	\(0\)	\(-315\)	\(-686\)	$1$	$\mathrm{SU}(2)[C_{6}]$	\(q-12\zeta_{6}q^{2}+(-80+80\zeta_{6})q^{4}+(-210+\cdots)q^{5}+\cdots\)
63.7.m.b	$63$	$7$	63.m	7.d	$6$	$4$	$2$	$14.493$	\(\Q(\sqrt{2}, \sqrt{-3})\)	None					7.7.d.b	$2$	$0$	\(8\)	\(0\)	\(150\)	\(280\)	$3$	$\mathrm{SU}(2)[C_{6}]$	\(q+(-4\beta _{1}-2\beta _{2}+\beta _{3})q^{2}+(30+30\beta _{1}+\cdots)q^{4}+\cdots\)
63.7.m.c	$63$	$7$	63.m	7.d	$6$	$8$	$4$	$14.493$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					21.7.f.b	$2$	$0$	\(5\)	\(0\)	\(42\)	\(748\)	$3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{1}+\beta _{2})q^{2}+(-43+43\beta _{2}-\beta _{5}+\cdots)q^{4}+\cdots\)
63.7.m.d	$63$	$7$	63.m	7.d	$6$	$8$	$4$	$14.493$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					21.7.f.a	$2$	$0$	\(5\)	\(0\)	\(294\)	\(-656\)	$2^{2}\cdot 3\cdot 7$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1+\beta _{1}-\beta _{2})q^{2}+(1-43\beta _{2}+4\beta _{3}+\cdots)q^{4}+\cdots\)
63.7.m.e	$63$	$7$	63.m	7.d	$6$	$16$	$8$	$14.493$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		✓	✓		63.7.m.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(380\)	$2^{12}\cdot 3^{8}\cdot 7^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{1}q^{2}+(21\beta _{2}+\beta _{3}+\beta _{7})q^{4}+(-\beta _{11}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(63, [\chi])\) into lower level spaces

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