Space of modular forms of level 63, weight 10, and trivial character

• Label: 63.10.a
• Level: $63$
• Weight: $10$
• Character orbit: 63.a
• Rep. character: $\chi_{63}(1,\cdot)$
• Character field: $\Q$
• Dimension: $23$
• Newform subspaces: $8$
• Sturm bound: $80$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	63.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(80\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(63))\).

	Total	New	Old
Modular forms	76	23	53
Cusp forms	68	23	45
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
\(+\)	\(+\)	\(+\)		\(18\)	\(4\)	\(14\)		\(16\)	\(4\)	\(12\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(20\)	\(6\)	\(14\)		\(18\)	\(6\)	\(12\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(20\)	\(7\)	\(13\)		\(18\)	\(7\)	\(11\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(18\)	\(6\)	\(12\)		\(16\)	\(6\)	\(10\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(36\)	\(10\)	\(26\)		\(32\)	\(10\)	\(22\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(40\)	\(13\)	\(27\)		\(36\)	\(13\)	\(23\)		\(4\)	\(0\)	\(4\)

Trace form

23 * q - 17 * q^2 + 6913 * q^4 - 3868 * q^5 + 2401 * q^7 - 26061 * q^8 + 8524 * q^10 + 46268 * q^11 - 193804 * q^13 - 88837 * q^14 + 2002885 * q^16 - 173862 * q^17 - 640366 * q^19 + 37228 * q^20 - 2882468 * q^22 - 3389484 * q^23 + 12940933 * q^25 + 11955052 * q^26 + 2028845 * q^28 - 2654674 * q^29 - 11653428 * q^31 + 18792971 * q^32 + 16073970 * q^34 - 2900408 * q^35 + 23499342 * q^37 - 16958594 * q^38 + 29638296 * q^40 + 19572730 * q^41 + 78496000 * q^43 - 36287984 * q^44 - 122760588 * q^46 + 30219588 * q^47 + 132590423 * q^49 - 59006731 * q^50 + 279723028 * q^52 + 205623882 * q^53 - 83464384 * q^55 - 90548913 * q^56 - 110681894 * q^58 - 20252898 * q^59 + 111929792 * q^61 + 300180468 * q^62 + 880496801 * q^64 - 241267048 * q^65 - 398108068 * q^67 - 55490850 * q^68 + 435166844 * q^70 - 654890532 * q^71 - 168514938 * q^73 + 1445227566 * q^74 - 3137968430 * q^76 - 374287088 * q^77 + 1630627936 * q^79 + 1308476548 * q^80 + 2292914570 * q^82 + 823710246 * q^83 + 284293944 * q^85 - 1906052056 * q^86 - 1813059348 * q^88 - 1616470786 * q^89 - 326948972 * q^91 - 3553586664 * q^92 - 2237074068 * q^94 + 69506528 * q^95 - 4558886910 * q^97 - 98001617 * q^98

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(63))\) 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
63.10.a.a	$63$	$10$	63.a	1.a	$1$	$1$	$1$	$32.447$	\(\Q\)	None	✓				21.10.a.a	$1$	$1$	\(24\)	\(0\)	\(144\)	\(2401\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+24q^{2}+2^{6}q^{4}+12^{2}q^{5}+7^{4}q^{7}+\cdots\)
63.10.a.b	$63$	$10$	63.a	1.a	$1$	$2$	$2$	$32.447$	\(\Q(\sqrt{345}) \)	None	✓				21.10.a.c	$1$	$1$	\(-30\)	\(0\)	\(-1128\)	\(4802\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-15-\beta )q^{2}+(58+30\beta )q^{4}+(-564+\cdots)q^{5}+\cdots\)
63.10.a.c	$63$	$10$	63.a	1.a	$1$	$2$	$2$	$32.447$	\(\Q(\sqrt{2353}) \)	None	✓				21.10.a.b	$1$	$0$	\(-9\)	\(0\)	\(-1170\)	\(-4802\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{2}+(92+9\beta )q^{4}+(-620+\cdots)q^{5}+\cdots\)
63.10.a.d	$63$	$10$	63.a	1.a	$1$	$2$	$2$	$32.447$	\(\Q(\sqrt{193}) \)	None	✓				7.10.a.a	$1$	$0$	\(6\)	\(0\)	\(2238\)	\(-4802\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{2}+(-310+6\beta )q^{4}+(1119+\cdots)q^{5}+\cdots\)
63.10.a.e	$63$	$10$	63.a	1.a	$1$	$3$	$3$	$32.447$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		7.10.a.b	$1$	$1$	\(-21\)	\(0\)	\(-1554\)	\(7203\)		$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-7+\beta _{2})q^{2}+(519+7\beta _{1}-8\beta _{2})q^{4}+\cdots\)
63.10.a.f	$63$	$10$	63.a	1.a	$1$	$3$	$3$	$32.447$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		21.10.a.d	$1$	$0$	\(13\)	\(0\)	\(-2398\)	\(-7203\)		$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(4+\beta _{1})q^{2}+(555+7\beta _{1}-5\beta _{2})q^{4}+\cdots\)
63.10.a.g	$63$	$10$	63.a	1.a	$1$	$4$	$4$	$32.447$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	63.10.a.g	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-9604\)		$+$	$+$	$+$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(446+\beta _{3})q^{4}+(-20\beta _{1}+\cdots)q^{5}+\cdots\)
63.10.a.h	$63$	$10$	63.a	1.a	$1$	$6$	$6$	$32.447$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	63.10.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(14406\)		$+$	$-$	$-$	$2^{5}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(357+\beta _{3})q^{4}+(24\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(63))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(63)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)