Space of modular forms of level 126, weight 10, and trivial character

• Label: 126.10.a
• Level: $126$
• Weight: $10$
• Character orbit: 126.a
• Rep. character: $\chi_{126}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $15$
• Sturm bound: $240$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(240\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(126))\).

	Total	New	Old
Modular forms	224	22	202
Cusp forms	208	22	186
Eisenstein series	16	0	16

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(26\)	\(2\)	\(24\)		\(24\)	\(2\)	\(22\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(30\)	\(2\)	\(28\)		\(28\)	\(2\)	\(26\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(29\)	\(4\)	\(25\)		\(27\)	\(4\)	\(23\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(27\)	\(3\)	\(24\)		\(25\)	\(3\)	\(22\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(28\)	\(2\)	\(26\)		\(26\)	\(2\)	\(24\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(28\)	\(2\)	\(26\)		\(26\)	\(2\)	\(24\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(29\)	\(3\)	\(26\)		\(27\)	\(3\)	\(24\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(27\)	\(4\)	\(23\)		\(25\)	\(4\)	\(21\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(110\)	\(10\)	\(100\)		\(102\)	\(10\)	\(92\)		\(8\)	\(0\)	\(8\)
Minus space			\(-\)		\(114\)	\(12\)	\(102\)		\(106\)	\(12\)	\(94\)		\(8\)	\(0\)	\(8\)

Trace form

22 * q + 5632 * q^4 + 4742 * q^5 + 2528 * q^10 - 64172 * q^11 + 191014 * q^13 + 76832 * q^14 + 1441792 * q^16 - 643352 * q^17 + 1848578 * q^19 + 1213952 * q^20 + 247744 * q^22 - 77744 * q^23 + 8192814 * q^25 - 47776 * q^26 - 5883688 * q^29 + 12769780 * q^31 - 19548224 * q^34 - 3202934 * q^35 - 42102376 * q^37 + 89184 * q^38 + 647168 * q^40 - 18702432 * q^41 - 48768100 * q^43 - 16428032 * q^44 + 22897792 * q^46 + 28040100 * q^47 + 126825622 * q^49 - 32535616 * q^50 + 48899584 * q^52 - 43802340 * q^53 + 45599560 * q^55 + 19668992 * q^56 + 72779840 * q^58 + 359458910 * q^59 - 238402262 * q^61 - 224753728 * q^62 + 369098752 * q^64 + 502575108 * q^65 + 523412496 * q^67 - 164698112 * q^68 - 201530336 * q^70 + 334697152 * q^71 + 78556164 * q^73 + 38415552 * q^74 + 473235968 * q^76 + 401917796 * q^77 - 1023944792 * q^79 + 310771712 * q^80 - 514093248 * q^82 - 2304938618 * q^83 - 438131228 * q^85 + 1383069888 * q^86 + 63422464 * q^88 + 2096357988 * q^89 + 893772250 * q^91 - 19902464 * q^92 - 764224320 * q^94 - 3274066520 * q^95 + 5138870712 * q^97

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(126))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	7
126.10.a.a	$126$	$10$	126.a	1.a	$1$	$1$	$1$	$64.895$	\(\Q\)	None	✓				14.10.a.b	$1$	$0$	\(-16\)	\(0\)	\(-544\)	\(-2401\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}-544q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.b	$126$	$10$	126.a	1.a	$1$	$1$	$1$	$64.895$	\(\Q\)	None	✓				42.10.a.e	$1$	$1$	\(-16\)	\(0\)	\(474\)	\(2401\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+474q^{5}+7^{4}q^{7}+\cdots\)
126.10.a.c	$126$	$10$	126.a	1.a	$1$	$1$	$1$	$64.895$	\(\Q\)	None	✓				42.10.a.f	$1$	$0$	\(-16\)	\(0\)	\(1634\)	\(-2401\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+1634q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.d	$126$	$10$	126.a	1.a	$1$	$1$	$1$	$64.895$	\(\Q\)	None	✓				42.10.a.b	$1$	$0$	\(16\)	\(0\)	\(-1590\)	\(2401\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}-1590q^{5}+7^{4}q^{7}+\cdots\)
126.10.a.e	$126$	$10$	126.a	1.a	$1$	$1$	$1$	$64.895$	\(\Q\)	None	✓				14.10.a.a	$1$	$1$	\(16\)	\(0\)	\(-560\)	\(-2401\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}-560q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.f	$126$	$10$	126.a	1.a	$1$	$1$	$1$	$64.895$	\(\Q\)	None	✓				42.10.a.a	$1$	$1$	\(16\)	\(0\)	\(76\)	\(-2401\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+76q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.g	$126$	$10$	126.a	1.a	$1$	$1$	$1$	$64.895$	\(\Q\)	None	✓				42.10.a.d	$1$	$0$	\(16\)	\(0\)	\(624\)	\(2401\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+624q^{5}+7^{4}q^{7}+\cdots\)
126.10.a.h	$126$	$10$	126.a	1.a	$1$	$1$	$1$	$64.895$	\(\Q\)	None	✓				42.10.a.c	$1$	$1$	\(16\)	\(0\)	\(2290\)	\(-2401\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+2290q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.i	$126$	$10$	126.a	1.a	$1$	$2$	$2$	$64.895$	\(\Q(\sqrt{66739}) \)	None	✓	✓	✓	✓	126.10.a.i	$1$	$1$	\(-32\)	\(0\)	\(-1064\)	\(-4802\)		$+$	$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(-532+\beta )q^{5}+\cdots\)
126.10.a.j	$126$	$10$	126.a	1.a	$1$	$2$	$2$	$64.895$	\(\Q(\sqrt{243601}) \)	None	✓		✓		42.10.a.h	$1$	$1$	\(-32\)	\(0\)	\(-534\)	\(4802\)		$+$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(-267-\beta )q^{5}+\cdots\)
126.10.a.k	$126$	$10$	126.a	1.a	$1$	$2$	$2$	$64.895$	\(\Q(\sqrt{474769}) \)	None	✓		✓		42.10.a.g	$1$	$0$	\(-32\)	\(0\)	\(142\)	\(-4802\)		$+$	$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(71-\beta )q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.l	$126$	$10$	126.a	1.a	$1$	$2$	$2$	$64.895$	\(\Q(\sqrt{211}) \)	None	✓	✓	✓	✓	126.10.a.l	$1$	$0$	\(-32\)	\(0\)	\(2184\)	\(4802\)		$+$	$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(1092+13\beta )q^{5}+\cdots\)
126.10.a.m	$126$	$10$	126.a	1.a	$1$	$2$	$2$	$64.895$	\(\Q(\sqrt{211}) \)	None	✓	✓	✓	✓	126.10.a.l	$1$	$1$	\(32\)	\(0\)	\(-2184\)	\(4802\)		$-$	$+$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(-1092+13\beta )q^{5}+\cdots\)
126.10.a.n	$126$	$10$	126.a	1.a	$1$	$2$	$2$	$64.895$	\(\Q(\sqrt{66739}) \)	None	✓	✓	✓	✓	126.10.a.i	$1$	$0$	\(32\)	\(0\)	\(1064\)	\(-4802\)		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(532+\beta )q^{5}-7^{4}q^{7}+\cdots\)
126.10.a.o	$126$	$10$	126.a	1.a	$1$	$2$	$2$	$64.895$	\(\Q(\sqrt{2305}) \)	None	✓		✓		14.10.a.c	$1$	$0$	\(32\)	\(0\)	\(2730\)	\(4802\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(1365-7\beta )q^{5}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(126))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(126)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)