Space of modular forms of level 126, weight 8, and trivial character

• Label: 126.8.a
• Level: $126$
• Weight: $8$
• Character orbit: 126.a
• Rep. character: $\chi_{126}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $13$
• Sturm bound: $192$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(126))\).

	Total	New	Old
Modular forms	176	18	158
Cusp forms	160	18	142
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
\(+\)	\(+\)	\(+\)	\(+\)		\(24\)	\(2\)	\(22\)		\(22\)	\(2\)	\(20\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(20\)	\(2\)	\(18\)		\(18\)	\(2\)	\(16\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(21\)	\(2\)	\(19\)		\(19\)	\(2\)	\(17\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(23\)	\(3\)	\(20\)		\(21\)	\(3\)	\(18\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(22\)	\(2\)	\(20\)		\(20\)	\(2\)	\(18\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(22\)	\(2\)	\(20\)		\(20\)	\(2\)	\(18\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(21\)	\(3\)	\(18\)		\(19\)	\(3\)	\(16\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(23\)	\(2\)	\(21\)		\(21\)	\(2\)	\(19\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(90\)	\(10\)	\(80\)		\(82\)	\(10\)	\(72\)		\(8\)	\(0\)	\(8\)
Minus space			\(-\)		\(86\)	\(8\)	\(78\)		\(78\)	\(8\)	\(70\)		\(8\)	\(0\)	\(8\)

Trace form

18 * q + 1152 * q^4 - 394 * q^5 - 3120 * q^10 + 6244 * q^11 - 17754 * q^13 - 5488 * q^14 + 73728 * q^16 - 23456 * q^17 + 3558 * q^19 - 25216 * q^20 - 52128 * q^22 + 144376 * q^23 + 394098 * q^25 - 143728 * q^26 - 391840 * q^29 - 565092 * q^31 + 297504 * q^34 - 144746 * q^35 + 162384 * q^37 + 1325808 * q^38 - 199680 * q^40 + 734328 * q^41 + 568452 * q^43 + 399616 * q^44 - 68544 * q^46 + 159084 * q^47 + 2117682 * q^49 - 2014432 * q^50 - 1136256 * q^52 + 615924 * q^53 - 2161272 * q^55 - 351232 * q^56 + 1034592 * q^58 + 8634890 * q^59 + 2943066 * q^61 + 907232 * q^62 + 4718592 * q^64 - 3431340 * q^65 + 13239600 * q^67 - 1501184 * q^68 - 1300656 * q^70 - 3554984 * q^71 - 6246084 * q^73 + 2636064 * q^74 + 227712 * q^76 - 5664988 * q^77 + 2286312 * q^79 - 1613824 * q^80 + 8628576 * q^82 + 356290 * q^83 + 9288564 * q^85 + 2107104 * q^86 - 3336192 * q^88 + 13108188 * q^89 - 10827138 * q^91 + 9240064 * q^92 - 12376992 * q^94 - 4505864 * q^95 + 2555472 * q^97

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(126))\) 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
126.8.a.a	$126$	$8$	126.a	1.a	$1$	$1$	$1$	$39.361$	\(\Q\)	None	✓				42.8.a.f	$1$	$1$	\(-8\)	\(0\)	\(-470\)	\(-343\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}-470q^{5}-7^{3}q^{7}+\cdots\)
126.8.a.b	$126$	$8$	126.a	1.a	$1$	$1$	$1$	$39.361$	\(\Q\)	None	✓				42.8.a.e	$1$	$0$	\(-8\)	\(0\)	\(-30\)	\(343\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}-30q^{5}+7^{3}q^{7}-2^{9}q^{8}+\cdots\)
126.8.a.c	$126$	$8$	126.a	1.a	$1$	$1$	$1$	$39.361$	\(\Q\)	None	✓				14.8.a.b	$1$	$1$	\(-8\)	\(0\)	\(400\)	\(-343\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+20^{2}q^{5}-7^{3}q^{7}+\cdots\)
126.8.a.d	$126$	$8$	126.a	1.a	$1$	$1$	$1$	$39.361$	\(\Q\)	None	✓				14.8.a.a	$1$	$0$	\(8\)	\(0\)	\(-448\)	\(-343\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}-448q^{5}-7^{3}q^{7}+\cdots\)
126.8.a.e	$126$	$8$	126.a	1.a	$1$	$1$	$1$	$39.361$	\(\Q\)	None	✓				42.8.a.d	$1$	$1$	\(8\)	\(0\)	\(-270\)	\(343\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}-270q^{5}+7^{3}q^{7}+\cdots\)
126.8.a.f	$126$	$8$	126.a	1.a	$1$	$1$	$1$	$39.361$	\(\Q\)	None	✓				42.8.a.b	$1$	$1$	\(8\)	\(0\)	\(18\)	\(343\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+18q^{5}+7^{3}q^{7}+2^{9}q^{8}+\cdots\)
126.8.a.g	$126$	$8$	126.a	1.a	$1$	$1$	$1$	$39.361$	\(\Q\)	None	✓				42.8.a.c	$1$	$0$	\(8\)	\(0\)	\(122\)	\(-343\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+122q^{5}-7^{3}q^{7}+\cdots\)
126.8.a.h	$126$	$8$	126.a	1.a	$1$	$1$	$1$	$39.361$	\(\Q\)	None	✓				42.8.a.a	$1$	$0$	\(8\)	\(0\)	\(410\)	\(-343\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+410q^{5}-7^{3}q^{7}+\cdots\)
126.8.a.i	$126$	$8$	126.a	1.a	$1$	$2$	$2$	$39.361$	\(\Q(\sqrt{1969}) \)	None	✓		✓		14.8.a.c	$1$	$0$	\(-16\)	\(0\)	\(-126\)	\(686\)		$+$	$-$	$-$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(-63-\beta )q^{5}+7^{3}q^{7}+\cdots\)
126.8.a.j	$126$	$8$	126.a	1.a	$1$	$2$	$2$	$39.361$	\(\Q(\sqrt{499}) \)	None	✓	✓	✓	✓	126.8.a.j	$1$	$0$	\(-16\)	\(0\)	\(56\)	\(-686\)		$+$	$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(28+\beta )q^{5}-7^{3}q^{7}+\cdots\)
126.8.a.k	$126$	$8$	126.a	1.a	$1$	$2$	$2$	$39.361$	\(\Q(\sqrt{3691}) \)	None	✓	✓	✓	✓	126.8.a.k	$1$	$1$	\(-16\)	\(0\)	\(168\)	\(686\)		$+$	$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(84+\beta )q^{5}+7^{3}q^{7}+\cdots\)
126.8.a.l	$126$	$8$	126.a	1.a	$1$	$2$	$2$	$39.361$	\(\Q(\sqrt{3691}) \)	None	✓	✓	✓	✓	126.8.a.k	$1$	$0$	\(16\)	\(0\)	\(-168\)	\(686\)		$-$	$+$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(-84+\beta )q^{5}+7^{3}q^{7}+\cdots\)
126.8.a.m	$126$	$8$	126.a	1.a	$1$	$2$	$2$	$39.361$	\(\Q(\sqrt{499}) \)	None	✓	✓	✓	✓	126.8.a.j	$1$	$1$	\(16\)	\(0\)	\(-56\)	\(-686\)		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(-28+\beta )q^{5}-7^{3}q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(126))\) into lower level spaces

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